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NCERT-Hindi-Class-12-Mathematics-Part-2

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① ✁✂✄
❞☎✆✆ ✝✞ ✟✠✡ ☛☞✌ ✍✆✎✏✑✍✒✓✔❞
❍✆✆✕ ✖ II
✈ ✁✂✄
❥☎✆✝✞✟✠ ✡☎☛☞✠✌✠☎✍ ✎✟ ✏✡❥✑✒☎☎ ✓✔✕✕✖✗ ✘✙✚☎✛✟ ✜✢ ✣✎ ✤✥✌☎✑✦ ✧✑★ ✩✧✪★✫✟ ✬✟✧✭ ✎☎✑ ✤☎✜❥ ✧✑★ ✬✟✧✭
✘✑ ✬☎✑❧✮☎ ✬☎✭☎ ✌☎✣✜✯✰ ✠✜ ✣✘✱☎✦✛ ✣✎✛☎✤✟ ✲☎✭ ✎✟ ✳✘ ✣✧❥☎✘✛ ✧✑★ ✣✧✡❥✟✛ ✜✢ ✣✬✘✧✑ ★ ✡✴✵☎☎✧✧✶☎
✜❣☎❥✟ ✷✠✧✩✸☎☎ ✹☎✬ ✛✎ ✩✧✪★✫ ✹☎✢❥ ✺☎❥ ✧✑★ ✤✟✌ ✹✦✛ ❥☎✫ ✤✭☎✯ ✜✙✯ ✜✢✰ ✭✠✟ ❥☎✆✝✞✟✠ ✡☎☛☞✠✌✠☎✍ ✡❥
✹☎✻☎✣❥✛ ✡☎☛☞✠✼❣ ✹☎✢❥ ✡☎☛☞✠✡✙✩✛✧✑✦★ ✽✘ ✤✙✣✭✠☎✾✟ ✣✧✌☎❥ ✡❥ ✹❣✫ ✎❥✭✑ ✎☎ ✡✴ ✠☎✘ ✜✢✰ ✽✘ ✡✴✠☎✘
❣✑✦ ✜❥ ✣✧✆☎✠ ✎☎✑ ✯✎ ❣❡☎✤✪✛ ✾✟✧☎❥ ✘✑ ✺☎✑❥ ✾✑✭✑ ✹☎✢❥ ✬☎✭✎☎❥✟ ✎☎✑ ❥✝☎ ✾✑✭✑ ✎✟ ✡✴✧✣✿ ❀☎ ✎☎ ✣✧❥☎✑✻ ✶☎☎✣❣✫
✜✢✰ ✹☎✶☎☎ ✜✢ ✣✎ ✠✑ ✎✾❣ ✜❣✑✦ ❥☎✆✝✞✟✠ ✣✶☎❁☎☎ ✭✟✣✛ ✓❂❃❄❅✗ ❣✑✦ ✧✣❆☎✍✛ ✤☎✫❇✧✑✦★✣✾✴✛ ✷✠✧✩✸☎☎ ✎✟ ✣✾✶☎☎
❣✑✦ ✎☎❈✮ ☎★✟ ✾✪❥ ✛✎ ✫✑ ✬☎✯❦❉✑✰
✽✘ ✡✴ ✠❜✭ ✎✟ ✘✡★✫✛☎ ✹✤ ✽✘ ✤☎✛ ✡❥ ✣✭✵☎✍ ❥ ✜✢ ✣✎ ✩✧✪★✫☎✑✦ ✧✑ ★ ✡✴☎✌☎✠✍ ✹☎✢❥ ✹❊✠☎✡✎ ✤✥✌☎✑✦
✎☎✑ ✎❞✡✭☎✶☎✟✫ ❉✣✛✣✧✣✻✠☎✑✦ ✹☎✢❥ ✘✧☎✫☎✑✦ ✎✟ ❣✾✾ ✘✑ ✘✟✒☎✭✑ ✹☎✢❥ ✘✟✒☎✭✑ ✧✑★ ✾☎✢ ❥☎✭ ✹✡✭✑ ✹✭✙✵☎✧
✡❥ ✣✧✌☎❥ ✎❥✭✑ ✎☎ ✹✧✘❥ ✾✑✛✑ ✜✢✦ ✰ ✜❣✑✦ ✠✜ ❣☎✭✭☎ ✜☎✑❉☎ ✣✎ ✠✣✾ ✬❉✜✐ ✘❣✠ ✹☎✢❥ ✹☎✬✮☎✾✟ ✾✟ ✬☎✯
✛☎✑ ✤✥✌✑ ✤❧✮ ☎✑✦ r☎❥☎ ✘☎✢✦✡✟ ❉✽✍ ✘✪✌✭☎❇✘☎❣❉✴✟ ✘✑ ✬✙❧✮✎ ❥ ✹☎✢❥ ✬✪✚✎❥ ✭✯ ✲☎✭ ✎☎ ✘✿✬✭ ✎❥ ✘✎✛✑ ✜✢✦✰
✣✶☎❁☎☎ ✧✑★ ✣✧✣✧✻ ✘☎✻✭☎✑✦ ✯✧✦ ✘✴☎✛
✑ ☎✑✦ ✎✟ ✹✭✾✑✒☎✟ ✣✎✯ ✬☎✭✑ ✎☎ ✡✴❣✒
✙ ☎ ✎☎❥❆☎ ✡☎☛☞✠✡✙✩✛✎ ✎☎✑ ✡❥✟❁☎☎
✎☎ ✯✎❣☎❋☎ ✹☎✻☎❥ ✤✭☎✭✑ ✎✟ ✡✴✧✣✿ ❀☎ ✜✢✰ ✘✬✍✭ ☎ ✹☎✢❥ ✡✜✫ ✎☎✑ ✣✧✎✣✘✛ ✎❥✭✑ ✧✑★ ✣✫✯ ✬✮✏❥✟ ✜✢ ✣✎
✜❣ ✤✥✌☎✑✦ ✎☎✑ ✘✟✒☎✭✑ ✎✟ ✡✴✣✼✠☎ ❣✑✦ ✡✪❥☎ ✵☎☎❉✟✾☎❥ ❣☎✭✑✦ ✹☎✢❥ ✤✭☎✯❦✐ ✳●✜✑✦ ✲☎✭ ✎✟ ✣✭✻☎✍✣ ❥✛ ✒☎✙❥☎✎ ✎☎
❉✴☎✜✎ ❣☎✭✭☎ ①☎✑❧✮ ✾✑✦✰
✠✑ ✳❍✑✶✠ ✩✧✪★✫ ✎✟ ✾✢✣✭✎ ✣❡☎✦✾❉✟ ✹☎✢❥ ✎☎✠✍✶☎✢✫✟ ❣✑✦ ✎☎✡★✟ ✡✑ ★❥✤✾✫ ✎✟ ❣☎❦❉ ✎❥✛✑ ✜✢✦✰ ✾✢✣✭✎
✘❣✠❇✘☎❥❆☎✟ ❣✑✦ ✫✌✟✫☎✡✭ ✳✛✭☎ ✜✟ ❡☎✏❥✟ ✜✢✐ ✣✬✛✭☎ ✧☎✣✆☎✍✎ ✎✢ ✫✑✦❧❥ ✧✑★ ✹❣✫ ❣✑✦ ✌✙✩✛✟✐ ✣✬✘✘✑
✣✶☎❁☎❆☎
✧✑ ★
✣✫✯
✣✭✠✛
✣✾✭☎✑✦
✎✟
✘✦ ✒ ✠☎
✜✎✟✎✛
✤✭
✘✧✑★ ✰
✣✶☎❁☎❆☎
✹☎✢ ❥
❣✪❞✠☎✦✎✭
✎✟
✣✧✣✻✠☎❦ ✵☎✟ ✽✘ ✤☎✛ ✎☎✑ ✛✠ ✎❥✑❉
✦ ✟ ✣✎ ✠✜ ✡☎☛☞✠✡✙✩✛✎ ✩✧✪★✫ ❣✑✦ ✤✥✌☎✑✦ ✧✑★ ✬✟✧✭ ✎☎✑ ❣☎✭✣✘✎ ✾✤☎✧
✛✸☎☎ ✤☎✑✣❥✠✛ ✎✟ ✬❉✜ ✒☎✙✶☎✟ ✎☎ ✹✭✙✵☎✧ ✤✭☎✭✑ ❣✑✦ ✣✎✛✭✟ ✡✴✵☎☎✧✟ ✣✘✱ ✜☎✑✛ ✟ ✜✢✰ ✤☎✑✚ ✎✟ ✘❣✩✠☎
✘✑ ✣✭✡✝✭✑ ✧✑★ ✣✫✯ ✳✡✫■✻ ✘❣✠ ✎☎ ❊✠☎✭ ❥✒☎✭✑ ✎✟ ✡✜✫✑ ✘✑ ✹✣✻✎ ✘✌✑✛ ✎☎✑✣✶☎✶☎ ✎✟ ✜✢✰ ✽✘
✎☎✑✣✶☎✶☎ ✎☎✑ ✹☎✢❥ ❉✜❥☎✭✑ ✧✑ ★ ✠❜✭ ❣✑✦ ✠✜ ✡☎☛☞✠✡✙✩✛✎ ✘☎✑✌❇✣✧✌☎❥ ✹☎✢❥ ✣✧✩❣✠✐ ①☎✑✝✑ ✘❣✪✜☎✑✦ ❣✑✦ ✤☎✛✌✟✛
✯✧✦ ✤✜✘ ✹☎✢ ❥ ✜☎✸☎ ✘✑ ✎✟ ✬☎✭✑ ✧☎✫✟ ❉✣✛✣✧✣✻✠☎✑✦ ✎☎✑ ✡✴☎✸☎✣❣✎✛☎ ✾✑✛ ✟ ✜✢✰
✯✭❏✘✟❏✽✍❏✹☎❥❏✝✟❏ ✽✘ ✡✙✩✛✎ ✎✟ ❥✌✭☎ ✧✑★ ✣✫✯ ✤✭☎✽✍ ❉✽✍ ✡☎☛☞✠✡✙✩✛✎ ✣✭❣☎✍❆☎ ✘✣❣✣✛ ✧✑★ ✡✣❥❑❣
✧✑★ ✣✫✯ ✧✿★✛✲✛☎ ✷✠♦✛ ✎❥✛✟ ✜✢✰ ✡✣❥✆☎✾☞ ✽✘
✡☎☛☞ ✠✡✙✩✛✎ ✧✑★ ✘✫☎✜✎☎❥ ✘❣✪✜ ✧✑★ ✹❊✠❁☎
✑ ✮✑ ★✘❥ ✡✧✭ ✧✙★❣☎❥ ✬✢✭ ✎✟ ✣✧✶☎✑ ✆☎
▲▼◆❖P◗◆❖❘❙❚ ✬✠✦✛ ✣✧✆❆☎✙ ✭☎❥✫✟✎❥ ✹☎✢❥ ✽✘ ✡✙✩✛✎ ✧✑★ ✘✫☎✜✎☎❥ ✡✴☎✡
✹☎✵☎☎❥✟ ✜✢ ✰ ✽✘ ✡☎☛☞ ✠✡✙✩✛✎ ✧✑★ ✣✧✎☎✘ ❣✑✦ ✎✽✍ ✣✶☎❁☎✎☎✑✦ ✭✑ ✠☎✑❉✾☎✭ ✣✾✠☎❯ ✽✘ ✠☎✑❉✾☎✭ ✎☎✑ ✘✦ ✵☎✧ ✤✭☎✭✑
✧✑★ ✣✫✯ ✜❣ ✳✭✧✑★ ✡✴☎✌☎✠☎✑❱ ✧✑ ★ ✹☎✵☎☎❥✟ ✜✢✦ ✰ ✜❣ ✳✭ ✘✵☎✟ ✘✦✩✸☎☎✹☎✑✦ ✹☎✢❥ ✘✦❉☛✭☎✑✦ ✧✑ ★ ✡✴✣✛ ✧✿ ★✛✲ ✜✢✦
iv
❢ ✁✂✄☎✆✝☎ ✞✟✝☎ ✠✆✠✄✡✝✄☎✆☛ ✠✄☞✌✍✎ ✏✑✄✄ ✠✂✒✄☎❢✌✒✄☎✆ ✓✎ ☞✔✔ ✕☎✝☎ ☞☎✆ ✂☞☎✆ ✖✔✄✗✏✄✟✘✙✚✓ ✠✂✒✄☎✌ ❢✔✒✄✛ ✂☞☛
❢✙✜✄☎✢✄ ✣✟ ✠☎ ☞✄✤✒❢☞✓ ✥✙✆ ✖✦✧✏✗ ❢✜✄★✄✄ ❢✙✩✄✄✌☛ ☞✄✝✙ ✠✆✠✄✡✝ ❢✙✓✄✠ ☞✆✪✄✄✕✒ ✫✄✗✄ ✟✍✄☎✬ ☞✭✮✄✄✕
❢☞✗✎ ✞✄✯✗ ✐✰✱✲✳✴✱✲❦✵✶ ✎✬✟✎✬ ✔☎✜✄✟✄✆✷☎ ✓✎ ✞✤✒★✄✏✄ ☞☎✆ ✌❢✸✏☛ ✗✄✢✹✺✎✒ ☞✄✝✎✹❢✗✆✌ ✠❢☞❢✏ ✫✄✗✄ ✟✍✔✻✄ ✼✂✽☞✘✾✒
✠☞✒ ✥✙✆ ✒✄☎✌✔✄✝ ✙☎❧ ❢✕✥ ✙✭❧✏✿ ✂✯✛✆ ❀✒✙❁✑✄✄✌✏ ✠✽✡✄✗✄☎✆ ✞✄✯✗ ✞✟✝☎ ✟✍✓✄✜✄✝✄☎✆ ☞☎✆ ❢✝✗✆✏✗ ❢✝❂✄✄✗ ✕✄✝☎
✙☎❧ ✟✍❢✏ ✠☞❢✟✚✏ ✥✝✬✠✎✬♦✚✬✞✄✗✬✹✎✬ ❢✹❃✟❢✮✄✒✄☎✆ ✥✙✆ ✠✽❄✄✙✄☎✆ ✓✄ ❁✙✄✌✏ ✓✗☎✌✎ ❢ ✝✠☎ ✩✄✄✙✎ ✠✆✜✄✄☎✡✝✄☎✆ ☞☎✆
☞✔✔ ✕✎ ✄ ✠✙☎❧✛
✝✒✎ ❢✔✾✕✎
❊❋ ❆●❍■✶ ❊❋❋❏
❅❆❇✲❈✱❉
✗✄✢✹✺✎✒ ✜✄✯❢★✄✓ ✞✝✽✠✡✆ ✄✝
✞✄✯✗ ✟✍❢✜✄★✄✮✄ ✟❢✗✢✄✔✈
✐ ✁ ✂
✄☎ ✆ ✄
❥✝ ✞✟✠✡ ☛
☞✝ ✌✍ ✎✝ ✏
✑
☛ ☛
✒
✈
✔
✩
✕
✱✛
✍
✣
✹
✥
✍
✒
✍
✥
✮
✲
✍✏
☛ ✝
☞✝
✛✞✝
✜★
✛
✗
❥
✮
✕ ✌
✲
✪
✯
✒
✌
✲
①
✍
✟
✡
✚☞
✏
✗✝
☛
✫✬
❥
✗
★
✥
✮
✏
✡
✍✒
✺
✺
✝
✓
✒
✫
✚
✏
✡
✥ ✏
✟
✡
☛
✝ ✛✏
✔
✜✕
✩
❂
✍
✙✝
✒
✸
✔✡
✏✝ ✛
✝✝
✔
✛
✔✕✤ ❧
✟
✡
❂
✝
❥
✝
✝
✛✕
✕❊
✱
✗
✣
✜★
✛
✍
✜
✞
✝
①
✛★
✜★
✛
✑ ✕
✝☛
❇★ ✥
✔
✝
✥
❅ ✝✝
✥ ✏
Committee
✹
❡
✛
✝
♠✣ ✰
✕
✱✛
•
✾
●
✏
✡
✏
❥
✥ ✛
✓
☛
✱
✰
✦ ✝ ✿✍
✕
✱✛
✥
✏
✒
✛
✕ ✌
✲
①
❉
✔
✑
✈
✔✕✔
❡
✙✝ ✝
✑
✗✣
✖
▼
✔
✑ ✝
❥
✌
✡
✱
✍
✜
✎✝ ✝
✍ ☞✝ ✎✝ ✝
✥
✛
✓
✍
✜
✔
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vi
♠ ✁✂✄☎✆✝✞✟✄ ✠✡☛ ☞✌ ✁✆✍✄✎✏ ✑✄✒✓ ✑✔✕ ✟
✖ ✄✡✗ ✘✄✡✔✄✡✎ ✙✄✄✡✎ ✓ ✚✌ ✘✡✏✡ ☞✕✛ ✟✄ ✏✄✁✠✜☛✢✣ ✚☞✕✁✠✤
✆✄✤✔✣ ❧☞✄✥ ✦✄✧ ★✩☞✡✎ ✑ ✔✄✔✡ ✢✧ ✑✄✠✪✟✢✏✄ ✫✬✧✣ ✑ ✔✄✟✄ ☞✒✭
• ❚✟✄✁✝✁✏✟ ✘✮✁✯✰✢✄✡✱✄☎✆✎✢✲ ✔✄✑✄✡✎ ✢✄ ✞✖ ✏✕✏✧✢✓✱✄ ✑✄✠✪✟✢ ☞✄✡✔✡ ✓ ✁✘✟✄ ✗✟✄ ☞✒✭
• ✗✁✱✄✏✧✟ ✑✠✤✄✓✱✄✄✑✄✡✎ ✑✄✒✓ ★✆✠✡☛ ✆☞①✁✠✯✄✟✄✡✎ ❧✒✆✡✳ ✁✠✴✄✔ ✛✠✎ ✆✄✝✄✁❧✢ ✁✠✴✄✔ ✆✡ ✦✄✧ ❧✄✡✫✬✄
✗✟✄ ☞✒✭
• ✁✠✯✄✟ ✠✡☛ ✖❢✟✡✢ ✵✄✎✫ ✝✡✎ ✟✄✜✶✏ ✑✄✒✓ ✁✠✁✠✤ ♠✘✄☞✓✱✄☎✑✦✟✄✆ ✁✘✛ ✗✛ ☞✒✎✭
• ✆✝✞✟✄✑✄✡✎ ✢✄✡ ☞✌ ✢✓✔✡ ✢✧ ✙✄✝✏✄ ✟✄ ✢✄✒✪✄✌ ✛✠✎ ✑✔✕ ✟✖ ✄✡✗ ✢✓✔✡ ✢✧ ✆✝✷ ✢✄✡ ✠✡✎☛✁✘✖✏ ✛✠✎
✝❧✚❡✏ ✢✓✔✡ ☞✡✏✕ ✑✸✟✄✟ ✠✡ ☛ ✑✎✏ ✝✡✎ ✘✄✡ ✟✄ ✘✄✡ ✆✡ ✑✁✤✢ ✆✎✢✲ ✔✄✑✄✡✎ ✢✄✡ ✆✝✄✠✡✁✪✄✏ ✢✓✔✡ ✠✄✌✡
♠✘✄☞✓✱✄✄✡✎ ✏✹✄✄ ✑✦✟✄✆① ✖✪✔✄✡✎ ✢✄ ✆✝✄✟✄✡❧✔ ✁✢✟✄ ✗✟✄ ☞✒✣ ❧✒✆✄ ✁✢ ✓✄✯✰✺✧✟ ✄✻✼✟①✽✟✄✜ ✾ ✓✡✵✄✄
✿❀❀❁ ✝✡✎ ✢☞✄ ✗✟✄ ☞✒✣ ★✆✧ ✠✡☛ ✑✔✕✾ ✝✡✤✄✠✧ ❂✄❃✄✄✡✎ ✠✡☛ ✁✌✛ ✦✄✧ ✄✻✼✟ ✞✕ ✏✢ ✝✡✎ ✽✕✔✄✒✏✧ ❡✱✄✜
✆✝✞✟✄✑✄✡✎ ✢✄✡ ✪✄✄✁✝✌ ✁✢✟✄ ✗✟✄ ☞✒✭
• ✁✠✯✄✟ ✢✄✡ ✑✄✒✓ ✑✁✤✢ ✓✡✖ ✱✄✄✘✄✟✢ ✚✔✄✔✡ ✠✡☛ ♠✘✼✘✡✪✟ ✆✡ ✁✠✯✄✟ ✢✧ ✆✎✁✙✄✶✏ ✛✡✁✏☞✄✁✆✢ ✯✮ ✻✦✄❡✁✝
✄✻ ✠✡☛ ✑✎✏ ✝✡✎ ✘✧ ✗★✜ ☞✒ ✑✄✒✓ ✖❢✟✡✢ ✄✻ ✠✡☛ ✖✄✓✎✦ ✄ ✝✡✎ ✆✎ ✚✁✎ ✤✏ ✢✹✄✔ ✛✠✎ ✆✕ ✁✖ ✆✍ ✗✁✱✄✏✴✄✡✎ ✠✡☛
✁✽❃✄ ✁✘✛ ✗✛ ☞✒✎ ✁❧✩☞✄✡✔✎ ✡ ✁✠✪✄✡ ✯✄✏✟✄ ✁✠✯✄✟①✠✞✏✕ ✢✄✡ ✁✠✢✁✆✏ ✑✄✒✓ ✆✕✚✄✡✤ ✚✔✄✔✡ ✠✡☛ ✁✌✛ ✑ ✔✄
✟✄✡ ✗✘✄✔ ✁✘✟✄✭
• ✑✎✏✏✳ ✁✠✯✄✟ ✢✧ ✆✎✢✲ ✔✄✑✄✡✎ ✠✡☛ ✆❡❃✄ ✛✠✎ ✁✓✱✄✄✝ ✠✡☛ ✖❢✟✙✄ ✆✄✓①✢✹✄✔ ✠✡☛ ✁✌✛ ✄✻ ✢✄
✆✎✁✙✄✶✏ ✆✄✓✄✎✪✄ ✦✄✧ ✞✖ ✏✕✏ ✁✢✟✄ ✗✟✄ ☞✒✭
✝✒✎ ✁✠✪✄✡✯✄ ✾ ✆✡ ✓✄✯✰✺✧✟ ✪✄✒✁✙✄✢ ✑✔✕✆✤
✎ ✄✔ ✑✄✒✓ ✁✖ ✪✄✙✄✱✄ ✁✓✯✄✘✼ ✠✡☛ ✁✔✘✡✪✄✢ ✖✄✡❄ ✠✮☛✯✱✄ ✠✕☛✝✄✓ ✢✄
✑✄✦✄✄✓✧ ☞❡✥ ✁❧✩☞✄✡✎✔✡ ✝✕✷✡ ✁✔✝✎✁❃✄✏ ✢✓ ✗✁✱✄✏ ✁✪✄✙✄✄ ✠✡☛ ✓✄✯✰✺✧✟ ✟
✖ ✄✆ ✢✧ ✢✫✬✧ ✆✡ ❧✄✡✫✬✄ ☞✒✭ ♠✩☞✄✡✎✔✡ ☞✝✡✎
★✆ ☞✡✏✕ ✚✄✒✁✍✢ ✁✓ ✡✖✙✟ ✏✹✄✄ ✞✠✞✹✟ ✠✄✏✄✠✓✱✄ ✖✘✄✔ ✁✢✟✄✭ ★✆ ✞✕ ✏✢ ✢✄✡ ✏✒✟✄✓ ✢✓✔✡ ✢✄ ✢✄✟✜
✑❢✟✎✏ ✆✕ ✵✄✘ ✛✠✎ ✖✪✄✎✆✔✧✟ ✓☞✄✭ ✝✒✎✣ ✁✠✴✄✔ ✛✠✎ ✗✁✱✄✏ ✢✧ ✆✌✄☞✢✄✓ ✆✝❡☞ ✠✡☛ ✑✸✟✙✄ ✖✄❄✡ ❧✡❄✠✧❄
✔✄✓✌✧✢✓ ✢✄ ✠✮☛✏✴ ☞❡✥ ✁❧✩☞✄✡✔✎ ✡ ✆✝✟①✆✝✟ ✓ ★✆ ✞✕ ✏✢ ✠✡☛ ✁✌✛ ✑ ✔✡ ✁✠✪✄✡✯✄ ✆✕✷✄✠ ✛✠✎ ✆☞✟✄✡ ✗
✘✡✢✓ ✞✕ ✏✢ ✠✡☛ ✆✕✤✄✓ ✝✡✎ ✢✄✟✜ ✁✢✟✄✭ ✝✒✎ ✁✓✯✄✘✼ ✠✡☛ ✆✎✟✕♥✏ ✁✔✘✡✪✄✢ ✖✄✡❄ ❧✧❄✓✠✧✩✘✖✄ ✢✄✡ ✦✄✧ ✤✩✟✠✄✘ ✘✡✏✄
☞✥❡ ✁❧✩☞✄✡✔✎ ✡ ✆✝✟①✆✝✟ ✓ ✄✻✼✟ ✞✕ ✏✢ ✆✡ ✆✎✚✁✎ ✤✏ ✁❣✟✄①✁✠✁✤ ✢✄✡ ✆✎✽✄✁✌✏ ✢✓✔✡ ✝✡✎ ✟✄✡✗✘✄✔ ✁✢✟✄✭
✝✒✎ ✖✄✡❄ ☞✕✠✕☛✝ ✁✆✎☞✣ ✝✕✵✟ ✆✎✟ ✄✡ ❧✢ ✛✠✎ ✑✸✟✙✄✣ ✁✠✴✄✔ ✛✠✎ ✗✁✱✄✏✣ ✫✄❅❄ ✠✧❄ ✧❄✁✆✎☞✣ ✆✎✟✄✡❧✢ ✏✹✄✄
✖✄✡❄ ✛✆❄✠✡☛❄✁✆✎☞ ✗✄✒✏✝ ✠✡☛ ✁✖ ✏ ✆✐✘✟ ✤✩✟✠✄✘ ❆✟♥✏ ✢✓✏✄ ☞❅✥❡ ✁❧✩☞✄✎✔✡ ✡ ★✆ ✁✓✟✄✡❧✔✄ ✢✄✡ ✆ ☛✌ ✚✔✄✔✡
☞✡✏✕ ✪✄✒✁✙✄✢ ✑✄✒✓ ✖✪✄✄✆✁✔✢ ✾ ✆✡ ✆✎✌❇✔ ✓☞✡✭ ✝✒✎ ★✆ ✔✡✢ ✢✄✟✜ ✆✡ ✆✎✚✍ ✆✦✄✧ ✰✧✝ ✠✡☛ ✆✘✞✟✄✡✎ ✑✄✒✓
✁✪✄✙✄✢✄✡✎ ✢✧ ✖✪✄✎✆✄ ✢✓✏✄ ☞✥❡ ✏✹✄✄ ♠✩☞✡✎ ✤✩✟✠✄✘ ✘✡✏✄ ☞❡✥ ❧✄✡ ★✆ ✢✄✟✜ ✝✡✎ ✁✢✆✧ ✦✄✧ ✾ ✝✡✎ ✟✄✡✗✘✄✔
✁✢✟✄ ☞✄✡✭
•
❈❉❊ ❉❋●❍ ■❏❊
✝✕✵✟ ✆✌✄☞✢✄✓
✄✻✼✟ ✞✕ ✏✢ ✆✎✠✤✜✔ ✆✁✝✁✏
✐ ✁✂✄✐☎✆✝✞ ✟✠✞ ✡ ✡✟☛✟✝
❢☞✌✍✎ ✏☞✑ ✒❢✓✍✔ ✕✖✍✗✘✍✙ ✕✚✛✗ ☞✜✢ ✣✤✥✦✍
t✧★✩ ✪✫✬✭✮✯ ✰✮✱✲✳✴✱ ❜✵✶✷✸✹✺ ✻✼✽✻
✾ ❀✾✿ ✺✸❪ ❁❂✧❃✮❪ ❁✮❄❅❆✧❇❆ ❈✳❆❉❆❪ ❊❇✰✮ ✪✫❋✫✪✫●✮✲✧❪ ❊❇✰✮❍
✚❡■✥ ✕✖✍✗✘✍✙
❊✳❆✫❏❑❆ t▲✰❪ ✻✼✽✾✻❀✿✾ ✺✸ ▼✪✭✮✩ ✪✫◆✮✮▼❪ ✪❖P✲✳ ✪✫❋✫✪✫●✮✲✧❪ ✪❖P✲✳❍
✚❡■✥ ✕✚◗☞✥✘
❣✯✫✯❑❘ ✪❈★❣❪ ✻✼✽✾✻❀✿✾ ✺✸ ❉✫★ ✷❙❚✽✽❯✽❱❲❳✽❪ ❨✳❆❄❅❆❉❈❆❉❘❆❪ ❉✰❆❈✳❆❄❅❆ ❁✮✱❆❩✳❆❪ ✰✧✳ ✪❖P✲✳❍
✕❧❬✥
❁✈✭✮ ❊✮✲ ✪❈★❣❪ ✺✶✷❭❲✸ ✻✼❙❫❴✽❪ ▼✪✭✮✩ ✪✫◆✮✮▼❪ ❖✧✮✲ ✪❈★❣ ✴✮❵✲❏t❪ ✪❖P✲✳ ✪✫❋✫✪✫●✮✲✧❪ ✪❖P✲✳❍
❉❆✫❏❑❆ ✱✮t❊❇✩❪ ✸✶❥✸❪ ❃✮❏❆ ✪❋✮❆❈❆ ❉✰❆❈✳❆❄❅❆❁✮✱❆❩✳❆❪ ◆✮✮❏❊✮✲❍
❇ ❪ ✴✰✮❅❩✴❍
❝✳❆❉❈❆❊✳❆ ✱✮t❇❪ ✻✼✽✾✻✾❀✿ ✺✸ ❃✮❏❆ ✪❋✮❆❈❆ ❉✰❆❈✳❆❄❅❆ ❁✮✱❆❩✳❆❪ ❘▲❈✱
❈✳❆❁✮✱❆ ❊❛❖ ✳❊❪ ✺❞✽❲❤ ✻✼✽✻
✾ ❀✿✾ ✺✸❪ ▼✪✭✮✩ ✪✫◆✮✮▼❪ ◆✮✮✱✩✳✧ ✪✫❦✮✰ ❈★♠♥✮✮✰❪ ❝★▼✲✮▲✱❪ ✴✰✮❅❩✴❍
❁✮✱❆❨✳❆ ❋✮❘✮❅❪ ✻✶♦♣✶♦✹✶♦❪ t✫✮❣✱ ✰✫✮❏❖✧ ✪✫●✮✲✧❪ ❘✯★▼❏❋✮❊✯✱❪ ✪❖P✲✳❍
✱✮❘ ❁✫✩✮✱❪ ✻✼✽✾✻✿❀✾ ✺✸ q❁✫✴✮❋✮❊❛✮ r✩s ❉✫★ ❈✲✮❣✴✮✱❪ ❨✳❆❄❅❆❉❈❆❉❘❆❪ ❉✰❆❈✳❆❄❅❆ ❁✮✱❆❩✳❆❪ ✰✧✳ ✪❖P✲✳❍
❁✮✱❆❊✳❆ ❘✮▲✧❅❪ ✸✶❥✸❪ ❨✳❆❄❅❆❉❈❆❉❘❆❪ ❉✰❆❈✳❆❄❅❆ ❁✮✱❆❩✳❆❪ ✰✧✳ ✪❖P✲✳❍
❉❈❆❉❈❆ ✉✮❏✱❪ ✻✼✽✾✻❀✿✾ ✺✸ ❈❘ ✇❊ ✫✯❑✲❊✪✩❪ ❉✰❆❄❅❆❉❈❆✧❇❆❪ ✩✯✱✮ ✫▲★❑❊❈ ❘❏①✮✮✲✧❍
❉❈❆✫❏❑❆❉❈❆ ▼✮▲✩❘❪ ✻✼✽✾✻❀✾✿ ✺✸ ❨✳❆❄❅❆❉❈❆❉❘❆❪ ❉✰❆❈✳❆❄❅❆❁✮✱❆❩✳❆❪ ✰✧✳ ✪❖P✲✳❍
❉❈❆✫❏❑❆ ✴✮▲✪❋✮✴❪ ✸✶❥✸❪ ▼✪✭✮✩ ✪✫◆✮✮▼❪ ✪✴✱✮❏❨②✳❘✲ ✴✮❵✲❏t❪ ✪❖P✲✳ ✪✫❋✫✪✫●✮✲✧❪ ✪❖P✲✳❍
❈★▼✳✩✮ ❁✱✮❏❨✮② ❪ ✻✶♦♣✶♦✹✶♦❪ ❉❆❊✳❆t❏❆ ♠✫❇❑✲❪ ❈✮✫❏❑✩❪ ✰✧✳ ✪❖P✲✳❍
❋✮▲✲t✮ ✪✩✫✮✱✳❪ ✻✶♦♣✶♦✹✶♦❪ ✫❏★❑❖❛✳✧ ✪✫●✮✲✧❪ ❝✱✴✮✴✮✰✮❪ ❣t✮✱✳❝✮▼❪ ③✮✱✉✮★❨❍
✪✫✰✮✧✴ ❝✯t✮❨❏❪ ④✾❫⑤✸✸❪ ✪✫❖◆✮❅ ❝✯✪✰✧✮❖✳ t❇✪✰✧✱ ✴✮❵✲❏t❪ ❈⑥✴✱❖✮✱✮ ⑦✮▲✴❪ ✰✮▼❊✯✱❪ ❘❣✮✱✮✬❩⑧❍
❈✯✰✳✲ ❝t✮t❪ ✺✶✷❭❲✸ ⑨✻✾⑩✽✷④⑨✹❪ ❉❈❆❈✳❆❄❅❆❁✮✱❆❩✳❆❪ ▼✯❨▼✮❶✫❪ ❣✪✱✧✮✭✮✮❍
✕❧❬✥ ✕✚◗☞✥✘
✫✳❆❊✳❆ ✪❈★❣❪ ✸✶❥✸❪ ❨✳❆❄❅❆❉❈❆❉❘❆❪ ❉✰❆❈✳❆❄❅❆ ❁✮✱❆❩✳❆❪ ✰✧✳ ✪❖P✲✳❍
viii
❢ ✁✂✄ ☎✆✝✁✞✟✠✝✡✞✝☛
▼☞✌✍✎✏✌ ✑✎✒✎✓✔ ✐✕✖✗✕✖✘✕✖✔ ❪✙✎✚✏ ✛✙✎✜✢✣ ✤✙✥✎✦✣✔ ✒✧✩★ ✑✜ ✎✪✧✏✔ ✤✢✫✦☞✬
✪☞✌✙✜✭✌ ✤✮✙✎✏☞✔ ❧✯✰✱✲ ✳✰✱✴✵✶ ✷✍✌✪✸✎✌✹ ✙✜✭✢★ ✸☞✣ ✤✙✥✎✦✣ ✺★✩✻✛✬
✼✺✌✽☞✌ ✤✾✎✪✎✻☞✔ ②✿✵❀❁❁ ✷✩✤❂✎✮✹ ✏✎❪❃☞✣ ✪✸✤✮❄✎✎ ✤✙❃✎✺ ✤✙✥✎✦✣✔ ✺❅✏❪✒✦ ✤✙✚✎✏✔ ✤✢✫✦☞✬
✼✌✙✜✭✌ ✏✎❪✪❅✮✔ ❁✕❥❁ ✷✩✤❂✎✮✹✔ ❆✎✜✌✤✑✎✌✺✌ ✼✛✌✺☞✌❇✓✌✍✎✏✌❈☞✌✔ ❄✎✎✜✪✎✦✔ ✒❉✣ ✪✸ ✢✑✜ ✎✬
✙☞✌✪☞✌ ✤✺★✚✔ ❁✕❥❁ ✷✩✤❂✎✮✹✔ ▼☞✌❇✓✌✼✺✌✼✒✌✔ ✼✛✌✺☞✌❇✓✌✍✎✏✌❈☞✌✔ ✛✣☞ ✤✢✫✦☞✬
❊❋●❍■ ❏❑▲◆❖P
✼✺✌✙✜✭✌ ✤✺★✚ ✩✎◗✮✒✔ ✐❘✰✿❙❚✰✿❦❧❁✔ ▼☞✌❇✓ ✌✼✺✌✼✒✌✔ ✼✛✌✺☞✌❇✓✌✍✎✏✌❈☞✌✔ ✛✣☞ ✤✢✫✦☞✬
✈ ✁ ✂
✐✄☎✆✝✞✟ ✠✡ ✐✝☛✟☞✐✌✍✎✏ ✡✑✒✓✝✝ ✏✝☞✝✔✕✝✝✖✝ ✗✘✙ ✄✚✛✚✄✖✄✜✝✎ ✐✢✄✎✣✝✝✄✤☞✝✘✥ ✗✘✙ ✦✧✌✑✩
★ ☞ ✡✧☞✝✘✤ ✗✘✙ ✄✖✪
✫✐✚✝
✧✝✄✞✔ ✏
✫✝✣✝✝☎
✬☞✭✎
✏☎✎✒
✧✮✯
✰✤✞✒✕✝
✡☎✚✱
✲✳✴✵✲✷✶✵ ✸✹✱
✡✝✥✄ ✜☞✏✒☞
✄✗✣✝✝✤✱
✄✞✩✖✒
✫✝✰✑✤✿❀✱ ❁❂❪✐✢❪❃✺ ✐✒❪✗✘✙❪
✄✗✕✗✄✗❢✝✖☞✺ ✗✌✙✻★✡ ✜✝✝✚✱ ②✵✼✽✹✹✱ ✄✕✝✦✖✒ ✚✘✕✝✚✖ ✐✒❪✰✒❪ ✏✝✾✖ ✰
✘
✄✎✗✝☎✒✱ ✸❧✴❄❅ ❆✴❄❇✼❈ ❁✫❪✐✢✝❪❃✱ ✗✘✙✥ ✞✢✒☞ ✄✗❢✝✖☞ ✡✥✤☛✚✺ ✪✡❪✦✒❪ ✄❉✝✐✝☛✒✱ ②✵✼ ✽✹✹✱ ✫✝☎❪✐✒❪✦✒❪
✄✗❪ ✡★☎✰✑✖ ✄✗✧✝☎✱ ✄✞✩✖✒✺ ✫✝✘❪✪✚❪ ✄✡✥✧✱ ✹❥❊✹✱ ✫✝☎❪✫✝✠✔❪✠✔❪ ✣✝✌✗✚✘✕✗☎✱ ❂❋❀✒ ✡✝✺ ✗✌✙✑✝☎✒ ✡☎✝✘✰✱
②✵✼✽✹✹✱ ✤✗✚✔✑●
✘✥ ✤✩✡✔ ✡✒✄✚☞☎ ✡✘✗✘✙✥ ❋☎✒ ✍✗★ ✙✖✱ ✚❪ ❍✱ ■✐✚✤☎✱ ✄✞✩✖✒✺ ✐✒❪✣✝✝✍✏☎ ✗✌✙✑✝☎✱ ✲❥❏❑❥❏
❱❥❏✱ ✰✗✝✧☎ ✚✗✝✘✞☞ ✄✗❢✝✖☞✱ ✖✘✐✝✓✝✒✱ ✫✚✥✎✐✌☎✱ ❁✫✝✥▲✢ ✐✢ ✞✕
✘ ✝❃✺ ▼✒✑✎✒ ✏✩✐✝✤✑✟✱ ✲❥❏❑❥❏❱❥✱ ✗✘✙❪✗✒❪
✚✝✖ ✗✮✙✐
✥ ✡✱ ✦✮✥✤✖✝✘☎✺ ☎✝✧✌✖ ✡✝✘✐✙✎✱ ②✵✼ ✽✹✹✱ ✪✫☎ ✐✙✝✘✡✔ ✤✝✘✩❋✚ ✰✌✦✖✒ ✠✥✄✍●●✟☞★●✱ ✡✌✦✎
✢ ✝✘ ✐✝✗✔✙✱ ✚☞✒
✄✞✩✖✒✺ ✗✥✄ ✞✎✝ ✏✝✖☎✝✱ ✡✗✝✘✔✞☞ ✏◆☞✝ ✄✗❢✝✖☞✱ ✄✗✏✝✡✐✌☎✒ ✰✚✐✞ ✗✘✥✙✞✢✱ ✚☞✒ ✄✞✩✖✒✺ ✰✚✝✞✔✚
✄❉✝✐✝☛✒✱ ②✵✼ ✽✹✹✱ ✤✗✚✔✑●
✘✥ ✫✝☎❪✪❖❪✪✡❪✪✡❪ ✪✘✰✝✬✖✱ ✄✑✰✝✘☎✑ ✫✝✮☎ ✡✌▼✒ ✡✌✆✝✑✝ ✰☞☎P✝✱ ✹❥❊✹✱ ❋✒❪
❋◗✖☞★❪✪✡❪✱ ✪✚❪✡✒❪✠✔❪✫✝☎❪●✒✱ ✚☞✒ ✄✞✩✖✒❘
✐✄☎✆✝✞✟ ✪✚❪✡✒❪✠✔❪✫✝☎❪●✒❪ ✑✘✥ ✄✧✥✞✒ ■✐✝✎✥☎❙✝ ✗✘✙ ✐✌✚ ☎✝✗✖✝✘✏✚ ✧✘✎✌ ✏✝☞✔✕ ✝✝✖✝ ✑✘✥ ✄✚✛✚✄✖✄✜✝✎
✐✢✄✎✣✝✝✄✤☞✝✘✥ ✏✒ ✦✧✌✑★✩☞ ✄●❚✐✄❙✝☞✝✘✥ ✗✘✙ ✄✖✪ ✫✝✣✝✝☎✒ ✧✮✺ ✰✒❪❋✒❪✿✖✱ ✫✗✏✝✕✝✐✢✝❚✎ ✹❥❊✹✱ ✪✚❪✡✒❪✠✔❪✫✝☎❪
●✒❪✱ ✚☞✒ ✄✞✩✖✒✺ ✰✒❪✪✡❪☎✝☛✝✮☎✱ ❆❯✸❲❱✵❱
❳ ✲✳✴✵✲✷✵✶ ✸✹✱ ✤✄❙✝✎ ✪✗✥ ✡✥✝✄✜☞✏✒ ✄✗✣✝✝✤✱ ✪✑❪✪✖❪ ✡✌✜✝✝✄❋❀☞✝
✄✗✕✗✄✗❢✝✖☞✱ ❂✞☞✐✌☎✱ ☎✝✰✍P✝✝✚✺ ✑✚✝✘✰ ✗✌✙✑✝☎ ☛✝✗✌✙☎✱ ❋✒❪✪❪✗✒❪ ✐✄◗✖✏ ✍✗★✙✖✱ ☎✝✰✘✞
✥ ✢ ✚✤☎✱
✡✝✄✧✦✝✦✝✞✱ ✤✝✄✰☞✝✦✝✞ ❁❂❪✐✢❪❃✺ ☎✝✑✘✕✗☎ ✞☞✝✖ ✕✝✑✝✔✱ ☎✝✰✏✒☞ ✠✥●☎ ✏✝✾✖ ✘✰✱ ✑P✝✌☎✝ ❁❂❪✐✢❪❃✺
❋✝✾❪ ✫✝☎❪✐✒❪ ✄✤✧✝☎✘✱ ◗✖✝✾✏ ✄☎✡✝✘✔✡ ✏✝✘✫✝✄❋✔✚●
✘ ☎✱ ✰✚✐✞ ✄✕✝✓✝✝ ✗✘✙✥ ✞✢✱ ✄❖❖✝✮✖✒✱ ✦✘✎✖
✌
❁✑❪✐✢❪❃✺ ✡✌✚✒✖
✦✰✝✰✱ ✪✡❪✡✒❪✠✔❪✫✝☎❪●✒❪✱ ✤✌❋ ✤
❀ ✝❝✗✱ ✧✄☎☞✝❙✝✝✺ ▼✒✑✎✒ ✗✒✚✝ ▲✒✥✤☎✝✱ ✡☎ ✖✓✑✒ ✦✝✄✖✏✝ ✡✒✄✚☞☎
✡✘✗✘✥✙❋☎✒ ✍✗★ ✙✖✱ ✜✝✝☎✒ ✦✝✗✖✒✱ ✄✞✩✖✒✺ ✪❪✗✘✙❪ ✗❨✖✗✝☎✱ ✹❥❊✹✱ ✪✚❪✡✒❪✠✔❪✫✝☎❪●✒✱ ✚☞✒ ✄✞✩✖✒❘
✐✄☎✆✝✞✟ ✄❖❉✝✝✥✏✚ ✫☎✄✗✥✞☎ ❖✝✗✖✝✱ ❅❳❞❄❩❱✹ ❲❱✵❬✴❭ ✲✳❫✴✴✹❥ ✞✒✐✏ ✏✐★☎✺ ☎✝✗✘✙✕✝ ✗✌✙✑✝☎ ✪✗✥ ✡❴✰✝✞
❩ ✷✶ ✹❥❊✹✱ ■✦✒
✧✮✞☎ ✫✥✡✝☎✒✱ ❊❥❏❱❥❏✲❥❏ ❆✴❵✲✹✵❱✹✺ ✗✘✙❪✐✒❪✪✡❪☞✝✞✗✱ ❡❭✴✵❑ ❡✴✵❧❭❛ ❅✴❵✲❥ ❜❯❊❱✹ ✎P✝✝ ✲✳✲
✗✌✙✑✝☎✒✱ ✫✄✣✝✑◆☞✌ ✑✧✝✄◆✎ ✎P✝✝ ☎❙✝▲✒☎ ☛✝✗✌✙☎ ♦✝☎✝ ✄✏✪ ✤✪ ✐✢☞ ✝✡✝✘✥ ✗✘✙ ✐✢✄✎ ✫✐✚✝ ✫✝✣✝✝☎ ✐✢✏● ✏☎✎✒
✧✮❘ ✪❪✐✒❪✡✒❪ ✫✝✾✄✐✙✡✱ ✄✗❣✝✚ ✪✗✥ ✤✄❙✝✎ ✄✕✝✓✝✝ ✄✗✣✝✝✤ ✪✗✥ ✐✢✏✝✕✝✚ ✄✗✣✝✝✤ ✣✝✒ ✫✐✚✘ ✡✧☞✝✘✤ ✗✘✙ ✄✖✪
✫✝✣✝✝☎ ✗✘✙ ✐✝❉✝ ✧✮ ✥❘
x
❢ ✁✂✄☎✆✝✞✟
❍✂✂✠ ☎ I
1.
2.
3.
4.
✈✡☛☞✌✡
✐✍✎✏✡✑✒✡
❧✓✔✕✓ ✖✑✓ ✐✗✘✒
1.1 ✙✚✛✜✢✣✚
1.2 ✤✥✦✧
✥ ✚❦✥ ★❦✩ ✪✫✣✚✬
1.3 ✪✩✭✮✚❦✥ ★❦✩ ✪✫✣✚✬
1.4 ✪✩✭✮✚❦✥ ✣✚ ✤✥✯✚❦✰✮ ✱✲✚✚ ✳✯✴✵✶✢✷✚✸✯ ✪✩✭✮
1.5 ✜✹✺✻✚✧✚✬✸ ✤✥✜✶✯✚✼✽
✐✍✾✏✘✡✿☛ ✾❀✡❁✡✿❂✡✾☛✏❃❄ ✐✗✘✒
2.1 ✙✚✛✜✢✣✚
2.2 ✻✚✧✚✬✙✚✛✱ ✤✥✣❅✪✮✚✼✽
2.3 ✪✫✜✱✭✚❦✢ ✜❆✚✣✚❦✷✚✜✢✱✸✯ ✪✩✭✮✚❦✥ ★❦✩ ❇✴✷✚✧✢❈
✈✡❉❄❊❋
3.1 ✙✚✛✜✢✣✚
3.2 ✻✚✳✯✛●
3.3 ✻✚✳✯✛●✚❦✥ ★❦✩ ✪✫✣✚✬
3.4 ✻✚✳✯✛●✚❦✥ ✪✬ ✤✥✜✶✯✚✼✽
3.5 ✻✚✳✯✛● ✣✚ ✪✜✬★✱❈
3.6 ✤✢✜✢✱ ✱✲✚✚ ✜★■✚✢ ✤✢✜✢✱ ✻✚✳✯✛●
3.7 ✻✚✳✯✛● ✪✬ ✪✫✚✬✥✜✙✚✣ ✤✥✜✶✯✚ ❏✻✚✳✯✛● ❑✪✚✥✱✬✷✚▲
3.8 ✳✯✴✵✶✢✷✚✸✯ ✻✚✳✯✛●
❧✡▼✾❂✡❁
4.1 ✙✚✛✜✢✣✚
4.2 ✤✚✬✜✷✚✣
4.3 ✤✚✬✜✷✚✣✚❦✥ ★❦✩ ❇✴✷✚✧✢❈
4.4 ✜❆✚✙✚✴✰ ✣✚ ◆✚❦❆✚✪✩✭
iii
v
1
1
2
8
13
22
38
38
38
48
62
62
62
67
71
91
93
98
99
112
112
113
119
131
xii
4.5
4.6
4.7
♠ ✁✂✄☎✆✂✝ ✞✂✟✄ ✁✠✡✂☛☞
✞✂✈✌✍✠ ✎✏✑ ✁✠✡✂☛☞✒ ✞✂✟✄ ✈✌✓✔✕✖
✁✂✄☎✆✂✝✂✏☛ ✞✂✟✄ ✞✂✈✌✍✠✂✏☛ ✎✏✑ ✞❧✓ ✗✌✂✏✘
133
137
144
5.
✙✚✛✜✢✣ ✜✤✚✚ ✥✦✧★✩✪✣✜✚
5.1 ❍✂✍☎✖✝✂
5.2 ✁✂☛✫✔✌
5.3 ✞✎✝✬❧✭✌✫✂
5.4 ♣✄✮✂✂✫✂☛✝✭ ✫✯✂✂ ✬✮✂✓✘✆✂✝✭✌ ✑✬❧
5.5 ✬✮✂✓✘✆✂✝✭✌ ✞✎✝✬❧
5.6
✑✬❧✂✏☛ ✎✏✑ ✂✗ ♣☎✬✝ ✐ ✂✏☛ ✎✏✑ ✞✎✝✬✒
5.7 ☎❢✫✭✌ ✝✂✏☎✰ ✝✂ ✞✎✝✬✒
5.8 ✖✂❡✌✖✂❧ ✗✖✏✌
160
160
160
176
185
191
195
197
200
6.
✥✦✧★✱ ✦✲✳ ✥✩✴✵✣
✶ ✚✲✷
6.1 ❍✂✍☎✖✝✂
6.2 ✄✂☎❥✂✌✂✏☛ ✎✏✑ ☎✄✎✫✸❧ ✝✭ ✹✄
6.3 ✎♦✸✖✂❧ ✞✂✟✄ ✺✂✁✖✂❧ ✑✬❧
6.4 ▲ ❥✂✸ ✄✏✡✂✂✻✼ ✞✂✟✄ ✞☎❍✂✬☛✽
6.5 ✁☎✾❧✝✰❧
6.6 ♠✿♣✫✖ ✞✂✟✄ ☎❧❀❧✫✖
210
210
210
215
223
229
233
✵❁❂❁❃✚❄❅❆❇ ✷❁❈✚✜ ❉✲✛ ❊✵✵❁❋✚✣✚●
A.1.1 ❍✂✍☎✖✝✂
A.1.2 ♠ ☎■✂ ❏✌✂ ✠✟?
265
265
265
✵❁❂❁❃✚❄❅ 2: ✷❁❈✚✜✪✣ ❁✩①❃✚❑✩
A.2.1 ❍✂✍☎✖✝✂
A.2.2 ✘☎✆✂✫✭✌ ☎❧✹❥✂✸❧ ❏✌✂✏☛▼
A.2.3 ✘☎✆✂✫✭✌ ☎❧✹❥✂✸❧ ✎✏✑ ☎✁◆✂☛ ✫
274
274
274
275
❖P◗❘❙◗❚◗
❯❱❲
❳❨❘❩ ❳◗❬❭❪ ❫◗❙❴❵❛
❜❝❜
❢ ✁✂✄☎✆✝✞✟
❍✂✂✠ ☎ II
7.
8.
9.
✈✡❡☛☞✡
✐✌✍✎✡✏✑✡
❧❡✡✒✓✑
7.1 ✔✕✖✗✘✙✕
7.2 ✚✘✕✙✛✜ ✙✕✢ ✣✤✙✛✜ ✤✢✥ ✦✧★✩✪✘ ✫✬✪✘ ✤✢✥ ✭✫ ✘✢✮
7.3 ✚✘✕✙✛✜ ✙✯ ✗✤✗✰✧✕✱
7.4 ✤★✥♦ ✗✤✗✲✕✳✴ ✫✥✛✜✕✢ ✮ ✤✢✥ ✚✘✕✙✛✜
7.5 ✣✕✮✗✲✕✙ ✗✔✕✵✜✕✢✮ ✶✕✷✕ ✚✘✕✙✛✜
7.6 ❬✕✮✸✲✕✹ ✚✘✕✙✛✜
7.7 ✗✜✗✲✺✻ ✚✘✕✙✛✜
7.8 ✙✛✜ ✙✯ ✣✕✰✕✷✔✕✖✻ ✫✬✘✧
✢
7.9 ✫✬✗✻✼✽✕✕✫✜ ✶✕✷✕ ✗✜✗✲✺✻ ✚✘✕✙✛✜✕✢✮ ✙✕ ✘✕✜ ✾✕✻ ✙✷✜✕
7.10 ✗✜✗✲✺✻ ✚✘✕✙✛✜✕✢✮ ✤✢✥ ✤★✥♦ ✿★❀✕✰✘❁
❧❡✡✒✓✑✡❂❃ ✏❂❄ ✈✑☛✐✌❅✡❂❆
8.1 ✔✕✖✗✘✙✕
8.2 ✚✕✰✕✷❀✕ ✤✪✕✢✮ ✤✢✥ ✣✮✻✿❁✻ ❇✕✢❈✕✫✥✛
8.3 ♥✕✢ ✤✪✕✢✮ ✤✢✥ ✘❉✧✤✻✯❁ ❇✕✢❈✕ ✙✕ ❇✕✢❈✕✫✥✛
✈✏✒✓ ❧❡❊✒❋●✡
9.1 ✔✕✖✗✘✙✕
9.2 ✣✕✰✕✷✔✕✖✻ ✚✮✙■✫✜✕❏✱
9.3 ✣✤✙✛ ✚✘✯✙✷❀✕ ✙✕ ✦✧✕✫✙ ❏✤✮ ✗✤✗✲✕✳✴ ❑✛
9.4 ✗♥❏ ❑★❏ ✦✧✕✫✙ ❑✛ ✤✕✛✢ ✣✤✙✛ ✚✘✯✙✷❀✕ ✙✕ ✗✜✘✕❁❀✕
9.5 ✫✬✽✕✘ ✙✕✢✗✴ ❏✤✮ ✫✬✽✕✘ ▲✕✕✻ ✤✢✥ ✣✤✙✛ ✚✘✯✙✷❀✕✕✢✮ ✙✕✢ ❑✛ ✙✷✜✢ ✙✯ ✗✤✗✰✧✕✱
iii
v
303
303
304
316
324
333
340
347
351
355
357
376
376
376
383
395
395
396
399
402
408
xiv
10.
❧ ✁✂✄ ☎✆✝✞ ✟✄✠
10.1
10.2
10.3
10.4
10.5
10.6
11.
11.6
11.7
11.8
11.9
11.10
✖☞✢✣✡✡✤✗ ♦✤✏ ✙✥✍ ✡✔
✖☞✢✣✡✡✤✗ ✍✡ ✦✡✤✧✙✏★
✛✍ ✒☞✢✣✡ ✖✤ ✖☞✢✣✡ ✍✡ ✧✎✱ ✡✚
✢✡✤ ✖☞✢✣✡✡✤✗ ✍✡ ✧✎✱ ✡✚✙✏★
❍✡☛☞✌✍✡
✔✤❥✡✡ ♦✤✏ ☞✢♦✮✏✯✍✡✤✖✡✰✚ ✒✡✲✔ ☞✢♦✮✏✯✒✚✎✙ ✡✕
✒✗✕☞✔✈✡ ✌✤✗ ✔✤ ❥✡✡ ✍✡ ✖✌✳✍✔✱✡
✢✡✤ ✔✤ ❥✡✡✒✡✤✗ ♦✤✏ ✌♥✦ ✍✡✤✱ ✡
✢✡✤ ✔✤❥✡✡✒✡✤✗ ♦✤✏ ✌♥✦ ✴✦☛✚✕✌ ✢☛✔✳
✖✌✕★
✢✡✤ ✔✤❥✡✡✒✡✤✗ ✍✡ ✖✵✯✕★✳✦ ✵✡✤✚✡
✢✡✤ ✖✌✕★✡✤✗ ♦✤✏ ✶✳✷ ✍✡ ✍✡✤✱ ✡
✖✌✕★ ✖✤ ☞✢✛ ✧✛ ✸✶✢✎ ✍✳ ✢☛✔✳
✛✍ ✔✤ ❥ ✡✡ ✒✡✲✔ ✛✍ ✖✌✕★ ♦✤ ✏ ✶✳✷ ✍✡ ✍✡✤ ✱ ✡
✹✺ ✻✄✼ ✽✾✄✿✞✾✄✫❀
12.1
12.2
12.3
13.
♦✎✏✑ ✒✡✓✡✔❍✡☛✕ ✖✗✍✘✙✚✡✛✜
❢✄✩ ✪✫✆✬ ✭✬✄ ✫ ✠
11.1
11.2
11.3
11.4
11.5
12.
❍✡☛☞✌✍✡
❍✡☛☞✌✍✡
✔✲☞❥✡✍ ✙✥✡✤✧ ✥✡✌✚ ✖✌❁✦✡ ✒✡✲✔ ❂✖✍✡ ✧☞✱✡✕✳✦ ✖☛❃✡✳✍✔✱✡
✔✲☞❥✡✍ ✙✥✡✤✧✥✡✌✚ ✖✌❁✦✡✒✡✤✗ ♦✤✏ ☞❍✡✴✚ ✙✥✍✡✔
✽✾✄ ✬✼✠✄
13.1
13.2
13.3
13.4
13.5
13.6
13.7
❍✡☛☞✌✍✡
✖✙✥☞✕✶✗✓ ✙✥✡☞✦✍✕✡
✙✥✡☞✦✍✕✡ ✍✡ ✧✎✱ ✡✚ ☞✚✦✌
❁♦✕✗❃✡ ▲✡❄✚✡✛✜
✶✤❝✡✯✙✥✌✤✦
✦✡✢❅☞❆✑✍ ✷✔ ✒✡✲✔ ✰✖♦✤✏ ✙✥✡☞✦✍✕✡ ✶✗❄✚
✶✔✚✡✲★✳ ✙✔✳✈✡✱✡ ✒✡✲✔ ☞❇✙✢ ✶✗❄✚
440
440
440
443
445
448
456
477
477
477
482
485
487
493
501
503
505
506
519
519
520
529
547
547
547
556
558
565
574
588
♠❈❉❊❋❉●❉
■❏❑
✐▼❊◆ ✐❉❖P◗ ❘❉❋❙❚❯
■❱❲
❁❂❃❄❃
7
❧❅❆❇❈❉ Integrals
Just as a mountaineer climbs a mountain – because it is there, so
a good mathematics student studies new material because
it is there. – JAMES B. BRISTOL
❍✁✂✄☎✆✁ (Introduction)
✈✝✞✟ ✠✡☛☞✌ ✈✝✞✟✍ ✞✎ ✏✑✞✒✓✔☞ ✓✕ ✝✖✗✑ ✡✘✙✌ ✚✛✜ ✓✗✟✔☞✖✑ ✝✖✗
✈☞✟✖✢☞☞✖✑ ✝✖✗ ✡✟✣ ✤✓✥☞✦ ✕✖✢☞ ✣✧ ✓✡✕★☞☞✡✩☞✌ ✞✕✔✖ ✞✎ ✏✪✤✫☞ ✣✝✑ ✬✏
✓✙✞☞✕ ✞✎ ✕✖✢☞☞✈☞✖✑ ✞✎ ✓✙✝☛☞✌☞ ✞☞ ✓✡✕✞✟✔ ✞✕✔☞ ✈✝✞✟✍ ✝✖✗
✡✟✣ ✪❢✟ ✈✡★☞✓✙✖✕☛☞ ✭☞☞✜ ✏✪☞✞✟✔ ✠✡☛☞✌✮ ✓✗✟✔☞✖✑ ✝✖✗ ✈☞✟✖✢☞ ✏✖ ✡✯☞✕✖
④☞✖✰☞ ✝✖✗ ④☞✖✰☞✓✗✟ ✞☞✖ ✓✡✕★☞☞✡✩☞✌ ✞✕✔✖ ✣✝✑ ✬✏✝✖✗ ④☞✖✰☞✓✗✟ ✞☞
✓✡✕✞✟✔ ✞✕✔✖ ✞✎ ✏✪✤✫☞ ✏✖ ✓✖✡✙ ✕✌ ✚✛✜
✫✡✘ ✣✞ ✓✗✟✔ f ✡✞✏✎ ✈✑✌✕☞✟ I ✪✖✑ ✈✝✞✟✔✎✫ ✚✛ ✈✭☞☞✦✌❡ I
✝✖✗ ✓✙♦✫✖✞ ✱✲✘✳ ✓✕ ✓✗✟✔ ✝✖✗ ✈✝✞✟✍ f ✴ ✞☞ ✈✡✤✌♦✝ ✚✛✮ ✌✲ ✣✞
✤✝☞★☞☞✡✝✞ ✓✙✥✔ ▲✵✌☞ ✚✛ ✡✞ ✫✡✘ I ✝✖✗ ✓✙♦✫✖✞ ✱✲✘✳ ✓✕ f ✴ ✡✘✫☞
✚✳✈☞ ✚✛ ✌☞✖ ❣✫☞ ✚✪ ✓✗✟✔ f ❑☞✌ ✞✕ ✏✞✌✖ ✚✛✶✑ ✝✖ ✏★☞✎ ✓✗✟✔
✡✍✔✏✖ ✚✪✖✑ ✣✞ ✓✗✟✔ ▲✔✝✖✗ ✈✝✞✟✍ ✝✖✗ ✷✓ ✪✖✑ ✓✙☞✸✌ ✚✳✈☞ ✚✛✮ ✬✏
✓✗✟✔ ✝✖✗ ✓✙✡✌✈✝✞✟✍ ✐✓❢✝✠✦ ✹ ✞✚✟☞✌✖ ✚✛✜✑ ✈✠✙✌✺ ✝✚ ✏❢✰☞ ✡✍✏✏✖
✫✖ ✏★☞✎ ✓✙✡✌✈✝✞✟✍ ✓✙☞✸✌ ✚☞✖✌✖ ✚✛✑✮ ✓✗✟✔ ✞☞ ✈✡✔✡✥❀✌ ✏✪☞✞✟✔ ✞✚✟☞✌☞ ✚✛ ✈☞✛✕ ✓✙✡✌✈✝✞✟✍
❑☞✌ ✞✕✔✖ ✞☞ ✫✚ ✓✙✻✪ ✏✪☞✞✟✔ ✞✕✔☞ ✞✚✟☞✌☞ ✚✛✜ ✬✏ ✓✙✞☞✕ ✞✎ ✏✪✤✫☞✣✧ ✈✔✖✞ ✼✫☞✝✚☞✡✕✞
✓✡✕✡✤✭☞✡✌✫☞✖✑ ✪✖✑ ✈☞✌✎ ✚✛✜✑ ▲✘☞✚✕☛☞✌✺ ✫✡✘ ✚✪✖✑ ✡✞✏✎ ④☞☛☞ ✓✕ ✡✞✏✎ ✝✤✌✳ ✞☞ ✌☞♦④☞✡☛☞✞ ✝✖✠ ❑☞✌ ✚✛✮
✌☞✖ ✤✝☞★☞☞✡✝✞ ✓✙✥✔ ✫✚ ▲✵✌☞ ✚✛ ✡✞ ❣✫☞ ✚✪ ✡✞✏✎ ④☞☛☞ ✓✕ ▲✏ ✝✤✌✳ ✞✎ ✡✤✭☞✡✌ ❑☞✌ ✞✕ ✏✞✌✖
✚✛✑✶ ✬✏ ✓✙✞☞✕ ✞✎ ✈✔✖✞ ✼✫☞✝✚☞✡✕✞ ✣✝✑ ✏✛✽☞✑✡✌✞ ✓✡✕✡✤✭☞✡✌✫☞✧ ✈☞✌✎ ✚✛✑✮ ✍✚☞✧ ✏✪☞✞✟✔ ✞✎ ✏✑✡✻✫☞
✡✔✡✚✌ ✚☞✖✌✎ ✚✛✜ ✏✪☞✞✟✔ ✠✡☛☞✌ ✞☞ ✡✝✞☞✏ ✡✔✾✔✡✟✡✢☞✌ ✓✙✞☞✕ ✞✎ ✏✪✤✫☞✈☞✖✑ ✝✖✗ ✚✟ ✞✕✔✖ ✝✖✗
✓✙✫☞✏☞✖✑ ✞☞ ✓✙✡✌✓✗✟ ✚✛✜
(a) ✫✡✘ ✣✞ ✓✗✟✔ ✞☞ ✈✝✞✟✍ ❑☞✌ ✚☞✖✮ ✌☞✖ ▲✏ ✓✗✟✔ ✞☞✖ ❑☞✌ ✞✕✔✖ ✞✎ ✏✪✤✫☞✮
(b) ✡✔✡✥❀✌ ✓✙ ✡ ✌✲✑✿☞✖ ✑ ✝✖ ✗ ✈✑ ✌✠✦ ✌ ✓✗✟✔ ✝✖ ✗ ✈☞✟✖ ✢☞ ✏✖ ✡✯☞✕✖ ④☞✖ ✰ ☞ ✞☞ ④☞✖ ✰ ☞✓✗✟ ❑☞✌ ✞✕✔✖
✞✎ ✏✪✤✫☞✜
7.1
G .W. Leibnitz
(1646–1716)
304
① ✁✂✄
♠☎✆✝✞✟✠ ✡☛☞✌☛☞ ✍✎✏✆☛✑✒
✢✌✢✥✦✠
✚✢✌✢✥✦✠
☎✜✎ ☞✆
✍☞
✖☞✗
✘☎
✚✌☞✓
✭☛✌☛
✎☞✕
✧✍
✢✖✢✬✆☛☞✕
✓✙
✑✖✕
✭☛✠☛
✚☛✤✲☛☛✛
☎✜✓☛✛
✭☛✠☛
✢✓✆☛
✡☛☞✌☛☞✕ ✓☛
✍✎☛✓✔✌
✱✆☛✖✣☛✢✛✓
✑✓
✧✌
✍✎☛✓✔✌❢
✍✎☛✓✔✌☛☞✕ ✖☞✗
✡☛☞ ✘☎☛☞✕ ✓✙ ✚☛☞✛
✍✢★✎✢✔✠
✢✌✢✥✦✠
✣✤❢
✆✣
✘☎
☎✜✎ ✆
☞
✩✢✪☛✠ ✓✣✔☛✠☛
✍✎☛✓✔✌
✍✎☛✓✔✌
✖☞✗
✢✌✢✥✦✠
☎✜✢
☞ ✛✠ ✓✛✠✙ ✣✤❪
✕
✚✢✌✢✥✦✠
✍✎☛✓✔✌
✓☛☞
✖☞✗ ✘☎ ✎☞✕ ✠✤ ✆☛✛ ✓✛✠✙ ✣✤❢ ✚✳☛✞ ✥☛☛✏✰☛❪
❧✢✦✓✛
✍✎✏✆☛✚☛☞✕
✓☛☞
✣✔
✍✕✫✬
✕
✎✈✆ ✑✓
✓✛✌☞
✖☞✗
✓✙ ✚☛✬☛✛✮☛✯ ✠
✑✖✕ ✚✢✮☛✆☛✕✢✰☛✓✙
✑✖✕ ☎✜☛ ✢✆✓✠☛
✮☛✙
✑✖✕
✣✤❢
✢✭✍☞ ✓✔✌
✢✖✐☛✌
✢✖✴☛
✢✔✑
✣✤
✍✎☛✓✔✌
✢✌✢✥✦✠
✖☞✗
✢✔✑
✭✤✍☞ ✢✖✢✮☛✵✌
✶☛☞✰☛☛☞✕
✍✎☛✓✔✌
✓☛
♠☎✆☛☞✩
✣✤❢
✚✈✆☛✆
✍✢✣✠
✎☞✕ ❪ ✣✎
✚☎✌☞
☎✜☛✛✕ ✢✮☛✓
♠✌✖☞✗
✚☛☎✓☛☞
✚✢✌✢✥✦✠
✩✝✪☛✬✎☛☞✷
✖☞✗
✑✖✕
✚✈✆✆✌
7.2
Inverse Process of Differentiation
✹✺✻✼✽✾
✼✻✿
❀❁✼✽✾
❁✿❂
❃❄❅❆❇✺
✢✌✢✥✦✠
✠✓
✍✎☛✓✔✌☛☞ ✕ ✑✖✕
✍✙✢✎✠
❈❉❇✺
❁✿❂
✍✎☛✓✔✌
✓✙
✖✝✗❜
✛✸☛☞✕✩☞ ❢
❊❈
✺✿ ❋
●
Integration as the
❍
✚✖✓✔✌ ✖☞✗ ✱✆✝■ ❏✎
☎✛
✣✎☞✕
✢✔✑
☎✗✔✌
✓✣☛
✚☛✧✑
✩✆☛
✚✖✓✔✭ ✢✡✆☛
✣✤ ❢
✢✌★✌✢✔✢✸☛✠
✭☛✌✠☞
✣✎
✓☛
✣✤✕
☎✜❏✎ ✓☛☞ ✍✎☛✓✔✌ ✓✣✠☞ ✣✤✕ ❢ ✢✓✍✙ ☎✗✔✌ ✓☛ ✚✖✓✔✌ ✐☛✠ ✓✛✌☞ ✖☞✗ ✏✳☛☛✌
✆✣
☎✜❏✎
♠✡☛✣✛✪☛☛☞✕
✣✝✚☛
✍✎☛✓✔✌
☎✛
✢✖✦☛✛
❖ ◆
✚☛✤✛
☎✜☞✢✶☛✠ ✓✛✠☞ ✣✤✕
✢✓
✧✍
☎✜✓☛✛ ✮☛✙ ✓✣✠☞ ✣✤✕ ✢✓
✑✖✕
❣❳❱
x2
✚☛✤✛
✧✍✢✔✑
☎✜✢✠▲✚✖✓✔✌
✌✣ ✙✕
✣✤✕ ❢ ✖✏✠✝✠❨
✐☛✠ ✓✛✌☞
✚☛✤✛
ex
✖☞✗
✣✎
❣❯❱❪
✣✤❢
❘
d x
(e ) = e x
dx
... (3)
✓☛
3
❚
❙
❣❯❱ ✎☞✕ ☎✗✔✌
☎✜ ✢✠✚✖✓✔✭
☎✜✢✠✚✖✓✔✭
❣❲❱
✣✎
✧✌
✡ ☞ ✸☛ ✠☞
✣✤✕
✢✓
✖☞✗
... (2)
◗
P
✍✎✙✓✛✪☛
cos x
3
❣✚✳☛✖☛
C
✚☛✤✛
❣❳❱
cos x
♠☎ ✆✝ ✞ ✟✠
☎✗✔✌☛☞✕ ✎☞✕ ✍☞ ☎✜■ ✆☞✓
☎✗✔✌
❣✚✳☛✖☛
sin x
✍✎☛✓✔✌❱
✍✎☛✓✔✌❱
❏✎✥☛❨
✓☛
✚✖✓✔✭ ✣✤❢
sin x
x3
3
✧✍☞ ✣✎
✣✤❢ ✧✍✙ ☎✜✓☛✛ ❣❲❱
✚☛✤✛
ex
✣✤❢
☎✝✌❨
✣✎
❪ ✢✭✍☞ ✚✦✛ ☎✗✔✌ ✎☛✌☛ ✭☛✠☛ ✣✤❪ ✓☛ ✚✖✓✔✭ ✥☛✯✵✆
✓☛☞
✢✌★✌✢✔✢✸☛✠
❩
☎✜✓ ☛ ✛
✓✣✔☛✠☛
☎✗✔✌
... (1)
d
d x3
( + C)
(sin x + C) cos x ,
dx 3
dx
✧✍
✖☛✏✠✢✖✓
✓✛☞✕ ❪
✌☛☞✉ ✓✛✠☞ ✣✤✕ ✢✓ ✢✓✍✙ ✮☛✙ ✖☛✏✠✢✖✓ ✍✕ ✸✆☛
✣✤❪
✚✳☛☛✞ ✠❑
= x2
▼◆
✍☞
✚✳☛✖☛
☎✯ ✖✞✩
d
(sin x) = cos x
dx
✢✓
▼
✣✎
✧✍✓☛
✣✤ ✚☛✤✛
☎✗ ✔✌☛ ☞ ✕
☎✗✔✌
✖☞✗
✖☞✗
❬
x2
✘☎
✚☛✤✛
✎☞✕
✢✔✸☛
✍✓✠☞
✣✤❨
✕
d x
(e + C) e x
dx
☎✜ ✢ ✠✚✖✓ ✔✭
❩
✚✳☛ ✖☛
✚☎✢✛✢✎✠ ☎✜ ✢✠✚✖✓✔✭ ✣✤❪
✕
✍✎☛✓ ✔✌
✚✢❭✠✙✆
✢✭✵✣☞✕ ✣✎
✖☛✏✠✢✖✓
❧ ✁✂✄☎
305
✆✝✞✟✠✡✠☛✝ ☞☛✌ ✆✍✎✏✑✟ ✆☛ ✒☞☛✏✓ ✡✑✔ C ❞✠☛ ❞✠☛✕✖ ✍✠✗ ✘✙✚✠✗ ❞✔☞☛✌ ✘✙✠✛✜ ❞✔ ✆❞✜☛ ✢✣✝✤ ✟✢✥ ❞✠✔✦✠
✢✣ ❣❞ C ❞✠☛ ✘✙✧✠✠✗✎✆✠✔ ✒☞☛✏ ✓ ✡✑✔ ❞✢✜☛ ✢✣✤✝ ☞✒✜✎✜★ C ✱❞ ✘✙✠✑✩ ✢✣✪ ❣✫✆☞☛✌ ✍✠✗ ❞✠☛ ✘❣✔☞❣✜✖ ✜
❞✔☞☛✌ ✢✍ ❣✚✱ ✢✎✱ ✘✌✩✗ ☞☛✌ ❣☞❣✬✠✭✗ ✘✙❣✜✡☞❞✩✫✠☛✝ ✟✠ ✆✍✠❞✩✗✠☛✝ ❞✠☛ ✘✙✠✛✜ ❞✔✜☛ ✢✣✤✝ ✮✟✠✘❞✜★ ✟❣✚
✱❞ ✘✌✩✗ F ✱☛✆✠ ✢✣ ❣❞
d
F (x) = f (x) , ✯ x ✰ I ✲☞✠✒✜❣☞❞ ✆✝✞✟✠✡✠☛✝ ❞✠ ✡✝ ✜✔✠✩✳ ✜✠☛ ✘✙✴✟☛ ❞
dx
d
F (x) + C = f (x) , x ✰ I
dx
✕✆ ✘✙❞✠✔ {F + C, C ✰ R}, f ☞☛✌ ✘✙ ❣✜✡☞❞✩✫✠☛✝ ☞☛✌ ✘❣✔☞✠✔ ❞✠☛ ✮✟✵✜ ❞✔✜✠ ✢✣✪ ✫✢✠✶ C
✒☞☛✏✓ ✡✑✔ C, ☞☛✌ ❣✩✱
✆✍✠❞✩✗ ❞✠ ✡✑✔ ❞✢✩✠✜✠ ✢✣✤
❢✷✸✹✺✻✼ ✆✍✠✗ ✡☞❞✩✫ ☞✠✩☛ ✘✌✩✗✠☛✝ ✍☛✝ ✱❞ ✡✑✔ ❞✠ ✡✝✜✔ ✢✠☛✜✠ ✢✣✤ ✕✆❞✠☛ ✚✽✠✠✖✗☛ ☞☛✌ ❣✩✱✪ ✍✠✗
✩✥❣✫✱ g ✡✠✣✔ h ✱☛✆☛ ✚✠☛ ✘✌✩✗ ✢✣✝ ❣✫✗☞☛✌ ✡☞❞✩✫ ✡✝✜✔✠✩ I ✍☛✝ ✆✍✠✗ ✢✝ ✣
f (x) = g (x) – h (x), ✾ x ✰ I ⑥✠✔✠ ✘❣✔✬✠✠❣✿✠✜ ✘✌✩✗ f = g – h ✘✔ ❣☞✑✠✔ ❞✥❣✫✱
df
= f❀ = g❀ – h❀ ✆☛ f❀ (x) = g❀ (x) – h❀ (x) ✯ x ✰ I ✘✙✠✛✜ ✢✣✤
dx
✡✧✠☞✠ f❀ (x) = 0, ❁ x ✰ I ✲✘❣✔❞❂✘✗✠ ✆☛✳
✡✧✠✠✖ ✜✈ I ✍☛✝ x ☞☛✌ ✆✠✘☛♦✠ f ☞☛✌ ✘❣✔☞✜✖✗ ❞✥ ✚✔ ✽✠❃✭✟ ✢✣ ✡✠✣✔ ✕✆❣✩✱ f ✱❞ ✡✑✔ ✢✣✤
♠✘✟✎✖ ✵✜ ❣❄✛✘✦✠✥ ☞☛✌ ✡✗✎✆✠✔ ✟✢ ❣✗✿❞✿✠✖ ❣✗❞✠✩✗✠ ✭✟✠✟✆✝❅✜ ✢✣ ❣❞ ✘❣✔☞✠✔ {F + C, C ✰ R},
f ☞☛✌ ✆✬✠✥ ✘✙❣✜✡☞❞✩✫✠☛✝ ❞✠☛ ✘✙✚✠✗ ❞✔✜✠ ✢✣✤
✜✠☛
✡❆ ✢✍ ✱❞ ✗✱ ✘✙✜✥❞ ✆☛ ✘❣✔❣✑✜ ✢✠☛✜☛ ✢✣✝ ✫✠☛ ❣❞ ✘✙❣✜✡☞❞✩✫✠☛✝ ☞☛✌ ✘❃✔☛ ✘❣✔☞✠✔ ❞✠☛ ❣✗❇❣✘✜ ❞✔☛❅✠✤
✟✢ ✘✙✜✥❞ ❈ f (x) dx ✢✣✪ ✕✆☛ x ☞☛✌ ✆✠✘☛♦✠ f ❞✠ ✡❣✗❣✽✑✜ ✆✍✠❞✩✗ ☞☛✌ ❇✘ ✍☛✝ ✘❉❊✠ ✫✠✜✠ ✢✣ ✤
✘✙ ✜✥❞✜★ ✢✍ ❈ f (x) dx = F (x) + C ❣✩✞✠✜☛ ✢✣✤✝
❋●❍■❏❑▲ ❣✚✟✠ ✢✎✡✠ ✢✣ ❣❞
dy
▼ f (x) , ✜✠☛ ✢✍ y = ❈ f (x) dx ❣✩✞✠✜☛ ✢✣✝✤
dx
✆✎❣☞◆✠ ☞☛✌ ❣✩✱ ✢✍ ❣✗❖✗❣✩❣✞✠✜ ✘✙✜✥❞✠☛✝P✘✚✠☛✝P ☞✠✵✟✠✝✽✠✠☛✝ ❞✠☛ ♠✗☞☛✌ ✡✧✠✠☛◗ ✆❣✢✜ ✆✠✔✦✠✥ ❘❙❚ ✍☛✝
♠❂✩☛❣✞✠✜ ❞✔✜☛ ✢✝★✣
❯❱❲❳❱❨ 7.1
✐❩❬❨❭❪✐❫❪❴❱❵❛❱❜❝❱
❡❤❱❥
❈ f (x) dx
f ❞✠ x ☞☛✌ ✆✠✘☛♦✠ ✆✍✠❞✩✗
❈ f (x) dx ✍☛✝ f (x)
✆✍✠❞❂✟
306
① ✁✂✄
f (x) dx
☎
❧❡✞✟✠✡
f
✟✞
❡✆✝
x
❧❡✞✟✠✡
✟☞✡✞
❧❡✞✟✠✡
✱✟
❧❡✞✟✠✡
✟✞
☛☞
✌✞✍
✟☞✡✞
F
❢✏❧✑✆✎
✐✎✠✡
❢✠✱
F (x) = f (x)
✒
❧❡✞✟✠✡
❧✝ ❢✓✔✞
❧❡✞✟✠✡
❧❡✞✟✠✡
✟✞
✟✞✆❞✖
✈☛☞
✛❡
✐✛✠✆ ❧✆
❧❡✞✟✠✡
❢✏❧✟✞
✑✆✎
✛✘
❣✛✣✍
✐✕✞ ❡✞❢✧✞✟
✩✐✔✞✆✦
✫✬✭✮✯
✛❡
❧✆ ✐✕❡✣✚✞
✐✎✠✡✞✆✝
❧✤✥✞✞✆✝ ✟✞✆ ✍✣ ☞✍
✝
✪✤❧ ☞✆
✐✎✠✡✞✆✝
✑✆✎
❢✠✚✞
✑✆✎
❧❡✞✟✠✡✞✆✝
Derivatives
✛✜ ✝✢
✟✞✆
❞✡
✌✞✍
❧✤✥✞
✰✲✳✭✮✴
✐✕✓ ❡
❢✏❧✆
❧✝✚✔✞
✏✞✡✍✆ ✛✜✢
✝
✐✕✞❡✞❢✧✞✟
✟☞✡✆
✟✞
✈☛☞
✛✝✢
✜
✟✛✍✆
✈✑✟✠✏✞✆ ✝ ✑✆✎
❧✟✍✆
✟☞✡✆
✑✞✙✍❢✑✟
✗✞✘
✐✎✠✡
✌✞✍
❞✡ ❧✤✥✞✞✆✝
❧✤✥✞✞✆✝ ✟✘ ❧✤ ☛✘
✑✆✎
❧✝✦✍
❢✡★✡❢✠❢✚✞✍
✟☞✆✝ ✦✢
✆
❡✆✝
✵✶✷ ✸✹✫✬✭✮✯✺
Integrals (Antiderivatives)
(i)
d xn 1
dx n 1
✼
✻
✿
❂
❢✑❢❉✞❊❋
d
x
dx
❍
❁
✽
❀ ✾
❡✆✝
✛❡
(iii)
d
– cos x
dx
(iv)
d
tan x
dx
(v)
d
– cot x
dx
(vi)
d
sec x
dx
(vii)
d
– cosec x
dx
◆ ❖
❘
❘
▼
❘
❅
cos x
❙ ❏
❙ ❏
❯ ❏
sin x
sec2 x
◆ ❖
cosec2 x
sec x tan x
❙ ❏
❆
C, n
❈
–1
❆
✛✜ ✝
☎
d
sin x
dx
❚
✪✆ ✚✞✍✆
1
■ ❏
(ii)
▼
❇
x n dx
❃
●✐
xn 1
n 1
❄
xn
cosec x cot x
☎
☎
☎
☎
☎
☎
dx
❑
x
▲
C
cos x dx sin x
P
sin x dx
P
sec 2 x dx
◗
– cos x
P
cosec 2 x dx
tan x
C
◗
◗
C
C
– cot x
◗
C
sec x tan x dx sec x
◗
C
P
P
cosec x cot x dx
P
✛❡
– cosec x
◗
C
✛✜ ✝
❧ ✁✂✄☎
d
1
–1
(viii) dx ✆ sin x ✝ ✞
1 – x2
✠
dx
1 – x2
✞ sin
–1
x✟C
d
1
–1
(ix) dx ✡ – cos x ☛ ☞
1 – x2
✍
d
1
–1
(x) dx ✎ tan x ✏ ✑
1 ✒ x2
✓
dx
–1
✑ tan x ✒ C
2
1✒ x
1
d
–1
(xi) dx ✎ – cot x ✏ ✑
1 ✒ x2
✓
dx
–1
✑ – cot x ✒ C
2
1✒ x
d
1
–1
(xii) dx ✔ sec x ✕ ☞
x x2 – 1
✍
1
d
–1
–
cosec
x
✞
✖
✗
(xiii) dx
x x2 – 1
✠
(xiv)
d x
(e ) ✘ e x
dx
(xv)
d
log x
dx
d
ax
(xvi) dx log a
dx
1– x
2
☞ – cos
dx
☞ sec
2
307
–1
–1
x✌C
x✌C
x x –1
dx
✞ – cosec
2
–1
x✟C
x x –1
x
x
✛ e dx ✙ e ✚ C
1
x
1
dx
x
log x
C
ax
a x dx
ax
log a
C
✐★✩✪✫✬ ✭✫✮ ✯✭ ✐★✪✩✰ ✱✲ ✳✮✴✵✪✶ ✷✪ ✸✹✺ ✻✯✼✮ ✷✵✴✫ ✸✹✲✭✫✮ ✸✽✸✾✪✿✻ ✐❀✶✻
❢✢✣✤✥✦✧
✜
✐✸✵✾✪✪✸❁✪✴ ✯❂✮ ✴❃✪✪✸✐ ✸✷✲✼ ✾✪✼ ✸✽✸❄✪❁❅ ✐★❄✻ ✽✫❀ ✲✮❆✾✪❇ ✭✫✮ ❈✲✷✪✫ ✾✪✼ ❉✩✪✻ ✭✫✮ ✵❊✪✻✪ ❋✪✸✯●❍
7.2.1 ✈■❏■❑▲▼ ◆❖P◗❘❏ ◗P ❙❚P■❖▼❯❚ ■❏❱❲❳P (Geometrical interpretation of
indefinite integral)
✭✪✻ ✶✼✸✹● ✸✷ f (x) = 2 x ✴✪✫ ✛ f (x ) dx ✙ x 2 ✚ C ✴❃✪✪ C ✽✫❀ ✸✽✸✾✪✿✻ ✭✪✻✪✫✮ ✽✫❀ ✸✶● ✯✭ ✸✽✸✾✪✿✻
✲✭✪✷✶✻ ✐✪✴✫ ✯❂✮❍ ✐✵✮✴ ❨ ❩✩✪✸✭✴✼✩ ❆❬ ✸❁❅ ✲✫ ✩✫ ✲✾✪✼ ✲✭✪✷✶✻ ✲✭✪✻ ✯❂✮❍ ❈✲ ✐★✷✪✵
y = x2 + C, ✹✯✪t C ●✷ ❭✽✫❪❫ ✳❋✵ ✯❂❴ ✲✭✪✷✶✻✪✫✮ ✽✫❀ ●✷ ✐✸✵✽✪✵ ✷✪✫ ✸✻❵✸✐✴ ✷✵✴✪ ✯❂❍ C, ✷✪✫
✸✽✸✾✪✿✻ ✭✪✻ ✐★❆✪✻ ✷✵✽✫❀ ✯✭ ✐✸✵✽✪✵ ✽✫❀ ✸✽✸✾✪✿✻ ✲❆❭✩ ✐★✪❛✴ ✷✵✴✫ ✯❂✮❍ ❈✻ ✲❜✷✪ ✲✸❝✭✸✶✴ ❵✐
308
① ✁✂✄
✈☎✆☎✝✞✟ ✠✡☛☞✌✆ ✍✎✏ ✑✒✓✔✟✕☛ ✒✖ ✗✕✘☞ ✠✡☛☞✌✆ ✙☞ ✒✚✛✌✕ ☞☛✘ ☎✆✜☎✒✟ ☞✚✟☛ ✍✎ ☎✢✠☞☛ ✈✣☛
y-
✛✘✤
✈✣☛
✈✆✥☎ ✦✝☛
C=0
✑✒✓✔✟✕☛
C=1
✒✚ ✍✎✏
✡✘ ✐ ✑❡☛☛✆☛✐✟☎✚✟
y-
❞☞☛❞★
✈✣☛
✛✘ ✤
✒✖ ✗✕✘☞
☞✚✆✘
✛✘✤
☎✦✝☛☛
✛✘✤
☎✌✙
y-
✈✣☛
✡✘ ✐
y = x2
y = x2 + 1
y = x2
C = – 1,
☎✌✙
✈✆✥☎ ✦✝☛
✡✘ ✐
☞☛
✍✎
✍✎ ✐
☎✦✝☛☛
✒☎✚✛☛✚
y-
☞✧
✈✣☛
✛✘✤
✢☛✘
☎✦✝☛☛
✡✘ ✐
✙☞
✙✘✠☛
✍✎
yy=x –1
❞☞☛❞★
☞☛✘ ✙☞
☞✚✆✘
✒✚
☎✢✠☞☛
✝☛✧✓☛★
✡✩✌
✪✫✦✥
✈✣☛ ✛✘✤ ✈✆✥ ✬✆☛✗✡☞ ☎✦✝☛☛
2
☎✌✙✮ ✛♦
✑❡☛☛✆☛✐✟☎✚✟
✒✚✛✌✕
✒✖ ☛✭✟
✒✚✛✌✕
✍☛✘ ✟☛
✍✎ ✏
y = x2
❞✠
☞☛✘ ✙☞
✒✖ ☞☛✚
C,
✛✘ ✤
✯✰☛☛✗✡☞
✒✚✛✌✕
☞☛
✡✘ ✐
✝☛✧✓☛★
❞✆
✍✎✏
✈☛✛✲✤☎✟
☞☛✘
✛✘✤
☎✌✙✮
C
✈☛✎✚
✛✥✤✱
✠✘
✯✰☛☛✗✡☞
✝☛✧✓☛★
✒✖ ✗ ✕✘☞
✒☛✟✘
✍☛✘✟ ☛ ✍✎ ✏
✛✘✤
✡☛✆
✍✡
✒✚✛✌✕
✒✖ ☛✭✟
✯✰☛☛✗✡☞
☞✧
✒✚✛✌✕☛✘ ✐
✒✚
✒✚✛✌✕
✬✆☛✗✡☞
✡☛✆☛✘ ✐
✛✘✤
✛✘✤ ☎✌✙ ✛♦
✬✆☛✗✡☞
✒✖ ✗✕✘☞
✍✎✏
✳✴✵
✡✘ ✐
✦✝☛☛★✕☛ ♥✕☛ ✍✎✏
✈✫
✍✡
❞✆
✛✘✤
✒✚✛✌✕☛✘✐
x=a
✚✘ ✶☛☛
⑥☛✚☛ ✒✖☎✟✷✱✘✦✆ ✒✚ ☎✛✞☛✚ ☞✚✟✘ ✍✎✏
✐ ✈☛✛✲✤☎✟ ✳✴✵
a<0
x=a
y = x2, y = x2 + 1, y = x2 + 2,
y = x2 – 1, y = x2 – 2
P0, P1, P2, P–1, P–2
✡✘ ✐ ✍✡✆✘
✛✘ ✤
a>0
☎✌✙
✸☛✧
☎✌✕☛ ✍✎✏
✠✗✕
✕✍
✍✎ ✏
☎✆✓☞✓☛★
✚✘ ✶☛☛
✕☎✦
✒✚✛✌✕☛✘ ✐
♦✡✝☛✹
☞☛✘
❞✗✕☛☎✦
dy
dx
✟☛✘ ❞✆ ✠✸☛✧ ✪✫✦✥✈ ☛✘ ✐ ✒✚
✕✍
☎✆☎✦★✓✔
✒✚
✛♦☛✘✐
❞✠
✾✡☛✆
☞✚✟☛
☞✧
✒✖ ☞ ☛✚
✽
✒✖ ☎✟✷✱✘ ✦✆
✡☛✆☛✘ ✐ ✛✘✤
✠✘
❀
✻
✒✖ ☛✭✟
R,
✛✘✤
✒✚
❞✆
✠✐♥✟ ✍✡✘ ✐ ❞✠
x2
✍✎
2a
✍✎✏
✠✸☛✧
✍☛✘✟ ☛
✚✘ ✶☛☛
y
✪✫✦✥✈ ☛✘ ✐
✠✡☛✐✟ ✚
✼
✛♦☛✘ ✐
F (x) C
☞☛✔✟✧
☞☛ ✡☛✆
✚✘✶☛☛✙✺
2 x dx
✪✫✦✥✈ ☛✘ ✐
f (x) dx
☎☞
✑✒✝☛★
✌✧☎✢✙✿
y = FC (x), C
✍✎
✒✚
✪✫✦✥✈ ☛✘ ✐
✍✎✐✏
C FC (x)
✻
✍✎
☎☞
✛♦☛✘ ✐
x = a,
☞✧
✑✒✝☛★
✾✡☛✆
❃❄❅❆❇❈❉
7.1
⑥☛✚☛
✚✘ ✶☛☛✙✺
✌✧☎✢✙✿
✠✡☛✐✟✚
✛♦☛✘ ✐ ✛✘✤
✍✎✐
✢✍☛✺
✒☎✚✛☛✚
a
☞☛✘
✒☎✚✛☛✚ ✛✘✤ ☎✛☎✸☛❂✆ ✠✦✑✕ ✒✖☛✭✟ ✍☛✘✟✘ ✍✎ ✐ ✈☛✎✚
❀
R
✈♥✖✟✹
☎✆✜☎✒✟
☎✆❁✆☎✌☎✶☛✟
☞✚✟☛
✍✎✏
C
✛✘✤
☞❡☛✆
☎✛☎✸☛❂✆
❞✆ ✠✦✑✕☛✘✐ ✡✘ ✐ ✠✘ ✍✡ ☎☞✠✧ ✙☞
✠✦✑✕ ☞☛✘ ✑✛✕✐ ✛✘✤ ✠✡☛❂✟✚ ✑❡☛☛✆☛✐✟☎✚✟ ☞✚✛✘✤ ✒✖ ☛✭✟ ☞✚ ✠☞✟✘ ✍✎✏
✐
✈☎✆☎✝✞✟ ✠✡☛☞✌✆ ☞☛ ❧✕☛☎✡✟✧✕
☎✆✜✒✰☛
✕✍✧
✍✎✏
❧ ✁✂✄☎
309
7.2.2 ✥✆✝✆✞✟✠ ✡☛☞✌✍✝☞✎✏ ✑✎✒ ✑✓✒✔ ✕✓✖☞✗☛✘ (Some properties of indefinite integrals)
❜✙ ✚✛ ✛✜✢✣✤✦✧ ★✦✩ ✪★ ✫✜✬✜✭✮✯ ✙★✰✱✲✬ ✳✦✴ ✳✵✴✤ ✶✵✷✰✸★✰✦✹ ✱✰✦ ✺✻✵✼✛✽✬ ✱✢✦✩✶✾✦
(i) ✜✬❢✬✜✲✜✿✰✯ ✛✜✢✷✰✰★✰✦✩ ✳✦✴ ✙✩✧❀✰❁ ★✦✩ ✫✳✱✲✬ ❂✳✩ ✙★✰✱✲✬ ✳✦✴ ✛❃❄★ ❂✱ ✧❅✙✢✦ ✳✦✴ ✺✻✵✼❄★ ✪❆✩❇
d
f (x) dx = f (x)
dx
f (x ) dx = f (x) + C, t✪✰❈ C ❂✱ ❉✳✦✣✤ ✫✮✢ ✪❆✾
✫✰❆✢
♠❊❊❋●❍ ★✰✬ ✲❡✜t❂ ✜✱ F❪ f ✱✰ ❂✱ ✛❃✜✯✫✳✱✲t ✪❆✩ ✫❞✰✰❁✯■
d
F(x) = f (x)
dx
✯✰✦
❜✙✜✲❂
f (x) dx = F(x) + C
d
dx
f (x) dx =
=
d
F (x) + C
dx
d
F (x) = f (x)
dx
❜✙❡ ✛❃✱✰✢ ✪★ ✧✦✿✰✯✦ ✪❆✩ ✜✱
f ❏(x) =
✫✰❆✢ ❜✙✜✲❂
d
f (x)
dx
f (x ) dx = f (x) + C
t✪✰❈ C ❂✱ ❉✳✦✣✤ ✫✮✢ ✪❆ ✜t✙✦ ✙★✰✱✲✬ ✫✮✢ ✱✪✯✦ ✪❆✩✾
(ii) ❂✦✙✦ ✧✰✦ ✫✜✬✜✭✮✯ ✙★✰✱✲✬ ✜t✬✳✦✴ ✫✳✱✲t ✙★✰✬ ✪❆✩ ✳❄✰✦✩ ✳✦✴ ❂✱ ✪❡ ✛✜✢✳✰✢ ✱✰✦ ✛❃✜✦ ✢✯
✱✢✯✦ ✪❆✩ ✫✰❆✢ ❜✙ ✛❃✱✰✢ ✙★✯✵❑✻ ✪❆✾✩
♠❊❊❋●❍ ★✰✬ ✲❡✜t❂ f ❂✳✩ g ❂✦✙✦ ✧✰✦ ✛✴✲✬ ✪❆✩ ✜t✬★✦✩
d
d
f (x) dx =
g (x) dx
dx
dx
✫❞✰✳✰
d ▲
f (x) dx – ◗ g (x) dx ▼ ◆ 0
P
dx ❖ ◗
✫✯❇
❙ f (x) dx – ❙ g (x) dx ❘ C , t✪✰❈ C ❂✱ ✳✰❉✯✜✳✱ ✙✩ ✿✻✰ ✪❆✾ ❚❯✻✰✦✩❱❲
✫❞✰✳✰
❙ f (x) dx ❘ ❙ g (x) dx ❳ C
310
① ✁✂✄
❜☎✆✝✞
✟✠✡☛☞
✟☛✌
g (x) dx
✞✟☞
❜☎
f (x ) dx
✍✗✘✡✎
✧✡☛
☎❧✏✑ ✒ ✓✏✡
✘✡☛
✍✗ ★✡✡✩✑☎ ✡✎
✟♦✡✰ ✩
✔✕ ✖
✩✔❞☞
✪
d
dx
✹
✼✾
❜☎
✪
(iv)
✔❧☛☞
❂
❅❇
✆✘☎❞
d
k
dx
✹
✿✡✕ ✎
✼
❜☎✆✝✞
▼
✆✝✫✡✘ ✎
✬✓✭✏
✘ ✎✏☛
✔✕ ☞
✆✮☎❧☛ ☞
R
✘❞
✍✗ ✡ ✯✝
✘✡
✪
✷
ii
✸
(i)
(ii)
✟☛✌
✳
(iv)
... (1)
❧☛ ☞
✟☛✌
✷❊✸
❈
✆✝✞
✪
f (x) dx +
✪
✺ ✻
✽
k
d
dx
▲✍✓✡☛ ✵
✾
✿✡✕ ✎
d
g (x) dx
dx
❇
✷❋✸
☎☛
❈
✘✎✏☛
f1, f2, ..., fn
❈
= k1 f1 (x) dx k2 f 2 (x) dx
❈
✪
k f (x) dx
❄
❑
k
✪
✔✕
❈
g (x)
... (2)
✆✘
f (x) dx
k f (x)
✔✑✞
✔❧
✍✡✏☛
✔✕ ☞
✆✘
✪
k f (x) dx
✍✌✝✩✡☛☞ ✘❞ ✆✩✆✐✯✏ ☎☞ ✫✓✡
✆✘✓✡
✮✡
☎✘✏✡
◆
❈
✔✡☛ ✏✡
f (x)
f (x) dx = k f (x)
✬✓✡✍✘❞✘✎♦✡
❈
✍✗✡ ●✏
✳
g (x) dx
❇
✘✡
❉✡❞
☎☞ ✧❉✡✰
❇
d
k f (x) dx
dx
⑥✡✎✡
✘✡
d
dx
❃ ❄
❆
f (x) dx
✪
k,
☎☞ ✫✓✡
✆✝✞
f (x) + g (x)
✺ ✻
✽
k1 f1 (x) k2 f 2 (x) ... kn f n (x) dx
✪
g (x) dx + C2 ,C2
✞✟☞
✆✘
f (x) dx
✿✡✕ ✎
✟☛✌
✪
✾
✵✑♦✡✶❧✰
(iii)
k2, ..., kn
✍✗✵ ✑ ♦✡✡☛☞
✔✕
■
✵✑♦✡✶❧✰
R
i
g (x) dx
✟✡❏✏✆✟✘
✔✕☞ ✖
✷ ✸ ☎☛
✵✑♦✡✶❧✰
❈
✔✕☞ ✖
f (x) dx + g (x) dx
✪
❇
f (x)
♠✣✣❢✴✥
(v)
✳
f (x) dx + g (x) dx
✍✗ ✘✡✎
❍
❁✡✏
☎❧✏✑✒✓
✪
[ f (x) + g (x)] dx
✿❀✓★✡✡
d
dx
✵✑♦✡✶❧✰
g (x) dx
R
☎❧✏✑✒✓
f (x) dx = g (x) dx ,
✪
✲
♠✣✣❢✴✥
R
f (x) dx + C1 ,C1
f (x) + g (x) dx
✱
C1 , C1
C2 , C2
✈✙✚❥
✍✆✎✟✡✎✡☛☞
❢✜✢✣✤✥✦
✛
(iii)
f (x) dx
✍✆✎✟✡✎
... kn
❈
✪
f n (x) dx
✔✕
✮✕ ☎✡
✳
k
✪
f (x) dx
✿✡✕ ✎ ✟✡❏✏✆✟✘ ☎☞ ✫✓✡✿✡☛ ☞
✆✘
✩❞✯☛
✆✧✓✡
✵✓✡
✔✕
k1,
❧ ✁✂✄☎
311
❢✆✝ ✞✟✝ ✠✡☛☞ ✌✍ ✠✎❢✏✑✒✌☛✓ ✔✍✏ ✌✕☞✖ ✒✖✡ ❢☛✝ ✞✗ ✑✘✏✔✍✙☞ ✚✖ ✝✖✚✖ ✠✡☛☞ ✌✛ ✜✍✍✖✓ ✌✕✏✖
✞❣✘ ❢✓✚✌✍ ✑✒✌☛✓ ❢✆✢✍ ✞✟✑✍ ✠✡☛☞ ✞❣✣ ✑✤✍✛✥✦ ✠✡☛☞ ✌✛ ✧✚ ✠✎✌✍✕ ✌✛ ✜✍✍✖✓★ ✓✍✖ ❢✆✝ ✞✟✝ ✠✡☛☞
✒✖ ✡ ✠✎❢✏ ✑✒✌☛✓ ✔✍✏ ✌✕☞✖ ✒✖✡ ❢☛✝ ✌✛ ✓✍✏✛ ✞❣★ ✌✍✖ ❢☞✕✛♦✍✩✍ ✪✍✕✍ ✚✗✍✌☛☞ ✌✞✏✖ ✞❣✣✘ ✧✚✖ ✞✗
✒✟✡✫ ✬✆✍✞✕✩✍✍✖✘ ✚✖ ✚✗✭✏✖ ✞❣✣✘
♠✮✯✰✯✱✲✯ ✳ ❢☞✕✛♦✍✩✍ ❢✒❢✴ ✌✍ ✬✠✢✍✖✵ ✌✕✏✖ ✞✟✝ ❢☞✶☞❢☛❢✜✍✏ ✠✡☛☞✍✖✘ ✌✍ ✠✎❢✏✑✒✌☛✓ ✔✍✏ ✌✛❢✓✝✣
(ii) 3x2 + 4x3
(i) cos 2x
(iii)
1
,x✷0
x
✰✸
(i) ✞✗ ✝✌ ✝✖✚✖ ✠✡☛☞ ✌✛ ✜✍✍✖✓ ✌✕☞✍ ✹✍✞✏✖ ✞❣✘ ❢✓✚✌✍ ✑✒✌☛✓ cos 2x ✞❣
✞✗ ✓✍☞✏✖ ✞❣✘ ❢✌
✑✈✍✒✍ cos 2x =
d
(sin 2x) = 2 cos 2x
dx
d ✺1
✻
1 d
(sin 2x) =
✼ sin 2 x ✽
dx ✾ 2
2 dx
✿
✧✚❢☛✝ cos 2x ✌✍ ✝✌ ✠✎❢✏✑✒✌☛✓
1
sin 2 x ✞❣✣
2
(ii) ✞✗ ✝✌ ✝✖✚✖ ✠✡☛☞ ✌✛ ✜✍✍✖✓ ✌✕☞✍ ✹✍✞✏✖ ✞❣✘ ❢✓✚✌✍ ✑✒✌☛✓ 3x2 + 4x3 ✞❣✣
d 3
4
❁ x ❃ x ❂ = 3x2 + 4x3
dx
✧✚❢☛✝ 3x2 + 4x3 ✌✍ ✠✎❢✏✑✒✌☛✓ x3 + x4 ✞❣✣
✑❀
(iii) ✞✗ ✓✍☞✏✖ ✞❣✘
d
(log x)
dx
1
d
, x 0 ❄❅❥❙
[log ( – x)]
x
dx
✧☞ ✆✍✖☞✍✖✘ ✌✍✖ ✚✘❜✍❢✦✏ ✌✕☞✖ ✠✕ ✞✗ ✠✍✏✖ ✞❣ ✘
✧✚❢☛✝ ❋
1
( – 1)
–x
1
,x 0
x
d
1
❆ log x ❇ ❈ , x ❉ 0
dx
x
1
1
dx ❊ log x ★ ✓✍✖ ❢✌
✒✖✡ ✠✎❢✏✑✒✌☛✓✍✖ ✘ ✗✖✘ ✚✖ ✝✌ ✞❣✣
x
x
♠✮✯✰✱✲✯ ● ❢☞✶☞❢☛❢✜✍✏ ✚✗✍✌☛☞✍✖✘ ✌✍✖ ✔✍✏ ✌✛❢✓✝
x3 – 1
(i) ❍ 2 dx
x
(ii)
❋
2
(x 3
2
■ 1) dx
(iii)
✰✸ ✞✗ ✠✎✍❏✏ ✌✕✏✖ ✞❣✘❑
x3 – 1
dx ▲ ❍ x dx – ❍ x – 2 dx
❍
2
x
(✵✟✩✍✴✗✙ v ✚✖)
(x 3
2 ex –
1
) dx
x
312
① ✁✂✄
x1 1
=
1 1
☎
✆
✞
✡
(ii)
✠
☛
x– 2 1
C2 ; C , C
–
1
2
–2 1
☎
✆
C1 –
x2
2
✙
1
+ C1 – C 2
x
x2
=
2
✚
1
+C,
x
✑✌✫✪
✔☞
❧☞✌✍✎✏
✑✒✓
✔✕✗
✖
❧☞✌✍✎✏
✑✒✓
✔✕✗
☛
x– 1
– C2
–1
C = C 1 – C2
t✔✌✛
✬✪ ✭✬✎
✑✖✮✯☞
✰✲✌✓
✱✍
☞✪ ✖
✑✜✢
✔✳✴
✱✍
❧☞✌✍✎✏
✑✒✓
✢✔✌✛
✸
2
(x 3
✶
1) dx
✷
✸
2
x3
2
✹
dx
✶
✸
dx
1
x3
=2
1
3
5
x
✺
1
) dx
x
✼
✺
C =
✺
3
(iii)
✠
✞
✡
✘
❜❧❧✪
✝
✞
✟
x2
=
2
=
❢✤✥✦✧★✩
✣
C1
✞
✟
✝
✢✔✌✛
✽
(x 2
✻
2 ex –
3 3
x
5
✻
x
✻
C
3
✽
x 2 dx
3
x2
✾
✻
✽
2 e x dx –
✽
1
dx
x
1
= 3
1
2
✿
2 e x – log x + C
✻
2 e x – log x + C
✿
2
= x
5
♠❀★❁❂✧★
(i)
(iii)
❆
✽
❃
✮✏❄✏✮✎✮✵✌✯
❧☞✌✍✎✏✌✪✖
(sin x cos x) dx
✞
1 – sin x
dx
cos 2 x
✍✌✪
5
2
❅✌✯
(ii)
✍✳✮t✱
❆
cosec x (cosec x
✞
cot x) dx
✮✎✵✌✪ ✫
✖ ✗
✪
❧ ✁✂✄☎
313
❣✆
(i) ❀✝✞✟
☛ (sin x ✠ cos x) dx ✡ ☛ sin x dx ✠ ☛ cos x dx
= – cos x ✠ sin x ✠ C
(ii) ❀✝✞✟
2
☛ (cosec x (cosec x + cot x) dx ✡ ☛ cosec x dx ✠ ☛ cosec x cot x dx
= – cot x – cosec x ☞ C
(iii) ❀✝✞✟
☛
1 – sin x
1
sin x
dx ✡ ☛
dx – ☛
dx
2
2
cos x
cos x
cos 2 x
2
= ☛ sec x dx – ☛ tan x sec x dx
= tan x – sec x ☞ C
♠✌✍❣✎✏✍ ✑
f (x) = 4x3 – 6 ⑥✞✒✞ ✓✔✒✕✞✞✔✖✞✗ ✓✘✙✚ f ❞✞ ✓✛✔✗✜✢❞✙✣ F ❑✞✗ ❞✤✔✣✥ ✣✝✞✟
F(0) = 3 ✝✦ ❆
❣✆ f (x) ❞✞ ✥❞ ✓✛✔✗ ✜✢❞✙✣ x4 – 6x ✝✦
d 4
(x – 6 x) = 4x3 – 6, ❜★✔✙✥ ✓✛✔✗✜✢❞✙✣ F,
dx
F(x) = x4 – 6x + C, ⑥✞✒✞ ✩✪❀ ✝✦ ✣✝✞✟ C ✜♣✒ ✝✦ ❆
F(0) = 3
✔✩❀✞ ✝❢✜✞ ✝✦ ✔❞
3=0–6×0+C
❜★★✪ ✓✛✞✫✗ ✝✞✪✗✞ ✝✦
C=3
✜✈✞✢✞
4
✜✗✬ ✜✕✞✤✖✭ ✓✛✔✗✜✢❞✙✣✮ F (x) = x – 6x + 3 ⑥✞✒✞ ✓✔✒✕✞✞✔✖✞✗ ✥❞ ✜✔⑥✗✤❀ ✓✘✙✚ ✝✦ ❆
♣✧✟✔❞
✯✰✱✲✳✴✵
(i) ✝✶ ✩✪✷✞✗✪ ✝✦✸ ✔❞ ❀✔✩ f ❞✞ ✓✛✔✗✜✢❞✙✣ F ✝✦ ✗✞✪ F + C, ✣✝✞✟ C ✥❞ ✜♣✒ ✝✦✮ ✕✞✤ f ❞✞
✥❞ ✓✛✔✗✜✢❞✙✣ ✝✦❆ ❜★ ✓✛❞✞✒ ❀✔✩ ✝✶✪✸ ✓✘✙✚ f ❞✞ ✥❞ ✓✛✔✗✜✢❞✙✣ F ❑✞✗ ✝✦ ✗✞✪ ✝✶
F ✶✪✸ ❞✞✪❜❡ ✕✞✤ ✜♣✒ ✣✞✪✹✺❞✒ f ✢✪✘ ✜✚✸✗ ✓✛✔✗✜✢❞✙✣ ✔✙✷✞ ★❞✗✪ ✝✦✸ ✔✣♦✝✪✸ F (x) + C,
C ✻ R ✢✪✘ ✼✓ ✶✪✸ ✜✔✕✞✽❀✾✗ ✔❞❀✞ ✣✞ ★❞✗✞ ✝✦❆ ✜✚❢ ✓❀
✛ ✞✪✿✞✪✸ ✶✪✸ ★✞✶✞♦❀✗✬ ✥❞ ✜✔✗✔✒✾✗
✓✛✔✗✐✸❁ ❞✞✪ ★✸✗❢✖✭ ❞✒✚✞ ✜✞✢❂❀❞ ✝✞✪✗✞ ✝✦ ✔✣★★✪ C ❞✞ ✥❞ ✔✢✔❂✞✖✭ ✶✞✚ ✓✛✞✫✗ ✝✞✪✗✞ ✝✦ ✜✞✦✒
✔✣★✢✪✘ ✓✔✒❃✞✞✶❄✢✼✓ ✔✩✥ ✝❢ ✥ ✓✘✙✚ ❞✞ ✥❞ ✜✔⑥✗✤❀ ✓✛✔✗✜✢❞✙✣ ✓✛✞✫✗ ✝✞✪✗✞ ✝✦❆
314
(ii)
① ✁✂✄
F
❞☎✆✝✞❞☎✆✝
✈✆✒ ☛
✠✡✌✣✎✆✟✥
(iii)
✜✌✗
✌✎✭
x,
❞✆ ✢☛
✕✒☞✬
✑✆✣✟
y dy
★✟✍
❁✽❂
✗✆✟✏ ✆✟☞
✠✍✎✏✆✟☞
✗✆✟✏ ✆✟☞
☛✒ ✌❊✆❞✣✆
✮
❖
✠✕✎✟
✠✍✎✏
✕✒ ☞
5.
k1, k2
✕✝
✏✕✝☞
✕✒
✾❆
❇❈✿❀❅
✓☞ ✣✖❋●
❞✆✟
e – x dx
❞✆✟ ❑✆✣
✈★❞✎✑
✌✑✓❞✝
P
❞✝
✈★❞✎✑
✕✆✟✣✝✬
✓✥✆❞✎✏
✕✥
❞✝
▲ ◆
P
✣✆✟ ✓✥✆❞✎✏
★✟✍
✓✷ ✤✆ ✣✗✏✖ ✓✆☛ ✩✠✆☞ ✣✌☛✣
k1
❘
★✟✍
✕✒ ☞
✌❞
✓☎✆✝
✭✟✓✆
★✟✍
✙✆✆✣
✓✟
k1
✑✆✣✆
✢✢✆✶
✌✗✭
✠✌☛☎✆✆✌❋✆✣
✕✖✭
❞✆
✕✒
k2
◗
✣❍✆✆✌✠
✈★❞✎✏
P
✣✆✟
❞✝
✙✆✆✣
✥✟☞
✕✒
★✟
✏✕✝☞
✮
f 2 (x) dx
✕✒ ☞ ✬
✕✆✟ ✣✟
❣✝❞
✦✓✝
✈✆✒☛ ✈✓✥✆❞✎✏✝✜
★✟✍
✕✆✟✣✝
✠☛
✭❞
✑✆✣✆
✓✟ ✭❞
✈✌❯✣✝✜
✜✆✟❱✜
✌❞✓✝
✌❞✜✆
✠✡❞✆☛
✓☎✆✝
★✟ ✍
✌★❋✆✜
✌❞✓✝
✠✍✎✏
✠✍✎✏✆✟☞
❞✥
✭❞
✣✆✟
✕✆✟✣✝
✭✟ ✓✆
✕✒
✣❞
✓✝✌✥✣
✈✢☛
✈✢☛
✕✒
✕✆✟✣✆
❞✆
✈☞✣☛
✠☛☞ ✣✖
✕✆✟ ✣✆
✑✔
✔✕✖ ✠✗
✌❞✓✝
✠✡✆❳✣
✕✆✟✣✆
✕✆✟ ✣✟
✕✒ ✬
✭❞
✠✌☛✛✆✆✥❚★✩✠
✕✒ ✬
✈✌❯✣✝✜
✭✟✓✆
✔✕✖✠✗
✕✒
✔✕✖✠✗
✠✍✎✏
✌✑✓❞✝
P
✙✆✆✣
✕✒ ✬
❞☛✣✟
✕☞✒
✓✥✆❞✎✏
✑✒ ✓✆✌❞
★✕
✣✆✟
✕✥✟❲✆✆
✠✌☛✛✆✆✥❚★✩✠
❨✔✗✖
✠✍✎✏
✕✒
✈★❞✎✏✝✜
✈✌❚✣✧★
✕✒
✈✌✴❞
✭❞
✕✆✟✣✆
f1 (x) dx
✮
✈✏★❞✎✏✝✜
✓✥✆❞✎✏✆✟☞
✔✕✖ ✠✗
✭❞
▼
✠✍✎✏
❞✆
✏✕✝☞
✗✆✟
P
✠✍✎✏
d
d
f1 (x) k2
f 2 (x)
dx
dx
❞☛✟☞ ✚✟✬
✈★❞✎✑
✓✆❍✆
✈❍✆✆✶ ✣■
✕✒ ☞
✕✒✬
✈❙✜✜✏
✌❞✜✆
✙✆✆✣
✕✒
(Comparision between differentiation and
☎✆✝ ✏✕✝☞ ✕✆✟✣✟ ✕✒ ☞✬ ✕✥
✠✍✎✏
✔✕✖ ✠✗
❞✆✟✦✶
❞☛✣✟
k2 f 2 (x) dx
✈✢☛
✥✟☞
★✟✍
✓✥✆❞✎✏
◗
✑✆✏✣✟
✠✍✎✏
✌❞✓✝
✌❞✓✝
✔✕✖ ✠✗
✕✥
✵
C
✺
▼
❞✳✆✆✈✆✟☞
✈❍✆✆✶ ✣
❞✆
✚✖ ✛✆✴✥✶
k1 f1 (x)
✓✟
✌❞✓✝
✌✥✎✣✆
6.
★✟ ✍
✓✥✆❞✎✏
✜✌✗
✦✓✌✎✭
✕✒☞✬
✓☞✌♥✜✆✭❉
✓✥✆❞✎✏✝✜
✱❡✢
★✟✍
1 5
y
5
✹
❃❄❅✾✿❀
✠☛
✜✕✆❉
✜✌✗
C
❏
(ii)
4.
✈✌✣✌☛✫✣ ✈♦✜
d
k1 f1 (x) k2 f 2 (x)
dx
(i)
✥✟ ☞
✺
✺
✼✽✾✿❀
✕✥
✕✒ ✬
✕✆✟ ✣✆
✱✗✆✕☛✛✆✣✲
y4 1
4 1
✹
7.2.3
integration)
3.
✈✓☞☎ ✆★
❞☛✏✆
✌✏☛✝✳✆✛✆ ✓✟ ✕✥ ✭✟✓✆ ✠✍✎✏ ❑✆✣ ✏✕✝☞ ❞☛ ✓❞✣✟ ✌✑✓❞✆
✸
1.
2.
✈✌☎✆✪✜✫✣
✥✟☞
✕✒ ✬
4
✻
✩✠
✌✤✆❞✆✟✛✆✌✥✣✝✜✘
2
✓✥✆❞✎
❞☛
★✟✍
❑✆✣ ❞☛✏✆ ✈★✯✰ ✕✆✟ ✑✆✣✆ ✕✒ ✬ ✱✗✆✕☛✛✆✣✲ ✌✏☛✝✳✆✛✆ ✌★✌✴ ✓✟
❞☛✏✆ ✈✓☞ ☎✆★ ✕✒ ✫✜✆✟✌
☞ ❞
e– x
✦✧✜✆✌✗
✌✤✆❞✆✟ ✛✆✌✥✣✝✜✘
✢☛ ✙✆✆✣✆☞ ❞✝✘
2
f (x) dx
✮
❞✆✟ ✠✡✆☛☞✌☎✆❞ ✠✍✎✏✆✟☞ ✑✒ ✓✟ ✌❞ ✔✕✖ ✠✗✘ ✎✙✆✖ ✚✛✆❞✝✜✘
✕✥
✠☛☞✣✖
❞✝
✠✌☛❡❧✟✗
✓✥✆❞✎✏
✢✢✆✶
❩❬❩
✱✓
✥✟ ☞
❞✝
✢✢✆✶
✈☞ ✣☛✆✎
✢✢✆✶
❞☛✟ ✚
☞ ✬
✟
✠☛
✭❞
❨✔✗✖
❞☛✣✟
✕✒☞
✠☛
❞☎✆✝
✌✑✓
✠☛
❧ ✁✂✄☎
315
7. ✱✆ ✝✞✟✠ ✡☛✞ ☞✡✆✟✌ ✆✍ ✎✏✍✑✒✓✔✏ ☞✕✍✖ ✗✍✔ ✘✍☛✓✍ ✘✙ ✌✙✚☛ ✑✆ ✑✛✱ ✘✜✱ ✡✢ ✡☛✞ ✑✛✱ ✘✜✱
➥✣✛✜ ✝✤ ✥✝✦✍✖ ✤☛✧✍✍ ✆✔ ✝★✡✩✍✓✍✪ ✫✚ ➥✣✛✜ ✝✤ ✝✞✟✠ ✡☛✞ ☞✡✆✟✌ ✡☛✞ ✒✍✠ ✡☛✞ ✣✤✍✣✤ ✘✍☛✓ ✔
✘✙❣ ✬✚✔ ✝★✆✍✤ ✑✛✱ ✘✜✱ ✝✞✟✠ ✆✍ ☞✑✠✑✦✭✓ ✚✒✍✆✟✠ ✱✆ ✛✮✚✤☛ ✡☛✞ ✚✒✍✯✓✤ ✑✥✕✍✓ ✡✢✍☛✯ ✡☛✞
✝✑✤✡✍✤ ✆✍☛ ✑✠✐✑✝✓ ✆✤✓✍ ✘✙✪ ✑✌✚✒☛✯ ✚✒✍✆✟✠ ✡☛✞ ✭✤ ✆✍☛ ✑✠✐✑✝✓ ✆✤✠☛ ✡✍✟☛ ☞✰✍ ✡☛✞
☞✠✜✟ ✯✣ ✤☛✧✍✍ ✡☛✞ ✚✗✍✔ ✡✢✍☛✯ ✡☛✞ ✝★✑✓✈✲☛✛✠ ➥✣✛✜☞✍☛✯ ✝✤ ✥✝✦✍✖ ✤☛✧✍✍✱✳ ✚✒✍✯✓✤ ✘✍☛✓ ✔ ✘✙❣
8. ✡✜✞✲ ✗✍✍✙✑✓✆ ✒✍♦✍✍✱✳ ✴✍✓ ✆✤✠☛ ✒☛✯ ☞✡✆✟✌ ✆✍ ✫✝✏✍☛✵ ✘✍☛✓✍ ✘✙ ✫✛✍✘✤✩✍✓✶ ✑✆✚✔ ✆✩✍ ✷✍✤✍
✑✆✚✔ ✚✒✏ t ✒☛✯ ✓✏ ✆✔ ✵✬✖ ✛✮✤✔ ✏✑✛ ✴✍✓ ✘✙ ✓✍☛ ✑✛✱ ✵✱ ✚✒✏ ✣✍✛ ✡☛✵ ✴✍✓ ✆✤✠☛ ✒☛✯
☞✡✆✟✌ ✚✘✍✏✆ ✘✍☛✓✍ ✘✙❣ ✫✚✔ ✝★✆✍✤ ✑✆✚✔ ✚✒✏ t ✝✤ ✏✑✛ ✡☛✵ ✴✍✓ ✘✙ ✓✍☛ ✑✛✱ ✵✱ ✚✒✏
✒☛✯ ✓✏ ✛✮✤✔ ✴✍✓ ✆✤✠☛ ✡☛✞ ✑✟✱ ✚✒✍✆✟✠ ✆✍ ✫✝✏✍☛✵ ✘✍☛✓✍ ✘✙❣
9. ☞✡✆✟✌ ✱✆ ✱☛✚✍ ✝★✢✒ ✘✙ ✑✌✚✒☛✯ ✚✔✒✍ ✆✍ ✗✍✍✡ ✚✒✍✑✘✓ ✘✙ ✸✔✆ ✫✚✔ ✝★✆✍✤ ✆✍ ✗✍✍✡
✯ ☛❣
✚✒✍✆✟✠ ✒☛✯ ✗✍✔ ✚✒✍✑✘✓ ✘✙ ✑✌✚✡☛✞ ✣✍✤☛ ✒☛✯ ✘✒ ✝✑✤✈✲☛✛ ✹✺✹ ✒☛✯ ☞✻✏✏✠ ✆✤☛✵
10. ☞✡✆✟✠ ✱✡✯ ✚✒✍✆✟✠ ✡☛✞ ✝★✢✒ ✱✆ ✛✮✚✤☛ ✡☛✞ ✼✏✜✽✢✒ ✘✙ ✌✙✚✍ ✑✆ ✝✑✤✈✲☛✛ ✹✺✾✺✾ (i) ✒☛✯
✭✭✍✖ ✆✔ ✌✍ ✭✜✆✔ ✘✙❣
✿❀❁❂❃❄❅❆ ❇❈❉
✑✠❢✠✑✟✑✧✍✓ ✝✞✟✠✍☛✯ ✡☛✞ ✝★✑✓☞✡✆✟✌ ❊✚✒✍✆✟✠❋ ✑✠✤✔✰✍✩✍ ✑✡✑● ✷✍✤✍ ✴✍✓ ✆✔✑✌✱❣
1. sin 2x
4. (ax + b)2
3. e 2x
2. cos 3x
5. sin 2x – 4 e3x
✑✠❢✠✑✟✑✧✍✓ ✚✒✍✆✟✠✍☛✯ ✆✍☛ ✴✍✓ ✆✔✑✌✱✶
3x
6. ❍ (4 e + 1) dx
2
x
9. ❍ (2 x ■ e ) dx
12. ❍
x3 ■ 3x ■ 4
x
dx
1
2
7. ❍ x (1 – 2 ) dx
x
❏
10. ❍ ▲
◆
13. ❚
2
1 ❑
x–
▼ dx
x❖
2
8. ❍ (ax ■ bx ■ c) dx
x3 P 5 x 2 – 4
dx
11. ◗
x2
x3 ❘ x 2 ❙ x – 1
dx 14. ❍ (1 – x) x dx
x –1
2
15. ❍ x ( 3x ■ 2 x ■ 3) dx
x
16. ❍ (2 x – 3cos x ■ e ) dx
2
17. ❍ (2 x – 3sin x ■ 5 x ) dx
18. ❍ sec x (sec x ■ tan x) dx
sec 2 x
dx
19. ❯
cosec 2 x
2 – 3sin x
20. ❱
dx
cos 2 x
316
① ✁✂✄
✐☎✆✝
✞✟
✠✡☛
✛
21.
x
✣
✞✞
☞✌☛
1
(C)
2 2
x
3
✗✘❀
7.3
✕✏✘✙✠✚
✎✩✚
✪
✫
2x 2
2
✪
C
✫
C
d
f ( x) 4 x3
dx
✭
(B)
2 3
x
3
(D)
3 2
x
2
1
✪
1 2
x
2
✫
1 2
x
2
3
✬
3
x4
✘✙✍☞✌☛
f (2) = 0
✪
C
✫
C
1
f (x)
❞✓✌
✎✩✚
(A)
x4
✢
1
x3
✭
129
8
(B)
x3
✢
1
x4
✢
129
8
(C)
x4
✢
1
x3
✢
129
8
(D)
x3
✢
1
x4
✭
129
8
❧✮✯✰✱✲
✘✐❢★✌
✐✘✔✹❢✌❀
✍✔★❞✓✐✿✡✺ ✕
❬✓✓✌✙
✖✗✝
✐☎✘❞✧✡✕★✙
1
2x 2
3
22.
✕✓
✦
1
(A)
✕✓
✤
x
1 3
x
3
✑✒✓✔
✜
✢
✥
✍✎✏
✰✳
☞✌☛
✐☎✓❁❞
(Methods of Integration)
✴✵✴✶✷✯✸
✎☞✝✌
✘✕✠
✠✌✍ ✌
✙✓
✕✏ ✙✓❞✏ ✎✩ ✘✙✍✕✓
✍☞✓✕★✝✓✌☛
✍✕❞✌
✎✩☛ ❂
✧✡✕★✙
✗✎
f
✕✏
✖✖✓✺
✘✝✔✏❃✓❄✓
f
✎✩ ❆✍✍✌
✕✏
✐✔
✻✓✏✼
✙✓✌
✧✓❅✓✘✔❞
✡✌✾ ✍☞✓✕★✝
✡✽✾❢
✘✡✘❅
✐✾★✝✓✌☛
✻✓✏✼
✡✌ ✾
❆✍☞✌☛
✧✡✕★✙✓✌☛
✍✌
F
✕✏
✠✌✍ ✌
✐✾★✝
✕✏ ✐☎✓✘❁❞ ✎✓✌❞✏ ✎✩ ❂ ❞✻✓✓✘✐ ✘✝✔✏❃✓❄✓
✐✔ ✧✓❅✓✘✔❞ ✗✎ ✘✡✘❅ ✧✝✌✕ ✐✾★✝✓✌☛ ✕✏ ✘❇✻✓✘❞ ☞✌☛ ❈✎✽❞ ✑✘✖❞ ✝✎✏☛ ✎✩ ❂ ✧❞✚ ✍☞✓✕★✝✓✌☛ ✕✓✌ ✐☎✓☞✓✘❄✓✕
❉✐
☞✌☛
✐✘✔✡✘❞✺❞
✧✓✡✆✗✕❞✓
1.
2.
3.
7.3.1
❆✍
✧✓☛ ✘✆✓✕
❬✓☛❑✆✓✚
✘❏✓❊✝✓✌☛
✑❊✎✌☛
❋✓❞
✘✡✘❅✗✓❍
✕✔✝✌
✡✌ ✾
✘✝✈✝✘★✘❬✓❞
✘★✠
✐✔
✎☞✌☛
✧✘❞✘✔●❞
✧✓❅✓✘✔❞
✘✡✘❅✗✓❍
✘✡✕✘✍❞
✕✔✝✌
✕✏
✎✩✚
☛
✍☞✓✕★✝
☞✌☛
✘✡✗✓✌✙✝
■✓✔✓
✍☞✓✕★✝
✍☞✓✕★✝
✐✘✔✹❢✌❀
✕✔✝✌
✎✽✠
☞✽❬ ✗
■✓✔✓
▲▼◆❖P◗❘❘▲❙
✑✐
☞✌☛
❆✝☞✌☛
✐☎✘ ❞❇✻✓✓✐✝
✐✘✔✡✘❞✺ ❞
❉✐
✎✩ ❂
✕✔❞✌
☞✌☛
✡✌✾
✐✘✔✡✘❞✺❞
❚❘❯❘
✎☞
✘★✠
✘✕✗✓
(Integration by substitution)
❱❲❘❳❨❙
✐☎✘ ❞❇✻✓✓✐✝
x = g (t)
✙✓
✍✕❞✓
✘✡✘❅
■✓✔✓
✐☎✘ ❞❇✻✓✓✘✐❞
✍☞✓✕★✝
✕✔❞✌
✎✽✠
✐✔
✘❀✠
✘✡✖✓✔
❜✠
✎✩ ❂
I=
❭
f (x) dx
✐✔
✘✡✖✓✔
✕✏✘✙✠
✕✔✌☛❜ ❂
✌
✍☞✓✕★✝
❇✡❞☛ ❩✓
❭
✖✔
x
f (x) dx
✕✓✌
✕✓✌
t
☞✌☛
✧❊✗
❧ ✁✂✄☎
317
dx
= g✏(t)
dt
dx = g✏(t) dt ✞❢✒☛✟✓ ❣✔✖✕
✈✆ x = g(t) ✐✝✞✟✠✡☛☛✞✐✟ ☞✌✞✍✎ ✟☛✞☞
❣✑
❜✗ ✐✝☞☛✘
I=
f ( x) dx
f {g (t )} g (t ) dt
✐✝✞✟✠✡☛☛✐✙ ✚☛✘☛ ✗✑☛☞❢✙ ✛✓✜ ✞❢✎ ✢❣ ✣✘ ✐✞✘✛✟✤✙ ☞☛ ✗✥✦☛ ❣✑☛✘✓ ✐☛✗ ✧✐❢★✩ ✎☞ ✑❣✪✛✐✥✫☛✤
✗☛✩✙ ❣✔✖ ✧✐✢☛✓✬✌ ✐✝✞✟✠✡☛☛✐✙ ✭✢☛ ❣☛✓✬☛ ❜✗☞☛ ✈✙✮✑☛✙ ❢✬☛✙☛ ❣✑✓✯☛☛ ✑❣✪✛✐✥✫☛✤ ❣✔✖ ✗☛✑☛✰✢✟✱ ❣✑ ✎☞
✎✓ ✗✓ ✐✜❢✙ ✛✓✜ ✞❢✎ ✐✝✞✟✠✡☛☛✐✙ ☞✘✟✓ ❣✔✕ ✞✍✗☞☛ ✈✛☞❢✍ ✲☛✌ ✗✑☛☞✳✢ ✑✓✕ ✗✞✴✑✞❢✟ ❣☛✓✵✕ ✍✔✗☛ ✞☞
✞✙✴✙✞❢✞✒☛✟ ✧✶☛❣✘✫☛☛✓✕ ✚☛✘☛ ✠✐✷✸ ✞☞✢☛ ✬✢☛ ❣✔✖
♠✹✺✻✼✽✺ ✾ ✞✙✴✙✞❢✞✒☛✟ ✐✜❢✙☛✓✕ ☞☛ x ✛✓✜ ✗☛✐✓✿☛ ✗✑☛☞❢✙ ☞✌✞✍✎
(ii) 2x sin (x2 + 1)
(i) sin mx
(iv)
(iii)
tan 4
x sec2
x
x
sin (tan – 1 x )
1 ❀ x2
✻❁
(i) ❣✑ ✍☛✙✟✓ ❣✔✕ ✞☞ mx ☞☛ ✈✛☞❢✍ m ❣✔✖ ✈✟✱ ❣✑ mx = t ✐✝✞✟✠✡☛☛✐✙ ☞✘✟✓ ❣✔✵✕ ✟☛✞☞
mdx = dt
1
1
1
cos t + C = –
cos mx + C
❜✗✞❢✎ ❃ sin mx dx ❂ ❃ sin t dt = –
m
m
m
(ii) x2 + 1 ☞☛ ✈✛☞❢✍ 2x ❣✔✖ ✈✟✱ ❣✑ x2 + 1 = t ✛✓ ✜ ✐✝✞✟✠✡☛☛✐✙ ☞☛ ✧✐✢☛✓✬ ☞✘✟✓ ❣✔✕ ✟☛✞☞
2x dx = dt
2
❜✗✞❢✎ ❆ 2 x sin (x ❄ 1) dx ❅ ❆ sin t dt = – cos t + C = – cos (x2 + 1) + C
1
(iii)
1 –2
1
x ❅
❣✔✖ ✈✟✱ ❣✑
☞☛
✈✛☞❢✍
x
2
2 x
x
t ♦◗s ❇❈❉r▲❊❋❋❇✉ ●❋ ❍❇■❋①
s ●❥rs ❏❙❛ r❋❉●
1
2 x
dx
dt
✞✍✗✗✓ dx = 2t dt
✐✝☛❑✟ ❣☛✓✟☛ ❣✔ ✖
✈✟✱ ❆
tan 4 x sec2
x
x
dx ❅ ❆
tan 4t sec 2t 2t dt
4
2
= 2 ❆ tan t sec t dt
t
318
① ✁✂✄
❢☎✆✝
✞✟
✠✡ ♥☛✞✝☞
2 tan 4 t sec 2t dt
❜✞❢✔✕
✗
x
✧★✩✪✫✬✭
(iv) tan – 1 x
tan
x
t
❞☞ ✈✯❞✔✮
✠✒✓
2
tan 5 t
5
=
2
tan 5
5
2
tan 5
5
✖
☎✌ ❢✍✎✏☞☞☎✑
1
1 x2
=
x
✦
sec2 t dt = du
✍☞❢❞
u5
5
✗
dx
x
❞✝✍✟
2 u 4 du = 2
✖
✗
tan 4 x sec 2
✈✍✥
tan t = u
☎✌ ❢✍✎✏☞☞☎✑
✘
✙
x
C
C (
❉✚☞✟✓ ❢❞
C(
u = tan t)
✛✜ ❦s✣
✢ ✤
x)
t
C
❞✐❢✮✕
✠✒ ❣ ✈✍✥ ✠✡
tan–1 x = t
☎✌ ❢✍✎✏☞☞☎✑ ❞☞ ✰☎✚☞✟✱ ❞✝✍✟ ✠✒✓ ✍☞❢❞
dx
= dt
1 x2
✲
❜✞❢✔✕
✈✶
✵
✠✡
sin (tan – 1 x)
dx
1 x2
✯✷✆✸
✞✡☞❞✔✑☞✟✓
(i)
✗
☎☞✍✟
✠✒✓ ❢❞ ❂
✍✶
✗
tan x dx
✠✒✓
❢❞
❁
–
✦
cot x dx
❢❞✚☞
✱✚☞
✠✒ ✽
☎✝
☎✆✔✑☞✟✓
✾✾☞✻
✈☞✒✝
❞✝✍✟
sin x
dx
cos x
❂
❅
✍☞❢❞
dt
t
✖
❂
✡✟✓
❢✼☞❞☞✟✺☞❢✡✍✐✚
C
❀
❞✐❢✮✕
❃
❢✮✑✡✟✓
❢✯❢❧
❃
sin x dx = – dt
– log t
tan x dx log sec x
✖
☎☞✍✟
sin t dt = – cos t + C = – cos (tan –1x) + C
☎✌ ❢✍✎✏☞☞☎✑
cot x dx log sin x
✠✡
✵
✞✡☞❞✔✑☞✟✓
tan x dx
☎✌ ❢✍✎✏☞☞❢☎✍
❅
✈✏☞✯☞
✗
✰☎✚☞✟✱
✿
cos x = t,
(ii)
❞☞
✡✠✹✯☎☛✺☞✻
tan x dx log sec x
✠✡
✳
✴
✦
C
C
❁
❂
cos x
dx
sin x
❄
C
❃
– log cos x
❄
C
✰✑✯✟✆
✠✒✓ ❣
☎✌ ☞✡☞❢✺☞❞
❧ ✁✂✄☎
319
sin x = t ✐✆✝✞✟✠✡✡✝✐✞ ☛☞✝✌✍ ✞✡✝☛ cos x dx = dt
✞r
✏ cot x dx ✎ ✏
dt
t
= log t ✑ C
= log sin x ✒ C
(iii) ✔ sec x dx ✓ log sec x ✑ tan x ✑ C
❣✕✖✗ ✘✡✞ ❣✙ ✝☛, ✜ sec x dx ✛ ✜
sec x (sec x ✚ tan x)
dx
sec x + tan x
sec x + tan x = t ✐✆✝✞✟✠✡✡✝✐✞ ☛✢✣✖ ✐✢ sec x (tan x + sec x) dx = dt
dt
✎ log t + C = log sec x ✦ tan x ✦ C
❜✤✝✥✍ ✏ sec x dx ✎ ✏
t
(iv) ✔ cosec x dx ✧ log cosec x – cot x ✒ C
❣✕ ✐✡✞✖ ❣✙✗ ✝☛, ✪ cosec x dx ✩ ✪
cosec x (cosec x ★ cot x)
dx
(cosec x ★ cot x)
cosec x + cot x = t ✐✆✝✞✟✠✡✡✝✐✞ ☛☞✝✌✍
✞✡✝☛ – cosec x (cot x + cosec x) dx = dt
❜✤✝✥✍
dt
✭ cosec x dx ✫ – ✭ t ✫ – log | t | ✫ – log |cosec x ✬ cot x | ✬ C
cosec2 x ✮ cot 2 x
✑C
= – log
cosec x ✮ cot x
= log cosec x – cot x ✒ C
♠✯✰✱✲✳✰ ✴ ✝✣✵✣✝✥✝✶✡✞ ✤✕✡☛✥✣✡✖✗ ☛✡✖ ✘✡✞ ☛☞✝✌✍✷
sin x
(ii) ✪ sin (x ★ a) dx
3
2
(i) ✔ sin x cos x dx
1
(iii) ✪ 1 ★ tan x dx
✱✸
(i) ❀❣✡✹
3
2
2
2
✔ sin x cos x dx ✓ ✔ sin x cos x (sin x) dx
2
2
= ✔ (1 – cos x) cos x (sin x) dx
320
① ✁✂✄
t = cos x
❜✍✆✎✌
✑
✐☎✆✝✞✟✠✠✆✐✝
✡☛✆☞✌
dt = – sin x dx
✝✠✆✡
sin 2 x cos 2 x (sin x) dx
✏
– (1 – t 2 ) t 2 dt
✑
2
4
= – (t – t ) dt
✑
✏
–
✒
✕
✗
1
cos3 x
3
dx = dt
= –
(ii) x + a = t
❜✍✆✎✌
✤
✐☎✆✝✞✟✠✠✆✐✝
✡✚✛✜
sin x
dx
sin (x a)
✢
✣
=
✐✚
✤
✙
t3 t5
–
3 5
✓
✖ ✔
C
✘
1
cos5 x C
5
✙
sin (t – a)
dt
sin t
✥
sin t cos a – cos t sin a
dt
sin t
= cos a dt – sin a cot t dt
✑
✑
= (cos a) t – (sin a) log sin t
✦
✩
= (cos a) (x
= x cos a
✈✝✫
✤
dx
1 tan x
✣
C1
✧
✪
a ) – (sin a ) log sin (x
✦
✩
★
a cos a – (sin a ) log sin (x
a)
✔
★
C1
✧
✪
a) – C1 sin a
sin x
dx = x cos a – sin a log |sin (x + a)| + C
sin (x a)
✬
C = – C1 sin a + a cos a,
☞t✠✭
(iii)
✥
✔
★
★
✢
✤
✌✡
✈✱✮
✞✯✜✰✲
✈✳✚
t✴✵
cos x dx
cos x sin x
✣
1 (cos x + sin x + cos x – sin x) dx
= 2
cos x sin x
✤
✣
1
= 2 dx
✥
x
= 2
✣
✬
C1
2
1 cos x – sin x
dx
2 cos x sin x
✣
✥
✬
1 cos x – sin x
dx
2 cos x sin x
✤
✣
... (1)
❧ ✁✂✄☎
cos x – sin x
dx ✐✠ ✡☛☞✌✠ ✍✎✡✏✑ ❆
cos x ✞ sin x
✈✆
I✝✟
✈✆
cos x + sin x = t ✐✒✡✓✔✕✌✌✡✐✓ ✍✎✡✏✑ ✓✌✡✍ (–sin x + cos x) dx = dt
❜✖✡✗✑
I✘✚
dt
✘ log t ✙ C2 = log cos x ✛ sin x ✛ C 2
t
I ✍✌✥ ✜✢✣ ✤✥✦ ✠✧✌★✥ ✐✠ ✩✤ ✐✌✓✥ ✩✪✦
dx
x
C
C
1
1
2
✟ 1 ✞ tan x ✝ 2 + 2 + 2 log cos x ✞ sin x ✞ 2
=
C C
x 1
+ log cos x ✫ sin x ✫ 1 ✫ 2
2 2
2
2
=
C C ✭
x 1
✬
+ log cos x ✮ sin x ✮ C , ✰ C ✯ 1 ✮ 2 ✱
2 2
2
2 ✳
✲
✴✵✶✷✸✹✺✻ ✼✽✾
✢ ✖✥ ✿❀ ✓✍ ☛✥❁ ✐✒❂★✌✥✦ ✤✥✦ ✐✒❃❄✥✍ ✐❁✗★ ✍✌ ✖✤✌✍✗★ ❅✌✓ ✍✎✡✏✑❆
2x
1.
1 ✞ x2
4. sin x sin (cos x)
6.
ax ❊ b
2.
❇ log x ❈
2
x
5. sin (ax ❉ b) cos (ax ❉ b)
7. x x ❊ 2
1
9. (4 x ✛ 2) x 2 ✛ x ✛ 1 10.
x– x
12. (x
15.
3
1
– 1) 3
x
5
x
9 – 4 x2
–1
etan x
18.
1 ❍ x2
2
21. tan (2x – 3)
1
3. x ✞ x log x
8. x 1 ✛ 2 x 2
11.
x2
13.
(2 ❋ 3 x 3 )3
14.
16. e 2 x ● 3
17.
e2 x – 1
19. 2 x
e ■1
20.
2
22. sec (7 – 4x)
23.
x
x✛4
,x>0
1
,x>0
x (log x)m
x
ex
2
e2 x – e – 2 x
e2 x ✮ e – 2 x
sin – 1 x
1 – x2
321
322
① ✁✂✄
24.
2cos x – 3sin x
6cos x 4sin x
25.
27.
sin 2x cos 2 x
28.
1
cos x (1 – tan x)2
2
cos x
cos x
26.
x
29. cot x log sin x
1 sin x
☎
30.
sin x
1 cos x
sin x
31.
1 cos x
✆
1
33.
1 – tan x
36.
✐✎✏✑
38.
39.
☎
✠
✔✕✖
✒✗
☛
37.
1 x
✍
✘✙✖
✚✛✜
✢✣✤✥
✦✤
10 x9 10 x log e10 dx
✧★✑
1 log x
35.
x3sin tan – 1 x 4
2
✡
x
✒✓
✆
✞
tan x
34.
sin x cos x
(x 1) x log x
☎
✟
✝
1
1 cot x
32.
2
2
x
☞
✌
✦✜✩✪✔✫
☎
x10 10 x
(A) 10x – x10 + C
(C) (10x – x10)–1 + C
✬
❝✥✤❝✥
✛✭✫
☎
(B) 10x + x10 + C
(D) log (10x + x10) + C
dx
sin 2 x cos 2 x
(A) tan x + cot x + C
(C) tan x cot x + C
✮✯✰✮✯
✱✲
❙
(B) tan x – cot x + C
(D) tan x – cot 2x + C
7.3.2
trigonometric identities)
❢✳✴✵✴✶✷✴❢✸✹✺✻
✼✽✾✿ ✼❢✸✵✴❀✴✶❁
✽✶❂
❃❄✻✴✶❅
❆✴❇✴
✼✸✴✵❈❉
(Integration using
✪❝ ✚✘✤✦t★ ✘✙✖ ✕❊❋● ✩❍✤✦✤✙■✤✩✘❏✜★ ✐❋❑✑ ✩✑✩✛❏ ✛✤✙❏ ✙ ✛✭▲
✖
❏✤✙ ✛✘ ✚✘✤✦❑✑ ▼✤❏ ✦✥✑✙ ✕✙❋ ✩❑✔ ✕❊❋ ●
▼✤❏
✚✕◆✚✩✘✦✤❖✤✙✖
♠❳❨❩❬❭❨
(i)
✬
❪
✦✤
✢✐★✤✙P
✩✑◗✑✩❑✩❘✤❏
✦✤✙
cos 2 x dx
✦✥❏✙
▼✤❏
(ii)
✛✭✖
✪✭✚ ✤
✩✦
✩✑◗✑✩❑✩❘✤❏
✢❚✤✛✥■✤✤✙✖
✚✕◆✚✩✘✦✤
cos 2 x
✬
sin 2 x cos 3 x dx
cos 2x = 2 cos2 x – 1
1 cos 2 x
2
✐✎✤ ❫❏
❯✤✥✤
✚✘❱✤★✤
✦✜✩✪✔
(iii)
❩❣
(i)
✕✙❋
✛✤✙❏✤
✦✤✙
✛✭❲
❞✘✥■✤
✦✜✩✪✔
✩✪✚✚✙
✬
sin 3 x dx
P★✤
✛✭❲
❧ ✁✂✄☎
1
1
2
323
1
✡ cos x dx ✠ 2 ✡ (1 + cos 2x) dx = 2 ✡ dx ☛ 2 ✡ cos 2 x dx
❜✆✝✞✟
=
x 1
☞ sin 2 x ☞ C
2 4
1
[sin (x + y) + sin (x – y)] , ✏✑❞ ✒✎✓✔✑ ✏✕✝✖✟
2
1
sin 2 x cos 3x dx
sin 5 x dx – sin x dx
2
(ii) ✆✌✍✆✝✎✏✑ sin x cos y =
r✗
=
1✘ 1
✙
– cos 5 x ✚ cos x ✜ ✚ C
✛
2✢ 5
✣
1
1
cos 5 x ☛ cos x ☛ C
10
2
3
(iii) ✆✌✍✆✝✎✏✑ sin 3x = 3 sin x – 4 sin x ✆❞ ✤✎ ✥✑r❞ ✤✦✧ ✝✏
= –
sin 3 x ✠
3sin x – sin 3x
4
❜✆✝✞✟
3
3
= –
✝✌✏❢✥r✪
1
✩ sin x dx ★ 4 ✩ sin x dx – 4 ✩ sin 3x dx
3
3
1
cos x ☞ cos 3x ☞ C
4
12
2
2
✬ sin x dx ✫ ✬ sin x sin x dx = ✬ (1 – cos x) sin x dx
cos x = t ✓❥✑✭❞ ✥✓ – sin x dx = dt
❜✆✝✞✟
t3
✱ sin x dx ✰ – ✱ ✮1 – t ✯ dt = – ✴ dt ✲ ✴ t dt ✳ – t ✲ 3 ✲ C
3
2
2
= – cos x ☞
1
cos3 x ☞ C
3
✵✶✷✸✹✺✻ ✝✼✑✏✑❞✔✑✝✎r✕✽ ✆✌✍✾✆✝✎✏✑✿✑❞✧ ✏✑ ❀✥✽✑❞❁ ✏✓r❞ ✤❂✟ ✽✤ ❃❄✑✑✍✽✑ ✖✑ ✆✏r✑ ✤✦ ✝✏ ❃✑❞✭✑❞✧ ❀❅✑✓
✆✎r❂❢✽ ✤✦✧❆
✐❇❈❉❊❋●❍ ■❏❑
▲ ✆❞ ▼▼ r✏ ✌❞◆ ✥❖❄✭✑❞✧ ✎❞✧ ✥❖P✽❞✏ ✥◆✞✭ ✏✑ ✆✎✑✏✞✭ ◗✑r ✏✕✝✖✟❆
1. sin2 (2x + 5)
4. sin3 (2x + 1)
2. sin 3x cos 4x
5. sin3 x cos3 x
3. cos 2x cos 4x cos 6x
6. sin x sin 2x sin 3x
324
① ✁✂✄
1 – cos x
8. 1 cos x
7. sin 4x sin 8x
☎
4
cos 2 x – cos 2
cos x – cos
✝
22.
1
cos (x – a) cos (x – b)
✏✑✒
✍✓
✔✕ ✒
20.
✖✗✘
✙✚✛✜
sin 2 x cos 2 x
dx
sin 2 x cos 2 x
✟
cos x sin x
✡
✢✛
✣✤✌
❝★✩❝★
✪✫
❙
e x (1 x)
dx
cos 2 (e x x)
✬✭✮✬✭
(B) tan x + cosec x + C
(D) tan x + sec x + C
✯✰
✱
(A) – cot (exx) + C
(C) tan (ex) + C
(B) tan (xex) + C
(D) cot (ex) + C
♦✲✳✴ ✵♦✵✶✷✸✹ ✺✳✻✼✷✽✾ ♦✽✳ ✿❀✷❁✻✼
✐✥✜❂❃✕ ❄
✖✒▼ ✒✥ ❧❈
(1)
(3)
✔✕ ✒
✗✔
✐☛ ✛✔✛✥❋✛✢
❘
dx
x – a2
2
P
x
2
☎
a
2
✢✛✕
❖✛❈
x–a
1
log
2a
x a
◗
✢✜✌✕
C
(Integrals of Some Particular Functions)
✖✔✛✢❆✌
✔✕✒
✙✌✢✛
(2)
◗
dx
❱
✔✗❉✑✐❊ ❋ ✛●
✥✌❅✌✥❆✥❇✛❈
✖✔✛✢❆✌✛✕✒
❯
21. sin – 1 (cos x)
2
✠
✢✘✥✦✏✧
(A) tan x + cot x + C
(C) – tan x + cot x + C
❜✖
✆
cos 2 x
19.
7.4
cos 2 x 2sin 2 x
18.
cos 2 x
✆
1
sin x cos3 x
24.
15. tan3 2x sec 2x
✞
sin 3 x cos3 x
17.
sin 2 x cos 2 x
16. tan x
23.
✆
cos x – sin x
1 sin 2 x
14.
4
✍✎
sin 2 x
12.
1 cos x
11. cos 2x
✝
✐☛☞✌
☎
4
10. sin x
13.
cos x
9. 1 cos x
1
x
tan – 1
a
a
☎
C
❘
✖❊ ❍ ✛✛✕ ✒
✐☛✤✛✕❏
✢✘
■✤✛❇✤✛
❨
dx
a – x2
2
▼✗◆ ❈
✖✕
a x
1
log
+C
2a
a x
◗
P
❚
❲
2
❑✛▲ ✜
✢✜✕ ✒❏✧
✕
dx
(4)
✢✜✕ ✒ ❏ ✕
x –a
2
log x
❳
x2 – a2
❳
C
❄❊✖✜✕
❧ ✁✂✄☎
(5)
dx
✞
2
a –x
2
✆ sin
–1
x
✝C
a
(6)
✞
dx
2
x ✝a
2
✆ log x ✝
x2 ✝ a2 ✝ C
✈✟ ✠✡ ☛☞✌✍✎✏✑ ☞✒✓✔✕✕✡✕✖✗ ✘✕✖ ✒✙✚ ✘✓✑✖ ✠✛✗✜
(1) ✠✡ ❣✕✢✑✖ ✠✛✗ ✒✘
1
1
✣
2
(x – a ) (x ✤ a)
x –a
2
1 ✥ (x ✧ a) – (x – a) ✦ 1 ✥ 1
1 ✦
= 2a ✩ (x – a) (x a) ✪ ★ 2a ✩ x – a – x a ✪
✧
✧ ✬
✫
✫
✬
❜✙✒✭✮
✞
dx
1 ✯ dx
dx ✰
–✞
✆
✱✞
✲
2
2a ✳ x – a
x ✝ a✴
x –a
2
=
1
✵ log | (x – a )| – log | (x ✷ a )|✶ ✷ C
2a
=
x–a
1
log
✧C
2a
x✧a
(2) ☛☞✌✍✎✏✑ ✸✹✺ ✻✖✼ ✈✢✍✙✕✓ ✠✡ ☞✕✑✖ ✠✛✗ ✒✘
1
1 ✥ (a ✧ x) ✧ (a ✽ x) ✦
1 ✾ 1
1 ✿
★
❀
✩
✪ =
2
❁
2
(
)
(
)
a
a
x
a
x
✽
✧
a –x
2a ❄ a ❃ x a ❀ x ❂❅
✫
✬
2
❜✙✒✭✮
❢▲▼◆❖P◗
❑
dx
dx ❇
1 ❆ dx
✣
✤
❍ 2
a – x 2 2a ❈❋ ❍ a ❊ x ❍ a ✤ x ❉●
=
1
[■ log | a ■ x | ❏ log | a ❏ x |] ❏ C
2a
=
1
a❀ x
log
❀C
2a
a❃x
✸✹✺ ✡✖✗ ☛☞✌✕✖❘ ✘❙ ❘❜✎ ✒✻✒❚ ✘❙ ❯✌✕❱✌✕ ☞✒✓❲❳✖❨ ❩❬❭ ✡✖✗ ✘❙ ❣✕✮❘❙✜
(3) x = a tan ❪ ✓❱✕✢✖ ☞✓ dx = a sec2 ❪ d❪
❜✙✒✭✮
a sec 2 ❫ d❫
a 2 tan 2❫ a 2
dx
x2
a2
=
1
1
1
x
d❴ ❵ ❴ ❏ C ❵ tan – 1 ❏ C
❛
a
a
a
a
325
326
(4)
① ✁✂✄
❡☎✆
✝✞✟✠✡
x = a sec
☛ r☞
x
2
✍
a
2
2
a sec
❡☎✆
a
(6)
❡☎✆
✝✞✟✠✡
2
✢
x
✜
✎
✏
✏
C1
x2 – a2
= log x
✑
x2 – a2 + C ,
☛ r☞
dx = a cos d
☛
✔
2
2
✔
2
a – a sin
✘
r☞
2
log a
✒
✑
✠t☎✓
C = C1 – log |a|
–1
= d = + C = sin
✕
✗
✕
✔
x
a
✖
C
dx = a sec2 d
✙
✙
a sec 2 d
✚
a
2
✢
2
a tan
✎
= log
2
✚
✚ ✜
a2
✎
x
a
✤
x2
a2
✤
1
= log x
✑
x2
✑
a
✥
= log x
✑
x2
✑
a
✥
✣
✤
tan )
C1
✣
C1
✒
log | a | C1
✑
C,
✑
✠t☎✓
C = C1 – log |a|
✦✧✯☎✫ ✰ ✌✫ ✱☞ t❡ ✭✲✮ ✳ ✱☎✴✵ ✌✩ ✪☎ ✦✧ ☎✶r ❢✵r✫ t✴✬ ✠☎✫ ✱✆✲ ✦✧ ✯☎✫✰
✌✫ ❧✦✯☎✫✰ ✞ t✴✬ ✱☎✴✵ ✷✩ ✌ ✵✫ ✌❡☎❢✝✆☎✫✬ ❢☎ ❡☎✆ ✻☎r ❢✵✆✫ ✭✫✮ ✟✝✡
✌❢r☎ t✴✽
C1
☛
= sec d = log (sec
❜✆ ✦✧☎❡☎✟★☎❢ ✌✩ ✪☎☎✫✬ ✭✫ ✮
✎
✑
✛
2
✎
= log x
dx
❜✌✟✝✡
a2
✍
a cos d
x
✍
x2
–1
a2
x = a tan
✟❢
✍
2
x
= log a
dx
❜✌✟✝✡
☛
log sec + tan + C1
x = a sin
✝✞✟✠✡ ✟❢
d
☛
= sec d
✎
(5)
tan
☛
a sec tan d
dx
❜✌✟✝✡
dx = a sec
❢✞ ✷✸✟✹✺
❜✆❢☎ ✌✞✼☎ ✦✧✯☎✫ ✰ ✟❢✯☎ ✠☎
❧ ✁✂✄☎
327
dx
(7) ✆✝✞✟✠✡ ☞ 2
, ❑✞✌ ✟✍✡✎ ✏✎✑ ✒✠✓ ✔✝
ax ☛ bx ☛ c
2
✕✗
c✖
b ✘ ✗ c b2
✕ 2 b
ax + bx + c = a ✚ x ☛ x ☛ ✛ ✙ a ✚✜ x ☛
✢ ☛✜ –
2a ✦ ✥ a 4a 2
a
a✤
✣
✚✣ ✥
2
✈✪ x ✫
✘✖
✢ ✛ ✒✠❢✞✌✎ ✔✧★✩
✦ ✛✤
c b2
b
– 2 ✭ ✮ k 2 ✒✠❢✞✌✎ ✔✯✓ ✔✝ ❥✞✌✎ ✔✧★ ✒✟
✬ t ✍❢✞✡✎ ❥✍ dx = dt ✓✏★
a
4a
2a
✗c
b2 ✘
1
dt
✜ –
✏✎✑ ✵❥ ✝✎★ ❥✒✍✏✒✌✲✌
2 ✢ ✏✎✑ ✒♦✰ ❥✍ ✒✡✱✞✲✍ ✟✍✌✎ ✔✯✓ ✳✔ ✆✝✞✟✠✡
2
☞
a t ✴ k2
✥ a 4a ✦
✔✞✎ ❣✞✌✞ ✔✧ ✈✞✧ ✍ ✶✆ ❥✷✟✞✍ ✶✆✟✞ ✝✞✡ ❑✞✌ ✒✟✳✞ ❣✞ ✆✟✌✞ ✔✧ ✩
dx
(8)
ax 2
bx c
, ✏✎✑ ❥✷✟✞✍ ✏✎✑ ✆✝✞✟✠✡ ✟✞✎ ❑✞✌ ✟✍✡✎ ✏✎✑ ✒✠✓ ✸✹✺ ✟✻ ✱✞✞✼✒✌ ✈✞✽✎ ✪✾✿✌✎
✔✯✓ ❥✷✞✝✞✒❀✞✟ ✆❁❂✞✞✎★ ✟✞ ❃❥✳✞✎✽ ✟✍✏✎✑ ✆✝✞✟✠✡ ❑✞✌ ✒✟✳✞ ❣✞ ✆✟✌✞ ✔✧ ✩
px ☛ q
dx , ❣✔✞✼ p, q, a, b, c ✈♦✍ ✔✧★❄ ✏✎✑ ❥✷✟✞✍ ✏✎✑ ✆✝✞✟✠✡ ❑✞✌ ✟✍✡✎ ✏✎✑ ✒✠✓
(9) ☞ 2
ax ☛ bx ☛ c
✔✝ ✓✎ ✆✻ ❅✞✎ ✏✞❆✌✒✏✟ ✆★❢✳✞✓✼ A ✌r✞✞ B ❑✞✌ ✟✍✌✎ ✔✧★ ✌✞✒✟
px + q = A
d
(ax 2 ❇ bx ❇ c) + B = A (2ax ❇ b) + B
dx
A ✌r✞✞ B, ❑✞✌ ✟✍✡✎ ✏✎✑ ✒✠✓ ✔✝ ❅✞✎ ✡✞✎★ ❥❈✞✞✎★ ✆✎ x ✏✎✑ ✽✯❀✞✞★✟✞✎★ ✓✏★ ✈♦✍✞✎★ ✟✞✎ ✆✝✞✡ ✟✍✌✎ ✔✧★✩
A ✌r✞✞ B ✏✎✑ ❑✞✌ ✔✞✎ ❣✞✡✎ ❥✍ ✆✝✞✟✠✡ ❑✞✌ ❥✷ ✞✝✞✒❀✞✟ ✵❥ ✝✎★ ❥✒✍✏✒✌✲✌ ✔✞✎ ❣✞✌✞ ✔✧✩
(10) ☞
( px ☛ q) dx
ax 2 ☛ bx ☛ c
, ✏✎✑ ❥✷✟✞✍ ✏✎✑ ✆✝✞✟✠✡ ✟✞ ✝✞✡ ❑✞✌ ✟✍✡✎ ✏✎✑ ✒✠✓ ✔✝ ✸❉✺ ✟✻ ✱✞✞✼✒✌
✈✞✽✎ ✪✾✿✌✎ ✔✧★ ✈✞✧ ✍ ✆✝✞✟✠✡ ✟✞✎ ❑✞✌ ❥✷ ✞✝✞✒❀✞✟ ✵❥✞✎★ ✝✎★ ❥✒✍✏✒✌✲✌ ✟✍✌✎ ✔✧★✩
✈✞✶✓ ❃❥✳✯❊
✲ ✌ ✒✏✒❋✳✞✎★ ✟✞✎ ✏✯✑● ❃❅✞✔✍❀✞✞✎★ ✟✻ ✆✔✞✳✌✞ ✆✎ ✆✝❍✌✎ ✔✧★✩
♠■❏▲▼◆❏ ❖ ✒✡P✡✒✠✒❢✞✌ ✆✝✞✟✠✡✞✎★ ✟✞✎ ❑✞✌ ✟✻✒❣✓
dx
(i) ☞ 2
x ◗ 16
(ii) ☞
dx
2 x ◗ x2
328
① ✁✂✄
❣☎
(i)
(ii)
❀✆✝✞
✒
✡
dx
x 16
2
✡
✠
dx
2x x2
2
☞
C
[7.4 (1)
✥✌
]
☞
dx
✏
✒
✑
x–1=t
dx
x–4
= log
2
8
x 4
x –4
☛
✟
1– x –1
dx = dt
dx
✍
❥✓✝✔✌
❜✥✖✗✘
2
✎
✕❥
✡
2x x
✠
✟
2
dt
✡
1– t
–1
= sin (t ) C
[7.4 (5) ]
✙
2
✥✌
–1
= sin (x – 1) C
✙
♠✚✛❣✜✢✛
(i)
✡
✣
x2
✖✔✤✔✖✗✖✓✝✦
✠
✥✧✝★✗✔✝✌ ✩
dx
6 x 13
(ii)
★✝✌
✡
✙
✪✝✦
3x 2
★✫✖✬✘✭
✙
dx
dx
13x 10
(iii)
✡
✠
5x2
✠
2x
❣☎
(i)
❀✆✝✞
x2 – 6x + 13 = x2 – 6x + 32 – 32 + 13 = (x – 3)2 + 4
❜✥✖✗✘
✧✝✔
✴
dx
6 x 13
✲
✡
✱
✴
✳
✗✫✖✬✘
❜✥✖✗✘
1
dx
2
x–3
22
x–3=t
dx = dt
1
dx
dt
t
tan – 1
C
2
2
2
2
x 6 x 13
t 2
1
–1 x – 3
C
= tan
2
2
x
✰
✮
✳
✯
✦r
✵
✟
✠
✙
[7.4 (3) ]
✙
✟
✡
✥✌
✙
✶
(ii)
✖❢❀✝
✥✌
✆✷✸ ✝
✖✗✓✝✦✌
✥✧✝★✗✔
✹✺✻✼✹✽
❁✕
★✝
✆❂ ✭ ✆✧
✥✧✝★❃❀
✾✌ ✿
✆❥
★✝✌
✆❂✩
❅
3x 2 13x – 10 3 x 2
❇
❈
❉
❇
❋
❍
=3
▲
❏
◆
x
✡
3x
✵
✙
13x 10
–
3
3
✙
P
▲
❘
❜✥✖✗✘
✾✌ ✿
13
6
dx
13x 10
✠
❆
❊
●
2
❑
–
❖
◗
✟
❏
◆
P
1
3
17
6
2
❑
❖
◗
■
(
▼
✕✐❚✝❯
✾❱❯ r✔✝✔✌
▼
❙
dx
✡
❏
◆
P
13
x
6
✙
2
❑
❖
◗
❏
✠
◆
P
17
6
2
❑
❖
◗
✕❥
)
✖✔✤✔✖✗✖✓✝✦
✕❄ ★✝❥
❧ ✁✂✄☎
✈✆
x✝
❜✌✍✎✏
✛
329
13
✞ t ✥✟✠✡☛ ☞✥ dx = dt
6
dx
1
✒ ✛
3x ✓ 13x ✔ 10 3
✑
dt
✕ 17 ✖
✘
✙ 6 ✚
2
t2 ✔ ✗
17
1
6 ✓C
=
log
1
17
17
t✓
3✜ 2 ✜
6
6
t–
[7.4 (i) ✌☛]
13 17
x✓ –
1
6x ✢ 4
1
6 6 ✓C
log
✣ C1
log
=
1 =
13 17
17
6 x ✣ 30
17
x✓ ✓
6 6
(iii) ❀★✠✩ ✳
dx
5 x✪ ✮ 2 x
✫✳
=
1
3x ✤ 2
1
1
log
log
✦ C1 ✦
17
x✦5
17
3
=
1
3x ✢ 2
1
1
✣ C , where C = C1 ✧
log
log
17
x✣5
17
3
dx
✬
5 ✯ x2 –
✱
=
2x ✭
5 ✰✲
1
5✺ ✴
dx
2
1✵ ✴1✵
✶x – ✷ –✶ ✷
5✹ ✸ 5✹
✸
✈✆
x–
❜✌✍✎✏
✺
2
1
✿ t ✥✟✠✡☛ ☞✥ dx = dt
5
dx
5 x❁ ❃ 2 x
❂
1
5✺
dt
✴1✵
t –✶ ✷
✸ 5✹
2
2
(☞✐✻✠✼ ✽✾✼ ✆✡✠✡☛ ☞✥)
330
① ✁✂✄
1
=
5
(i)
✒✓
✥✔✕✔✥✖✥✗✘✙
t2 –
✝
❧✚✘✛✖✔✘☛ ✜
✛✘☛
✢✘✙
★
2
✆
C
[7.4 (4) ]
❧☛
✡
x2 –
☞
✝
✟
2x
5
C
☞
x 3
x 2
dx
2x 6x 5
✧
✞
1
5
✛✣✥✤✦
★
★
✩
☎
✠
1
1
log x –
5
5
=
♠✌✍✎✏✑✍
log t
(ii)
✩
5 4x
★
✪
x2
★
dx
✎❣
(i)
❧✫✬✘
✭✮✯✰✱✲
✴❃✘✘☛ ✜
4A = 1
✳✴✵✘☛ ✶
d
2 x2
dx
x
x+2= A
♥✘☛ ✔✘☛ ✜
✛✘
❁
✿
♦☛ ❄
❧☛
✙r✘✘
✛✷✙☛
6x 5
❁
✶✹❅✘✘✜ ✛✘☛ ✜
6A + B = 2
✩
★
✚☛ ✜❡
❜❧✥✖✦
I1
❊
✺✘✾✷
I2
●
✺✈
x
❉
❋
■
3
2
dt
t
❊
t,
log t
✷✗✘✔☛
✛✘☛
❧✚✘✔
1
4
A=
✩
✸✾ ✜
❂
✛✷✔☛
✺✘✾✷
✧
★
✴✷
★
★
❉
✚✘✔
✸✚
✴✘✙☛
✸✾✜❇
1
2
B=
dx
1
2 2x 6x 5
✩
1
1
I1
I2 (
4
2
(4x + 6) dx = dt
✖✣✥✤✦
✧
★
★
)
... (1)
✴✷
❉
dx
2
2x 6x 5
❊
❂
1
4x 6
dx
4 2x 6x 5
✷✗✘✔☛
❍
✛✷✙☛
★
❈
★
2x2 + 6x + 5 = t,
✺✥✻✘✼✵✽✙
B = A (4 x 6) B
✺❆✷✘☛ ✜
=
I1
✸✚
✺r✘♦✘
x 2
2x 6x 5
✧
❀ ❁
✦♦✜
★
❜❧✥✖✦
✸✹✦
❍
✴✷
C1 = log | 2 x 2
●
1
dx
2 x2 3x
★
■
❍
dx = dt,
✸✚
❍
✴✘✙☛
6 x 5 | C1
★
5
2
✸✾✜
=
... (2)
★
1
2
dx
✩
❏
▲
◆
x
★
3
2
2
❑
▼
❖
❏
★
▲
◆
1
2
2
❑
▼
❖
❧ ✁✂✄☎
1
2✌
I2 ✆
dt
✝1✞
t ✟✠ ✡
☛ 2☞
2
=
2
1
1
2✎
2
tan –1 2t ✍ C2
331
[7.4 (3) ✥✏]
3✒
–1
–1 ✑
= tan 2 ✔ x ✓ ✕ + C2 = tan 2 x 3 + C 2
2✗
✖
... (3)
✭✘✙ ✚✛✜✢ ✭✣✙ ✤✛ ✦✧★✛✏✩ ✭✪✙ ✫✏✬ ✤✢✮✏ ✧✢ ✯✫ ✧✛✰✏ ✯✜✬
✌
1
1
x✟2
dx ✆ log 2 x 2 ✟ 6 x ✟ 5 ✟ tan – 1 ✱ 2 x ✟ 3✲ ✟ C ,
4
2
2x ✟ 6x ✟ 5
✳
t✯✛✴
C✵
C1 C2
✶
4
2
(ii) ★✯ ✥✫✛✤❀✮ ✷✸✹ ✭✪✺✙ ✻✏✼ ✽✧ ✫✏✬ ✯✜✾ ✚✛✿❁ x 3 ✤✛✏ ❞✮❂✮❞❀❞❃✛✰ ✽✧ ✫✏✬ ✚❞❄✛❅★❆✰
✤✢✰✏ ✯✜✬
d
(5 – 4 x – x 2 ) + B = A (– 4 – 2x) + B
dx
♥✛✏✮✛✏✬ ✧❇✛✛✏✬ ✥✏ x ✻✏✼ ✩♦❈✛✛✬✤✛✏✬ ❁✻✬ ✚❉✢✛✏ ✬ ✤✛✏ ✥✫✛✮ ✤✢✮✏ ✧✢ ✯✫ ✧✛✰✏ ✯✜✬
– 2A = 1 ✚✛✜✢ – 4 A + B = 3,
x ✶3✵ A
✚✈✛✛❊✰❋
A= –
✿✥❞❀❁
▲
1
✚✛✜✢ B = 1
2
x■3
5 ❑ 4 x ❑ x2
dx ❏ –
dx
1 ● – 4 – 2 x ❍ dx
■▲
▲
2
5 ❑ 4 x ❑ x2
5 ❑ 4 x ❑ x2
1
I +I
2 1 2
I1, ✫✏ ✬ 5 – 4x – x2 = t, ✢❃✛✮✏ ✧✢ (– 4 – 2x) dx = dt
= –
✿✥❞❀❁
I1 ❏ ▲
▼ – 4 ❑ 2 x ◆ dx
5 ❑ 4x ❑ x2
❏▲
... (1)
dt
= 2 t ✟ C1
t
= 2 5 – 4 x – x 2 ❖ C1
✚P
I2 ❏ ▲
dx
5 ❑ 4 x ❑ x2
x + 2 = t ✢❃✛✮✏ ✧✢ dx = dt
❏▲
... (2)
dx
9 – (x ■ 2)2
✧✢ ❞✻❉✛✢ ✤✐❞t❁
332
① ✁✂✄
I2
❜☎✆✝✞
dt
✟
✡
32
✠
–1
= sin
☎❧☞✌✍✎✏✏✥ ✑
✒✓✔
x 3
✤
✡
5 – 4x – x
✟
2
✞✕✑
✟
t2
t
sin – 1 + C2
3
x 2
C2
3
☛
✒✖✔
✌✏✥
1.
✗
☎✥
✓✖
✚✌
✕✥ ✵
✒✗✔
❧✥✑
✘✙✆✚✛✜✏✏✆✘✚
– 5 – 4x – x 2 + sin – 1
✘✵✝✢✏✥ ✑
✌✏
2.
x 2
C
3
1
✤
9 – 25 x 2
x
8.
x6
✤
11.
2x 2
✤
9x
2
✽
x –1 x – 2
✾✽
✾
2x
✿
6x 7
22.
❂
x–5 x–4
❃❂
x 3
2
x – 2x 5
✤
✠
2– x
✼
9.
a6
❃
1
6x 5
✤
12.
✤
8 3x – x
15.
18.
2
x –1
1
1
7 – 6 x – x2
❀
x–a x–b
✠
✤
x 2
21.
✤
✤
❁❀
5x 2
1 2 x 3x 2
✤
4 x – x2
x2
✺
1
2
5x 3
23.
2
✹
✼
✤
20.
★
tan 2 x 4
x 2
✼
19.
C C2 –
sec2 x
✤
x–3
t✣✏✧✑
✻
x 2
17.
,
x2
6.
1 x6
✿
✿
2
✸
1
14.
4x 1
16.
✣✦✑
✱✲✳
x2
1
13.
✌✍✚✥
3.
✤
x2 – 1
2
✘✙✏✐✚
1
3x
5.
1 2 x4
1
10.
✣❧
✌☞✆t✞✶
1 4x 2
x –1
7.
✘✍
✤
✤
✷
4.
☎❧✏✌✝✢
✌✍✢✥
1
3x 2
x6 1
☎✥
]
... (3)
☛
✩✪✫✬✭✮✯✰
✘✙ ✴✢
[7.4 (5)
4 x 10
✤
x2
2x 3
❁
C1
2
❧ ✁✂✄☎
333
✐✆✝✞ ✟✠ ✡☛☞ ✟✌ ✍✎☞ ✏✑✒ ✓✔✕✖ ✗✕ ✘✙✞ ✗✒✚✛✡✜
24.
dx
❝✢✣❝✢ ✥✤ ✜
x 2x 2
(A) x tan–1 (x + 1) + C
(C) (x + 1) tan–1x + C
2
dx
25.
9 x 4 x2
7.5
(B) tan–1 (x + 1) + C
(D) tan–1x + C
❝✢✣❝✢ ✥✤ ✜
(A)
1 –1 ✧ 9 x ✦ 8 ★
sin ✪
✫✩C
9
✬ 8 ✭
(B)
1
✧ 8x ✦ 9 ★
sin –1 ✪
✫✩C
2
✬ 9 ✭
(C)
1 –1 ✧ 9 x ✦ 8 ★
sin ✪
✫✩C
3
✬ 8 ✭
(D)
1
✧ 9x ✦ 8 ★
sin –1 ✪
✫✩C
2
✬ 9 ✭
✈✮✯✰✱✮✲ ✰✳✮✴✵✮✶✯ ✷✮✸✮ ✹✺✮✲✻✵
(Integration by Partial Fractions)
P(x)
Q(x) , ♥✕✎ ✿✑❀✐♥✕✎☞ ☛✎✽ ❁✞❀✐✕❂ ☛✎✽ ❃✐ ✍✎☞ ✐✚✖❄✕✕✚❅✕❂ ✚✗✙✕ ✛✕❂✕
✑❣ ✛✑✕❆ P(x) ✡☛☞ Q(x), x ✍✎☞ ✿✑❀✐♥ ✑❣☞ ❂❡✕✕ Q(x) ❇ 0. ✙✚♥ P(x) ✗✒ ❞✕✕❂ Q(x) ✗✒ ❞✕✕❂ ✏✎ ✗✍
▲✍✖✼✕ ✗✒✚✛✡ ✚✗ ✡✗ ✐✚✖✍✎✙ ✐✽✾✞
✑❣❈ ❂✕✎ ✐✚✖✍✎✙ ✐✽✾✞ ✓✚✘❂ ✐✚✖✍✎✙ ✐✽✾✞ ✗✑✾✕❂✕ ✑❣ ❁❉✙❡✕✕ ✚☛❅✕✍ ✐✚✖✍✎✙ ✐✽✾✞ ✗✑✾✕❂✕ ✑❣❊ ✚☛❅✕✍
✐✚✖✍✎✙ ✐✽✾✞✕✎☞ ✗✕✎ ✾❋✿✒ ❄✕✕● ✚☛✚❍ ■✕✖✕ ✓✚✘❂ ✐✚✖✍✎✙ ✐✽✾✞ ☛✎✽ ❃✐ ✍✎☞ ✐✚✖☛✚❂❏❂ ✚✗✙✕ ✛✕ ✏✗❂✕
P(x)
P (x)
P(x)
, ✛✑✕❆ T(x) x ✍✎☞
✚☛❅✕✍ ✐✚✖✍✎✙ ✐✽✾✞ ✑❣❈ ❂✕✎
▼ T(x ) ◆ 1
Q(x)
Q(x)
Q(x)
P1 (x)
✡✗ ✓✚✘❂ ✐✚✖✍✎✙ ✐✽✾✞ ✑❣❊ ✑✍ ✛✕✞❂✎ ✑❣☞ ✚✗ ✡✗ ✿✑❀✐♥ ✗✕ ✏✍✕✗✾✞
✡✗ ✿✑❀✐♥ ✑❣ ❁✕❣✖
Q(x)
✑❣❊ ❑✏ ✐✆✗✕✖ ✙✚♥
☛❣✽✏✎ ✚✗✙✕ ✛✕❂✕ ✑❣❈ ❁❂✜ ✚✗✏✒ ❄✕✒ ✐✚✖✍✎✙ ✐✽✾✞ ✗✕ ✏✍✕✗✾✞ ✚✗✏✒ ✓✚✘❂ ✐✚✖✍✎✙ ✐✽✾✞ ☛✎✽
✏✍✕✗✾✞ ✗✒ ✏✍▲✙✕ ☛✎✽ ❃✐ ✍✎☞ ✐✚✖☛✚❂❏❂ ✑✕✎ ✛✕❂✕ ✑❣❊ ✙✑✕❆ ✐✖ ✑✍ ✚✛✞ ✐✚✖✍✎✙ ✐✽✾✞✕✎☞ ☛✎✽ ✏✍✕✗✾✞
✐✖ ✚☛✘✕✖ ✗✖✎☞●❈✎ ✓✞☛✎✽ ✑✖ ✖❣✚❖✕✗ ❁✕❣✖ ✚■❞✕✕❂ ●❀✼✕✞❖✕☞P✕✎☞ ✍✎☞ ✚☛❞✕✚◗❂ ✑✕✎✞✎ ☛✕✾✎ ✑✕✎●
☞ ✎❊
P(x)
P(x)
dx ✗✕ ✍✕✞ ❙✕❂ ✗✖✞✕ ✘✕✑❂✎ ✑❣☞ ✛✑✕❆
✡✗ ✓✚✘❂ ✐✚✖✍✎✙
✍✕✞ ✾✒✚✛✡ ✚✗ ✑✍ ❘
Q(x)
Q(x)
✐✽✾✞ ✑❣❊ ✡✗ ✚☛✚❍❈ ✚✛✏✎ ❁✕☞✚✝✕✗ ✚❄✕❉✞✕✎☞ ✍✎☞ ✚☛✙✕✎✛✞ ☛✎✽ ✞✕✍ ✏✎ ✛✕✞✕ ✛✕❂✕ ✑❣❈ ✗✒ ✏✑✕✙❂✕ ✏✎ ✚♥✡
✑❀✡ ✏✍✕✗❚✙ ✗✕✎ ✏✕❍✕✖✼✕ ✐✚✖✍✎✙ ✐✽✾✞✕✎☞ ☛✎✽ ✙✕✎● ☛✎✽ ❃✐ ✍✎ ✚✾❖✕✕ ✛✕✞✕ ✏☞❄✕☛ ✑❣❊ ❑✏☛✎✽ ✐✝✘✕❂❯
✐❱☛❏ ❙✕❂ ✚☛✚❍✙✕✎☞ ✗✒ ✏✑✕✙❂✕ ✏✎ ✏✍✕✗✾✞ ✏✖✾❂✕✐❱☛❏✗ ✚✗✙✕ ✛✕ ✏✗❂✕ ✑❣❊ ✚✞❋✞✚✾✚❖✕❂
✏✕✖✼✕✒ ❲❳✟ ✚✞✚♥❏❅◗ ✗✖❂✒ ✑❣❈ ✚✗ ✚☛✚❄✕❉✞ ✐✆✗✕✖ ☛✎✽ ✐✚✖✍✎✙ ✐✽✾✞✕✎☞ ☛✎✽ ✏✕❡✕ ✚✗✏ ✐✆✗✕✖ ☛✎✽ ✏✖✾
❁✕☞✚✝✕✗ ✚❄✕❉✞✕✎☞ ✗✕✎ ✏☞✿❨ ✚✗✙✕ ✛✕ ✏✗❂✕ ✑❣❊
334
① ✁✂✄
❧☎✆✝☎✞
Ø☛☎☞✌
✐✍✆☛✎✏
✐✑✒✓
✌☎
✔✐
✈☎☞ ✍✕☎✌
✍❢☎✖✓☎✎☞
✌☎
✔✐
A
x–a
✘
A
x–a
✛
3.
px 2 qx r
(x – a ) (x – b) (x – c )
A
x–a
✢
B
x–b
4.
px 2 qx r
(x – a ) 2 (x – b)
A
x–a
✤
B
(x – a)2
5.
px 2 qx r
(x – a ) (x 2 bx c)
A
x–a
✥
Bx + C ,
x bx c
1.
px q
,a
(x – a) (x – b)
2.
px q
(x – a) 2
✜
✣
♠✺✶✮✻✼✸
✷✧✫✯✧✴
✣
❃❄❅❆❇❈❅
❆❣
✵ ❋ ✶✧
✷ ✧✫ ✯ ✧ ✴
❉❉
❊
✦✮✩✧
✣
2
x + bx + c
A, B
✽✭✲
❞✧
C
✾✿✲
✥
✩✧✬✭
✷✲ ✱✶✧✾★
✽✧✰
❞✧
✬✮✯ ✧✰✱✧✲ ✳
❂✧✸
✾❞
♠✺✶✧✭ ✬
♠✵❀ ✸
❞✫✸✭
✦✮ ✾❖
✥
A
✩✧✪ ✫
B
✵❞✶✧
✵t✰❞✧✭
t✧
♠✵❀✸
B
✙
x–a
2
2
✚
C
x–c
✢
✥
✤
C
x–b
✥
✷❞✸✧✹
✵✿✵❁
✷✭
❂✧✸
❞✫✸✭
✦✪✲ ✹
❞✴✵t✾✹
✺✵ ✫ ✽✭ ✶
✿✧♦✸✵✿❞
✷✲ ✱✶✧✾★
✦✪ ✲
✺❍ ■✰
✦✪
❏ ✷✵ ■✾
✩ ✧✲ ✵ ❑ ✧❞
✵ ▲ ✧▼ ✰ ✧ ✭ ✲
✿ ✭❍
A
P
✥
✵t✰❞✧✭
x 1
✥
✦✽✭✲
♠✵❀✸
✥
B ,
x 2
✵■✱✧✸✭
◗◗◗
✦✪✲
✥
✵✿✵❁
✷✭
❂✧✸
❞✫✰✧
✦✪ ✹
✦✽
✺✧✸✭
✦✪✲
1 = A (x + 2) + B (x + 1)
x
✿✭❍
✬✮✯ ✧✧✲❞ ✧✭✲
✾✿✲
✩❀✫
✺❋✧✭✲
❞✧✭
✾✿✲
❏✰
❏✷
✷✽✴❞✫✯✧✧✭✲
✺❯❞✧✫
❞✧✭
✦■
✷✽✧❞●✶
❞✫✰✭
✺✫
✵✰❜✰✵■✵✱✧✸
✷✽✧✰
❞✫✰✭
✺✫
A+B=0
2A + B = 1
A=1
✦✽✭✲
◆✺
✩✧✪✫
✽✭✲
◆✺
✦✽
1
(x 1) (x 2)
t✦✧★
✦✪✲
✰✦✴✲
B
x–b
✥
✷ ✽✧❞ ● ✶
❞✧
✩✧✪✫
✿✧♦✸✵✿❞
dx
(x 1) (x 2)
7.2 (i)],
b
✣
✣
t✦✧★
✗
✜
✣
[
✟✠✡
✺❯✧❱✸
✦✽
✺✧✸✭
✦✪✲
B=–1
✦✧✭ ✸✧
✦✪
✺❯✧❱✸
✦✧✭ ✸✧
✦✪ ✹
1
1
=
x 1
(x 1) (x 2)
–1
x 2
❘❙❚
❧ ✁✂✄☎
❜✆✝✞✟
dx
=
(x 1) (x 2)
335
dx
dx
–
x 1
x 2
= log x ✠ 1 ✡ log x ✠ 2 ✠ C = log
x ☛1
☛C
x☛2
❢☞✌✍✎✏✑ ♠✒✓✔✕✖✗ ✆✘✙✚✛✜✢ ✣✤✥ ✟✚ ✆✦✕✆✝✘✚✢ ✧★ ✩✪✢✢✕✗✫ ✟✚ ✟✬✆✢ ✚✪✢✭ ✮✢✬ x ✦✬♦ ✆✯✢✙ ✰✦✙✚✢✓✕
✆✯✢✙ ✘✢✭✢✬✱ ✦✬♦ ✝✞✟ ✆✲✓ ✧★✳ ✦✔♦✴ ✞✬✵✢✚ ✆✱✦✬♦✗ ✶ ✚✢ ♠✒✓✢✬❞ ✓✧ ✷✸✢✢✕✭✬ ✦✬♦ ✝✞✟ ✚✛✗✬ ✧★✱ ✝✚ ✝✷✓✢
✧✔✩✢ ✚✪✢✭ ✟✚ ✆✦✕✆✝✘✚✢ ✧★ ✩✢★✛ ✆✱✦✬♦✗ = ✚✢ ♠✒✓✢✬❞ ✓✧ ✷✸✢✢✕✭✬ ✦✬♦ ✝✞✟ ✚✛✗✬ ✧★✱ ✝✚ ✝✷✓✢ ✧✔✩ ✢
✚✪✢✭ ✟✚ ✆✘✙✚✛✜✢ ✧★ ✩✪✢✢✕✗✫ ✓✧ ✷✸✢✢✕✭✬ ✦✬♦ ✝✞✟ ✝✚ ✝✷✓✢ ✧✔✩✢ ✚✪✢✭ x ✦✬♦ ✝✭✝✸✹✗ ✘✢✭✢✬✱ ✦✬ ♦ ✝✞✟
✆✲✓ ✧★✳
x2 ❂ 1
dx ✚✢ ✘✢✭ ❅✢✗ ✚✙✝✮✟✳
✺✻✼✽✾✿✼ ❀❁ ❄ 2
x ❃ 5x ❂ 6
x2 ❂ 1
✽❣ ✓✧✢❆ ✆✘✢✚❇✓ 2
✟✚ ♠✝✹✗ ✒✝✛✘✬✓ ✒♦✞✭ ✭✧✙✱ ✧★ ❜✆✝✞✟ ✧✘ x2 + 1 ✚✢✬
x – 5x ❂ 6
x2 – 5x + 6 ✆✬ ✯✢✢❞ ✚✛✗✬ ✧★✱ ✩✢★✛ ✧✘ ✒✢✗✬ ✧★✱ ✝✚
x2 ❈ 1
5x – 5
5x – 5
❉ 1❈ 2
❉ 1❈
2
(x – 2) (x – 3)
x – 5x ❈ 6
x – 5x ❈ 6
✘✢✭ ✞✙✝✮✟ ✝✚
5x – 5
A
B
❊
❂
(x – 2) (x – 3) x – 2 x – 3
✗✢✝✚
5x – 5 = A (x – 3) + B (x – 2)
✷✢✬✭✢✬✱ ✒♥✢✢✬✱ ✆✬ x ✦✬♦ ❞✔✜✢✢✱ ✚✢✬✱ ✟✦✱ ✩✹✛ ✒✷✢✬✱ ✚✢✬ ✆✘✢✭ ✚✛✭✬ ✒✛ ✧✘ ✒✢✗✬ ✧★✱ A + B = 5 ✩✢★ ✛
3A + 2B = 5.
❜✭ ✆✘✙✚✛✜✢✢✬✱ ✚✢✬ ✧✞ ✚✛✭✬ ✒✛ ✧✘
A = – 5 ✩✢★✛ B = 10 ✒✐✢❋✗ ✚✛✗✬ ✧★✱✳
✩✗✈
❜✆✝✞✟
x2 ☛ 1
5
10
● 1❍
☛
2
x–2 x–3
x – 5x ☛ 6
x2 ☛ 1
dx
1
dx ● ■ dx ❍ 5 ■
dx ☛ 10■
■ 2
x–2
x–3
x – 5x ☛ 6
= x – 5 log | x – 2 | + 10 log | x – 3| + C
336
① ✁✂✄
3x 2
dx
(x 1)2 (x 3)
☛
♠☎✆✝✞✟✆
✝❣
✔❢✘✍
✠✡
✌
✙✚✛✍
☞
✜✎✍❞✢✘
3x – 2
(x 1)2 (x 3)
✴
❞✍
✎✍✏
✑✍✒
❞✓✔✕✖✗
✎✫✬
✔❢✖
✙✚✖
☞
✜✍✣✤✍✓
✥✦✧★✩✪
A
B
x 1 (x 1)2
✳
✴
✭✫✮
✯✰
❞✍
✙✱✗
✛✒✲
✙✎
C
✴
✴
✜✎✍❞✢✘
✴
x 3
✴
✴
✔✵✶✍✒✫
✙✱ ✬
3x – 2 = A (x + 1) (x + 3) + B (x + 3) + C (x + 1)2
= A (x2 + 4x + 3) + B (x + 3) + C (x2 + 2x + 1 )
x2
,x
A + C = 0, 4A + B + 2C = 3
3A + 3B + C = – 2
✒✍✔❞
❢✍ ✫✏ ✍ ✫✬
✰ ♥✍✍ ✫ ✬
✜✫
♦ ✚✤✍✍ ✬❞ ✍ ✫ ✬
✭ ✫✮
✭✮
✫
♦ ✚✤✍✍ ✬❞ ✍ ✫✬
✖✭
✛✷✣
✛✍✱✣
A
11
,B
4
–5
2
C
✈✸❙❥
–11
4
❞✓
✰ ❢✍ ✫ ✬
❜✏
✰✍✒✫ ✙✱✬ ✗ ❜✜
✰✐❞✍✣
✜✎✍❞✢✘
✒ ✚ ✵✏✍
❞ ✣✏ ✫
❞✍✫
✜✎✓❞✣✤✍✍✫✬
✔✏✹✏✔✵✔✶✍✒
✰✣
✙✵
✯✰
✎✫✬
✰✍✒ ✫
❞✣✏✫
✰✐✍✺✒
✙ ✱✬
✔❞
✰✣
✙✎
✙✍✫✒✍
✙✱✗
3x 2
11
5
11
–
–
=
2
2
4 (x 1) 2 (x 1)
4 (x 3)
(x 1) (x 3)
3x 2
11 dx 5
dx
–
=
2
4 x 1 2 (x 1) 2
(x 1) (x 3)
❜✜✔✵✖
11
= 4 log x +1
11
x +1
= 4 log x + 3
♠☎✆✝✞✟✆
✝❣
✒r
( x2
✠✽
✾
✿
(x 2
x2
1) ( x 2
✾
✾
x2
1) (x 2
❞✍✫
4)
(x
2
✾
✾
4)
dx
✵✓✔✕✖
x2
1) (x 2
✾
4)
❞✍
✛✍✱✣
❀
✎✍✏
x2 = y
✒✍✔❞
☞
5
11
log x 3
2 (x + 1) 4
✻
✼
5
+C
2 (x + 1)
❞✓✔✕✖✗
✣✔✶✍✖
y
(y 1) (y 4)
✾
A
y
=
y 1
(y 1) (y 4)
☞
✑✍✒
✴
11 dx
4 x 3
☞
✾
B
☞
y 4
✭✫✮
✯✰
✎✫ ✬
☞
y = A (y + 4) + B (y + 1)
✔✵✔✶✍✖
✴
✴
C
❧ ✁✂✄☎
337
♥✆✝✞✆✝✟ ✠✡✆✆✝✟ ☛✝ y ♦✝☞ ✌✍✎✆✆✟✏✆✝✟ ✑♦✟ ✒✓✔ ✠♥✆✝✟ ✏✕ ✖✍✗✞✆ ✏✔✞✝ ✠✔ ✘✙ ✠✆✖✝ ✘✚✟ A + B = 1 ✒✆✚✔
4A + B = 0, ❢✛☛☛✝ ✠✜✆✢✖ ✘✆✝✖✆ ✘✚
A= –
✒✖✤
(x
❜☛❢✗✑
x2
1) (x 2
2
(x
x 2 dx
1) (x 2
2
4)
4)
= –
= –
1
3
✈✣❥❙ B
1
2
3 (x ✥ 1)
✥
1
dx
2
3 x 1
1 –1
= – tan x
3
4
3
4
2
3 (x ✥ 4)
4
dx
2
3 x 4
4 1
x
tan – 1
3 2
2
C
1 –1
2
–1 x
✦C
= – tan x ✦ tan
3
3
2
♠✠✧✍★✩✖ ♠♥✆✘✔✎✆ ✙✝✟ ♦✝☞♦✗ ✒✆✟❢✪✆✏ ❢✫✆✬✞ ♦✆✗✝ ✫✆✆✌ ♦✝☞ ❢✗✑ ✠✜❢✖✭✮✆✆✠✞ ❢✏✧✆ ✌✧✆ ✮✆✆ ✞ ❢✏
☛✙✆✏✗✞ ♦✆✗✝ ✫✆✆✌ ♦✝☞ ❢✗✑✯ ✒✰ ✘✙ ✑✏ ✑✝☛✝ ♠♥✆✘✔✎✆ ✏✕ ✓✓✆★ ✏✔✖✝ ✘✚✟ ❢✛☛✙✝✟ ☛✙✆✏✗✞ ♦✝☞
❢✗✑ ✠✜❢✖✭✮✆✆✠✞ ❢♦❢✱ ✑♦✟ ✒✆✟❢✪✆✏ ❢✫✆✬✞ ❢♦❢✱ ♥✆✝✞✆✝✟ ✏✆✝ ☛✟✧✍✩✖ ✲✠ ☛✝ ✠✜✧✍✩✖ ❢✏✧✆ ✌✧✆ ✘✚✯
✳✴✵✶✷✸✵ ✹✺
✾
✻ 3 sin ✽ – 2 ✼ cos ✽
5 – cos 2 ✽ – 4 sin ✽
d ✽ ✏✆ ✙✆✞ ❞✆✖ ✏✕❢✛✑✯
✶❣ ✙✆✞ ✗✕❢✛✑ y = sin✿
✖✰
dy = cos✿ d ✿
❜☛❢✗✑
3 sin – 2 cos
5 – cos
2
– 4 sin
(3y – 2) dy
5 – (1 – y 2 ) – 4 y
d =
3y – 2
dy =
= ❀ 2
y – 4y ✥ 4
✒✰ ✘✙
❜☛❢✗✑
♥✆✝✞✆✝✟ ✠✡✆✆✝✟ ☛✝
3y – 2
❃ y – 2❄
2
❅
A
y❇2
❆
B
❢✗❈✆✖✝ ✘✚✟
(y ❇ 2) 2
3y – 2
y–2
2
I
❁✙✆✞ ✗✕❢✛✑❂
[☛✆✔✎✆✕ 7.2 (2) ☛✝]
3y – 2 = A (y – 2) + B
y ♦✝☞ ✌✍✎✆✆✟✏ ✑♦✟ ✒✓✔ ✠♥✆✝✟ ✏✕ ✖✍✗✞✆ ✏✔✞✝ ✠✔ ✘✙ ✠✆✖✝ ✘✚✟❉ A = 3 ✑♦✟
B – 2A = – 2, ❢✛☛☛✝ ✘✙✝✟ A = 3 ✑♦✟ B = 4 ✠✜✆✢✖ ✘✆✝✖✆ ✘✚✯
338
① ✁✂✄
❜☎✆✝✞
✟✠✡☛☞✌
I
✢
✣
☎✍✡✎✝✏
[
✆✏✑✏✆✝✆✒✡✓
✔✕
1
✥
★
4 –
✩
= 3 log (2 sin )
✰
✧
✱
✲
4
2 sin
✰
x 2 x 1 dx
(x 2) (x 2 1)
★
✹❀
✆❁✳✡
✆❢❊✡✆✌✓
✚✛✗
☎✡❆❧✡☛
✕♥✡✡✖✗
☎✖
x2
✟✡✛❆
☎✍☛✎❆❧✡✡✖✗
✍✡✏
❄✆❅✓
2
✯
✬
+C (
❉✳✡✖✗ ✆✎
2 – sin
4
2 – sin
✴ ✚✍✖❣ ✡✡
C
✯
✭
)
✵✏✡✶✍✎
✚✛
✱
❞✡✓
✎☛✆✿✞✜
✕✆❆✍✖ ✳
✕❇✝✏
A
x 1
=
x 2
1) (x 2)
✎✡✖
,x
❢✖❇
♦❂ ❧✡✡✗ ✎✡✖✗
A + 2C = 1
✚✝
☎✍✡✎❃✳
(x
✎❆✏✖
2
2
❢✖❇
✕✘ ✡✙✓
✕❆
✆✏✑✏✆✝✆✒✡✓
x
❜☎✆✝✞
2
✕✆❆✍✖✳
✚✛✜
✕❇✝✏
✟✡✗ ✆❣✡✎
✎✡✖
✆✠✡❈✏✡✖✗
✍✖✗
Bx + C
(x 2 1)
x2 + x + 1 = A (x2 + 1) + (Bx + C) (x + 2)
2B + C = 1
✕✘ ✎✡❆
✭ ✮
✜
❜☎✆✝✞
❜☎
C = 3 log sin
2.2(5)]
(x
❜✏
✎✡
✞✎
x2
❁✡✖✏ ✡✖✗
★
★
☎✍✡✎❃✳
[
✪
★
★
✚❂ ✟✡
✎❆✓✖
✚✛✜
✤
✦
y 2
✫
✾
✚✡✖✓ ✡
✤
✧
✼✽
✕✘ ✡✙✓
3
4
dy
+
] dy = 3 dy + 4
2
y – 2 (y – 2)
y–2
(y – 2)2
= 3 log y 2
♠✷✸✹✺✻✸
✍✖✗
♦❂ ❧✡✡✗ ✎✡✖✗
✎❆✓✖
✚✍
A
✔✕
✟❅❆
3
,B
5
✕✘ ✡✙✓
❋
2
,C
5
✚✡✖✓ ✡
✎☛
✓❂✝✏✡
✎❆✏✖
❋
1
5
✕✡✓✖
✕❆
✚✍
A + B =1,
✚✛✗✜
✚✛
2
1
x
3
5
5
=
2
5 (x 2)
x 1
x 1
3
=
5 (x 2)
1) (x 2)
●
3 dx
x2 x 1
dx =
2
5 x 2
(x +1) (x 2)
=
✕❁✡✖✗
✚✛✗ ✜
❋
✍✖✗
✞❢✗
1
2x
dx
2
5 x 1
3
log x 2
5
◆
◆
1 2x 1
5 x2 1
❍
●
●
❏
▲
●
■
❑
▼
1
1
dx
2
5 x 1
1
log x 2 1
5
◆
◆
1
tan –1 x C
5
◆
❧ ✁✂✄☎
339
✐✆✝✞✟✠✡☛ ☞✌✍
✶ ✎✏ ✑✶ ✒✓ ✔✏✕ ✖✗✘✙✚✏✛ ✜✏✛ ✖✢✣✜✏✤ ✖✕✥✙✚✏✛ ✓✚ ✎✜✚✓✥✙ ✓✦✢✧★✩
1.
x
(x ✪ 1) (x ✪ 2)
4.
x
(x – 1) (x – 2) (x – 3)
6.
1 – x2
x (1 – 2 x)
9.
12.
15.
2.
1
x –9
2
3.
3x – 1
(x – 1) (x – 2) (x – 3)
5.
2x
x ✫ 3x ✫ 2
2
7.
x
(x ✬ 1) (x – 1)
8.
3x ✪ 5
3
x – x2 ✭ x ✪ 1
10.
2x ✮ 3
(x – 1) (2x ✬ 3)
11.
5x
(x ✬ 1) (x 2 ✮ 4)
x3 ✪ x ✪ 1
x2 ✭ 1
13.
2
(1 ✭ x) (1 ✪ x 2 )
14.
3x – 1
(x ✪ 2) 2
1
16.
4
x ✯1
2
2
1
x (x n ✪ 1)
x
(x – 1) (x ✬ 2)
2
[✎✛✔✕✏ ✒✰ ✱✛✘✚ ★✔✛ ✲✣ ✓✚✏ x n – 1 ✎✏ ✳✴✵✚✚ ✓✦✢✧★ ✱✚✷✣
xn = t ✣✢❥✚★ ]
17.
cos x
(1 – sin x) (2 – sin x)
[✎✛✔✕✏ ✒✰ sin x = t ✣✢❥✚★]
18.
(x 2 ✸ 1) (x 2 ✸ 2)
(x 2 ✸ 3) (x 2 ✸ 4)
2x
(x ✬ 1) (x 2 ✬ 3)
21.
1
[✎✛✔✕✏ ✒✰ ex = t ✣✢❥✚★]
(e – 1)
19.
2
20.
1
x (x 4 – 1)
x
✖✗✘✙ ✑✑ ★✔✛ ✑✹ ✜✏✛ ✎✲✦ ✺✻✚✣ ✓✚ ✼✤✙ ✓✦✢✧★✩
x dx
22. ✽
❝✣✚❝✣ ✲✷ ✰
( x ✭ 1) ( x ✭ 2)
( x ✯ 1) 2
✫C
(A) log
x✯2
( x ✯ 2) 2
✫C
(B) log
x ✯1
2
✿ x ✾1 ❀
❃ ❁C
❄ x✾2❅
(C) log ❂
(D) log ( x ❆ 1) ( x ❆ 2) ❇ C
340
① ✁✂✄
23.
dx
x ( x 2 1)
✆
(A) log x
(C)
7.6
☞
❬✍✎✏✑✍✒
❜✘
✙✚✝✛✜✢✣
✦✞
✘✤✞✦✧★
✟✠ ✪
✭✢✳
1
log (x 2 +1) + C
2
✡
log x
✌
✤✢✥
✦✝★✢
✬★✴✘✞✝
✤✢✥
x
✦✩
✫✦
✶✙✷✞✢✱✩
❝✟✴❞
✙✞❞✢
✟✠ ✥
✻
✙♥✞✞✢ ✥
✦✞
✘✤✞✦✧★
uv
✬✈✞✭✞
✤✞★
✾
✧✩✚✲✫
u
✼
✦✝★✢
✽
u
✙✝
✹❁✺
❀
✬✈✞✞✰ ❞❆
❀
✷✚✣
✤✢✥
[(
✟✤
f
❡✷❊❞
✚✦✷✞
“
✙✳✧★✞✢ ✥
✣✞✢
✙❇ ✈✞✤
✙✳✧★
✦✞✢
✦✞
✦✩
✯✯✞✰
u
v
✬✞✠ ✝
✦✝✢✱
✥ ✢
✲✞✢
✚✦
✣✞✢
✙✳✧★✞✢✥
✭✢✳
✱✴✵✞★✙✳✧
✣✞✢ ✬✭✦✧★✩✷ ✙✳✧★ ✟✠ ❞✞✢ ✬✭✦✧★ ✭✢✳ ✱✴✵✞★✙✳✧
v
☛
du
dx
✙✞❞✢
✽
✟✠ ✥
✚✦
du
dx
dx
v
... (1)
✾
✬✞✠ ✝
dv
= g (x)
dx
✬✞✠ ✝
v=
✚★❂★✚✧✚❃✞❞
❀
❞❝
g (x) dx
✤✢✥
❄✙
✚✧❃✞✞
✲✞
❀
✘✦❞✞
✟✠
❀
❅
❀
f (x) g (x) dx = f (x) g (x) dx – [ f (x) g (x) dx] dx
✘✦❞✞
✭✢✳
✚✭✚✮
f (x) g (x) dx = f (x) g (x) dx – [ g (x) dx f (x)] dx
✦✞✢ ✙❇ ✈✞✤
✲✞
✌
✻
✿
✘✤✩✦✝✵✞
log (x 2 +1) + C
✟✠✸
✟✤
dv
dx
dx
du
= f (x)
dx
❜✘✚✧✫
✌
dv
du
dx uv – v dx
dx
dx
u = f (x)
✚✦
1
log x
2
1
log (x 2 +1) + C
2
✚✦
d
dv
(uv) u
dx
dx
✣✞✢★✞✢✥
✬✞✠ ✝
✹✤✞★ ✧✩✚✲✫✺ ✤✢✥
✟✤
☛
(Integration by Parts)
✘✤✞✦✧★
✟✤
(B) log x
1
log (x 2 +1) + C (D)
2
✓✔✍✕✖✗
✷✚✣ ✫✦✧ ✯✝
✚★✷✤
❝✝✞❝✝
☎
❀
✙✳✧★
✬✞✠ ✝
g
✦✞✢
❅
❀
✣❈ ✘✝✞
✙✳✧★
✤✞★
❀
✧✢✥
❞✞✢ ❜✘
✘❈ ❉✞
✦✞✢
✚★❂★✚✧✚❃✞❞
❄✙
✟✠✸
✱✴✵✞★✙✳✧
✬✭✦✧★
✦✞
✘✤✞✦✧★
)×(
✱✴✵✞✞✥✦
=(
✚❢❞✩✷
✙❇ ✈✞✤
✙✳✧★
✙✳✧★
✦✞
)×(
✚❢❞✩✷
✘✤✞✦✧★
)]
✙✳✧★
✦✞
✦✞
✘✤✞✦✧★
✘✤✞✦✧★
”
)—
❧ ✁✂✄☎
341
♠✆✝✞✟✠✝ ✡☛ ☞ x cos x dx ❞✌ ✍✌✎ ✏✌✑ ❞✒✓✔✕✖
✞❣ f (x) = x (✐✗✘✌✍ ✐✙✚✎) ✈✌✛✜ g (x) = cos x (✓❢✑✒✢ ✐✙✚✎) ✜✓❥✌✕✖ ✑✣ ❥✌✤✥✦✌✧ ★✍✌❞✚✎ ★✩
✐✗✌✪✑ ✫✌✩✑✌ ✫✛ ✓❞
d
✭ x cos x dx ✬ x ✭ cos x dx – ✭ [ dx (x) ✭ cos x dx] dx
= x sin x – ☞ sin x dx = x sin x + cos x + C
✍✌✎ ✚✒✓✔✕ ✓❞ ✫✍ f (x) = cos x ✕✱✤ g (x) = x ✚✩✑✩ ✫✛✤ ✑✣
d
✭ x cos x dx ✬ cos x ✭ x dx – ✭ [ dx (cos x) ✭ x dx] dx
= (cos x)
x2
2
sin x
x2
dx
2
❜★ ✐✗❞✌✜ ✫✍ ✮✩❥✌✑✩ ✫✛✤ ✓❞ ★✍✌❞✚✎ ☞ x cos x dx , ✑r✚✎✌✯✍❞ ✮✰✓✲✳ ★✩ x ❞✒ ✈✓✴❞ ✵✌✌✑
✱✌✚✩ ✈✓✴❞ ❞✓♦✎ ★✍✌❞✚✎ ✍✤✩ ✐✓✜✱✓✑✶✑ ✫✌✩ ✔✌✑✌ ✫✛✖ ❜★✓✚✕ ✐✗✘✌✍ ✐✙✚✎ ✕✱✤ ✓❢✑✒✢ ✐✙✚✎ ❞✌
✷✓✸✑ ✸✢✎ ✍✫✯✱✐✹✺✌✶ ✫✛✖
✻✼✽✾✿❀❁
1. ✢✫ ✱✺✌✶✎✒✢ ✫✛❂✤ ✓❞ ❥✌✤✥✦✌✧ ★✍✌❞✚✎ ✮✌✩ ✐✙✚✎✌✩✤ ✱✩✙ ❃r✺✌✎✐✙✚ ❞✒ ★❄✌✒ ✓❅✘✌✓✑✢✌✩✤ ✍✩✤ ✐✗✢r❆✑
✎✫✒✤ ✫✛❂ ✷✮✌✫✜✺✌✑✢✌ ☞
x sin x dx ❞✒ ✓❅✘✌✓✑ ✍✩✤ ✢✫ ✓✱✓✴ ❞✌✍ ✎✫✒✤ ❞✜✑✒ ✫✛✖ ❜★❞✌
❞✌✜✺✌ ✢✫ ✫✛ ✓❞ ✕✩★✌ ❞✌✩❜✶ ✐✙✚✎ ✈✓❅✑✯✱ ✍✩ ✫✒ ✎✫✒✤ ✫✛ ✓✔★❞✌ ✈✱❞✚✔
x sin x ✫✛✖
2. è✢✌✎ ✮✒✓✔✕ ✓❞ ✓❢✑✒✢ ✐✙✚✎ ❞✌ ★✍✌❞✚✎ ✏✌✑ ❞✜✑✩ ★✍✢ ✫✍✎✩ ❞✌✩❜✶ ★✍✌❞✚✎ ✈✸✜
✎✫✒✤ ✔✌✩✥✉✌ ✘✌✌✖ ✢✓✮ ✫✍ ✓❢✑✒✢ ✐✙✚✎ cos x ✱✩✙ ★✍✌❞✚✎ ❞✌✩ sin x + k, ✱✩✙ ❇✐ ✍✩✤ ✓✚❥✌✑✩
✫✛❂✤ ✔✫✌❈ k ❞✌✩❜✶ ✈✸✜ ✫✛❂ ✑✣
☞ x cos x dx ❉ x (sin x ❊ k ) ❋ ☞ (sin x ❊ k ) dx
= x (sin x
k)
sin x dx
k dx
= x (sin x k ) + cos x – kx C = x sin x ● cos x ● C
✢✫ ✮✦✌✌✶✑✌ ✫✛ ✓❞ ❥✌✤✥✦✌✧ ★✍✌❞✚✎ ✓✱✓✴ ✱✩✙ ✐✗✢✌✩❃ ★✩ ✈✤✓✑✍ ✐✓✜✺✌✌✍ ✏✌✑ ❞✜✎✩ ✱✩✙ ✓✚✕
✓❢✑✒✢ ✐✙✚✎ ✱✩✙ ★✍✌❞✚✎ ✍✩✤ ✈✸✜ ❞✌ ✔✌✩✥✎
✉ ✌ ❍✢✘✌✶ ✫✛✖
3. ★✌✍✌■✢✑✧ ✢✓✮ ❞✌✩❜✶ ✐✙✚✎ x ❞✒ ✵✌✌✑ ✱✩✙ ❇✐ ✍✩✤ ✫✛ ✈✘✌✱✌ x ❞✌ ✣✫r✐✮ ✫✛ ✑✌✩ ✫✍ ❜★✩ ✐✗✘✌✍
✐✙✚✎ ✱✩✙ ❇✐ ✍✩ ✚✩✑✩ ✫✛✖✤ ✑✘✌✌✓✐ ✕✩★✒ ✓❅✘✌✓✑ ✍✩✤ ✔✫✌❈ ✮✹★✜✌ ✐✙✚✎ ✐✗✓✑✚✌✩✍ ✓❏✌❞✌✩✺✌✓✍✑✒✢ ✐✙✚✎
✈✘✌✱✌ ✚✵✌r❃✺✌❞✒✢ ✐✙✚✎ ✫✛❂ ✑✌✩ ✫✍ ✷✎❞✌✩ ✐✗✘✌✍ ✐✙✚✎ ✱✩✙ ❇✐ ✍✩ ✚✩✑✩ ✫✛✤✖
342
① ✁✂✄
♠☎✆✝✞✟✆
✝❣
✠✡
✐✔☞✕✖✗☞
log x
✢✫✓
x
log x dx
✍✕✘✙
✚✙✛
log x
✢✣
✤✣☞✍✜✘
☛
❑☞✌
✏✜✒
✍☞✙
✢✣
✐✔✩ ☞✣
✍✎✏✑✒✓
✒✙✤✙
✐ ✛✜✘
✐✛✜✘
✒✚★
✍☞
✥❞✕
✥✘✦ ✣ ☞✘
✐✛✜✘
✜✧ ☞✘✙
✬
✍☞✙
✏✭✌✎✮
(logx 1) dx = log x 1 dx
= log x . x –
x
✝❣
✠✴
✐✔✩ ☞✣
✯✰✤✕✙
☛
x e x dx
✐✛✜✘
✐✛✜✘
☛
❑☞✌
ex
✒✚★
✍☞
❜✤✏✜✒
✐✛✜✘
✢✫ ★
✜✙ ✌✙
✏✑✤✍☞
✢✫✓
★
✯✰ ✤✕✙
✥✚ ✍✜✑
✐✛✜✘
✍☞
✸✹
✻
✍☞✙
x e x dx
1 x2
[
✳
d
(log x) 1 dx] dx
dx
1
x dx
x
✱
x log x – x C
✲
✍✎✏✑✒✓
✏✭✌✎✮
✤✣☞✍✜✘
x sin – 1 x
♠☎✆✝✞✟✆
✥✤✣✩☞✪
✢✫✓
✥✌✈
♠☎✆✝✞✟✆
✣✙★
✶
dx
✐✛✜✘
✚✙✛
✵✐
✣✙★
✜✎✏✑✒
= ex
x ex
✷
❑☞✌
✍✎✏✑✒✓
☛
1 . e x dx = xex – ex + C
✺
✝❣
✣☞✘
✜✎✏✑✒
✐✔✩ ☞✣
✐✛✜✘
= sin – 1x,
✥☞✫✕
✏✭✌✎✮
x
=
✐✛✜✘
1 x2
✷
x dx
✥✼
✢✣
✏✭✌✎✮
✐✛✜✘
✍☞
✤✣☞✍✜✘
❑☞✌
✍✕✌✙
✢✫★
✥✩☞☞✪ ✌✽
☛
❑☞✌
✍✕✌✙
✢✫★ ✓
1 x2
✷
t = 1 – x2
dt = – 2x dx
✕✏❥☞✒
✌✼
x dx
❜✤✏✜✒
1 x
x sin – 1 x
✥✌✈
1 x
2
2
= –
1
2
dt
= – t
t
✾ ✿
1 x2
✿
1
dx = sin – 1 x – 1 x 2
1 x
2
( – 1 x 2 ) dx
2
1
= – 1 x 2 sin 1 x x C = x – 1 x sin x C
❀
✿
❢❂❃❄❅❆❇
✗☞✎
❜✤
sin–1 x =
✤✣☞✍✜✘
✍☞✙
❈ ✐✔ ✏✌❉✩☞☞✏✐✌
✢✜
✏✍✮☞
✑☞
✍✕✘✙ ✐✕
✤✍✌☞
✢✫✓
❁
✥☞✫✕
✌✼
❀
✿
❁
❥☞★ ❊❋☞✈
✤✣☞✍✜✘
✍☞
❁
●✐✮☞✙ ✧
✍✕✌✙
✢✦ ✒
❧ ✁✂✄☎
343
x
♠✆✝✞✟✠✝ ✡☛ ☞ e sin x dx ✥✌✍ ✎✏✑✒✓✔
✞❣ ex ✎✌❞ ✕✖✗✌✘ ✕✙✚✛ ✓✜✢ sin x ✎✌❞ ✑✣✍✏✤ ✕✙✚✛ ✜❞✙ ✦✕ ✘❞✢ ✚✏✑✒✓✔ ✍✧ ★✌✢✩✪✌✫ ✬✘✌✎✚✛ ✬❞
✭✘ ✕✌✍❞ ✭✮✢ ✑✎
x
x
I = e sin x dx e ( – cos x)
= – ex cos x + I1 (✘✌✛ ✚✏✑✒✓)
e x cos x dx
... (1)
I1 ✘❞✢ e ✓✜✢ cos x ✎✌❞ ✯✘✪✌✫ ✕✖✗✌✘ ✓✜✢ ✑✣✍✏✤ ✕✙✚✛ ✘✌✛✍❞ ✭✰✓ ✭✘ ✕✌✍❞ ✭✮✢ ✑✎
x
x
x
I1= e sin x – e sin x dx
I1 ✎✌ ✘✌✛ (1) ✘❞✢ ❡★✌✛❞ ✕❡ ✭✘ ✕✌✍❞ ✭✮ ✢ ✑✎
I = – e x cos x e x sin x – I ✈✗✌✜✌ 2I = ex (sin x – cos x)
I = e x sin x dx
✈✍✫
ex
(sin x – cos x ) + C
2
✑✜✎❢✕✍✫ sin x ✎✌❞ ✕✖✗✌✘ ✕✙✚✛ ✓✜✢ ex ✎✌❞ ✑✣✍✏✤ ✕✙✚✛ ✚❞✛❞ ✕❡ ✱✌✏ ✲✕✤✰✴
✳ ✍ ✬✘✌✎✚✛ ✎✌❞
✥✌✍ ✑✎✤✌ ✒✌ ✬✎✍✌ ✭✮✔
x
7.6.1 ☞ e [ f (x) + f ✵ (x)] dx ♦✶✷ ✸✹✺✻✼ ✺✻ ✽✾✻✺✿❀
x
x
I = e x [ f (x ) + f (x )] dx = ☞ e f (x) dx + ☞ e f ✵(x) dx
✭✘❞✢ ✥✌✍ ✭✮ ✑✎
= I1
e x f (x) dx, t❁❦→ I1 = e x f (x) dx
... (1)
I1 ✘❞✢ f (x) ✓✜✢ e ✎✌❞ ✯✘✪✌✫ ✕✖✗✌✘ ✓✜✢ ✑✣✍✏✤ ✕✙✚✛ ✚❞✍❞ ✭✰✓ ✓✜✢ ★✌✢✩✪✌✫ ✬✘✌✎✚✛ ✣✌❡✌ ✭✘
x
x
✕✌✍❞ ✭✮✢ I1 = f (x) ex – ☞ f ✵(x) e dx ❂ C
I1 ✎✌❞ (1) ✘❞✢ ✕✖✑✍❃✗✌✌✑✕✍ ✎❡✛❞ ✕❡ ✭✘ ✕✌✍❞ ✭✮✢
x
I = e f (x )
e x f ( x)
❄❅❆
f (x) e x dx
e x f (x) dx C = ex f (x) + C
f ( x) dx = e x f ( x) C
♠✆✝✞✟✠✝ ✡✡ ✥✌✍ ✎✏✑✒✓
(i)
e x (tan – 1 x
1
) dx
1 x2
(x 2 + 1) e x
(ii) ❇
dx
(x + 1) 2
✞❣
1
x
–1
) dx
(i) ✤✭✌❈ I = ☞ e (tan x ❂
1 ❂ x2
✈✧
f (x) = tan– 1x, ✚✏✑✒✓② ✍✧ f ❉(x) =
1
1 ❂ x2
344
① ✁✂✄
✈☎✆
✝✞✟✠
✡☛✈✠
ex [ f (x) + f (x)]
1
x
–1
) dx = ex tan– 1x + C
I = e (tan x
1 x2
☞✌✠✍✎✟
❜☞✝✘✙
(ii)
✌✠❡
✘✚✝✛✙
(x 2 + 1) e x
dx
(x + 1) 2
I=
✝✍
✏
= ex [
✌✠❡
✘✚✝✛✙
f (x)
✝✍
x2 – 1
(x + 1) 2
✜
☎r
✝✞✟✠
✡☛✈✠
✌✑✕
x 2 – 1 + 1+1)
] dx
e [
(x + 1) 2
✢
2
] dx
(x +1) 2
★
✡✖ ✗
x
✣
✤
ex [
x –1
2
+
] dx
x + 1 (x +1)2
✩
✪
✧
✈☎✆
✓✔
2
x 1
f (x)
(x 1)2
x 1
ex [f (x) + f (x)]
✥
✦
♦✑✒
☞✌✠✍✎✟
✏
♦✑ ✒
✓✔
✌✑✕
✡✖ ✗
2
x 1 x
x 1 x
e dx =
e
2
(x 1)
x 1
❜☞✝✘✙
C
✐✫✬✭✮✯✰✱
✶
☞✑
✵✵
☎✍
♦✑ ✒
✔✷✸ ❡✠✑✕
1. x sin x
5. x log2 x
♦✑ ✒
✔✒✘❡✠✑✕
✍✠
☞✌✠✍✘❡
✲✳✴
✍✚✝✛✙✗
3. x2 ex
7. x sin– 1x
2. x sin 3x
6. x2 log x
4. x log x
8. x tan–1 x
x cos 1 x
✹
–1
–1
10. (sin x)
9. x cos x
2
11.
12. x sec2 x
1 x2
15. (x2 + 1) log x
✺
14. x (log x)2
13. tan –1x
x ex
16. e (sinx + cosx) 17.
(1 x ) 2
x 1 sin x
18. e 1 cos x
✼
x
✻
19. e
x
❃
❅
❇
1 1
–
x x2
–1
22. sin
❃
❅
❇
✔✷ ✸ ❡
23.
✵❋
✙♦✕
✌✑ ✕
❄
❆
❈
☞✡✚
❍■✠❏
✍✠
❑✟❡
✍✚✝✛✙✗
3
▲
x 2 e x dx
(A)
(C)
1 x3
e
3
1 x3
e
2
r❏✠r❏
✾
21. e2x sin x
❉
❈
❊
✵●
❉
❆
2x
1 x2
❁
(x 3) e x
20.
(x 1)3
❄
✾
✿
✡✖ ✆
✧
C
(B)
▼
C
(D)
1 x2
e
3
1 x2
e
2
✧
C
✧
C
✽
❀
❂
❧ ✁✂✄☎
345
24. ✝ e x sec x (1 ✆ tan x) dx ❝✞✟❝✞ ✠✡☛
(A) ex cos x + C
(B) ex sec x + C
(C) ex sin x + C
(D) ex tan x + C
7.6.2 ♦☞✌✍ ✎✏✑ ✒✓✔✕✖ ♦✗✌ ✘✙✕✔✚✛ (Integrals of some more types)
❀✠✟✜ ✠✢ ✣✟✤✥✦✟☛ ✧✢✟★✩✪ ✫✬✫✭ ✮✞ ✯✟✭✟✫✞✰ ✬✱✲✳ ✫✬✫✦✟✴✵ ✮✶★✟✞ ✬✷✲ ✮✶✟✢✟✫✸✟★ ✧✢✟★✩✪✟✷✤ ★✹ ✺✺✟✻
★✞✷✤❞✼✷ ✽✡✧✷ ✫★
(ii) ❂ x 2 ❁ a 2 dx
x 2 ✾ a 2 dx
(i)
✿
(i)
✢✟✪ ✩✹✫✽❡ ✫★ I ❃ ❂
(iii)
2
2
✿ a ✾ x dx
x 2 ❄ a 2 dx
✯✺✞ ✮✲✩✪ ✈ ★✟✷ ✫❅✰✹❀ ✮✲✩✪ ✢✟✪✰✷ ✠✱❡ ✯✟✡✞ ✣✟✤✥✦✟☛ ✧✢✟★✩✪ ❅✟✞✟ ✠✢ ✮✟✰✷ ✠✡✤
2
I= x x
2
1
2
a2
2x
x
a2
x2
2
= x x ❆a ❆❇
2
2
x ❆a
2
x dx
2
2
dx = x x ❆ a ❆ ❇
2
2
2
2
2
= x x ❉ a ❉ ✝ x ❉ a dx ❉ a ✝
2I = x x 2
✯❊✟✬✟
2
I=
a2
2
x – a dx
2
x ❆a
2
dx
dx
x2 ❉ a2
dx
2
2
2
= x x ❉ a ❉I❉a ✝
✯❊✟✬✟
x2 ❆ a2 ❈ a2
x2 ❉ a2
dx
a2
x2
x
2
a2
a2
log x
x –a –
2
2
2
x2 – a2
C
❜✧✹ ✮✶★✟✞ ❋●✧✞✷ ❋✟✷ ✧✢✟★✩✪✟✷✤ ✢✷✤ ✯✺✞ ✮✲✩✪ ✈ ★✟✷ ✫❅✰✹❀ ✮✲✩✪ ✩✷★✞ ❡✬✤ ✣✟✤✥✦✟☛
✧✢✟★✩✪ ✫✬✫✭ ❅✟✞✟ ✠✢ ✮✟✰✷ ✠✡✤
1
a2
2
2
2
2
log x ❍ x 2 ❍ a 2 ❍ C
(ii) ❏ x ❍ a dx ■ x x ❍ a ❍
2
2
1
a2
x
2
2
2
2
sin –1 ❍ C
(iii) ❏ a ❑ x dx ■ x a ❑ x ❍
2
2
a
✫✬★❢✮✰☛ ✧✢✟★✩✪✟✷✤ (i), (ii) ❡✬✤ (iii) ✢✷✤ ▲✢✦✟☛ x = a sec▼ , x = a tan▼ ✯✟✡✞
x = a sin▼, ✮✶✫✰✐❊✟✟✮✪ ★✞✪✷ ✮✞ ◆✟✹ ❜✪ ✧✢✟★✩✪✟✷✤ ★✟✷ ❖✟✰ ✫★❀✟ ✽✟ ✧★✰✟ ✠✡✼
346
① ✁✂✄
♠☎✆✝✞✟✆
✝❣
✠✡
è✕✌✖
x2
☞
✗✏✑✒✓
☛
✑✎
2 x 5 dx
☛
x2
✚
x+1=y
✈✛
x2
✠✫
è✕✌✖
✣❥
y2
✗✏✑✒✓
✑✎
=
1
(x 1) x 2
2
y2
4
✥✌✍
✭
✘
✩✢
❂
♦✢✦
1.
4 x2
4.
x2
7.
✣✮ ❃✖✌✢ ❄
❇
❉
4x 1
1 3x
❈
❉
❇
x
2
♦✢✦
✣✦❅✖✌✢❄
✤
y2
✪
✙
❥✜✌✖✢
✪
✪
✤
C [7.6.2 (ii)
♦✢✦ ✧✣✕✌✢ ★ ✩✢
x2
✪
✪
2x 5
C
✪
✪
♦✢✦
✧✣✕✌✢★
4 (x 1)2 dx
✭
✚
✣❥
✘
dx = dy
=
1
y 4 y2
2
=
1
(x 1) 3 2 x x 2
2
✤
✯
✳
✰
✩❆✌✎❅✖
2.
1 4x 2
5.
1 4x
8.
4
y 2 dy
4
✎✌
✤
✎✏✑✒✓✔
✭
3 2x x 2 dx =
✍✎
4 dx
2 x 5 2 log x 1
✪
✐✸✹✺✻✼✽✾
❁
✘
✍✛
4
log y
2
✤
3 2 x x 2 dx
x+1=y
✣✮✎ ✌❥
✤
✪
✬
✚
(x 1) 2
✚
22 dy
1
y
2
✬
✙
dx = dy,
=
✈✛
❜✩
✘
3 2x x 2 dx
☞
✎✏✑✒✓✔
2 x 5 dx
❥✜✌✖✢
2 x 5 dx =
♠☎✆✝✞✟✆
✝❣
✘
✥✌✍
x
2
❈
❊
3x
✳
✰
✤
C [7.6.2 (iii)
2 sin –1
✱
✴
✶
x 1
2
✰
✲
✵
✷
✿❀✿
✎✏✑✒✓✔
❇
❊
y
4
sin –1
2
2
x2
3.
x2
❈
4x 6
6.
x2
❉
4x 5
9.
1
❋
❈
x2
9
❊
✰
C
✩✢
]
]
❧ ✁✂✄☎
347
✐✆✝✞ ✟✠ ✡☛☞ ✟✟ ✌✍☞ ✎✏✑ ✒✓✔✕ ✖✔ ✗✘✞ ✖✑✙✚✡✛
10. ✢ 1 ✜ x 2 dx ❝✕✔❝✕ ✏✣✤
(A)
1
x
1 ✧ x 2 ✧ log ✥ x ✧ 1 ✧ x 2 ✦ ✧ C
2
2
(C)
2
x (1 ★ x 2 ) 2 ★ C
3
3
(D)
3
(B)
2
(1 ★ x 2 ) 2 ★ C
3
x2
1
1 ✩ x 2 ✩ x 2 log x ✩ 1 ✩ x 2 ✩ C
2
2
11. ✢ x 2 ✪ 8 x ✜ 7 dx ❝✕✔❝✕ ✏✣
7.7
(A)
1
( x ✫ 4) x 2 ✫ 8 x ✧ 7 ✧ 9log x ✫ 4 ✧ x 2 ✫ 8 x ✧ 7 ✧ C
2
(B)
1
( x ✧ 4) x 2 ✫ 8 x ✧ 7 ✧ 9log x ✧ 4 ✧ x 2 ✫ 8 x ✧ 7 ✧ C
2
(C)
1
( x ✬ 4) x 2 ✬ 8 x ✭ 7 ✬ 3 2 log x ✬ 4 ✭ x 2 ✬ 8 x ✭ 7 ✭ C
2
(D)
1
9
( x ✫ 4) x 2 ✫ 8 x ✧ 7 ✫ log x ✫ 4 ✧ x 2 ✫ 8 x ✧ 7 ✧ C
2
2
❢✮❢✯✰✱ ✲✳✴✵✶✮
(Definite Integral)
✙✐✷✸✍ ✐✙✕✹✷✍✺✔✍☞ ✌✍☞ ✏✌✞✍ ✻✙✞✙✝✗✼ ✎✌✔✖✸✞✔✍☞ ☛✍✽ ❝✔✕✍ ✌✍☞ ✻✾✘✘✞ ✙✖✘✔ ✏✣ ✻✔✣✕ ☛✿✽✷ ✙☛✙✝✔❀❁ ✐✽✸✞✔✍☞
☛✍✽ ✎✌✔✖✸✞✔✍☞ ✎✙✏✼ ✻✙✞✙✝✗✼ ✎✌✔✖✸✞✔✍☞ ✖✔✍ ♦✔✼ ✖✕✞✍ ✖✑ ☛✿✽✷ ✙☛✙❂✘✔✍☞ ✐✕ ✗✗✔❃ ✖✑ ✏✣✛ ❄✎
✐✙✕✹✷✍✺ ✌✍☞ ✏✌ ✙✖✎✑ ✐✽✸✞ ☛✍✽ ✙✞✙✝✗✼ ✎✌✔✖✸✞ ✖✔ ✻✾✘✘✞ ✖✕✍❅
☞ ✛✍ ✙✞✙✝✗✼ ✎✌✔✖✸✞ ✖✔ ✡✖
✻✙✈✼✑✘ ✌✔✞ ✏✔✍✼✔ ✏✣✛ ✡✖ ✙✞✙✝✗✼ ✎✌✔✖✸✞ ✖✔✍
b
❆ a f (x) dx , ✎✍ ✙✞✙✺❃❀❁ ✙✖✘✔ ✚✔✼✔ ✏✣ ✚✏✔❇
b❪ ✎✌✔✖✸✞ ✖✑ ✒✹✗ ✎✑✌✔ ✼❈✔✔ a, ✎✌✔✖✸✞ ✖✑ ✙✞❉✞ ✎✑✌✔ ✖✏✸✔✼✑ ✏✣☞✛ ✙✞✙✝✗✼ ✎✌✔✖✸✞ ✖✔
✐✙✕✗✘❪ ✘✔ ✼✔✍ ✘✔✍❅✔✍☞ ✖✑ ✎✑✌✔ ☛✍✽ ❊✐ ✌✍☞ ✖✕✔✘✔ ✚✔✼✔ ✏✣ ✻❈✔☛✔ ✘✙✺ ✻☞✼ ✕✔✸ [a, b] ✌✍☞ ❄✎✖✔ ✖✔✍❄❃
✐✆✙✼✻☛✖✸✚ F ✏✣ ✼✔✍ ✙✞✙✝✗✼ ✎✌✔✖✸✞ ✖✔ ✌✔✞ ✻☞✙✼✌ ❣❝✺✿✻✔✍☞ ✐✕ F ☛✍✽ ✌✔✞✔✍☞ ☛✍✽ ✻☞✼✕ ✻❈✔✔❃✼❋
F(b) – F(a) ☛✍✽ ❝✕✔❝✕ ✏✔✍✼✔ ✏✣❪ ☛✍✽ ❊✐ ✌✍☞ ✖✕✔✘✔ ✚✔✼✔ ✏✣✛ ✙✞✙✝✗✼ ✎✌✔✖✸✞ ☛✍✽ ❄✞ ✺✔✍✞✔✍☞ ❊✐✔✍☞
✖✑ ✏✌ ✻✸❅❞✻✸❅ ✗✗✔❃ ✖✕✍❅
☞ ✍✛
7.7.1 ●❍■❏❑▲▼ ◆❖ P❖◗❍ ❘■▲ ❙❑ ◗■❚ ❯❱❯❲❳❨ P◗❍◆▼❱ (Definite integral as the limit
of a sum)
✌✔✞ ✸✑✙✚✡ ✙✖ ✡✖ ❝☞✺ ✻☞✼ ✕✔✸ [a, b] ✐✕ ✡✖ ✎☞✼✼ ✐✽✸✞ f ✐✙✕❩✔✔✙❀✔✼ ✏✣✛ ✌✔✞ ✸✑✙✚✡ ✙✖
✐✽✸✞ ☛✍✽ ✎❩✔✑ ✌✔✞ ❬❭✔✍✓✔✕ ✏✣☞ ❄✎✙✸✡ ✐✽✸✞ ✖✔ ✻✔✸✍❫✔ x-✻❴✔ ✎✍ ❵✐✕ ✡✖ ☛❛ ✏✣✛
348
① ✁✂✄
y = f (x), x = a, x = b
♦☎
b
✙
a
f (x) dx
x=b
♦✠✏
✒❣ ✚
✥✓✖
✛✟
✡☛✞☞✠
♠✎✈✆✗☞✞✑✞✠✆
✘✠✆
2, 3, …, n
✎☞
✈✞♦✧✏ ✡✗
✈✞✤✗
✍✞✠
✍✓✡❞✱
(ABLC)
✒✘
♦ ✠✏
✑✓✡❞✱
✝✞✠✌✞
✡✑✱✢
✍✞
✛✟
✝✞✠✌✞✎✏✑
♦☎ ✢
x-
✈✝✞
✒✓
✡✔✡✕✖✗
✟✘✞✍✑✔
✱♦ ✆
✍ ✞✠ ✡ ✣✤✞✠ ✆
x=a
✦✈✞♦✧✏ ✡✗
★✩✪
è✤✞✔
♠✎✝✞✠✌✞✞✠✆ ✍✞
✫✓✡❞✱
✤✞✠✴
✤✡✫
n
✒❣ ❞✒✞✰ ✎✵✶ ✤✠✍
✱♦ ✆
✫✠✡✬✞✱✭✚
✡✔✡✫❧✮ ✣
✟✘✞✔
❞✒✞✰
b a
h
n
✈r✞♦✞
✟✠
✍☞ ✔✠
✡☛✞☞✠
✟✠
✎✡☞✯✞✞✡✮✞✗
★✩✪
✜ ✞✗
✟✠
PRSQP
[x0, x1], [x1, x2] ,..., [xr – 1, xr], ... [xn – 1, xn],
n
x0 = a, x1 = a + h, x2 = a + 2h, ... , xr = a + rh
PRSQP, n
✝✞✠✌✞
✍ ✞✠
✈✝✞
✍✞✠
✡♦✯✞✞✡❞✗
xn = b = a + nh
✖✡✖❧✗
✝✞✠ ✌✞
[a, b]
✈✆✗☞✞✑
✝✞✠✌✞✎✏ ✑
x-
✱♦✆
✲ ✳ ✗✞✠
h
✲
✗r✞✞
0
♠✎✝✞✠✌✞ ♠✎✈✆✗☞✞✑✞✠✆
[xr – 1, xr], r = 1,
✒❣✚
✎✞✗✠
✒❣ ✆
<
✍✞ ✝✞✠✌✞✎✏✑
✡✍
(ABDCA)
✝✞✠✌✞
✍✞ ✝✞✠✌✞✎✏✑
<
(ABDM)
✈✞✤✗
... (1)
✍✞ ✝✞✠✌✞✎✏✑
✷✸✹✺✻✼✽ ✾✿❀
▲✎✮✣✗❁
✱✍
✫❅✟☞✠
♦✠✏
xr – xr–1
✤✡✫
✑✴✯✞✴
✟✘✞✔
✲
✒✞✠
0
✈r✞✞❧✗❂
❞✞✗✠
✒❣✚
✆
h
✲
✈✥
0,
✒✘
✗✞✠
✟✘✓✍☞❃✞
✡✔❆✔✡✑✡✬✞✗
✦❄✭
✘✠
✤✞✠✴✎✏✑✞✠✆
✫✕✞✞❧✱
✍✞
✴✱
✗✓✔✞✠✆
✝✞✠✌✞✎✏✑
✡✔✘✞❧❃ ✞
✍☞✗✠
✒❣✆
n ❇1
sn = h [f(x0) + … + f(xn - 1)] = h
❉
f ( xr )
... (2)
r❈0
n
Sn = h [f (x1 )
✈✞❣☞
f (x2 )
f (xn )] h
f (xr )
... (3)
r 1
sn
✤✒✞✰
✈✞✤✗✞✠ ✆
✈✆✗☞✞✑
♦✠✏
✱♦✆
Sn
♠✎✈✆✗☞✞✑✞✠✆
✝✞✠ ✌✞✎✏✑✞✠✆
[xr–1, xr]
♦✠✏
♦✠✏
✤✞✠✴
✡✑✱
sn <
✍✞✠
✒✘
✝✞✠ ✌✞
[xr–1, xr] r = 1, 2, 3, …, n,
✡✔✡✫❧✮✣
✎✞✗✠
✒❣✆
PRSQP
✍☞✗✞
✒❣✚
✎☞
✈✟✡✘✍✞
✥✔✠ ☎✘✕✞❁
✦❄✭
♦✠✏
✡✔❆✔ ✈✞✤✗✞✠✆ ✱♦✆ ♠✐✖
✟✆✫✯✞❧
✘✠✆
✡✍✟✓
▲♦✠✐ ❊
♠✎
✡✍
✍✞
✝✞✠✌✞✎✏✑
< Sn
... (4)
❧ ✁✂✄☎
349
❀✆✝ n ✞ ✟ ❪ ✠✡☛ ☞✌✍✆✌❀✡✎ ✏✑✒✓✔✡✕ ✏☛ ✏✑✒✓✔✡✕ ✖✡☛✠✓ ✗✘✓ ✙✡✠✓ ✖✚✑ ✛✡✚✜ ❀✖ ✢✡✣ ✆✘❀✡ ✙✡✠✡ ✖✚✑
✆✒ ❢✤✥ ✛✡✚✜ ❢✦✥ ✧☛★ ✏✓✆✢✠ ✢✡✣ ✩✒ ✏✢✡✣ ✖✚✑ ✠✪✡✡ ✫✬✡❀✆✣✭✮ ✏✓✆✢✠ ✢✡✣ ✖✓ ✧✯ ✧☛★ ✛✰✠✕✱✠
✛✬✡✓✭✌ ✈✡☛✲✡☞★✘ ✖✚✳
✏✑✡✧☛★✆✠✒ ✬✡✡✭✡✡ ✢☛✑ ✖✢ ✴✏☛ ✆✣✵✣✆✘✆✶✡✠ ☞✷✒✡✜ ✆✘✶✡✠☛ ✖✚✑
lim Sn = lim sn
n
n
b
✈✡☛✲✡ PRSQP ✒✡ ✈✡☛✲✡☞★✘ = ✸ f (x ) dx
a
... (5)
✴✏✏☛ ❀✖ ☞✠✡ ✗✘✠✡ ✖✚ ✆✒ ✛✬✡✓✭✌ ✈✡☛✲✡☞★✘ ✧✯ ✧☛★ ✣✓✗☛ ✧☛★ ✛✡❀✠✡☛✑ ✩✧✑ ✧✯ ✧☛★ ❜☞✜ ✧☛★
✛✡❀✠✡☛✑ ✧☛★ ✹✓✗ ✧☛★ ✆✒✏✓ ✈✡☛✲✡☞★✘ ✒✡ ✏✓✆✢✠ ✢✡✣ ✬✡✓ ✖✚✳ ✏✺✆✧✻✡ ✧☛★ ✆✘✩ ✖✢ ☞✷✼❀☛✒ ✫☞✛✑✠✜✡✘
✧☛★ ✹✡❀☛✑ ✆✒✣✡✜☛ ☞✜ ✧✯ ✒✓ ✫✎✗✡✴✕ ✧☛★ ✹✜✡✹✜ ✫✎✗✡✴✕ ✧✡✘☛ ✛✡❀✠✡☛✑ ✒✡☛ ✘☛✱
✑ ☛✳ ✛✠♦ ✖✢ ❢✽✥ ✒✡☛ ✝✺✹✡✜✡
✆✣✵✣✆✘✆✶✡✠ ✾☞ ✢☛✑ ✆✘✶✡✠☛ ✖✚✑✳
b
h [ f (a ) ❂ f (a ❂ h) ❂ ... ❂ f (a ❂ (n – 1) h]
✿ a f ( x)dx = lim
h❁ 0
✛✪✡✧✡
1
b
[ f (a) ❅ f (a ❅ h) ❅ ... ❅ f (a ❅ (n – 1) h]
✿ a f ( x)dx = (b – a ) nlim
❃❄ n
✙✖✡✎
h=
b–a
n
... (6)
0 ❆❇♥ n
✫☞❀✺♠
✕ ✠ ❈❀✑✙✒ ❢❉✥ ❀✡☛✱☞★✘ ✒✓ ✏✓✢✡ ✧☛★ ✾☞ ✢☛✑ ✆✣✆❊✗✠ ✏✢✡✒✘✣ ✒✓ ☞✆✜✬✡✡✭✡✡ ✒✖✘✡✠✡ ✖✚✳
❋●❍■❏❑▲ ✆✒✏✓ ✆✧✆❊✡✭✌ ✛✑✠✜✡✘ ☞✜ ✩✒ ☞★✘✣ ✧☛★ ✆✣✆❊✗✠ ✏✢✡✒✘✣ ✒✡ ✢✡✣ ☞★✘✣ ✩✧✑ ✛✑✠✜✡✘
☞✜ ✆✣✬✡✕✜ ✒✜✠✡ ✖✚ ☞✜✑✠ ✺ ✏✢✡✒✘✣ ✧☛★ ✫✏ ✗✜ ☞✜ ✣✖✓✑ ✆✙✏✒✡ ✗❀✣ ✖✢ ✐✧✠✑ ✲✡ ✗✜ ✒✡☛ ✆✣✾✆☞✠
✒✜✣☛ ✧☛★ ✆✘✩ ✒✜✠☛ ✖✚✑✳ ❀✆✝ x ✧☛★ ✐✪✡✡✣ ☞✜ ✐✧✠✑ ✲✡ ✗✜ ✒✡☛ t ✛✪✡✧✡ u ✏☛ ✆✣✆✝✕✭✌ ✆✒❀✡ ✙✡✠✡ ✖✚
b
b
b
✛✪✡✧✡ ✸ f (u ) du
✠✡☛ ✖✢ ✏✢✡✒✘✣ ✸ a f ( x) dx ✧☛★ ✐✪✡✡✣ ☞✜ ✧☛★✧✘ ✏✢✡✒✘✣
✸ a f (t ) dt
a
✆✘✶✡✠☛ ✖✚✑ ✳ ✛✠♦ ✆✣✆❊✗✠ ✏✢✡✒✘✣ ✧☛★ ✆✘✩ ✏✢✡✒✘✣ ✗✜ ✩✒ ✢▼✒ ✗✜ ✒✖✘✡✠✡ ✖✚✳
2
2
◆❖P◗❘❙P ❚❯ ❀✡☛✱☞★✘ ✒✓ ✏✓✢✡ ✧☛★ ✾☞ ✢☛✑ ✸ ( x ❱ 1) dx ✒✡ ✢✡✣ ❞✡✠ ✒✓✆✙✩✳
0
◗❣ ☞✆✜✬✡✡✭✡✡ ✧☛★ ✛✣✺✏✡✜
1
b
✸ a f ( x) dx = (b – a ) nlim n [ f (a )
f (a h) ...
b–a
n
✙✖✡✎
h=
✴✏ ✫✝✡✖✜✔✡ ✢☛✑☛
a = 0, b = 2, f (x) = x2 + 1, h ❲
2–0 2
❲
n
n
f (a
(n – 1) h]
350
① ✁✂✄
2
❜☎✆✝✞
✠
0
( x 2 1) dx = 2 lim
✟
n✡☛
1
[ f (0)
n
2
f( )
n
☞
1
22
= 2 lim [1 ( 2
n
n
n
✑
✑
✌✍
= 2 lim
n
☞
4
f ( ) ...
n
☞
42
1) ( 2
n
✑
1
[(1 1 ... 1)
n
n
✖✗
✘✗
✙
✐✚
✑
1) ...
✑
1 2
(2
n
✣
22 2
(1
n
= 2 lim
1
[n
n
✑
4 (n 1) n (2n – 1)
]
6
n
= 2 lim
1
[n
n
☞
2 (n 1) (2n – 1)
]
n
3
n✌✍
n✡☛
= 2 lim [1
✩
n✧★
✣
✑ ✒
2 (n – 1)
)]
n
(2n – 2) 2
n2
✏
✑
1 ]
✓
✕
42 ... (2n – 2) 2 ]
1
[n
n
✛
✎
✔
= 2 lim
n✜✢
f(
☞
22 ... (n – 1)2 ]
✣
✣
✥
✤
✦
14
2
1
1
4
(1 ) (2 – )] = 2 [1
] =
3
3
n
n
3
✪
☞
2 x
♠✫✬✭✮✯✬
✭❣
✰✱
✵✆❃❄✲✲❅✲✲
❀✲✳✴✵✶✝
✺✳✶
✷✸
☎✸✹✲
✺✳✶
✻✵
✹✳✼
✠
✠
P❙ ✼
❑✳■✲✸
✺✳✶
e dx
✷✲
✹✲✽
✾✲✿
✷✸✆❁✞❂
❆✽❇☎✲❃
1 0
e dx = (2 – 0) lim
e
0
n
n
❈
2 x
✴❇■✲✲✳❏✲❃
0
n
❊
✌✍
✵▲✲✳✼
✺✳✶
❀✲✳✴✵✶✝
✺✳✶
☎▼ ◆✲
2
n
✑ e
4
n
✑ e
2n – 2
n
✑ ... ✑ e
❊
●
✷✲
❉
❋
❋
❍
❖✵❀✲✳✴
✷❃✿✳
P❇✞
❁P✲◗
a = 1, r
❘
2
n
e
,
P✹
✵✲✿✳
✆✷
1
e dx = 2 lim [
0
n
n
2 x
❚
2n
e n –1
✌✍
2
en✥
❲
❳
❨
❩
❨
❩
❬
❭
1 e2– 1
] = 2 lim
n
n n2
1
e –1
❯❱
2
2 (e – 1)
=
lim
n
2
en –1
2
n
= e2 – 1
2
(e h 1)
0
h
[ lim
h❪
❫
❴
1
]
✺✳✶ ❖✵❀✲✳✴ ☎✳
❧ ✁✂✄☎
351
✐✆✝✞✟✠✡☛ ☞✌✍
❀✎✏✑✎✏✒ ✓✔ ✕✔✖✎ ✗✏✘ ✙✚ ✖✏✒ ✛✜✢✜✛✣✛✤✎✥ ✛✜✛✦✧✥ ✕✖✎✓✣✜✎✏✒ ✓✎ ✖✎✜ ★✎✥ ✓✔✛✩✪✫
2. ✮ 0 ( x ✭ 1) dx
4
x
5. ✮ 1 e dx
✰
2
4. ✬ 1 ( x ✯ x) dx
7.8
5
b
1. ✬ x dx
a
❞✲✳ ❞✴ ✵✶✷✶✸✹✶✺✻ ✼✽✾❁
✿
3
2
3. ✮ 2 x dx
1
4
6. ✬ ( x ✱ e2 x ) dx
0
(Fundamental Theorem of Calculus)
7.8.1 ④❂❃❄❂❅❆❇ ❅❆❇❈ (Area function)
b
❣✖✜✏ ✬ f ( x) dx ✓✎✏ ✗❉ y = f (x), x-✈❊✎❋ ✪✗✒
a
✓✎✏✛●❀✎✏✒ x = a ✥r✎✎ x = b ✕✏ ✛❍✎■✏ ❊✎✏❏✎ ✗✏✘ ❊✎✏❏✎✚✘✣ ✗✏✘
✙✚ ✖✏✒ ✚✛■❑✎✎✛▲✎✥ ✛✓❀✎ ❣▼✫ ✖✎✜ ✣✔✛✩✪ [a, b] ✖✏✒ x ✓✎✏◆❖
x
✈✎✗❙✘✛✥ ❚❯❱ ✖✏✒ s✎❀✎✒✛✗✘✥ ❊✎✏❏✎
✬ a f ( x) dx
✗✏ ✘ ❊✎✏❏✎✚✘✣ ✓✎✏ ✛✜✙✛✚✥ ✓■✥✎ ❣▼✒ [❀❣✎❲ ❀❣ ✖✎✜ ✛✣❀✎
➥P◗❘ ❣▼ ✥P
✑❀✎ ❣▼ ✛✓ x ❳ [a, b] ✗✏ ✘ ✛✣✪
f (x) > 0 ❣▼✫
✛✜✢✜✛✣✛✤✎✥ ✓r✎✜ ✕✎✖✎❢❀✥❨ ✈❢❀ ✚✘✣✜✎✏✒ ✗✏ ✘ ✛✣✪ ❑✎✔
✕❩❀ ❣▼✫ ◆✕ s✎❀✎✒✛✗✘✥ ❊✎✏❏✎ ✓✎ ❊✎✏❏✎✚✘✣ x ✗✏✘ ✖✎✜ ✚■
✛✜❑✎❖■ ❣▼✫
❸❹❺❻❼❽❾ ❿➀➁
◗♥✕■✏ ✦✎❬◗✎✏✒ ✖✏✒ ◆✕ s✎❀✎✒✛✗✘✥ ❊✎✏❏✎ ✓✎ ❊✎✏❏✎✚✘✣ x ✓✎ ✪✓ ✚✘✣✜ ❣▼✫ ❣✖ x ✗✏✘ ◆✕ ✚✘✣✜ ✓✎✏ A(x)
✕✏ ✛✜✛◗❖▲● ✓■✥✏ ❣▼✒✫ ◆✕ ✚✘✣✜ A(x) ✓✎✏ ❣✖ ❊✎✏❏✎✚✘✣ ✚✘✣✜ ✓❣✥✏ ❣▼✒ ✈✎▼■ ❀❣ ❣✖✏✒ ✛✜✢✜✛✣✛✤✎✥ ✕♥❏✎
✕✏ ✚❭✎❪✥ ❣✎✏✥✎ ❣▼✫
x
A (x) = ✬ a f ( x ) dx
... (1)
✏ ❣▼✫✒ ✥r✎✎✛✚ ❣✖ ❀❣✎❲ ✚■ ✗✏✘✗✣ ◆✜✓✔ ❫❀✎✤❀✎ ✓■✏✒✑✏
◆✕ ✚✛■❑✎✎▲✎✎ ✚■ ✈✎❜✎✛■✥ ◗✎✏ ✈✎❜✎■❑✎♥✥ ✚❭✖❀
❴❀✎✏✒✛✓ ◆✜✓✔ ❵✚✚✛❛✎ ◆✕ ✚✎❝❡❀ ✚❘❤✥✓ ✓✔ ✕✔✖✎ ✗✏✘ P✎❣■ ❣▼✫
7.8.2 ❅❥❦❃♠ ♦ ♣❦❂q❇❈ t✉✇❂① q② ❅❥③❂❦ ⑤❂⑥❂⑦⑧❂⑨① ❅❥❦❃♠ (First fundamental theorem
of integral calculus)
✖✎✜ ✣✔✛✩✪ ✛✓ P✒ ◗ ✈✒✥■✎✣ [a, b] ✚■ f ✪✓ ✕✒✥✥ ✚✘✣✜ ❣▼ ✈✎▼■ A (x) ❊✎✏❏✎✚✘✣ ✚✘✣✜ ❣▼✫ ✥P
✕❑✎✔ x ❳ [a, b] ✗✏✘ ✛✣✪ A⑩(x) = f (x)
7.8.3 ♣❦❂q❇❈ t✉✇❂① q② ✉❶①②♠ ⑤❂⑥❂⑦⑧❂⑨① ❅❥❦❃♠ (Second fundamental theorem of
integral calculus)
❣✖ ✜✔✧✏ ✪✓ ✪✏✕✏ ✖❣❩✗✚♥❷✎❖ ✚❭✖❀
✏ ✓✔ ❫❀✎✤❀✎ ✓■✥✏ ❣▼✒ ✛✩✕✓✔ ✕❣✎❀✥✎ ✕✏ ❣✖ ✚❭✛✥✈✗✓✣✩ ✓✎
❵✚❀✎✏✑ ✓■✥✏ ❣❘✪ ✛✜✛✦✧✥ ✕✖✎✓✣✜✎✏✒ ✓✎ ✖✎✜ ★✎✥ ✓■✥✏ ❣▼✫✒
352
✐☎✆✝✞
F
① ✁✂✄
✟
❡✠✡
☛☞✌✍✎
b
✘✙❣
✕✑
✛
✌✏
✑✒✓
✔✒✕✖✠☛
[a, b]
f
✥✖
✎✏
✱✒✕✕
✥✗☛✡
✘✙
✔✠✙✖
f
✏✠
✥❞✌ ✕✔✚✏☛✍
f ( x ) dx = [F( x )] ba = F (b) – F(a)
a
❢✜✢✣✤✦✧
1.
b
✬✠★✓✠✩ ✒ ❡✩✒ ✘❡
F
2.
✏✠
✪✘
✲✳✴
✥❞ ❡✩✪
✫ ✏✠✩ ✭✱
b
✱☞❡✠
✔❀✪✒✕
✱❡✠✏☛✡
3.
✥❞❡✩✪
✏✠✩
✥✖
❡✠✡✵
✲✥✪✠✩✸☞
✹✠✕
✥❞✏✠✖
✏✖✡✩
✘✙
–
✮✪✯✕ ✏✖✕✩ ✘✙ ✒ ✌✏
✶✲✱☞
✯✪✠✩✌
✒ ✏
✏☞
✥❞✌ ✕
✪✘
✔✠✱✠✡
✔✚✏☛✍
✘❡✩✒
✌✚✌❧
✰
✪✠✩✸✥✗☛
✥❞✓✠✡
f ( x) dx = (f
a
✏✠
✌✡✷✡
✏☞
✏✖✕☞
✱☞❡✠
✱☞❡✠
✹✠✕
✚✩ ✗ ✥❞✌ ✕ ✔✚✏☛✍
a
)
❡✠✡ ❣
✥✖
✌✏✎
✌✑✡✠
✌✡✌✬✴✕
✘✙❣
✎✏ ✌✡✌✬✴✕ ✱❡✠✏☛✡ ✹✠✕ ✏✖✡✩ ❡✩✒ ✍✌✺☛ ✱✒✌✻✪✠ ✎✏ ✎✩✱✩ ✥✗☛✡ ✏✠ ✥❞✠✼✕ ✏✖✡✠ ✘✙ ✌✍✱✏✠
✔✚✏☛✍
✌✓✪✠
✸✪✠
✘✙❣
✱❡✠✏✈✪
✪✘
✔✚✏☛✡
✔✠✙✖
✱❡✠✏☛✡
✚✩✗
✑☞✴
✱✒✑❧
✒
✏✠✩
✔✠✙✖
❡✍✑✽ ✕ ✏✖✕✠ ✘✙❣
4.
b
✛
a
f ( x) dx
❡✩✾
✒
[a, b]
3
✌✡✌✬✴✕
✱❡✠✏☛✡
[– 2, 3]
✚✩✗
❆
b
(i)
✔✌✡✌✬✴✕
C
✏✠✩
✏✖✩✒
b
✰
✕✠✩
✰
a
✱❡✠✏☛✡
☛✩ ✡✩ ✏☞
✥✠✕✩
✘✙✒
❈
✭✱
♠●❍■❏❑❍
(i)
✏✠
✟▲
2
✘✙ ✯✪✠✩✌
✒ ✏
✥❞✏✠✖
✌✡✌✬✴✕
❋
✱❡✠✏☛✡
✲✓✠✘✖❃✠✠✩✒
✌✡✷✡✌☛✌▼✠✕
x dx
P
❈
4 sin 3
0
✪✌✓
2
1
– 1) 2
✘✙✒
⑥✠✖✠
✯✪✠✩✌
✒ ✏
❡✠✡
☛☞✌✍✎
✔✌❁✠✮✪✯✕
F(x)
✘❡
F(x)
✪✘
✚✩✗
♦❉✠✠✡
✥✖
✑✒✓
✘✙ ❣
b
a
✔✒✕✖✠☛
✥✗☛✡
❋
❡✠✡
2 t cos 2 t dt
❋
✹✠✕
✏✖✡✩
❡✩✒
✱❡✠✏☛✡
F(x) + C
✥✖
✔✴✖
✌✚✴✠✖
✥✖
(ii)
✏☞✌✍✎✾
✌✚✴✠✖
❈
✏✖✕✩
✏✠
❡✠✡
✍✠✩
✌✏
✰
❊
♦✚✩✳❜
a
✔✴✖
f ( x) dx
✌✚☛✿✼✕
✏✠
❡✠✡
✘✠✩
✍✠✕✠
✘✙❣
✘✙✒ ❣
✹✠✕
9
x
4
3
x 2 )2
✏☞✌✍✎❣
dx
(iii)
2
❖
1
x dx
( x 1) ( x 2)
◆
◆
f
f ( x ) dx )
[F(b) C] – [F(a) C] F(b) – F( a)
(30 –
(iv)
❁✠❞✠✌
✒ ✕❡✽ ☛✏
(Steps for calculating
b
✹✠✕
✱❡✠✏☛✡✠✩✒
3 2
✛
✡✘☞✒
✏✖✡✠
f (x) = x( x
✌☛✎
✹✠✕ ✏☞✌✍✎❣
❊
✚✿✗❜
✴✴✠❇
✹✠✕ ✏✖✡✩ ✚✩✗ ✴✖❃✠
f ( x) dx [F ( x) C] ba
✘❡
✏☞
✌✏
b
(ii) [F ( x)]a = F(b) – F(a)
✔✑
dx
✚✩✗
f ( x ) dx
✔✠✚✬✪✏✕✠
✏✠ ✱✿ ✥✌✖❁✠✠✌❂✠✕ ✎✚✒ ✱✒✕✕ ✘✠✩✡✠ ✔✠✚✬✪✏ ✘✙❣ ✲✓✠✘✖❃✠✕❄
1
– 1) 2
f ( x) dx
❊
a
x( x
2
2
–1<x<1
❁✠✠✸
✥✌✖❁✠✠✌❂✠✕ ✡✘☞ ✘✙❣
❅
f
✥✖ ✥✗☛✡
✘✙❣
❧ ✁✂✄☎
353
❣✆
3 2
x3
(i) ❡✝✞ ✟✠✡☛☞ I ✌ ✍ 2 x dx ✎✏ ❆ ✑✒✝✓✡✔ ✕ ✗ x 2 dx ✖
✖ F ( x)
3
❜✘✡✟☞ ✡✙✚✠✒ ✛✝✜✝✢✣✝✤✚ ✥✦❡✒
✓ ✘✓ ✎❡ ✥✝✚✓ ✎✏✔ ✡✕
I ✧ F (3) – F (2) ✧
27 8 19
– ✧
3 3 3
x
9
(ii) ❡✝✞ ✟✠✡☛☞ ✡✕ I ★ ✩
4
(30 –
30 –
3
x2
❜✘ ✥✦✕✝✢
t ❥✯❦✉s ✐❥ –
3
x 2 )2
3
x dx
2
dx ✘✪✫✥✬✦ ✝❡ ✎❡ ✘❡✝✕✭✒ ✕✝ ✥✦✡✚✛✪✕✟☛ ✮✝✚ ✕✢✚✓ ✎✏✔ ❆
dt ✛✬✝✪✝
2
x dx ✧ – dt
3
✶
✷
2 ✰ 1✱
2✸
1
2 dt
✹
✺ F ( x)
–
dx
★
=
=
✲
✳
3 ✹
✸
✩
✩
3
2
3
t
3
✴
✵
3
t
✸✻ (30 – x 2 ) ✹✼
(30 – x 2 ) 2
x
❜✘✡✟☞ ✕✟✞ ✕✠ ✡✙✚✠✒ ✛✝✜✝✢✣✝✤✚ ✥✦❡✓✒ ✘✓ ✎❡ ✥✝✚✓ ✎✏✔✽
9
✾
✿
2❀
1
❁
2❅
1
1 ❇ 2 ❍ 1 1 ■ 19
I ❂ F(9) – F(4) ❂ ❀
❈
❏
❑
3 ❁ =
❉
❊ =
3
3 ❋ (30 – 27) 30 – 8 ● 3 ▲◆ 3 22 ▼❖ 99
❀❃ (30 – x 2 ) ❁❄
4
2
x dx
(iii) ❡✝✞ ✟✠✡☛☞ I ❑ ◗
1 ( x P 1) ( x P 2)
✛✝✔✡✈✝✕ ✡✣✝❘✞ ✕✝ ❙✥✒✝✓❚ ✕✢✚✓ ✎❯☞ ✎❡ ✥✝✚✓ ✎✏✔ ✡✕
x
–1
2
❱
❲
( x ❲ 1) ( x ❲ 2) x ❲ 1 x ❲ 2
❜✘✡✟☞
x dx
◗ ( x P 1) ( x P 2) ❑ – log x P 1 P 2log x P 2 ❑ F( x)
✛✚✽ ✕✟✞ ✕✠ ✡✙✚✠✒ ✛✝✜✝✢✣✝✤✚ ✥✦❡✒
✓ ✘✓ ✎❡ ✥✝✚✓ ✎✏✔ ✡✕
I = F(2) – F(1) = [– log3 + 2 log4] – [– log2 + 2 log3]
❳ 32 ❨
= – 3 log3 + log2 + 2 log4 = log ❩ ❬
❭ 27 ❪
354
① ✁✂✄
☞
(iv)
❡☎✆
I
✝✞✟✠✡☛
sin 2t = u
✌
✍
✑❥☎✆✕
✈✗✘
✚
4
sin 3
0
2t cos 2 t dt .
2 cos 2t dt = du
✐✑
sin 3 2t cos 2 t dt
✙
=
❜✢✟✝✡
✔✝✆
✔✞
✟✣✗✞✤
✩
I F ( ) – F (0)
4
✪
✈✎
✈✖☎✒☎
✢✕
1.
✸✹
✗✔
1
❁
1
✿
4
4.
0
✒✕✺
✐★ ✻✆☎✕✼
❡✕ ✼
( x 1) dx
4
❄ ❃
5.
cosec x dx
9.
1 4 1 4
[u ]
sin 2t
8
8
✐★ ❡✤
✕
✢❡☎✔✝✆☎✕ ✼
31
✚
2
2
0
1
✏
☞
16.
✍
2
❈
1
❊
18.
20.
❋
0
1
❋
0
5x2
17.
x2 4x 3
❇
(sin 2
( x ex
❇
✏
sin
x
) dx
4
✩
✛❦✜ ②❤❢t✱
0
x
✔☎
✴✵✶
❡☎✆
✔✞✟✠✡✾
✽☎✗
2
dx
3.
cos 2x dx
dx
6.
10.
1 – x2
x dx
2 x2 1
❁
1
❂
❅
✏
❈
4
(2sec2
0
x
❇
x3
❇
dx
0 1 x2
11.
2x 3
dx
0 5x2 1
15.
✍
✏
❅
❅
❅
2
✏
❀
7.
4
2) dx
19.
6 x 9) dx
e dx
1
✏
❀
☞
5 x
1
14.
(4 x3 – 5 x 2
❉
x
x
– cos 2 ) dx
2
2
●
F (t )
✢✕
3
13.
x dx
1
du
2
✪
✟✆✟✻✓✗
sin 2x dx
2
cos 2
0
✔✞✟✠✡
1
1
[sin 4 – sin 4 0]
8
2
8
6
12.
✟✒✓☎✑
✚
✈☎✥☎✑✦☎✧✗
❃
8.
cos 2t dt =
✐✑
✩
✪
2.
❀
sin 3 2t cos 2 t dt
1 3
u du
2
✫✬✭✮✯✰✲✳
✷
✏
0
6x 3
dx
x2 4
❅
❅
❁
4
0
tan x dx
3
dx
2
x
2
1
x e x dx
0
❆
2
1
❧ ✁✂✄☎
355
✐✆✝✞ ✟✠ ✡☛☞ ✟✟ ✌✍☞ ✎✏✑ ✒✓✔✕ ✖✔ ✗✘✞ ✖✑✙✚✡✛
3 dx
21. ✢
❝✕✔❝✕ ✏✣✤
1 1 ✜ x2
(A)
22.
2
3
✧0
(A)
✥
(B)
3
2✥
3
(C)
✥
(D)
6
✥
12
dx
❝✕✔❝✕ ✏✣✤
4 ✦ 9x2
✥
(B)
6
✥
12
7.9 ★✩✪✫✬✭✮✮★✯ ✰✮✱✮ ✪✯✪✲✳✫ ✴✵✮✶✷✯✮✸✹
Integrals by Substitution)
(C)
✥
(D)
24
✶✮ ✵✮✯ ✺✮✫ ✶✱✯✮
✥
4
(Evaluation of Definite
✙✐❢✻✍ ✐✙✕✼❢✍✽✔✍ ☞ ✌✍ ☞ ✏✌✞✍ ✾✙✞✙✝✗✿ ✎✌✔✖✻✞ ❀✔✿ ✖✕✞✍ ✖✑ ✾✞✍✖ ✙☛✙❁✘✔✍☞ ✖✑ ✗✗✔❂ ✖✑ ✏✣✛
✾✙✞✙✝✗✿ ✎✌✔✖✻✞ ❀✔✿ ✖✕✞✍ ✖✑ ✌✏✈☛✐❃❄✔❂ ✙☛✙❁✘✔✍☞ ✌✍ ☞ ✡✖ ✙☛✙❁ ✐✆✙✿❅❆✔✔✐✞ ✙☛✙❁ ✏✣✛
b
✐✆✙✿❅❆✔✔✐✞ ✙☛✙❁ ✎✍ ❇ f ( x) dx , ✖✔ ✌✔✞ ❀✔✿ ✖✕✞✍ ☛✍❞ ✙✻✡ ✾✔☛✝✘✖ ✗✕❄✔ ✙✞❈✞✙✻✙❉✔✿ ✏✣✤
a
1. ✎✌✔✖✻✞ ☛✍❞ ❝✔✕✍ ✌✍☞ ✎✑✌✔✾✔✍☞ ☛✍❞ ✙❝✞✔ ✙☛✗✔✕ ✖✑✙✚✡ ✾✔✣✕ y = f (x) ✾❆✔☛✔ x = g (y)
✐✆✙✿❅❆✔✔✙✐✿ ✖✑✙✚✡ ✿✔✙✖ ✙✽✘✔ ✏❊✾✔ ✎✌✔✖✻✞ ✡✖ ❀✔✿ ❋✐ ✌✍☞ ✐✙✕☛✙✿❂✿ ✏✔✍ ✚✔✡✛
2. ✎✌✔✖✻✞ ✾✗✕ ✖✑ ●✘✔❉✘✔ ✙✖✡ ✙❝✞✔ ✞✡ ✎✌✔✖❍✘ ✖✔ ✞✡ ✗✕ ☛✍❞ ✎✔✐✍■✔ ✎✌✔✖✻✞
✖✑✙✚✡✛
3. ✞✡ ✗✕ ☛✍❞ ❅❆✔✔✞ ✐✕ ✐❊✞✤ ✐✆✙✿❅❆✔✔✐✞ ✖✑✙✚✡ ✾✔✣✕ ✒✓✔✕ ✖✔✍ ✌❃✻ ✗✕ ☛✍❞ ❋✐ ✌✍☞ ✙✻✙❉✔✡✛
4. ✗✕❄✔ ♣❏❑ ✎✍ ✐✆✔▲✿ ✒✓✔✕ ✖✔ ✎✌✔✖✻✞ ✖✑ ✽✑ ✏❊▼❂ ✎✑✌✔✾✔✍☞ ✐✕ ✌✔✞ ❀✔✿ ✖✑✙✚✡ ✾✔✣✕ ✒✼✗
✎✑✌✔ ☛✔✻✍ ✌✔✞ ✎✍ ✙✞❈✞ ✎✑✌✔ ☛✔✻✍ ✌✔✞ ✖✔ ✾☞✿✕ ❀✔✿ ✖✑✙✚✡✛
✪❖P★◗✮❘ ▼✎ ✙☛✙❁ ✖✔✍ ✿✑☛✆✿✕ ❝✞✔✞✍ ☛✍❞ ✙✻✡ ✏✌ ✙✞❈✞✙✻✙❉✔✿ ✐✆✖✔✕ ✾✔❙✍ ❝❚❯ ✎✖✿✍ ✏✣✛☞
◆
✗✕❄✔ ♣✠❑ ✡☛☞ ♣✟❑ ✖✔✍ ✖✕✞✍ ☛✍ ❞ ❝✔✽ ✗✕❄✔ ♣❏❑ ✖✔✍ ✖✕✞✍ ✖✑ ✾✔☛✝✘✖✿✔ ✞✏✑☞ ✏✣✛ ✘✏✔❱
✎✌✔✖✻✞ ✖✔✍ ✞✡ ✗✕ ☛✍ ❞ ❋✐ ✌✍☞ ✕❉✔✔ ✚✔✿✔ ✏✣ ✾✔✣✕ ✎✌✔✖✻✞ ✖✑ ✎✑✌✔✾✔✍ ☞ ✖✔✍ ✞✡ ✗✕ ☛✍❞ ✾✞❊✎✔✕
✐✙✕☛✙✿❂✿ ✖✕ ✻✍✿✍ ✏✣☞ ✿✔✙✖ ✏✌ ✎✑❁✍ ✾☞✙✿✌ ✗✕❄✔ ✖✑ ✙❲✘✔ ✖✕ ✎☛✍❞☞ ✛
356
① ✁✂✄
✈☎✆✝
✆✞✟
✠✡
☛☞☎✠✌✍☎☎✟✎
1
♠✔✕✖✗✘✕
✖❣
✙✚
✢
t = x5 + 1,
1
✛
✞✟
✞✡✏✑✟
✠✒✎✓
5 x 4 x5 1 dx
✜
✌❥☎✣✟
❞☎
✡☎✣
✤☎✑
❞✥✦✧✝✓
dt = 5x4 dx
★✌
3
✆✞✦❜✝
✪
1
✈✑✭
✰
1
✮
5 x 4 x5 1 dx =
✩
5x
4
x
5
✫
✬
3 1
2
( x5 1) 2
1 dx =
3
✱
✯
3
2 2
2
t = ( x5 1) 2
3
3
t dt =
✲
✳
✴
✵
✴
✶
✵
✷
–1
3
2 5
(1 1) 2 – (– 1)5 1
=
3
✺
✼
✽
✼
✸
✹
3
2
✽
✿
3
2 2
2
=
3
❁
❄
3
2
❃ 0
❄
❆
✞❏❑ ★▲▼☎✡
❢❉❊❋●❍■
✞✥✡☎✈☎✟✎
✡☎✣
❏✟❧
✈✣◗ ✞☎✌
❜✥✦✧✝
✡☎✣
✞✡☎❞❜✣
✤☎✑
❞✌✑✟
✧✒✞✟❘✧✒✞✟
❞☎
◆★☎✎ ✑✌✍☎
✞✟
❞✌✑✟
✣☎✟✉
✈☎✒✌
✑❞
★✦✌❏✦✑❑✑
✠☎✟✑☎
5 x 4 x5 1 dx =
1
✯
✮
✠✒
2
✢
❯
1
✡☎✣
✈☎✒✌
❵
❜✥✦✧✝
0
tan – 1 x
dx
1 x2
❞☎
✑❖
◆★☎✎✑✦✌✑
✞✡☎❞❜✣
❞☎
✡☎✣
❞✥✦✧✝
✦❞
✤☎✑
✟ ❏✒✞✟
❏✒✞❘
❪
❙
✞✟
✑❞
❚
★✦✌❏✦✑❑✑
✠☎✟✑☎
✠✒✓
3
2
❱
❳
❨
❳
❩
❨
❬
3
❯
❲
0
3
2 2
2 – 02
3
❱
❳
❨
❳
❩
❨
❬
=
2
(2 2)
3
❭
4 2
3
❞✥✦✧✝✓
✬
t = tan – 1x,
✑❖
dt
❛
1
dx .
1 x2
✧❖
x=0
✑☎✟
t=0
✈☎✒✌
✧❖
x=1
✑☎✟
t
❝
✈✑✭
✟ ✧✒✞✟
✧✒✞❘
x, 0
✞✟
1
✑❞
★✦✌❏✦✑❑✑
✠☎✟✑☎
✠✒
✟ ❏✟ ✞
✒
✟
❏✒✞❘
t, 0
✐
✞✟
4
✑❞
★✦✌❏✦✑❑✑
♦
1
✆✞✦❜✝
✣P✥
t dt
0
2 2
t
=
3
✖❣
✠✒✎
✑☎✟
✧❖
1
✰
❫❴
❅
❇
4 2
3
❈
✠✒✎✓
✑❖
✆✞✦❜✝
♠✔✕✖✗✘✕
=
t = x5 + 1.
dt = 5 x4 dx
x=–1 t=0
x=1 t=2
x, – 1 1
t
✑☎✟
✧❖
✈✑✭
✠✡
✾
❀
2
(2 2)
3
❂
❅
✻
✾
♥
0
–1
tan x
dx =
1 x2
❦
♦
4
t
② 0
dt
♣
r
t
2
t
2
q
4
=
s
✇
0
1 2
–0
2 16
③⑤
④
⑥
⑦
⑧
⑨
⑤
❛
2
32
✠☎✟✑☎
✠✒✓
❡
❤
4
❧ ✁✂✄☎
357
✐✆✝✞✟✠✡☛ ☞✌✍✎
✶ ✏✑ ✒ ✓✔ ✕✑✖ ✗✘✙✚✛✑✜ ✏✢✛✔✣✚✛✑✜ ✔✛ ✢✛✚ ✗✘✤✓✥✦✛✛✗✚ ✔✛ ✧✗★✛✑✩ ✔✪✓✑ ✫✬✭ ✮✛✓ ✔✯✤✰✭✱
1
✴
x
1. ✳ 0 2
x ✲1
1
2. ✷ 2 sin ✵ cos5 ✵ d ✵ 3. ✿ 0 sin
0
dx
–1✸
2x ✹
dx
✺
2 ✻
✽1✼ x ✾
❂
2
4. ❁ 0 x x ❀ 2 dx (x + 2 = t2 ✪✤❥✛✭)
sin x
dx
5. ❄ 02
1 ❃ cos 2 x
2
dx
6. ✳ 0
x ✲ 4 – x2
2✸ 1
1 ✹ 2x
8. ✿ 1 ✺ – 2 ✻ e dx
✽ x 2x ✾
1
dx
7. ✳ ❅1 2
x ✲ 2x ✲ 5
✗✘✙✚ ❆ ✭✕✜ ✶❇ ✢✑ ✜ ✏✫✯ ✧❈✛✪ ✔✛ ❉★✚ ✔✯✤✰✭✱
1
9. ✏✢✛✔✣✚ ❋ 1
3
1
3 3
x )
(x ❊
x4
(A) 6
dx ✔✛ ✢✛✚ ✫●❍
(B) 0
(C) 3
(D) 4
(C) x cos x
(D) sin x + x cos x
x
10. ★✤■ f (x) = ❏ 0 t sin t dt , ✓r f ❑(x) ✫●❍
(A) cos x + x sin x
7.10
(B) x sin x
❢✞❢✝▲▼ ◆❖✟P✡✞✟◗❘ ✠◗❙ ✠❚❙❯ ❱❚❲✟❳❖❨
(Some Properties of Definite Integrals)
✤✚✤✙❉✓ ✏✢✛✔✣✚✛✑✜ ✕✑✖ ✕✬✖❩ ✢✫❬✕✗❭❪✛❫ ✩✬❪✛❴✢✛✑❵ ✔✛✑ ✫✢ ✚✯❉✑ ✏❭❉✯r❛ ✔✪✓✑ ✫●✜✱ ★✑ ✩✬❪✛ ❴✢❫ ✤✚✤✙❉✓
✏✢✛✔✣✚✛✑ ✜ ✔✛ ✢✛✚ ❜✛✏✛✚✯ ✏✑ ✮✛✓ ✔✪✚✑ ✢✑✜ ✧✗★✛✑✩✯ ✫✛✑✩
✜ ✑✱
b
b
P0 :
❏ a f ( x) dx ❝ ❏ a f (t ) dt
P1 :
f ( x) dx ❣ 0
❏ a f ( x ) dx ❝ – ❏ b f ( x ) dx , ✤✕✤✙✛❞❡✓★✛ ❁ a
P2 :
b
a
a
b
c
b
b
a
a
b
❏ a f ( x) dx ❝ ❏ a f ( x) dx ❤ ❏ c f ( x) dx ❦ a, b, c ✕✛✥✓✤✕✔ ✏✜❥★✛✭♠ ✫●✱✜
P3 :
❁ a f ( x) dx ❣ ❁ a f (a ❀ b ♥ x) dx
P4 :
❏ 0 f ( x) dx ❝ ❏ 0 f (a ♦ x) dx ♣q★✛✚ ■✯✤✰✭ ✤✔ P4, P3 ✔✯ ✭✔ ✤✕✤✙✛❞❡ ✤✥✦✛✤✓ ✫●s
P5 :
2a
❏0
a
a
0
0
f ( x) dx ❝ ❏ f ( x) dx ❤ ❏ f (2a ♦ x) dx
358
① ✁✂✄
2a
P6 :
0
f ( x) dx
a
2
= 0,
P7 :
a
(i)
☛
✠
✡
a
✢
✥☞✱✥☞
✛
a
☞✦✣✧ ✏
✓✍
✵✶✶✷✸✹
x=t
P1
❞✴
✵✶✶✷✸✹
✍✗✒
✎✗✙✧
✓✔ ✲
✎✐ ✞✙✺✖✗✗✎✒
✑✮✞✼✥
✞☞
✢
f
✞☞
☞✮
☞✦✒✧
f
✝✞✟
✥☞
✩✪ ✫✗✬✍✗✧ ✭
★✒
b
✓✍
f ( x) dx ,
✝✞✟
✜
❞✴
✌✧
0
f ( x) dx 0 ,
P0
✎✐✍✧✝
a
☛
f (2a
❀☎✆
✞✣✤✗✍
✌✮✬✧
P2
✓✍
❞✴
✎✧ ✐✞❃✗✙
☞✦✙✧
✵✶✶✷✸✹
✍✗✒
✓✔✲
✝✞✟
✑✮✞✼✥
✎✐✗ ✻✙
✎✐✞✙✕✣☞✑✼
✕✗✔✦
(2)
(3)
✕✗✔✦
☞✗✧
c
a
★✌✌✧
P3
✩✪ ✫✗✬✍✘
❞✴
✼❣
b
✢
P2
a
✙❣
✞✌❢
✍✗✒
t = a.
f ( x) dx =
=
❞✴
✼❣
☛
c
✎✦
✵✶✶ ✷✸✹
x = a, t = 0
☛
a
✕✖✗✗✘✙ ✚
✝✞✟
f (–x) = –f (x)
✓✔ ✳
✲
✓✗✧✙✮
✓✔ ✳
F
✓✔ ✳
✜
– [F (a) F (b)]
✙❣
☞✑✒
☞✮
❁
✞✽✙✮✝
a
✜ ❁
✢
b
✕✗✬✗✦✾✗✿✙
f ( x) dx ,
f ( x) dx 0
✡
✎✐✞✙✕✣☞✑✼
F
✓✔ ❄
✙❣
... (1)
f ( x) dx = F(c) – F(a)
... (2)
f ( x) dx = F(b) – F(c)
... (3)
✓✍
c
✓✗✧ ✙✗
☞✗
✙❣
✓✔
f (– x) = f (x)
f ( x) dx = F(b) – F(a)
b
✎✗✙✧
✓✔✲
✞☞
f ( x) dx = F(b) – F(a) =
b
a
f ( x) dx
✓✔ ✳
✑✮✞✼✥
✞☞
t = a + b – x.
✙❣
dt = – dx.
✼❣
x=a
✙❣
,t=b
✕✗✔ ✦
★✌✞✑✥
a
❁
✢
b
=
P4
a
b
f ( x) dx
✵✶✶✷✸✹
x=b
✼✗✧ ❅❆✒✧
✢
a = b,
f
✞☞
b
a
c
☞✦✙✧
✜
✞☞
☛
✎✏✑✒
f ( x) dx F (b) – F (a)
a
f ( x)
✥☞ ✌✍ ✎✏✑✒ ✓✔ ✕✖✗✗✘✙✚ ✝✞✟
✯✎✎✞✰✗
✎✦
☞✗
f
a
✝✓✗❂
x)
f (2a – x) = – f (x)
✝✞✟
f ( x) dx 2
a
(ii)
f ( x) dx,
0
☛
a
b
☛
a
b
f (a b – t ) dt
❇
f (a b – t ) dt (P1 )
❈
✌✧
f (a b – x) dx (P0 )
t=a–x
❈
✦ ✞❥✗ ✥
✌✧
✕✗✔✦
P3
☞✮
✙✦✓
✕✗✩✧
❣ ✞❉❆✥✳
✕❣
dt = – dx,
❧ ✁✂✄☎
359
P5 ❞✆ ✝✞✞✟✠✡ P2, ☛☞ ✌✍✎☞✏✑ ☛✒✓✏ ✔✕✖ ✔✗ ✍☞✓✏ ✔✘✙ ✚☛
2a
0
f ( x) dx =
a
0
f ( x) dx
2a
a
f ( x) dx
♥☞✖✛ ✍✜☞ ✢✏✣ ♥✤✥✒✏ ✥✗☞☛✦✧ ✗✏✙ t = 2a – x ✍✐✚✓★✩☞☞✚✍✓ ☛✪✚✫✖✬ ✓✭ dt = – dx ✈☞✘✒ ✫✭
x = a, ✓✭ t = a ✈☞✘✒ ✫✭ x = 2a, ✓✭ t = 0 ✈☞✘✒ x = 2a – t ❍☞✪ ✍✐☞✮✓ ✔☞✏✓☞ ✔✘✯
❜✥✚✦✖ ♥✤✥✒☞ ✥✗☞☛✦✧
2a
0
✰ a f ( x) dx = – ✱ a f (2a – t ) dt
a
a
= ✱ 0 f (2a – t ) dt = ✱ 0 f (2a – x) dx ✍✐☞✮✓ ✔☞✏ ✓☞ ✔✘✯
✈✓✲
2a
a
a
✰ 0 f ( x) dx = ✰ 0 f ( x) dx ✳ ✰ 0 f (2a ✴ x) dx
P6 ❞✆ ✝✞✞✟✠✡ P5, ☛☞ ✌✍✎☞✏✑ ☛✒✓✏ ✔✕✖ ✔✗ ✍☞✓✏ ✔✘✙ ✚☛
2a
0
a
f ( x) dx =
0
a
f ( x) dx
f (2a
0
x) dx
... (1)
f (2a – x) = f (x), ✓☞✏ r✵✶ ✚✧✷✧✚✦✚✸☞✓ ✹✍ ✗✏✙ ✍✚✒✢✚✓✺✓ ✔☞✏ ✫☞✓☞ ✔✘
✈✭ ✎✚♥
a
2a
✱ 0 f ( x) dx =
0
a
f ( x) dx
0
f ( x) dx
a
2
0
f ( x) dx
f (2a – x) = – f (x), ✓✭ r✵✶ ✚✧✷✧✚✦✚✸☞✓ ✹✍ ✗✏✙ ✍✚✒✢✚✓✺✓ ✔☞✏ ✫☞✓☞ ✔✘✙
✈☞✘✒ ✎✚♥
2a
✰0
a
a
f ( x) dx = ✰ 0 f ( x) dx ✴ ✰ 0 f ( x) dx ✻ 0
P7 ❞✆ ✝✞✞✟✠✡
0
a
a
P2 ☛☞ ✌✍✎☞✏✑ ☛✒✓✏ ✔✕✖ ✔✗ ✍☞✓✏ ✔✘✙ ✚☛ ✰ f ( x) dx = ✱ ✽ a f ( x) dx ✾ ✱ 0 f ( x) dx
✼a
♥☞✎✏✙ ✍✜☞ ✢✏✣ ✍✐✩☞✗ ✥✗☞☛✦✧ ✗✏✙ t = – x ✒✸☞✧✏ ✍✒
dt = – dx ✫✭ x = – a ✓✭ t = a ✈☞✘✒ ✫✭ x = 0, ✓✭ t = 0 ✈☞✘ ✒ x = – t ❍☞✪ ✍✐☞✮✓ ✔☞✏✓☞ ✔✘✯
❜✥✚✦✖
a
✱ ✽a
0
f ( x) dx =
a
a
f ( x) dx
0
a
f ( x) dx
a
= ✰ 0 f (– x) dx ✳ ✰ 0 f ( x) dx
(P0 ✥✏)
... (1)
(i) ✈✭ ✎✚♥ f ✖☛ ✥✗ ✍✣✦✧ ✔✘ ✓✭ f (–x) = f (x) ✓☞✏ r✵✶ ✥✏ ✍✐☞✮✓ ✔☞✏✓☞ ✔✘ ✚☛
a
a
a
a
✱ ✽ a f ( x) dx ✿ ✱ 0 f ( x) dx ✾ ✱ 0 f ( x) dx ✿ 2 ✱ 0 f ( x) dx
360
① ✁✂✄
(ii)
❀☎✆
f
☎✥✝✞✟
✠✡☛☞
a
✢
2
♠✣✤✦✧★✤
✦❣
✎✏
✌✟
✌✟
1
✙
✬
✆✑✱✞✎✑
☎☛✱✞
2
✢
✩✪
☎✘
✕✘✎✑
a
[ –1, 0]
✌✍✲
f (–x) = – f (x)
✎✏
f ( x) dx
x3 – x dx
1
✫
✌✍✲
✙
✌✍
✘✞
✠✐
a
✚ ✛
✟✞☞
x3 – x
✢
f ( x) dx
0
❞✞✎
0
✳
✎✞✑
✒✓✔
a
✜
✢
✕✑
✠✖ ✞✗✎
✌✞✑✎✞
✌✍
☎✘
f ( x) dx 0
✚
0
✘✭☎✮✯✰
[0, 1]
✈✞✍ ✐
x3 – x
✠✐
0
✴
✈✞✍ ✐
[1, 2]
✠✐
x3 – x
✳
0
☎✘
x3 – x dx =
0
✢
1
✙
0
=
✢
1
✙
✜
( x3 – x) dx
✜
x4 x2
= 4 – 2
✶
✸
✹
✺
✻
✾
❂
❄
= –
1 1
–
4 2
1
4
❆
1
2
✵
✷
✿
❃
✾
❁
❅
❆
❂
❄
1
2
✸
✺
–1
❇
✢
0
1
0
✵
= –
1
( x3 – x) dx
✢
0
( x – x3 ) dx
x2 x4
–
2
4
1 1
–
2 4
1
4
❆
2
– ( x3 – x) dx
2
✿
❁ ✼
❃
✵
✷
✹
✻
0
✸
✺
✢
1
4–2 –
✽
✾
❂
❄
1
4
❆
1
( x3 – x) dx
3
1
=
2
2
(P2 )
✕✑
( x3 – x) dx
x4 x2
–
4
2
❅
❇
✢
2
✜
1
✶
✜
2
✶
✹
✻
1 1
–
4 2
❇
1
✿
❃
❅
3
11
2
4
4
❆
❈
❊
♠✣✤✦✧★✤
✩❉
✦❣
✠✖✑ ☎●✞✎
✌✟
❋
4
sin 2
–❊
4
✘✐✎✑
✌✍ ✲
x dx
☎✘
✘✞
sin2 x
✟✞☞
✯✘
❞✞✎
✕✟
✘✭☎✮✯✰
✠✡☛☞
❍
❜✕☎☛✯
■
4
–❍
4
✌✍✰
❏
sin 2 x dx = 2
❑
4 sin 2
0
x dx
[P7 (1) ]
✕✑
▼
❏
(1 cos 2 x )
= 2 4
dx =
0
2
▲
❑
❖
❊
1
= x – sin 2 x
2
P
❘
❚
◗
❙
❯
4
0
=
❲ ❱
❩
❭
4
–
4
(1 ◆ cos
0
2 x) dx
1
sin
–0
2
2
❱ ❳
❬
❪
❱
❨
4
–
1
2
❧ ✁✂✄☎
♠✆✝✞✟✠✝ ✡✡
☛ x sin x
✌0
1 ☞ cos 2 x
✞❣ ✎✍✏ ❡✓✔✕✖ ✔✥
dx ✥✍ ✎✍✏ ✑✍✒ ✥✓✔✕✖✗
✘ (✙ ✚ x) sin (✙ ✚ x) dx
☛ x sin x
dx
I = ✌0
=
✜0
1 ✛ cos 2 (✙ ✚ x)
1 ☞ cos 2 x
☛ (✤ ✦ x) sin x dx
= ✌0
✈✧✍★✍
2I=
✈✧✍★✍
I=
361
2
1 ☞ cos x
0
✤
(P4 ✢✣)
☛ sin x dx
✦I
= ✤✌
0 1 ☞ cos 2 x
sin x dx
1 cos 2 x
☛ sin x dx
2 ✌ 0 1 ☞ cos 2 x
cos x = t ❥✩✍✏✣ ✪❥ – sin x dx = dt
✕t x = 0 ✒t t = 1 ✈✍✫❥ ✕t x = ✬ ✒t t = – 1 ✭✫✗ ❜✢✔❡✖ ✭✎ ✪✍✒✣ ✭✫✮ ✔✥
I=
– ✤ ✯1 dt
✤ 1 dt
✌1
2 =
2
2 ✌ ✯1 1 ☞ t 2
1☞ t
(P1 ✢✣)
1
1 dt
✖✥ ✢✎✪✱❡✏ ✭✫
= ✤✌0
2 ❉✰✍✣✮✔✥
1 t2
1☞ t
tan – 1 t
=
1
0
(P7 ✢✣)
2
tan – 1 1 – tan 1 0
4
–0
4
1
♠✆✝✞✟✠✝ ✡✲ ✴ sin 5 x cos 4 x dx ✥✍ ✎✍✏ ✑✍✒ ✥✓✔✕✖✗
✳1
1
5
4
✞❣ ✎✍✏ ❡✓✔✕✖ ✔✥ I = ✶ sin x cos x dx ✈✍✫❥ f (x) = sin5 x cos4 x
✵1
✒t f(– x) = sin5 (– x) cos4 (– x) = – sin5 x cos4 x = – f (x), ✈✧✍✍✷✒✸ f ✖✥ ✔★✹✍✎ ✪✱❡✏
✭✫ ❜✢✔❡✖ I = 0 [P7 (ii) ✢✣]
☛
♠✆✝✞✟✠✝ ✡✺ ✌ 2
0
sin 4 x
dx ✥✍ ✎✍✏ ✑✍✒ ✥✓✔✕✖✗
sin 4 x ☞ cos 4 x
☛
✞❣ ✎✍✏ ❡✓✔✕✖ ✔✥
sin 4 x
I= ✌ 2 4
dx
0 sin x ☞ cos 4 x
... (1)
362
① ✁✂✄
sin 4 (
✆
2
✠ 0
I=
r☎
☛
2
✌ 0
=
✭✍✎
✏✑✒✓
✭✔✎
✕✑✡
✖✑✡ ✗✙
✘ ✡
4
sin (
✱
✯✰
✳ ✱
1
6
✛✜
✜✑✙
❡✴✣✖✵
2
(P4 )
❧✡
x)
✞
... (2)
✚✑r✡
✛✒✢
✣✕
✤
✤
sin 4 x cos 4 x
dx
sin 4 x cos 4 x
✥
2
★ 0
2
dx
★0
✧
✥
✧
[ x] 2
0
✕✑
tan x
✲
✣✕
I=
✳ ✱
1
✜✑✙
❞✑r
✲
I=
r☎
cos x dx
3
✳ ✱
cos x
6
✺ ✹
✾
3
❀
3
✹
✼
6
✻
✽
x dx
✿
❁
(P3 )
❧✡
cos
✺ ✹
✾
❀
❃
3
✹
✼
6
✽
x
✻
✿
sin
✼
❁
✺ ✹
✾
❀
3
✹
✼
✖✑✡✗✙
✘ ✡
I=
x
✿
❁
... (2)
✚✓
✛✜
✚✑r✡
✛✒ ✢
✣✕
2I =
3
● ❈
❈
dx
❊ ❆
6
✏r✈
✽
❄
❈
✕✑✡
6
✻
sin x
dx
sin x
cos x
3
❅ ❃
6
✭✔✎
... (1)
sin x
✲
❂ ✸
6
=
✷
cos
✸
✏✑✒✓
2
✕✴✣✖✵✶
✱
dx
tan x
3
6
✭✍✎
✦
✧
4
✱
✫❣
✟
✝
dx
3
♠✩✪✫✬✮✪
dx
4
☞
✚✓
I=
✏r✈
x)
x) cos (
✞
2
✞
2
cos 4 x
dx
cos 4 x sin 4 x
✤
2I =
✝
✝
x
❇
3
❈
6
❉
❊
3
❉
❋
6
❉
❊
6
12
■
♠✩✪✫✬✮✪
✯❍
❏
2
log
0
sin x dx
✕✑
✜✑✙
❞✑r
✕✴✣✖✵✶
❑
✫❣
✜✑✙
❡✴✣✖✵
✣✕
I=
▲
2
0
log sin x dx
▼
r☎
I=
2
❲ 0
▼
log sin
❖ ◆
❙
❯
2
P
◗
x dx
❚
❱
❘
❲
2
log
0
cos x dx
(P4 )
❧✡
❧ ✁✂✄☎
363
I, ♦✆✝ ✞✟✆✠✟✆✡ ☛✟✠✟✆ ✡ ☞✟✆ ✌✟✆✍✠✎ ✆ ✏✑ ✒☛ ✏✟✓✆ ✒✔✡
✗
2I = ✙ 2 ✕ log sin x ✘ log cos x ✖ dx
0
✗
= ✙ 2 ✚ log sin x cos x ✘ log 2 ✜ log 2 ✛ dx ( log 2 ✌✟✆✍✠✎ ✆ t♦✡ ✢✟✣✟✠✆ ✏✑)
0
✗
=
✗
2
✙ 0 log
sin 2 x dx ✜ ✙ 2 log 2 dx
(❉✤✟✆ ✡?)
0
✏✐✥✟☛ ✦☛✟☞✧✠ ☛✆✡ 2x = t ✑❥✟✠✆ ✏✑ 2 dx = dt ✌★ x = 0 ✓✟✆ t = 0 ✈✟✔✑ ✌★ x =
❜✦✪✧t
2I =
✓✟✆ t = ✩
2
1 ✫
✬
log sin t dt ✭ log 2
✮
2 0
2
✯
2 2
✰
= ✲ 0 log sin t dt ✱ log 2 [P6 ✦✆ ❉✤✟✆✪✡ ☞ sin (✩ – t) = sin t)
2
2
✯
✰
= ✲ 2 log sin x dx ✱ log 2 (♣✑ t ☞✟✆ x ☛✆✡ ✏✪✑♦✪✓❡✓ ☞✑✠✆ ✏✑)
0
2
✳
= I ✴ log 2
2
✶
2
✈✓✵ ✷ 0 log sin x dx =
–✬
log 2
2
✸✹✺✻✼✽✾✿ ❀❁❂❂
✪✠✪❢♣✓ ✦☛✟☞✧✠✟✆✡ ♦✆✝ ❃❄❅✟❆☛✟✆❇ ☞✟ ❈✏✤✟✆❃ ☞✑✓✆ ✒❄t ❊ ✦✆ ❊❋ ✓☞ ♦✆✝ ✏✐❢✠✟✆✡ ☛✆ ✡ ✦☛✟☞✧✠✟✆✡ ☞✟ ☛✟✠
❑✟✓ ☞●✪✌t❍
✗
1. ✙ 2 cos 2 x dx
0
◆
cos5 x dx
4. P 02
sin 5 x ❖ cos5 x
■
2
2. ▲ 0
5
sin x
sin x ❏ cos x
5. ❙ ◗5 | x ❘ 2 | dx
✯
dx
2
3. ✲ 0
8
3
sin 2 x dx
3
sin 2
x
3
▼ cos 2
6. ❙ 2 x ❚ 5 dx
x
364
① ✁✂✄
1
7.
✆
✝
x (1 x)n dx
0
4
log
✟ 0
8.
☎
2
(1 tan x) dx
9.
✞
✡
☛
sin x log sin 2 x) dx
✒
15.
✗
4
18.
✡
13.
sin x cos x
dx 16.
1 sin x cos x
✖
0
17.
✗
0
cos5 x dx
log (1 cos x) dx
✘
x
0
x
✖
a x
✣✤
✆
0
f ( x) g ( x) dx 2
✥
✢✱✩
✵✷
✬❞ ✩
✸✲✹
✱❞♦
✺✻✚★
✪✫
✤✚
✬❞✩
a
✆
0
f ( x) dx ,
✫✣★✭✚✚✣✮✚✯
✼❀✴
❀✣♥
✣✤❀✚
✰❀✚
f
✦✚✧★
g
✤✚❞
f (x) = f (a – x)
✲✧✳
✤✹✣✽✢✳
✍
2
20.
✎ ✾✍
( x3
✿
x cos x tan 5 x 1) dx
✿
✿
✤✚
✬✚✴
✲✧ ❁
2
(A) 0
(B) 2
❃
21.
❋
2
log
0
❄
❇
❉
4 3 sin x
dx
4 3 cos x
(C)
❈
❆
✤✚
✬✚✴
✲✧ ❁
❊
(B)
3
4
(C) 0
❢●❢●❍
◗❣
❚❯
✗
cos 6 x 1 sin 6 x dx
✖
t = 1 + sin 6x,
(D) 1
❂
❅
❆
(A) 2
♠❖P◗❘❙P
★❥✚✴❞
✫★
❱✚✯
(D) –2
■❏❑▲▼◆❑
✤✹✣✽✢✳
dt = 6 cos 6x dx
1
✜✸✣❜✢
dx
✕
x 1 dx
♥✙✚✚✛✜✢
✵✶
✡
✠
0
g(x) + g(a – x) = 4
✫✐ ✙✴
✡
14.
x dx
a
✓
✕
a
19.
✠
2
sin 2
–✍
2
2✓
sin 7 x dx
2
–✍
2
✎
✑
2
0
✎
✍
x dx
0 1 sin x
✔
11.
☞
✏
12.
x 2 x dx
✍
2
(2log
✌ 0
10.
0
cos 6 x 1 sin 6 x dx =
1 2
t dt
6
3
1 2 2
(t )
=
6 3
❲
3
❳
C=
1
(1 sin 6 x) 2
9
❳
❳
C
✢✱✩
❧ ✁✂✄☎
365
1
( x 4 ☞ x) 4
♠✆✝✞✟✠✝ ✡☛
dx ❑✍✎ ✏✑✒✓✔✕
✌
x5
✞❣ ✖✗ ✘✙✍✚✎ ✏✛✎✜ ✖✢✣ ✒✏
✈✧
1
1
x3
( x4 ✤
✦
x5
1
x) 4
1 – x– 3
1
1
(1 ✤ 3 ) 4
x
dx ✥ ✦
dx
x4
t , ❥★❦✉s ✩❥
1
❜✪✒✫✔
( x4
x) 4
x5
3
dx
x4
dt
1
1 4
t dt
dx =
3
5
5
1 4
4✬
1 ✭4
= ✮ t 4 ✯ C = ✱1 ✰ 3 ✲ ✯ C
3 5
15 ✳
x ✴
x 4 dx
♠✆✝✞✟✠✝ ✵✶ ✹
❑✍✎ ✏✑✒✓✔✕
( x ✷ 1) ( x 2 ✸ 1)
x4
1
✺ ( x ✻ 1) ✻ 3
✞❣ ✖✗ ✘✙✍✚✎ ✏✛✎✜ ✖✢✣ ✒✏
2
2
( x ✼ 1) ( x ✻ 1)
x ✼ x ✻ x ✼1
= ( x ✽ 1) ✽
✈✧
1
( x ✾ 1) ( x 2 ✽ 1)
... (1)
1
A
Bx ✿ C
❀
✿ 2
♦✜❂ ❃✘ ✗✜✣ ✈✒❄✍❅❆❇✎ ✏✛✎✜ ✖✢✣ ... (2)
2
( x ❁ 1)( x ✿ 1) ( x ❁ 1) ( x ✿ 1)
❜✪✒✫✔
1 =A (x2 + 1) + (Bx + C) (x – 1)
=(A + B) x2 + (C – B) x + A – C
♥✍✜❈✍✜✣ ✘❉✍✍✜✣ ♦✜❂ ❊❋●✍✍✣✏✍✜✣ ✏✑ ✎❋✫ ❈✍ ✏✛❈✜ ✘✛ ✖✗ ✘✍✎✜ ✖✢✣ ✒✏ A + B = 0, C – B = 0 ✈✍✢✛
1
1
A – C = 1, ✒✓✪✪✜ ✘✙✍✚✎ ✖✍✜✎✍ ✖✢ ✒✏ A ❍ , B ❍ C ❍ –
2
2
A, B ✔♦✣ C ✏✍ ✗✍❈ ❞■❏ ✗✜✣ ✘✙✒✎▲▼✍✍✒✘✎ ✏✛❈✜ ✘✛ ✖✗ ✘✍✎✜ ✖✢✣ ✒✏
1
1
1
x
1
✾
✾
◆
2
2
2
( x ✾ 1) ( x ✽ 1) 2( x ✾ 1) 2 ( x ✽ 1) 2( x ✽ 1)
... (3)
366
① ✁✂✄
✭☎✆
✝✞✟
✭✠✆
✡✟☛
☞✌✍✎✏✑✞✞✍☞✎
✝✒✓✟
☞✒
✔✡
☞✞✎✟
✔✕ ☛
✍✝
4
x
( x 1) ( x 2
✘
x 1)
✗
( x 1)
✖
✗
✗
1
2( x 1)
✘
✘
✗
1
x
2
2 ( x 1)
✗
1
✘
2( x
2
✗
1)
❜✙✍✚✛
✜
x4
( x 1) ( x 2
✘
✖
✗
log (log x)
♠✢✣✤✥✦✣
✧★
✤❣
✚✩✍✪✛
✡✞✓
x 1)
✗
dx
I
x2
2
✗
x
log (log x)
✬
☞✞✎✟
✔✕ ☛
1
1
1
log x 1 – log ( x 2 1) – tan – 1 x C
2
4
2
✘
1
dx
(log x) 2
= log (log x) dx
✈✞❜✛✮ ☞✌✑✞✡
✗
✫
✝✩✍✪✛
✬
1
dx
(log x) 2
✙✡✞✝✚✓ ✡✟☛ ✠ ✝✞✟ ✍✯✎✩✰ ☞✱✚✓ ✲✟✱ ✳☞ ✡✟☛ ✚✟✎ ✟ ✔✕ ☛✴ ✎✵ ✶✞☛✷✸✞✹ ✙✡✞✝✚✓ ✙✟ ✔✡
✍✝
= x log (log x)
☞ ✐✓ ✹
1
x dx
x log x
✯✞✒✞
✻
dx
log x ,
✍ ✲✼✞✒
☞✒
✙✡✞✝✚✓
✝✩✍✪✛✮
✺
✝✩✍✪ ✛✮
❜✙
✬
✠
☞✌ ✝✞✒
dx
log x
✝ ✞✟
✔✡
✫
✬
✍ ✯✎ ✩ ✰
☞✞✎✟
✔✕ ☛
✝✞✟
(1),
✡✟☛
✒✶✞✓✟
☞✒
✔✡
☞✞✎✟
dx
(log x)2
dx
(log x)2
☞ ✱✚✓
... (1)
✲✟✱
✳☞
✡✟☛
dx
x
log x
= x log (log x)
✽
x
log x
dx
(log x)2
✾
C
✈ ✞✕✒
✶ ✞ ☛ ✷ ✸✞ ✹
... (2)
✔✕ ☛
I = x log (log x)
✚ ✩✍✪ ✛
✍✝
1
1
x
dx
– x –
=
2
log x
log x
(log x ) x
(2)
✗
1
dx
(log x) 2
I = x log (log x)
✍ ✲✍ ❢
❑✞✎
✗
dx
(log x)2
❧ ✁✂✄☎
♠✆✝✞✟✠✝ ✡☛
tan x dx ❑☞✌ ✍✎✏✑✒✓
cot x
✞❣ ✔✕ ✖☞✌✗ ✔✘✙ ✏✍ I ✜ ✥ ✚ cot x ✢
✣
✈✩
tan x ✛✤ dx ✦ ★ tan x (1 ✧ cot x) dx
tan x = t2, ❥✪☞✫✗ ✖❥ sec2 x dx = 2t dt
✈✬☞✭☞
dx =
✌✩
2t dt
1 t4
1
2t
dt
2
t (1 t 4 )
I= t 1
2
= 2
✌✩
I
dy
2
y
♠✆✝✞✟✠✝ ✡✾
2
2
1
dt
t2
2
1
t
2
t
1
1✱
✰
1
= y, ❥✪☞✫✗ ✖❥ ✳ 1 ✲ 2 ✴ dt = dy
t ✶
t
✵
t✯
✖✐✫✮
(t 1)
dt = 2
t4 1
1
dt
t2
= 2
1
2
t
t2
1
2
2 tan – 1
=
2 tan – 1 ✺✺
sin 2 x cos 2 x dx
9 – cos 4 (2 x)
✞❣ ✕☞✫ ❡✎✏✑✒ ✏✍ I ✦ ★
y
2
=
t
C = 2 tan – 1
✷ t2 ✹1✸
✻✻ ✧ C =
✼ 2t ✽
1
t
2
C
✷ tan x ✹ 1 ✸
✻✧C
✼ 2 tan x ✽
2 tan – 1 ✺
❑☞✌ ✍✎✏✑✒✓
sin 2 x cos 2 x
9 – cos 4 2 x
dx
✈✩
cos2 (2x) = t ❥✪☞✫✗ ✖❥ 4 sin 2x cos 2x dx = – dt
❜✿✏❡✒
I❅ –
1
1 –1 ❁ t ❂
1
dt
❄
2
❀1 ❃1
❅ – sin ❈ ❉ ❆ C ❅ ❇ sin ❊ cos 2 x ❋ ❆ C
▲
4
4
4
● 3❍
■3
❏
9 – t2
367
368
① ✁✂✄
♠☎✆✝✞✟✆
✠✠
3
2
☞
✡1
x sin ( x) dx
☛
✥✌
x sin
✝❣
❀✕✌✖
f (x) = | x sin x | =
✗
3
2
|
✣
✜1
❜✚✓✛✱
✍✌✎
✏✌✑
x, 1 x 1
x sin
1
x sin x | dx =
✢
☞ ✡
1
=
✭✮✌
✯✫ ✰
3
2
|
☞ ✡1
♥✌✫✎ ✌✫ ✬
✚✍✌✥✛✎✌✫ ✬
✚✍✌✥✛✎
✥✌
2
✪
0
✛✒✓✔✱
1
✴
✶
✹
✻
✶
✢
1
2
✶
x dx
a cos x b 2 sin 2 x
2
2
★
✭✲
✕✍
✭✌✑✫
✩
3
2
✪
1
✕✳✬
1
☛
x sin x dx
★
✓✥
x cos x
sin x
2
✓✥
I=
3
✵
✺
✢✼
✥✌
✷
1
1
✸
✢
✍✌✎
✢
2
✏✌✑
✥✒✓✔✱❆
x dx
2
2
a cos x b 2 sin 2 x
✣
0
(
❁
✷
✸
✣
x) dx
x) b 2 sin 2 (
✢ ✶
2
0
2
a cos (
✢ ✶
✸
(P4
=
✾
★
✪
0
dx
2
2
a cos x b 2 sin 2 x
★
✪
0
✾
✈✑❄
2I =
★
✪
0
✾
✩
✿
✾
=
3
2
✿
❁
✍✌✎
x sin x dx
x sin x dx
1
✾
✝❣
☛
3
2
✤
✦
☞1
2
✢
✠✽
♦s✘ ❢②✙
x sin x dx
sin x
☛
=
♠☎✆✝✞✟✆
1
✧
✥✲✎✫
– x cos x
x sin x | dx =
✪
♦s✘ ❢②✙
3
2
x, 1 x
1
♥✌❀✫✬
✥✒✓✔✱ ❆
dx
a cos x b 2 sin 2 x
2
2
✿
dx
a cos x b 2 sin 2 x
2
2
✿
✪
✩
0
I
✯✫✰
x dx
a cos x b 2 sin 2 x
2
2
✿
✢ ✶
x)
❂✭❀✌✫❃
✚✫
)
❧ ✁✂✄☎
✠
369
✟
✟
dx
dx
✠
✡ ☛2 ✌ 2 2
I = ✌0 2
2
2
2
2
0 a cos x ☞ b 2 sin 2 x
2
a cos x ☞ b sin x 2
(P6 ✞♦✍ ✎✏✑✝♦✒ ✓♦)
✈✆✝✞✝
✔
sec 2 x dx
(✈✘✙✝ ✚✞✘ ✛✜ ✢✝♦ cos2 x ✓♦ ✣✝✝✒ ✤♦✥♦ ✏✜)
=
a 2 ✖ b2 tan 2 x
b tan x = t, ✜❥✝✥♦ ✏✜ b sec2 x dx = dt
✕✗ 2
0
✈✦
t✦
x = 0 r✦ t = 0 ✈✝✧✜ t✦ x ✩
❜✓✬✭✚
I=
dt
b
a2
0
t2
★
2
r✦ t ✪ ✫
t
1
tan –1
b a
a
2
0
ab 2
0
2 ab
✮✯✰✱✰ ✲ ✳✴ ✵✶✵✶✷ ✳✸✹✺✱✶✻✼
✽ ✓♦ ✾✿ r✢ ✞♦✍ ✏❀✙✥✝♦✘ ✞♦✍ ✏✍✭✥✝♦✘ ✢✝ ✓❁✝✢✭✥ ✢❂✬t✚❃
1.
3.
5.
6.
9.
12.
1
x❄ x
1
x ax ❆ x 2
1
1
x2
❈
[✓✘✞✍♦ r : x =
[✓✘✞✍♦ r:
1
x3
5x
( x ✖ 1) ( x 2 ✖ 9)
cos x
4 ❄ sin 2 x
x3
1 P x8
15. cos3 x elog sinx
18.
1
2.
3
1
sin 3 x sin ( x ❙ ❚)
a
✜✬❥✝✚]
t
1
1
x2
●
x❅a ❅ x❅b
1
x3
❉
3
x 2 ( x 4 ❇ 1) 4
1
1
x3
1
4.
1
❊
❋
❍1 ● x 6 ■
❍
■
❏
❑
, x = t6 ✜✬❥✝✚]
e5 log x ▼ e 4 log x
e3 log x ▼ e2 log x
sin x
sin ( x ▲ a)
8.
sin 8 x ◆ cos8 x
10.
1 ◆ 2sin 2 x cos 2 x
11.
1
cos ( x ❖ a ) cos ( x ❖ b)
ex
(1 ◗ e x ) (2 ◗ e x )
14.
1
( x ✖ 1) ( x 2 ✖ 4)
16. e3 logx (x4 + 1)– 1
17.
f ❘ (ax + b) [f (ax + b)]n
7.
13.
19.
2
sin ❯ 1 x ❄ cos❯ 1 x
, (x ❲ [0, 1]❳
sin ❯ 1 x ❱ cos❯1 x
370
① ✁✂✄
20.
1
☎
x
1
✆
x
✡
✍
☞
24.
✑✒
✓✓
✔✕
✬
25.
ex
✴ ✬
✮
✰
✲
2
✼
28.
✖✒ ✗
✘✙✚✛✜✒ ✢
✣✒ ✢
✤✛✤✚✥✔
6
✵
4
✸ 0
✱
✳
✌
29.
❁
✕✜
✣✜✛
✧✜✔
✕★✤✩✪✫
sin x cos x
dx 27.
cos 4 x sin 4 x
✶
0
1 x
✿
✹
2
✻ 0
✹
dx
1
✽
✾ ✼
✑✣✜✕✦✛✜✒✢
✯
sin x cos x
dx
sin 2 x
3
☞
30.
x
❀
4
✻ 0
❂
31.
2
sin
❄ 0
4
33.
❋
❇
1
✤✛❢✛✤✦✤●✜✔
34.
❄
1
❋
❃
1
❑
32.
❉
✕✜✒
❊
✤✑❍
❉
❊
✕★✤✩✪
■
(
❉
40.
41
✑✒
✘✙✚✛
34
39
✑✒
41.
)
2
2
log
3
3
0
1
35.
✠
✺
✺
✝
P
✕★
✖✒ ✗
dx
e e
✺
❵
✑★✣✜
✘✙ ✚✛✜✒ ✢
x
✣✒ ✢
❝❴✜❝❴
◗
✖✒ ✗
❋
0
x e x dx 1
❏
▲
37.
tan 3 x dx 1 log 2
✔✕
x
sin x cos x
dx
9 16 sin 2 x
✫
❏
❳✜✒ ❨ ✘✗✦
✻
✔✕
x17 cos 4 x dx 0
4
2
❘ 0
44
✺
❈
❖
38.
❆
cos 2 x dx
cos 2 x 4 sin 2 x
x 1| | x 2 | | x 3 dx
✠
1
36.
☛
✎
x tan x
dx
sec x tan x
❅
2 x tan 1 (sin x) dx
dx
2
x ( x 1)
3
✞
x4
1 sin x
dx 26.
1 – cos x
✭
✞
✞
x 2 1 log ( x 2 1) 2 log x
✠
✷✏
22.
✝
✟
x2 x 1
( x 1) 2 ( x 2)
✞
✝
21.
1 x
1 x
–1
23. tan
2 sin 2 x x
e
1 cos 2 x
❩✘
✑r★
✣✒✢
❪❫✜❴
39.
1
❲
0
sin
1 2❬ 3x
❭
0
e
✕✜
dx
✥❳✛
❙
1
x dx
✕✜
❚
❯
✣✜✛
2
❱
✧✜✔
◆
2 sin 3
0
1
✕★✤✩✪✫
✕★✤✩✪✫
r❛❜
(A) tan–1 (ex) + C
(C) log (ex – e – x) + C
(B) tan–1 (e – x) + C
(D) log (ex + e – x) + C
x dx
▼
2
3
❧ ✁✂✄☎
371
cos 2 x
dx ❝✞✟❝✞ ✠✡☛
42. ✝
(sin x ✆ cos x) 2
(A)
–1
☞C
sin x ☞ cos x
(B) log |sin x ✌ cos x | ✌ C
(C) log |sin x ✍ cos x | ✎ C
(D)
1
(sin x ✏ cos x) 2
b
43. ❀✑✒ f (a + b – x) = f (x), r✟✓ ✔ x f ( x) dx ❝✞✟❝✞ ✠✡☛
a
(A)
a✕b b
f (b ✖ x) dx
2 ✗a
(B)
a✕b b
f (b ✕ x) dx
2 ✗a
(C)
b✘a b
f ( x) dx
2 ✙a
(D)
a✕b b
f ( x) dx
2 ✗a
1
✜ 2x ✛ 1 ✢
44. ✝ tan ✚1 ✣
dx ✧✟ ★✟✩ ✠✡☛
2 ✤
0
✥ 1✆ x ✛ x ✦
(A) 1
(B) 0
(C) –1
(D)
✪
4
✫✬✭✬✮✯✬
✰ ✱★✟✧✲✩✳ ✴✵✧✲✩ ✧✟ ✶❀✷✸✹★ ✺✻✹★ ✠✡✼ ✴✵✧✲✩ ✽✑✾✟r ★✓✿ ✠★✓✿ ❁✧ ✺❂✲✩ ✑✒❀✟ ✠✷✴✟
✠✟✓r✟ ✠✡ ✴✟✡✞ ✠★✓✿ ❣✱ ✺❂✲✩ ✧✟ ✴✵✧✲❃
✴❄✟✵✟ ✴✵✧✲ ❅✟r ✧✞✩✟ ✠✟✓r✟ ✠✡ ✺✞✿r✷
✱★✟✧✲✩ ✽✑✾✟r ★✓✿ ✠★✓✿ ❁✧ ❁✓✱✟ ✺❂✲✩ ❅✟r ✧✞✩✟ ✠✟✓r✟ ✠✡ ✑❃✱✧✟ ✴✵✧✲ ✑✒❀✟ ✠✷✴✟
✠✟✓r✟ ✠✡✼ ✴r☛ ✱★✟✧✲✩ ❁✧ ❁✓✱✟ ✺✻✹ ★ ✠✡ ❃✟✓ ✑✧ ✴✵✧✲✩ ✧✟ ✶❀✷✸✹★ ✠✡✼
d
F( x) ❆ f ( x) . r❝ ✠★ ❈ f ( x) dx ❇ F ( x) ✎ C ✑✲❢✟r✓ ✠✡✿✼ ❀✓
dx
✱★✟✧✲✩ ✴✑✩✑❉❊r ✱★✟✧✲✩ ✴❄✟✵✟ ✶❀✟✺✧ ✱★✟✧✲✩ ✧✠✲✟r✓ ✠✡✿✼ C ✱★✟✧✲✩ ✴❊✞
★✟✩ ✲❡✑❃❁ ✑✧
✧✠✲✟r✟ ✠✡✼ ❣✩ ✱❞✟❡ ✱★✟✧✲✩✟✓✿ ★✓✿ ❁✧ ✴❊✞ ✧✟ ✴✿r✞ ✠✟✓r✟ ✠✡✼
✰ ❚❀✟✑★✑r ✒❋✑●❍ ✱✓ ✴✑✩✑❉❊r ✱★✟✧✲✩ ✵✹✟✓✿ ✵✓❂ ✺✑✞✵✟✞ ✧✟ ✱★■✠ ✠✡ ✑❃✱★✓✿ ✺✻✸❀✓✧ ✱✒❏❀
y❑✴▲✟ ✵✓❂ ✴✩✷✑✒❉✟ ▼✺✞ ✧❡ r✞✺◆❂ ✴❄✟✵✟ ✩❡❊✓ ✧❡ r✞✺◆❂ ❏✵❀✿ ✵✓❂ ✱★✟✿r✞ ❏❄✟✟✩✟✿r✑✞r
✧✞✵✓❂ ✺✻✟❖r ✑✧❀✟ ❃✟ ✱✧r✟ ✠✡✼
372
☎
① ✁✂✄
✈✆✝✆✞✟✠
[ f ( x)
1.
2.
✡☛☞✌✍✝
✆✌✡❢
✎✏ ✑
✎✒✑✓
✔✒✕☞✖☛✗
g ( x)] dx
✎☞✣✠✆✎✌
✢☞❢
✈✆✖✌
★✥☞✩✌✠✪✫
✡✤✙✥☞♦❧
✚✛✤
✆✝✘✝✆✍✆✙☞✠
f ( x ) dx
k,
✡✤✙✥☞
✎✏ ✑
g ( x ) dx
✆✍♦
✧
k f ( x) dx
f1, f2, f3,..., fn ,
✥✆✬
✚✛✜
✦
✚✛✤
✩✑✍✝
k f ( x) dx
✧
k1, k2,...,kn ,
✠✐☞☞
✎☞✣✠✆✎✌
✠☞✏
[ k1 f1 ( x) k2 f 2 ( x) ... kn f n ( x)] dx
= k1
☎
✮✯✰✱
✧
f1 ( x) dx k2
✭
✲✳✴✵✴✶✷✴✸
f 2 ( x) dx ... kn
✭
✧
✭
✧
f n ( x) dx
✹✵✴✸✺✻
xn 1
C, n
n 1
✼
(i)
x n dx
✿
(ii)
✧
(iv)
✧
✽
✾
❀
– 1.
✆✎✆✞☞❁❂✠✪
✾
✧
dx
✦
cos x dx sin x C
(iii)
✧
sec 2 x dx
(v)
✧
– cosec x C (viii)
✧
✦
❃
❄
tan x C
✭
x C
❃
sin x dx
– cos x C
❄
✭
cosec 2 x dx
– cot x C
❄
✭
(vi) sec x tan x dx sec x C
❄
✧
✭
dx
(vii)
cosec x cot x dx
✧
✧
1 x
❆
(xi)
✧
✧
(xv)
☎
dx
1 x2
❉
❊✴❋ ✶●✴✸
✣☛▼✕☞
✈☞✛ ▼
❞☞ ☞✠
❄ ❆
2
❄ ❆
cos
cot
x x
2
a x dx
1 x
1
1
x C
✭
(x)
✧
Q (x), x
2
dx
1 x2
❄
sin
❄
tan
(xii)
✭
✧
1
1
x C
✭
x C
✭
✭
dx
x C
❅
❅
x x2 1
❄
sec
✭
C
❅
1
x C
✭
❆
❄ ❆
❆
❇
✶❍✴■✻✴❏❋
✌❢✆◆♦
✡✏
❅
❅
✭
dx
(xiii)
✭
❆
dx
(ix)
❄
cosec 1 x C
❅
(xiv)
✭
1
ax
log a
❑✴▲✴
✆✌
❈
♦✌
✚✛
e x dx e x
❄
1
dx log x C
x
(xvi)
✹✵✴✸✺✻
✎✏✑ ♥✚✒✩✬
✈ ✆✖✌
C
✧
✩✆▼☛✏✥
✚✛✤
✠ ☞✏
✈☞✛▼
P ( x)
,
Q ( x)
Q (x) 0.
P(x)
Q (x)
✩✑✍✝
✚☛
✬☞✏
❀
✥✆✬
✌☞ ✏
♥✚✒ ✩✬☞✏✤
♥✚✒✩✬
✡✏
✌☞
P(x)
✈✝✒✩☞✠
✌❢
❞☞☞✠
✆ ✎✢ ☞ ☞ ✆◆ ✠
✚✛
✆◆✡☛✏✤
P(x)
♥✚✒ ✩✬
Q(x),
✌ ▼✠ ✏
✚✛✤
✌❢
✠ ☞ ✆✌
❧ ✁✂✄☎
373
P ( x)
P ( x)
✆ T ( x) ✝ 1
♦✞✟ ✠✡ ☛✞☞ ✌✍✎✏✏ ✑✏ ✒♦✞✟ ✑✓✏✔ T (x), ✱✕ ✖✓✗✡✘ ✓✙ ✚✏✙✛
Q ( x)
Q( x)
P1 (x) ✕❞ ✜✏✏✢ Q(x) ✕❞ ✜✏✏✢ ✒✞ ✕☛ ✓✙✣ ✖✓✗✡✘ ✓✏✞✤✞ ♦✞✟ ✕✏✛✥✏ T (x) ✕✏ ✒☛✏✕✍✤
✚✏✒✏✤❞ ✒✞ ✈✏✢ ✌✕✦✏ ✑✏ ✒✕✢✏ ✓✙✣
P1 ( x)
✕✏✞ ✌✤✧✤✌✍✌✎✏✢ ✡★✕✏✛ ✕❞ ✚✏☞✌✩✏✕ ✌✪✏✫✤✏✞☞
Q( x)
♦✞✟ ✦✏✞✬✡✟✍ ♦✞✟ ✠✡ ☛✞☞ ✭✦✮✢ ✕✛✢✞ ✓✗✱ ✯✒✕✏ ✒☛✏✕✍✤ ✈✏✢ ✌✕✦✏ ✑✏ ✒✕✢✏ ✓✙✣
1.
A
B
px ✝ q
✲
=
,a✴b
x✳a x✳b
( x ✰ a ) ( x ✰ b)
2.
px ✵ q
( x ✶ a) 2
=
3.
px 2 ✷ qx ✷ r
( x ✸ a ) ( x ✸ b) ( x ✸ c )
=
4.
px 2 ✷ qx ✷ r
( x ✸ a ) 2 ( x ✸ b)
=
5.
px 2 ✹ qx ✹ r
( x ✺ a ) ( x 2 ✹ bx ✹ c)
=
A
x✶a
A
x✳a
A
x✰a
A
x✼a
✵
✲
B
( x ✶ a)2
B
x✳b
✲
C
x✳c
✝
B
C
✝
2
x✰b
( x ✰ a)
✻
Bx + C
,
x ✻ bx ✻ c
2
✑✓✏✔ x2 + bx + c ♦✞✟ ✚✏✬✞ ✚✏✙ ✛ ✬✗✥✏✤✎✏☞✽ ✤✓❞☞ ✌✕✱ ✑✏ ✒✕✢✞✣
✾ ✐✿❀❁❂❃❄❄✐❅ ❆❄❇❄ ❈❉❄❊❋❅
✒☛✏✕✍✤ ♦✞✟ ●✛ ☛✞☞ ✡✌✛♦✢❍✤ ✌✘✱ ✓✗✱ ✒☛✏✕✍✤ ✕✏✞ ✌✕✒❞ ✱✕ ✚✏■✏✛✪✏❏✢ ✒☛✏✕✍✤ ☛✞☞
✡✌✛♦✌✢❍✢ ✕✛ ✘✞✢✏ ✓✙✣ ✦✓ ✌♦✌■ ✌✑✒☛✞☞ ✓☛ ✱✕ ●✛ ✕✏✞ ✌✕✒❞ ✘❏✒✛✞ ●✛ ☛✞☞ ✡✌✛♦✌✢❍✢ ✕✛✢✞
✓✙ ☞ ✡★✌✢❣❑✏✏✡✤ ✌♦✌■ ✕✓✍✏✢❞ ✓✙✣ ✑✖ ✒☛✏✕▲✦ ☛✞☞ ♦✗✟▼ ✌◆✏✕✏✞✥✏✌☛✢❞✦ ✡✟✍✤ ✒✌✧☛✌✍✢
✓✏✞☞ ✢✏✞ ✓☛ ✒☛✏✕✍✤ ✈✏✢ ✕✛✤✞ ♦✞✟ ✌✍✱ ♦✗✟▼ ✒✗✡✌✛✌●✢ ✒♦❍ ✒✌☛✕✏✚✏✞☞ ✕✏ ❖✡✦✏✞✬ ✕✛✢✞
✓✙☞✣ ✡★✌✢❣❑✏✏✡✤ ✌♦✌■ ✕✏ ❖✡✦✏✞✬ ✕✛✢✞ ✓✗✱ ✓☛ ✌✤✧✤✌✍✌✎✏✢ ✡★✏☛✏✌✥✏✕ ✒☛✏✕✍✤✏✞☞ ✕✏✞ ✡★✏P✢
✕✛✢✞ ✓✙◗☞
(i) ❙ tan x dx ❘ log sec x ✻ C
(ii) ❙ cot x dx ❚ log sin x ❯ C
(iii) ❙ sec x dx ❘ log sec x ✻ tan x ✻ C
(iv) ❙ cosec x dx ❘ log cosec x ✼ cot x ✻ C
374
☎
① ✁✂✄
♦✆✝✞
✟♦✟✠✡☛☞
(i)
✘
✌✝✍✎✡✏✑
dx
x a2
2
✜
a
✗
C
1
a x
log
2a
a x
✙
C
✙
x
✛
1
x a
log
2a
x a
✗
✕
2
✚
2
✥
x
2
x
2
✦
a
2
a
2
✣
log x
dx
(vi)
☎
✥
❬✡✑✧ ✠✡★
❢✩✪
✘
✤
✣
x2
✤
log | x
x2
✤
✦
✤
a2
✤
✥
dx
x a2
2
✣
✤
1
tan
a
dx
C (v)
✥
a2
✦
✣
x2
sin
✢
✢
1
1
x
a
x
a
✤
✤
C
C
a2 | C
✤
✒✓✡✔✍✎
✫✬✪
f1
✭✮✯✰✱✲✳
f2 ,
r✴✱✱
f1 ( x) . f 2 ( x) dx
✵✲✮
f1 ( x)
✖
✘
❢✯✪
✹✱
✈✵✹✯
×
✐✬ ❄✱✱✳✹
✫✶
f 2 ( x) dx
✭✮✯✰✱✲✳ ✵✲ ✮ ✐✬❄✱✰✭✮✯ ✹✱ ❅✶✱✹✯✰
✭✮✯✰
(iii)
✛
dx
(iv)
✒✓✡✔✍✎
✕
✖
dx
(ii)
♦✏✝
=
❢❆r❇❈
✭✷✱✸r
✼
✕
✘ ✾
❀
✹✺r✲
✹✱
❢✹
d
f1 ( x) . f 2 ( x) dx dx ,
dx
✽
✘
×
✭✷✴✱✶ ✭✮✯✰
✭✮✯✰
✫✻✳
✿
❁
✈✴✱ ✱ ❂r❃
❢❆r❇❈ ✭✮✯✰ ✹✱ ❅✶✱✹✯✰
}
❅✶✱✹✯✰
✹✱
❅✶✱✹✯✰
.
✩ ✱✲
–{
✭✷✴✱✶
✭✷✴✱✶
✭✮✯✰
✪✵✳ ❢❆r❇❈ ✭✮✯✰ ✵✲ ✮ ❉❈✰ ✶✲✳ ❅✱✵❊✱✰❇ ✺❋✱✰❇ ❉✱❢✫✪● ❍✭■❏r❈✱ ✫✶✲✳ ✪✲❅✲ ✭✮✯✰ ✹✱✲ ❢❆r❇❈
✭✮✯✰
☎
☎
✥
✵✲ ✮
❑✭
e x [ f ( x)
♦✆✝✞
(i)
(ii)
(iii)
(iv)
◗
✟♦✟✠✡☛☞
❲
❲
❪
x
2
a2
ax
2
❉✱❢✫✪
✯✲ ✰✱
f ( x)] dx
✤
x2
✶✲✳
✌❘✔ ✡❙
❚
♦✏✝
a 2 dx
✣
✥
❢▲❅✹✱
▼✱❢✯◆▼✱✱❖❢ r
P✱r
✫✻ ●
✤
✒✓✡✔✍✎
❯
x 2
x
2
❱
a
❨
x2
❱
a dx
❯
❨
x 2 dx
❩
x 2
a
2
dx
bx
✫✶✲✳
e x f ( x) dx C
x 2
x
2
2
❅✶✱✹✯✰
❚
a2
2
❚
a2
log x
2
❱
x2
❚
a2
❱
a2
log x
2
❱
x2
❱
a2
❭
a2
sin
2
❳
1
x
a
❭
❱
❱
C
C
C
dx
❫❴❵❛❵
c
ax
2
✵✲ ✮ ✭✷✹✱✺ ✵✲ ✮ ❅✶✱✹✯✰✱✲ ✳ ✹✱✲ ✭✷✱✶✱❢❄✱✹
bx
c
❧ ✁✂✄☎
375
✿✆ ✝✞✟ ✠✡☛✡✠☞✠✌✍✎ ✠✏✠✑ ✒✍✓✍ ✆✠✓✏✠✎✔✎ ✠✕✖✍ ✗✍ ✘✕✎✍ ✙✚✛
✜
2
ax + bx + c = a ★ x ✥
2
✬
px
(v)
ax
2
q dx
bx
c
✈✰✱✲✱
b
x✥
a
✜✣
c✢
b ✤ ✣ c b2 ✤✢
a
x
✦
✥
★✪
✫✩
✫ ✥✪ ✧
a ✩✭
2a ✯ ✮ a 4a 2 ✯ ✩✭
★✬ ✮
2
px
q dx
ax 2
bx
c
✏✞♦ ✆✳✕✍✓ ✏✞♦ ✘✝✍✕☞✡✍✞✟ ✕✍✞ ✆✳✍✝✍✠✴✍✕
✿✆ ✝✞✟ ✆✠✓✏✠✎✔✎ ✠✕✖✍ ✗✍ ✘✕✎✍ ✙✚✛✟
px ✵ q ✶ A
d
(ax 2 ✵ bx ✵ c) ✵ B ✶ A (2ax ✵ b) ✵ B , A ✎r✍✍ B ✕✍ ✝✍✡ ❞✍✎
dx
✕✓✡✞ ✏✞♦ ✠☞✷ ✸✍✞✡✍✞✟ ✆✹✍✍✞✟ ✘✞ ✺✻✴✍✍✟✕✍✞✟ ✕✼ ✎✻☞✡✍ ✕✼ ✗✍✎✼ ✙✚✽
b
✾ ✙✝✡✞ ❀ a f ( x) dx ✕✍✞❁ ✏❂ y = f (x), a ❃ x ❃ b, x-❄✹✍ ✷✏✟ ✕✍✞✠❅✖✍✞✟ x = a ❄✍✚✓ x = b
✘✞ ✠❆✍✓✞ ✹✍✞❇✍ ✏✞♦ ✹✍✞❇✍✆♦☞ ✏✞♦ ✿✆ ✝✞✟ ✆✠✓❈✍✍✠❉✍✎ ✠✕✖✍ ✙✚✽ ✝✍✡ ☞✼✠✗✷ [a, b] ✝✞✟ x ✷✕ ❊❋✸✻
x
✙✚ ✎❋ ● f ( x) dx ✹✍✞ ❇✍✆♦☞ ✆♦☞✡ A (x) ✕✍✞ ✠✡✿✠✆✎ ✕✓✎✍ ✙✚✽ ✹✍✞❇✍✆♦☞ ✆♦☞✡ ✕✼
a
✘✟✕❍✆✡✍ ✙✝✞✟ ✕☞✡ ✕✼ ❄✍✑✍✓❈✍■✎ ✆✳✝✞ ✖ ✕✼ ❄✍✞✓ ✠✡☛✡✠☞✠✌✍✎ ✿✆ ✝✞ ✟ ✆✳✠✞ ✓✎ ✕✓✎✼ ✙✚✽
❪ ✝✍✡ ☞✼✠✗✷ ✠✕ ✹✍✞❇✍✆♦☞ ✆♦☞✡
✾ ❏❑▲▼◆❖ P◗❘▲❙ ▼❚ ❯❱❲▲❑ ❳▲❨▲❩❬▲❭❙ ❯❱❑❫
x
A(x) = ❀ f ( x) dx , ❴ x ❵ a, ✒✍✓✍ ✆✠✓❈✍✍✠❉✍✎ ✙✚ ✗✙✍⑥ ✆♦☞✡ f ❄✟✎✓✍☞ [a, b] ✆✓ ✘✟✎✎
a
✆♦☞✡ ✝✍✡✍ ✺✖✍ ✙✚✽ ✎❋ A❛ (x) = f (x) ❴ x ❜ [a, b]
❪
✾ ❏❑▲▼◆❖ P◗❘▲❙ ▼❚ ◗❝❙❚❫ ❳▲❨▲❩❬▲❭❙ ❯❱❑❫
✝✍✡ ☞✼✠✗✷ ✠✕✘✼ ❋✟✸ ❄✟✎✓✍☞ [a, b] ✆✓ f , x ✕✍ ✘✟✎✎ ✆♦☞✡ ✙✚ ❄✍✚ ✓ F ✷✕ ✸■✘✓✍
✆♦☞✡ ✙✚ ✗✙✍⑥
b
d
F( x) ❡ f ( x) , f ✏✞♦ ✆✳✍❢✎ ✏✞♦ ✘❈✍✼ x ✏✞♦ ✠☞✷ ✙✚❁ ✎❋
dx
b
● a f ( x) dx ✐ ❣ F( x) ❥ C❤a ✐ F (b) ❦ F ( a)
✖✙ ✆✠✓✘✓ [a, b] ✆✓ f ✕✍ ✠✡✠♠♥✎ ✘✝✍✕☞✡ ✕✙☞✍✎✍ ✙✚ ✗✙✍⑥ a ✎r✍✍ b ✘✝✍✕☞✡
✕✼ ✘✼✝✍✷⑥ ✕✙☞✍✎✼ ✙✚✟ a ✠✡☛✡ ✘✼✝✍ ✕✙☞✍✎✼ ✙✚ ❄✍✚✓ b ✕✍✞ ♣q♥ ✘✼✝✍ ✕✙✎✞ ✙✚✽✟
—s—
❖P◗❘◗
8
❙❯❱❲❳❨❱❩❬ ❭❩❪ ❫❨❴❵❛❜❱❩❝
(Application of Integrals)
One should study Mathematics because it is only through Mathematics that
nature can be conceived in harmonious form. – BIRKHOFF
❍✁✂✄☎✆✁ (Introduction)
❚✝✞✟✠✟✡ ✠☛☞✌ ✍✠✎☛ ✟✏✞✑✞✒✓✞☛☞ ✔✞✝✡✞☛☞✌ ✕✠✖☞✗ ✘✡✒✑✞✒✙✓✞☛☞ ✚✛☞ ✛✜✢✞✞☛☞
✕✟✍✡ ✟✛✟✑✞❧✎ ❚✝✞✟✠✡✣✝ ✔✞✛✜✤✟✡✝✞☛☞ ✛☛✤ ✥✞☛✏✞✦✤✖ ✛☛✤ ✦✟✧★✖✎
✛☛✤ ✟✖✚ ✕♦✏✞✞☛☞ ★✞ ✔✩✝✝✎ ✟★✝✞ ✍✪✫ ✛✞✬✡✟✛★ ✓✣✛✎ ★✣ ✔✎☛★
✕✠✬✝✞✔✞☛☞ ✛☛✤ ✟✖✚ ✭✟✮✞✡ ✛☛✤ ✔✎✒✦✯✝✞☛✭ ✠☛☞ ✰✕ ✦✯★✞✧ ✛☛✤ ✕♦✏✞
✠♦✖ ✍✞☛✡☛ ✍✪☞✫ ✦✯✞✧☞✟✑✞★ ❚✝✞✟✠✟✡ ✛☛✤ ✕♦✏✞✞☛☞ ★✣ ✕✍✞✝✡✞ ✕☛ ✍✠
✔✎☛★ ✕✞✈✞✧✮✞ ✔✞✛✜✤✟✡✝✞☛☞ ✛☛✤ ✥✞☛✏✞✦✤✖ ★✞ ✦✟✧★✖✎ ★✧ ✕★✡☛
✍✪☞✫ ✝❣✟✦ ✝☛ ✕♦✏✞ ✛✱✞☛☞ ✲✞✧✞ ✟✳✞✧☛ ✥✞☛✏✞✦✤✖ ✛☛✤ ✦✟✧★✖✎ ✛☛✤ ✟✖✚
✔✦✝✞✙✴✡ ✍✪☞ ✰✕✛☛✤ ✟✖✚ ✍✠☛☞ ✕✠✞★✖✎ ✭✟✮✞✡ ★✣ ✛✒✤✵ ✕☞★✶✦✎✞✔✞☛☞
★✣ ✔✞✛❞✝★✡✞ ✍✞☛✭✣✫
✟✦✵✖☛ ✔✩✝✞✝ ✠☛☞ ✍✠✎☛ ✝✞✭☛ ✦✤✖ ★✣ ✕✣✠✞ ✛☛✤ ❢✦ ✠☛☞ ✟✎✟❞✘✡
✕✠✞★✖✎✞☛☞ ★✞ ✦✟✧★✖✎ ★✧✡☛ ✕✠✝ ✛✱ y = f (x), ★✞☛✟✷✝✞☛☞
x = a, x = b ✚✛☞ x-✔✥✞ ✕☛ ✟✳✞✧☛ ✥✞☛✏✞✦✤✖ ★✞☛ ✸✞✡ ★✧✎☛ ★✞ ✔✩✝✝✎
✟★✝✞ ✍✪✫ ✰✕ ✔✩✝✞✝ ✠☛☞ ✍✠ ✕✞✈✞✧✮✞ ✛✱✞☛☞ ✛☛✤ ✔☞✡✭✙✡✌ ✕✧✖ ✧☛✹✞ ✔✞☛☞ ✚✛☞ ✛✜✢✞ ✌☛☞ ✦✧✛✖✝✞✌☛☞ ✡✺✞✞ ✻✣✳✞✛✜✢✞✞☛☞
✼✛☛✤✛✖ ✠✞✎★ ❢✦✽ ★✣ ✘✞✦✞☛☞ ✛☛✤ ✗✣✘ ✟✳✞✧☛ ✥✞☛✏✞✦✤✖ ★✞☛ ✸✞✡ ★✧✎☛ ✛☛✤ ✟✖✚ ✕✠✞★✖✎✞☛☞ ✛☛✤ ✚★
✟✛✟❞✞✾✷ ✔✎✒✦✯✝✞☛✭ ★✞ ✔✩✝✝✎ ★✧☛✭☞ ☛✫ ✿✦✧✞☛❀✡ ✛✱✞☛☞ ✕☛ ✟✳✞✧☛ ✥✞☛✏✞✦✤✖ ★✞☛ ✑✞✣ ✸✞✡ ★✧☛✭☞ ☛✫
8.2 ❁✁❂✁❃❄✁ ❅❆✁❇❈ ❅❇❉ ❊❈❋●■❋ ❏✁❇❑✁▲❉▼ (Area Under Simple Curves)
✟✦✵✖☛ ✔✩✝✞✝ ✠☛☞ ✍✠✎☛✌ ✝✞✭☛ ✦✤✖ ★✣ ✕✣✠✞ ✛☛✤ ❢✦ ✠☛☞ ✟✎✟❞✘✡ ✕✠✞★✖✎ ✚✛☞ ★✖✎ ★✣ ✔✞✈✞✧✑✞♦✡
✦✯✠☛✝ ★✞ ✿✦✝✞☛✭ ★✧✡☛ ✍✒✚ ✟✎✟❞✘✡ ✕✠✞★✖✎ ★✞ ✦✟✧★✖✎ ✛✪✤✕☛ ✟★✝✞ ✓✞✚✌ ★✞ ✔✩✝✝✎ ✟★✝✞
✍✪✫ ✔✗ ✍✠ ✛✱ y = f (x), x-✔✥✞ ✚✛☞ ★✞✟☛ ✷✝✞◆ x = a ✡✺✞✞ x = b ✕☛ ✟✳✞✧☛ ✥✞☛✏✞✦✤✖ ★✞☛ ✸✞✡ ★✧✎☛
8.1
A.L. Cauchy
(1789-1857)
❧ ✁✂✄☎✁✆✝ ✞✆✟ ✠☎✡☛✌☞ ✁✆✍
377
❞✎ ✏✑✒✑✓ ✔✕✖ ✏✖✗✘✑✙✓ ✒✚ ✛✜✑✢✗ ✣✕✣✤ ❞✎ ✥✥✑✙ ❞✦✗✚
❣✧✖★ ✏✑✕✩✪✣✗ ✫✬✭ ✒✚ ❣✮ ✕✯ ✕✚✪ ✏✖✗✰✙✗ ✱✑✚✲✑✛✪✳ ❞✑✚
❝❣✴✗ ✒✎ ✛✗✳✎ ✔✕✖ ✵✶✕✑✙✤✦ ❝❣✴✗ ✒✎ ✛✣❱✷ ❱ ✸✑✚ ✖ ✒✚
✣✓✣✮✙ ✗ ✮✑✓ ✒❞✗✚ ❣✧ ✖ ★ y ✵♠ ✥ ✑✹✙ ✔✕✖ dx ✥✑✧♣✺✑✹✙
✕✑✳✎ ✔❞ ▲✕✚✻✼ ✛❱✷❱✎ ✛✦ ✣✕✥✑✦ ❞✎✣✽✔✾ ✹✒✮✚✖ dA
(✛✜✑✦✖✣✐✑❞ ✛❱✷❱✎ ❞✑ ✱✑✚✲✑✛✪✳) = ydx, ✽❣✑♠ y = f(x) ❣✧★
✸❣ ✱✑✚✲✑✛✪✳ ✛✜✑✦✖✣✐✑❞ ✱✑✚✲✑✛✪✳ ❞❣✳✑✗✑ ❣✧ ✽✑✚
✣❞ ✱✑✚✲✑ ✕✚✪ ✐✑✎✗✦ ✣❞✒✎ ▲✕✚✻✼ ✣▲❢✑✣✗ ✛✦ ▲❢✑✑✣✛✗
❲❳❨❩❬ ❭❪ ❫❴❜
❣✧ ✔✕✖ a ✗❢✑✑ b ✕✚✪ ✮✶✸ x ✕✚✪ ✣❞✒✎ ✮✑✓ ✒✚ ✣✕✣✓✣♦✙✿❱ ❣✧ ★ ✕✯ y = f (x), ❞✑✚✣❱✸✑✚✖ x = a, x = b
✔✕✖ x❀✏✱✑ ✒✚ ✣❁✑✦✚ ✱✑✚✲✑ ✕✚✪ ✕✴✪✳ ✱✑✚✲✑✛✪✳ A ❞✑✚✾ ✱✑✚ ✲✑ PQRSP ✮✚✖ ✒✐✑✎ ✛✗✳✎ ✛✣❱✷❱✸✑✚✖ ✕✚✪ ✱✑✚ ✲✑✛✪✳✑✚✖
✕✚✪ ✸✑✚✰✛✪✳ ✕✚✪ ✛✣✦❂✑✑✮ ✕✚✪ ❃✛ ✮✚✖ ♦✚❄✑ ✒❞✗✚ ❣✧✖★ ✒✑✖✕✚✪✣✗❞
✐✑✑✿✑✑ ✮✚ ✖ ❣✮ ✹✒✚ ✹✒ ✛✜❞✑✦ ✏✣✐✑❍✸❅✗ ❞✦✗✚ ❣✧✖❆
b
b
b
A = ❈ a dA ❇ ❈ a ydx ❇ ❈ a f ( x) dx
✕✯ x = g (y), y-✏✱✑ ✔✕✖ ✦✚❄✑✑✔♠ y = c, y = d ✒✚ ✣❁✑✦✚ ✱✑✚✲✑
❞✑ ✱✑✚✲✑✛✪✳ ✣✓❉✓✣✳✣❄✑✗ ✒❊✲✑ ❋✑✦✑ ✛✜✑✢✗ ✣❞✸✑ ✽✑✗✑ ❣✧★
d
d
A = ■ c xdy ● ■ c g ( y ) dy
✸❣✑♠ ❣✮ ✱✑✧✣✗✽ ✛✣❱✷❱✸✑✚✖ ✛✦ ✣✕✥✑✦ ❞✦✗✚ ❣✧✖ ✽✧✒✑ ✣❞
✏✑✕✩✪✣✗ ✫✬✈ ✮✚ ✖ ♦❏✑✑✙✸✑ ✰✸✑ ❣✧ ★
❲❳❨❩❬❭❪ ❫❴❵
❑▼◆❖P◗❘ ✸✣♦ ✥✣✥✙✗ ✕✯ ❞✎ ✣▲❢✑✣✗ x-✏✱✑ ✕✚✪ ✓✎✥✚ ❣✧✾ ✗✑✚ ✽✧ ✒ ✑ ✣❞ ✏✑✕✩✪✣✗ ✫✬❙ ✮✚ ✖ ♦❏✑✑✙ ✸ ✑
✰✸✑ ❣✧✾ ✽❣✑♠ x = a ✒✚ x = b ✗❞
f (x) < 0 ✹✒✣✳✔ ✣♦✔ ❣✴✔ ✕✯✾
x-✏✱✑ ✔✕✖ ❞✑✚✣❱✸✑✚✖ x = a, x = b ✒✚
✣❁✑✦✚ ✱✑✚ ✲✑ ❞✑ ✱✑✚✲✑✛✪✳ ❚❂✑✑❯✮❞ ❣✑✚
✽✑✗✑ ❣✧✾ ✛✦✖✗✴ ❣✮ ✱✑✚✲✑✛✪✳ ✕✚✪ ✕✚✪✕✳
✒✖❄✸✑❯✮❞ ✮✑✓ ❞✎ ❣✎ ✥✥✑✙ ❞✦✗✚
❣✧✖★ ✹✒✣✳✔ ✸✣♦ ✱✑✚✲✑✛✪✳ ❚❂✑✑❯✮❞
❣✧ ✗✑✚ ❣✮ ✹✒✕✚✪ ✣✓✦✛✚✱✑ ✮✑✓✾ ✏❢✑✑✙✗✷
b
a
f ( x) dx ❞✑✚ ✳✚✗✚ ❣✧✖★
❲❳❨❩❬ ❭❪ ❫❴❛
378
① ✁✂✄
❧✌✍✌✎✏✑✒
✉✪✫✓
✔✖ ✬
✱✓❧✌
✔✌✓
✭✖❧ ✌
✤✥✌
✧✾✌✿✑
✱✘✳
✭✌✑✌
✗✕✏✌
✔✖
✤✌✘ ✮ ✛ ✗✑
✗✕
y = f (x), x⑥✌★✌
❧✕✑✌
✗✕
✘✙
✯✰✲
✍✓ ✳
✘✚✛✜
✴✵✌✌ ✶ ✏✌
x=a
✕✌✓✗✺✏✌✓✳
✕✌
✤✥✌
✔✖ ✷
✣ ✏✌
x=b
✑✩✌✌
x-
✢✌✌✣
❧✓
✏✔✌ ✸
✗✻✌★✓
✘✓✛
✦✧★
A1 < 0
✥✌✓✼✌
✕✌
✔✖
✑ ✩✌✌
✑✩✌✌
✘✚✛✜
A2 > 0
✥✌✓✼✌✧✛✹
A
✢✌✌✣
✔✖ ✬
❧✽✼✌
x-
✤✥✌
❣❧ ✗✹✱
✘✓✛
✘✙
A = A1 + A2
✔✖ ✷
✈☎✆✝✞✟✠ ✡☛☞
♠❀❁❂❃❄❁
❂❆
❅
2
2
x +y =a
✘✮♦✌
✤✌✘✮✛✗✑
2
✯✰❇
✍✓✳
✗✴✱
✧✗★✑✒
❧✍✗✍✑
✔✖
✭✖❧ ✌
✗✕
✗✴✱
✧✾✌ ✿✑
a2
0
x2 + y 2 = a2
❣❧✗✹✱
✧★
❏
✱✘✳
❊❋✘✌✶● ★
✧✗✺❍✺ ✏✌✸
y
✔✌✓✑✌
✥✌✓✼✌
✤✥✌
✘✚✛✹
✕✌
x=0
✑✩✌✌
✴✌✓✉✌✓✳
✘✓✛
y-
✤✥✌
✥✌✓✼✌✧✛✹
x=a
✘✮ ♦✌
❧✓
❧✓
✹✓✑ ✓
✔✚✱
AOBA
)
a2
y
✫✑✚ ✩✌✌✐✵ ✌
✧✾ ✩✌ ✍
✗✹✏✌
✗✻✌★✌
x2
✭✌✑✌
✥✌✓✼ ✌✧✛✹
✔✖✷
✧✾✌ ✿✑
✍✓✳
✔✌ ✑
✓ ✌
✔✖✷
❧✗❑✍✗✹✑
❧✍✌✕ ✹✉
✗✉ ❑✉✗✹✗▼✌✑
✔✖
✕ ★✉ ✓
◆✧
✍✓✳
✈☎✆✝ ✞✟✠ ✡☛❴
✔✖✒
= 4
❧✓ ✗✻✌★✓ ✥✌✓✼✌
x 2 dx ,
■
●✉✌▲✍✕
✕✌✓
✔✚✱
✘✮♦✌
AOBA
✥✌✓✼ ✌
✔✚✱
✤✥✌ ✱✘✳ ✕✌✓✗✺✏✌✓✳
ydx (
0
a
=4
❉✏✌✓✳ ✗✕
❈
✗✻✌★✓
✕✪✗✭✱✷
]
a
= 4
❧✓
x-
❉✏✌✓✳✗✕
❞✌✑
✥✌✓✼✌✧✛✹
x-
✗✴✱ ✔✚✱ ✘✙✬
[
✘✮ ♦✌
✔✚✱
=4(
✕✌
❖
❙
❯
= 4
x 2
a
2
❲
❬
❪
a2
2
❳
2
◗
x2
❘
a
x
sin –1
2
a
❲ ❨ ❳
❭ ❬
❫ ❪
2
❭❩❨
❫
a2
a
P
❚
❱0
2
= 4
a
a
0
sin 1 1
2
2
0
✕✌ ✥✌✓✼✌✧✛✹
)
❧ ✁✂✄☎✁✆✝ ✞✆✟ ✠☎✡☛✌☞ ✁✆✍
379
❢✎✏✑✒✓✔ ✥✕✖✗ ✘✙ ✚✗✛✜✢✘✣ ✤✦✧ ★✩✪ ✫✬✗✗✭✮✗ ✯✮✗ ✰✕
④✗✕✘✣✥ ✱✘✲✳✲✮✗✩✪ ✙✴ ✵✵✗✭ ✙✶✣✩ ✰✷✸ ✛✜✹✗ ✺✗✶✗ ✘✻✗✶✩ ④✗✩✼✗ ✙✗
✛✷✢♦ ④✗✩✼✗✱✢♦
a
a
= 4 ✽ 0 xdy = 4 ✿ 0 a 2 ✾ y 2 dy
❁y
= 4❅
❇2
= 4
a
a2
y❂
sin ❀1 ❆
a ❃y ❄
2
a ❈0
2
2
a2
sin 1 1
2
a
0
2
= 4
(❉✮✗✩✪?)
0
a2 ❊
2
❋ ❊a
2 2
➤➥➦➨➩➫➭ ➯➲➵
2
♠●❍■❏❑❍ ▲ ✫✴✻✗✭✛✜✹ ✗
2
x
a2
y
1 ✖✩ ✘✻✗✶✩ ④✗✩✼✗ ✙✗
b2
④✗✩✼✗✱✢♦ ✙✗ ▼✗✣ ✙✴✘✥✸◆
■❣ ✚✗✛✜✢✘✣ 8.7 ★✩✪ ✫✴✻✗✭✛✹✜ ✗ ✖✩ ✘✻✗✶✩ ④✗✩✼✗ ABA❖B❖A ✙✗ ④✗✩✼✗✱✢♦
P◗❘ ❙❘
q ❚Ø❯ x
= 4
❱❲❦❯ ❞❦Ps❳❨❦❛s
x 0, x
a ⑥❦❩❦ ✐❬❭❦❪ ♣r❭q❦❦❫➧ ❦ ❪s❛
P❴❦❩s ❲❦s ❵ ❦ AOBA ❞❦ ❲❦s ❵ ❦✐❜②
(❉✮✗✩✪✘✙
✫✴✻✗✭✛✜✹✗
x-✚④✗
✸✛✪
y-✚④✗
✫✗✩✈✗✩✪ ✛✩✢ ✱✘✶✣❝ ✖★✘★✣ ✰✕)
a
✉ ✇
= 4 ydx (❡❤❥♥t
①③⑤⑦ ③⑧♥→ ⑨⑩
❶ ❶ ❷❸
❹ )
0
x2
✚❺ 2
a
➀✖✘♦✸
y2
= 1 ✖✩ y
b2
y ➁✈✗➂★✙
a
= 4➆ 0
b 2
a
a
x 2 ✱❻✗❼✣ ✰✗✩✣✗ ✰✕❽ ✱✶✪✣✷ ④✗✩✼✗ AOBA ✱❻❾✗★ ✵✣✷❾✗✗❿✬✗ ★✩✪ ✰✕
✘♦✮✗ ✥✗✣✗ ✰✕❽ ➀✖✘♦✸ ✚➃✗✴➄✲ ④✗✩✼✗✱✢♦
b 2
a ➅ x 2 dx
a
a
x➈
4b ➇ x 2 2 a 2
a ➊ x ➋ sin –1 ➍ (❉✮✗✩?✪ )
➉
➌
a ➎2
a ➏0
2
➣
a 2 ➐1 ➔ ➒
4b ➑➓ a
sin 1➞ ➙ 0➜
➛➝ ↔ 0 ↕
a ➡➛➟ 2
2
➠
➢➜
❋
4b a 2 ❊
❋ ❊ ab ✰✕◆
a 2 2
➤➥➦➨➩➫➭ ➯➲➳
380
① ✁✂✄
❢☎✆✝✞✟✠ t✡☛☞
✍☞
✌✍ ✎☞✏✑ ✒✌✓ ✔✕✔ ✖✗✘ ✙✚☞☞✛✜☞ ✢✜☞ ✣✡ ✤☞✡✌✓t
✥✌✦✧✦✜☞✗✘ ✍★ ✩✩☞✛ ✍✪✓✗ ✣✫✬ ✙★✭☞✛✏✑✮☞
✤☞✗❞ ☞✥✒✯
b
= 4
0
xdy = 4
4a y 2
b
b 2
a
b
y
b
0
y 2 dy (
❉✜☞✗✘
4a b 2
b 2 2
ab
✱✳✵
0
0
✣✡✰
8.2.1
by a curve and a line)
✳✴
?)
b
b2
y
sin –1
2
b
2
b
b2
0
sin –1 1
2
2
4a
b
✱✲
b2
✱✲
❬❭❪❫❴❵❛ ❝❤❝
✶✷✸✹✹
✺✷
✻✼✹✶✷
✽✹✷✾✹
✲✹
✽✹✷✾ ✹✿❀❁
(The area of the region bounded
❜☛ ❂✥✥✌✪❃❄✗✙ ✖✗✘❅ ✣✖ ✬✍ ✪✗❆☞☞ ✎☞✡✪ ✬✍ ✏✑✮☞❅ ✬✍ ✪✗❆☞☞ ✎☞✡✪ ✬✍ ✥✪✏✯✜❅ ✓❇☞☞ ✬✍ ✪✗❆ ☞☞ ✎☞✡✪ ✬✍
✙★✭☞✛✏✑✮☞ ☛✗ ✌✭☞✪✗ ✤☞✗❞ ☞ ✍☞ ✤☞✗❞ ☞✥✒✯ ♥☞✓
✍✪✗✘ ✢✗ ❂✥✪☞✗❉✓
✩✌✓✛✓
✏❈☞✗✘ ✏✗✒ ☛✖★✍✪❊☞ ✏✗ ✒✏✯ ✥❋ ☞✖☞✌❊☞✍ ●✥
✖✗✘ ✣★ ✎❡✜✜❍ ✌✍✬ t☞✬■ ✢✗ ❉✜☞✗✌
✘ ✍ ✎❏✜ ●✥☞✗✘ ✏☞✯✗ ☛✖★✍✪❊☞ ✍☞ ❂✥✜☞✗✢ ❜☛ ✥☞❑✧✜✥✫▲✓✍ ✏✗✒ ✎❡✜✜❍
✤☞✗❞ ☞
☛✗
④☞✣✪
♠▼◆❖P◗◆
❖❣
✣✡✰
❘
❉✜☞✗✘✌✍
❜☛✌✯✬
2
4
0
✣✡✘✰
✏❈
✌✙✬
y = x2
✣✫✬
✎☞✏✑✒✌✓
xdy = 2
= 2
✪✗❆ ☞☞
☛✖★✍✪❊☞
✔✕❙
☛✗
❳✌✙ ✬
y=4
y = x2
AOBA
✤☞✗❞ ☞
✣✫✬
✏❈ ❅
y=0
BOND
✓❇☞☞
✤☞✗ ❞ ☞
✍☞
0
ydy (
❉✜☞✗✘
3 4
2 2
y
= 2
3
0
☛✗
⑥☞✪☞
✪✗ ❆ ☞☞✎ ☞✗✘
4
❩
✬✏✘
✌✭☞✪✗
✌❍●✌✥✓
✍☞
y–
✤☞✗❞ ☞
✎❚☞★❯✦
✎✤☞
y=4
✍☞
✏❈
✤☞✗❞ ☞✥✒✯
y-
✎✤☞
✤☞✗❞ ☞✥✒✯
♥☞✓
✏✗✒
✍★✌t✬✰
✥✌✪✓✈
✌❍❱❍✌✯✌❆☞✓
☛✖✌✖✓
●✥
✖✗✘
✬✏✘
☛✗ ✌ ✭☞✪✗
✤☞✗❞ ☞✥✒✯❨
?)
4
32
8
3
3
✜✣☞■ ✣✖❍✗ ✤☞✡✌✓t ✥✌✦✧✦✜☞■ ✯★ ✣✡✘ t✡☛☞ ✌✍ ✎☞✏✑ ✒✌✓ ✔✕❙ ✖✗✘
✙✚☞☞✛✜☞ ✢✜☞ ✣✡✰
❬❭❪❫❴ ❵❛ ❝❤✐
✬✍
✥❋ ☞❲✓
✥✪✏✯✜
✣☞✗✓ ☞
✣✡✈
❧ ✁✂✄☎✁✆✝ ✞✆✟ ✠☎✡☛✌☞ ✁✆✍
381
❢✎✏✑✒✓✔ ④✕✖✗✕ AOBA ❞✕ ④✕✖✗✕✘✙✚ ✘✛✕✜✢ ❞✣✤✖
♦✖✙ ✥✚✦ ✧★ PQ t✩✪✫ ✬✙✭♦✕✮✯✣ ✘✥✰✱✰✲✕✳ ✚✖ ✪❞✢✖
✧✩❣ t✩✪✕ ✥❞ ✴✕♦✵✙✥✢ ✶✷✸✹ ★✖❣ ✺✻✕✕✮✲ ✕ ✼✲✕ ✧✩✽ ✾✪♦✖✙
✥✚✦ ✧★ ✪★✫❞✣✿✕✕✖❣ x2 = y ✦♦❣ y = 4 ❞✕✖ ✧✚ ❞✣✢✖
✧✩❣ ✥t✪✪✖ x = –2 ✦♦❣ x = 2 ✘✛ ✕✜✢ ✧✕✖✢ ✕ ✧✩✽
✾✪ ✘✛❞✕✣ ④✕✖✗✕ AOBA ❞✕✖ ♦❀✕✖❣ y = x2, y = 4 ✦♦❣
❞✕✖✥✰✲✕✖❣ x = –2 ✢r✕✕ x = 2 ✪✖ ✥❁✕✣✕ ④✕✖✗✕ ✘✥✣❂✕✕✥❃✕✢
✥❞✲✕ t✕ ✪❞✢✕ ✧✩✽
✈❵❛❜❝❤❦ ♥♣qs
✾✪✥✚✦ ④✕✖✗✕ AOBA ❞✕ ④✕✖✗✕✘✙✚
2
= ❅ ❄ 2 ydx [ y = ❆❇❈✺❉ Q ❞✕ y ✥✤✺✖✻❊ ✕✕❣❞ – ❇❈✺❉ P ❞✕ y ✥✤✺✖✻✮ ✕✕❣❞❋ = 4 – x2]
2
2
= 2 ❅ 0 ● 4 ■ x ❍ dx
x3
2 4x
3
(❏✲✕✖❣?)
2
2 4 2
0
8
3
32
3
❑▲▼◆❖P◗ ✬✘✣✕✖❏✢ ✬✺✕✧✣✿✕✕✖❣ ✪✖ ✲✧ ✥✤❃❞❃✕✮ ✥✤❞✚✢✕ ✧✩ ✥❞ ✥❞✪✫ ④✕✖✗✕ ❞✕ ④✕✖✗✕✘✙✚ ♠✕✢ ❞✣✤✖ ♦✖✙ ✥✚✦
✧★ ✬✙✭♦✕✮✯✣ ✴r✕♦✕ ④✕✩✥✢t ✘✥✰✱✰✲✕✖❣ ★✖❣ ✪✖ ✥❞✪✫ ❞✕✖ ❂✕✫ ✚✖ ✪❞✢✖ ✧✩❣✽ ✾✪✪✖ ✴✕✼✖ ✧★ ✾✤ ✺✕✖✤✕✖❣ ✘✥✰✱✰✲✕✖❣
★✖❣ ✪✖ ✥❞✪✫ ✦❞ ❞✫ ❡❡✕✮ ❞✣✖✼
❣ ❘✖ ✬✙✭♦✕✮✯✣ ✘✥✰✱✰✲✕✖❣ ❞✕✖ ✪✕★✕❙✲✢❚ ✴✥✯❞ ✘✛✕r✕✥★❞✢✕ ✺✫ t✕✦✼✫✽
❯❱❲❳❨❩❲ ❬ ✘✛r✕★ ❡✢❉r✕✕❊✻✕ ★✖❣ ♦✵✐✕ x2 + y2 = 32, ✣✖❥✕✕
y = x, ✦♦❣ x-✴④✕ ✪✖ ✥❁✕✣✖ ④✕✖✗✕ ❞✕ ④✕✖✗✕✘✙✚ ♠✕✢ ❞✫✥t✦✽
❳❭ ✥✺✦ ✧❉✦ ✪★✫❞✣✿✕ ✧✩❚❣
y=x
✴✕✩✣
... (1)
x2 + y2 = 32
... (2)
✪★✫❞✣✿✕ ❆✸❋ ✢r✕✕ ❆❪❋ ❞✕✖ ✧✚ ❞✣✤✖ ✘✣ ✧★ ✘✕✢✖
✧✩❣ ✥❞ ✥✺✲✕ ✧❉✴✕ ♦✵✐✕ ✦♦❣ ✺✫ ✧❉✾✮ ✣✖❥✕✕ ✦❞ ✺❫✪✣✖ ❞✕✖ ✘✛r✕★
❡✢❉r✕✕❊✻✕ ★✖❣ B(4, 4) ✘✣ ✥★✚✢✖ ✧✩❣ ❆✴✕♦✵✙✥✢ ✶✷✸✸❋✽ x-✴④✕
♦✖✙ ❴✘✣ BM ✚②❈ ❥✕✫❣✥❡✦✽
✾✪✥✚✦❘ ✴❂✕✫❃✰ ④✕✖✗✕✘✙✚ = ④✕✖✗✕ OBMO ❞✕ ④✕✖✗✕✘✙✚
+ ④✕✖✗✕ BMAB ❞✕ ④✕✖✗✕✘✙✚
✈❵❛❜❝❤❦ ♥♣qq
382
✍✎✏
① ✁✂✄
✑✒✓✔✒
OBMO
❞✒
✑✒✓✔✒✕✖✗
4
=
✕✐✜✢
BMAB
✑✒✓ ✔✒
=
4 2
✣
4
❞✒
4 2
ydx =
✣
1
4 2
2
1
2
=8
✮
❧✯✰❞✱✲✒
♠❃❄❅❇❈❄
0
✶✷✸
✳✹✵
♥✰❊✒❋ ✷●❍ ✒
BOB RFSB
▲
✾❂ ✳✍✒✷● ✖■✽
♥✰■❏✶
= 2
ae
❘
0
■❞
✑✒✓ ✔✒
❨
... (3)
4 2
x
✦
4 2
✫
✭
4
4
32 16
2
32 sin –1 1
❞✒
1
2
1
32 sin –1
2
... (4)
✺✒✓✻✕✖✗
y2
1
b2
✼✒✽
✶✷✸
❞✱✜✓
❞✒✓■❁✺✒✓ ✸
e<1
✶✷✸
BOB RFSB
▲
❲
x
2
ae
❚
❳
❬
2b
ae a 2 a 2 e2
2a
✘
❪
❴
✚
2
= ab e 1 e
❯
sin –1
✩
✕✱
✾✯
x=0
✍✿✒✰❀❁
✍✒❂ ✱
✑✒✓ ✔✒✕✖✗
x = ae,
❧✓
■❊✒✱✓
A=4
✑✒✓✔✒
❲
❳
a2
0
❞✒
❙
■♥✶ ✾✐✶
♥✰❊✒❋ ✷❍
● ✒ ✶✷✸ ✱✓▼ ✒✒✍✒✓✸
x=0
❫
✑✒✓✔✒✕✖✗
ae
❱
❩
❭
0
a 2 sin –1 e
sin –1 e
✙
❵
✛
❱
❩
❭
❞✒
✽r✒✒
x 2 dx
a2
x
sin –1
a
2
✮ ✕✒✽✓
✾❂ ✸ ❆
✑✒✓✔✒✕✖✗
✼✒✽
✾❂ ❆
❞✒ ✍✿✒✰❀❁ ✑✒✓✔✒✕✖✗
ydx = 2 b
a
❯
❨
❬
=8
◆❖P◗✵❆
2b x 2
a
=
a 2
=
32
✩
b2 = a2 (1 – e2)
❅❣
è✺✒✜
0
✮
x2
a2
❏✾✒❑
✾✐✍ ✒
1
2
✮
❞✰■❏✶✏
✑✒✓✔✒
4
✙
✛
– (8 + 4 ) = 4 – 8
✳✴✵
❉
✘
✚
✤
=
✧
✪
✬
1 2
x
2
32 x 2 dx
4
★
✥
0
x dx =
✑✒✓✔✒✕✖✗
1
x 32 x 2
2
=
0
4
y dx
✈☎✆✝ ✞✟✠ ✡☛☞✌
x = ae
❧✓ ■❊✒✱✒
❧ ✁✂✄☎✁✆✝ ✞✆✟ ✠☎✡☛✌☞ ✁✆✍
383
✐✎✏✑✒✓✔✕ ✖✗✘
1. ♦✙ y2 = x, ❥✚✛✜✜✢✜✚✣ x = 1, x = 4 ✱♦✣ x-✢✈✜ ✤✚ ✥✦✜❥✚ ✈✜✚✧✜ ★✜ ✈✜✚✧✜✩✪✫ ✬✜✭ ★✮✥✯✱✰
2. ✩✲✳✜✴ ✵✭✶✳✜✜✷✸✜ ✴✚✣ ♦✙ y2 = 9x, x = 2, x = 4 ✱♦✣ x-✢✈✜ ✤✚ ✥✦✜❥✚ ✈✜✚✧✜ ★✜ ✈✜✚✧✜✩✪✫ ✬✜✭ ★✮✥✯✱✰
3. ✩✲✳✜✴ ✵✭✶✳✜✜✷✸ ✜ ✴✚✣ x2 = 4y, y = 2, y = 4 ✱♦✣ y-✢✈✜ ✤✚ ✥✦✜❥✚ ✈✜✚✧✜ ★✜ ✈✜✚✧✜✩✪✫ ✬✜✭ ★✮✥✯✱✰
x2
4. ♥✮✦✜✹♦✺✻✜
16
y2
1 ✤✚ ✥✦✜❥✚ ✈✜✚✧✜ ★✜ ✈✜✚✧✜✩✪✫ ✬✜✭ ★✮✥✯✱✰
9
x2
5. ♥✮✦✜✹♦✺✻✜
4
y2
1 ✤✚ ✥✦✜❥✚ ✈✜✚✧✜ ★✜ ✈✜✚✧✜✩✪✫ ✬✜✭ ★✮✥✯✱✰
9
6. ✩✲✳✜✴ ✵✭✶✳✜✜✷✸✜ ✴✚✣ ♦✺✻✜ x2 + y2 = 4, ❥✚✛✜✜ x = 3 y ✱♦✣ x-✢✈✜ ✼✜❥✜ ✥✦✜❥✚ ✈✜✚✧✜ ★✜ ✈✜✚✧✜✩✪✫ ✬✜✭
★✮✥✯✱✰
7. ◆✚♥★ ❥✚✛✜✜ x ✽
a
2
✼✜❥✜ ♦✺✻✜ x2 + y2 = a2 ♦✚✪ ◆✜✚✾✚ ✿✜✜❀ ★✜ ✈✜✚✧✜✩✪✫ ✬✜✭ ★✮✥✯✱✰
8. ❁✥♥ ♦✙ x = y2 ✱♦✣ ❥✚✛✜✜ x = 4 ✤✚ ✥✦✜❥✜ ❂✶✢✜ ✈✜✚✧✜✩✪✫ ❥✚✛✜✜ x = a ✼✜❥✜ ♥✜✚ ⑥❥✜⑥❥ ✿✜✜❀✜✚✣ ✴✚✣
✥♦✿✜✜✥✯✭ ❂✜✚✭✜ ❂❢ ✭✜✚ a ★✜ ✴✜❞ ✬✜✭ ★✮✥✯✱✰
9. ✩❥♦✫❁ y = x2 ✱♦✣ y = x ✤✚ ✥✦✜❥✚ ✈✜✚✧✜ ★✜ ✈✜✚✧✜✩✪✫ ✬✜✭ ★✮✥✯✱✰
10. ♦✙ x2 = 4y ✱♦✣ ❥✚✛✜✜ x = 4y – 2 ✤✚ ✥✦✜❥✚ ✈✜✚✧✜ ★✜ ✈✜✚✧✜✩✪✫ ✬✜✭ ★✮✥✯✱✰
11. ♦✙ y2 = 4x ✱♦✣ ❥✚✛✜✜ x = 3 ✤✚ ✥✦✜❥✚ ✈✜✚✧✜ ★✜ ✈✜✚✧✜✩✪✫ ✬✜✭ ★✮✥✯✱✰
✩✲✸❞ ❃❄ ✱♦✣ ❃❅ ✴✚✣ ✤❂✮ ❆✻✜❥ ★✜ ✵❁❞ ★✮✥✯✱❇
12. ✩✲✳✜✴ ✵✭✶✳✜✜✷✸✜ ✴✚✣ ♦✺✻✜ x2 + y2 = 4 ✱♦✣ ❥✚✛✜✜✢✜✚✣ x = 0, x = 2 ✤✚ ✥✦✜❥✚ ✈✜✚✧✜ ★✜ ✈✜✚✧✜✩✪✫ ❂❢❇
(A) ❈
(B)
❉
2
(C)
❉
3
(D)
❉
4
13. ♦✙ y2 = 4x, y-✢✈✜ ✱♦✣ ❥✚✛✜✜ y = 3 ✤✚ ✥✦✜❥✚ ✈✜✚✧✜ ★✜ ✈✜✚✧✜✩✪✫ ❂❢❇
(A) 2
8.3
(B)
9
4
(C)
❊✒❋ ✓●✒❋❍ ✓❋■ ❏❑▲✓▼✕❖ P✒❋◗✒ ❘✒ P✒❋◗✒✐■✔
9
3
(D)
9
2
(Area Between Two Curves)
✫❢⑥✥❞②✜ ★✮ ✵✚✭❞✜ ✱♦✣ ✢✣✭✬✜✹❞ ★✮ ✤❙✵✜❚✹ ♦✚✪ ✩✪✫❯♦❱✩ ✥★✤✮ ✈✜✚✧✜ ★✜✚ ✩✲✜❥✣✥✿✜★ ✈✜✚✧✜✩✪✫ ★✮ ⑥✺❂✭❲
✤✣✛❁✜ ✴✚✣ ✩✥✾❲✾❁✜❳ ★✜✾★❥ ✢✜❢❥ ❚❞ ✩✲✜❥✣✥✿✜★ ✈✜✚✧✜✩✪✫✜✚✣ ★✜ ❁✜✚❀✩✪✫ ✬✜✭ ★❥❨ ✈✜✚✧✜✩✪✫ ♦✚✪ ✩✥❥★✫❞
★✮ ✥✙❁✜ ✤✴✜★✫❞ ★❂✫✜✭✮ ❂❢✰ ★❩✩❞✜ ★✮✥✯✱❨ ❂✴✚✣ ♥✜✚ ♦✙ y = f (x) ✢✜❢❥ y = g (x) ✥♥✱ ❂✶✱
❂❢✣ ✯❂✜❳ [a, b]✴✚✣ f (x) ❬ g(x) ✯❢✤✜ ✥★ ✢✜♦✺✪✥✭ ❭❪❃❅ ✴✚✣ ♥✸✜✜✹❁✜ ❀❁✜ ❂❢✰ ✥♥✱ ❂✶✱ ♦✙✜✚✣ ♦✚✪ ✤✴✮★❥❫✜
✤✚ y ★✜ ❆✿✜❁✥❞❴❵ ✴✜❞ ✫✚✭✚ ❂✶✱ ❚❞ ♥✜✚❞✜✚✣ ♦✙✜✚✣ ♦✚✪ ✩✲✥✭❙◆✚♥★ ❛⑥♥✶ x = a ✭✳✜✜ x = b ✼✜❥✜ ♥✚❁ ❂❢✰✣
384
① ✁✂✄
❧☎✆✝✞✟
☎✡✕
✞✡✟✆
✠✡☛
❧☞✌✆
✦✧ ★
❧❡✒✠✛✆✥✟✝
f (x) – g (x)
✝✆
✵✆✧ ✷✸✆✶✚
✓✠✕
✍✎✆✆✏✟
✥✧❧ ✆
dx
✝✑✟✡
✒✞✓
✩✆✠✪☛✒✫
✒✝
✦✧ ❣
✠✡☛
✶❧✒✞✓
✏✔ ✆✑✕✒✖✆✝
✬✭✮✯
✏✔ ✆✑✕✒ ✖✆✝
✗✆✡✌✆✏☛✞
✰✱✆✆✚✣✆
☎✡✕
✲✣✆
✝✆✡
✦✧ ★
✘☛✙✠✆✚ ✛✑
✏✔✆✑✕✒ ✖✆✝
✏✒✜✢✜✣✆✡✕
✏✜✢✜✳
✠✡☛
✝✳
✤✏
✘✴☛✵✆✶✚
✗✆✡✌✆✏☛✞
✈✹✺✻✼✽✾ ✿❀❁❂
b
dA = [f (x) – g(x)] dx,
✫✎✆✆
✠❡ ☛✞
A=
✗✆✡✌✆✏☛✞
❄
a
[f ( x) g ( x)] dx
❃
❢❅❆❇❈❉❊
A=[
–[
✠♦
y = f (x), xy = g (x), x-
✩✗✆
✩✗✆
✠♦
b
=
[a, c]
✣✒ ✰
✩ ✆✠ ✪ ☛ ✒ ✫
❧✝✫✆
❄
✬✭✮❏
☎✡ ✕
a
f ( x) dx
f (x)
☎✡✕
✰✱✆✆✚ ✣✆
✫✎✆✆
b
❃
❍
✲✣✆
❄
a
✓✠✕
g ( x) dx =
g (x)
✦✧ ❣
✑✡❋✆✆✩✆✡✕
✫✆✡
✠♦✆✡✕
❧✡
❧✡
✑✡❋✆✆✩✆✡✕
b
f ( x) g ( x) dx
a
[c, b]
✫✎ ✆✆
x = a, x = b
x = a, x = b
✒●✆✑✡
☎✡✕
f (x)
✗✆✡✌✆✆✡✕
✝✆
■
❧✡
✥✦✆✴
=
ACBDA
✗✆✡ ✌✆
c
=
❄
a
▼
✝✆
✗✆✡✌✆✏☛✞
f ( x) g ( x) dx
❃
◆
b
❖
❄
c
▼
+
✗✆✡ ✌✆
BPRQB
g ( x) f ( x) dx
❃
✈✹✺✻✼✽✾ ✿❀❁P
◆
✝✆
✗✆✡ ✌✆✏☛✞
✝✆
✗✆✡✌✆
[a, b]
✗✆✡ ✌✆✏☛✞
✝✆
☎✡✕
f (x)
✏✔✝ ✆✑
]
✗✆✡ ✌✆✏☛✞
a<c<b
✒✟❑✟✒✞✒❋✆✫
✦✧ ▲
✗✆✡✌✆✏☛✞
✗✆✡✌✆
✒●✆✑✡
✥✦✆✴
g (x)
✗✆✡✌✆✏☛✞
✒●✆✑✡
❍
]
g (x)
✥✧❧✆
✒✞❋✆✆
✒✝
✥✆
❧ ✁✂✄☎✁✆✝ ✞✆✟ ✠☎✡☛✌☞ ✁✆✍
385
♠✎✏✑✒✓✏ ✔ ✥✕✖ ✗✘✙✚✛✕✖✜ y = x2 ✱✙✜ y2 = x ✢✖ ✣✤✕✘✖ ✦✕✖✧✕
❞✕ ✦✕✖✧✕✗★✚ ✩✕✪ ❞✫✣✬✱✭
✑❣ ✬t✢✕ ✣❞ ✮✕✙✯★✣✪ ✰✲✳✴ ✵✖✜ ✥✶✕✕✷✛✕ ✸✛✕ ✹t✺ ✻✼ ✥✕✖✼✕✖✜
✗✘✙✚✛✕✖✜ ✙✖★ ✗✐✣✪✽✾✖✥❞ ✿❀✥❁ O (0, 0) ✱✙✜ A (1, 1) ✹t✭
✛✹✕❂ y 2 = x ✮✈✕✙✕ y =
x = f (x) ✮✕t✘ y = x2 = g (x),
✬✹✕❂ [0, 1] ✵✖✜ f (x) ❃ g (x) ✹t ✭
✻✢✣✚✱ ✾✕✛✕✜ ✣✙★✪ ✦✕✖✧✕ ❞✕ ✮❜✕✫❄❅ ✦✕✖✧✕✗★✚
=
1
❛❝❡❢❤❥❦ ♥♣qr
❉ 0 ❆ f ( x) ❈ g ( x) ❇ dx
3
1
2 2
x
3
2❋
= ❉ 0 ❊● x ❈ x ❍ dx
x3
3
1
=
0
2 1
1
■
❏
3 3
3
♠✎✏✑✒✓✏ ❑ x-✮✦✕ ✙✖★ ▲✗✘ ✪✈✕✕ ✙✯▼✕ x2 + y2 = 8x ✱✙✜ ✗✘✙✚✛ y2 = 4x ✙✖★ ✵♦✛✙✪✫✷ ✦✕✖✧✕ ❞✕
✦✕✖✧✕✗★✚ ✩✕✪ ❞✫✣✬✱✭
✑❣ ✙✯▼✕ ❞✕ ✣✥✛✕ ✹❁✮✕ ✢✵✫❞✘◆✕ x2 + y2 = 8x, (x – 4)2 + y2 = 16 ✙✖★ ❖✗ ✵✖✜ ✮✣❜✕P✛◗✪ ✣❞✛✕
✬✕ ✢❞✪✕ ✹t✭ ✻✢ ✙✯▼✕ ❞✕ ✙✖★✜ ✥✐ ✿❀✥❁ (4, 0) ✹t ✪✈✕✕ ✣✧✕❘✛✕ ❙ ✻❞✕✻✷ ✹t✭ ✗✘✙✚✛ y2 = 4x ✙✖★ ✢✕✈✕
✻✢✙✖★ ✗✐✣✪✽✾✖✥ ✢✖ ✗✐✕❚✪ ✹✕✖✪✕ ✹t ❯
x2 + 4x = 8x
✮✈✕✙✕
x2 – 4x = 0
x (x – 4) = 0
x = 0, x = 4
✮✈✕✙✕
✮✈✕✙✕
✻✢ ✗✐❞✕✘ ✻✼ ✥✕✖ ✙❱✕✖✜ ✙✖★ ✗✐ ✣✪✽✾✖ ✥ ✿❀✥❁ O(0, 0)
✱✙✜ x-✮✦✕ ✢✖ ▲✗✘ P(4,4) ✹t✜✭
✮✕✙✯★✣✪ ✰✲✳❲ ✢✖ x-✮✦✕ ✢✖ ❳✗✘ ✻✼ ✥✕✖✼✕✖✜ ✙❱✕✖✜ ✙✖★ ✵♦✛
✢✣❨✵✣✚✪ ✦✕✖✧✕ OPQCO ❞✕ ✦✕✖✧✕✗★✚
= (✦✕✖✧✕ OCPO ❞✕ ✦✕✖✧✕✗★✚) + (✦✕✖✧✕ PCQP ❞✕ ✦✕✖✧✕✗★✚)
4
8
= ❉ ydx ❩ ❉ ydx
0
4
4
= 2❪
0
x dx ❬ ❪
8
4
42 ❭ ( x ❭ 4)2 dx ❫◗✛✕✖❴✜ ❵
❛❝❡❢❤ ❥❦ ♥♣qs
386
① ✁✂✄
3 4
2 2
x
3
= 2
32
=
3
32
3
=
♠✟✠✡☛☞✠
✌
✴✍✙✵ ✍✐✑✲
④✍✓
✡❣
✦✧★ ✍✩ ✎✏✪✍
x2
4
y2
1
36
42 t 2 dt ,
x 4 t
t☎❦✆
0
0
t
42 t 2
2
1
t
42 sin –1
2
4
4
2
42
1
2
0
✈✍✎✏ ✑✒✓
OA = 2
✒❢✰✘✙ ✚
4
❜❞✍❜✩
✔✕✖✗
✓✜✍✍
AOBA
OB = 6
0
32
3
sin –1 1
✘✙ ✚
4
✐✛✜ ✍✘
❜❞✍❜✩
✮✯✱
0 8
✢✓✣✜ ✍✍✤✥✍
✲★✍✣
✘✙ ✚
✢✍✐
32
3
2
✦✧★✍✩✎✏✪ ✍
AB
✫✎✚
4
=
9x2 + y2 = 36
AB
❢✧✎✍
✎✙ ✑
4
(8 3 )
3
✝
❞✍
✫❞
✘♦✳✎✓✧✩
✞
✬✍✍✭
✴✍✙ ✵✍
❞✧✒❢✫✱
❞✍
✒✦✳✍
✈✜✍✎✍
✰✘✧❞✶✷✍
✮✣ ✈ ✍
x2
22
y2
1
62
✎✙✑
9x2 + y2 = 36,
✹✐
✘✙ ✚
✈✒✬✍✺✳✻✓
✈✜✍✍✩✓ ✸
✒❞✳✍
❢✍
✰❞✓✍ ✮✯ ✈✍✯✶ ❜✰✒✲✫ ❜✰❞✍ ✈✍❞✍✶ ✈✍✎✏ ✑✒✓ ✔✕✖✗ ✘✙ ✚ ✒✦✫ ✮✣✫ ✈✍❞✍✶
❢✯✰✍ ✮✯✱
❜✰✎✙ ✑
✈✼✣✰✍✶✽
❢✧✎✍
AB
y–0=
❞✍
✰✘✧❞✶✷✍
6 0
( x 2)
0 2
✈✜✍✎✍
y = – 3 (x – 2)
✈✜✍✎✍
y = – 3x + 6
✈✍✎✏✑✒✓
✔✕✖✗
✘✙ ✚
✦✥✍✍✩✳✙
✮✯✾
✿✍✳✍✚ ✒✎✑✓
=3
2
❁
0
✴✍✙✵ ✍
❇❈❉❊❋●❍ ■❏❑▲
❞✍
4 x 2 dx
❀
x
4 x2
=3
2
=3
2
2
✴✍✙✵ ✍✐✑✲
0
2
❀
❁
0
(6 3x)dx
❀
4 –1 x
sin
2
2
2sin –1 (1)
2
6x
0
12
12
2
❂✻✳✍✙✚❃❄
3x 2
2
2
0
3 2
❅
2
6 =3 –6
❆
✮✯
❞✍
❧ ✁✂✄☎✁✆✝ ✞✆✟ ✠☎✡☛✌☞ ✁✆✍
387
♠✎✏✑✒✓✏ 9 ✔✕✖✗✘✙ ✗✖ ✚✛✜✖✢✣ ✗✤✥✢ ✦✧★ ★✗ ★✢✔✢
❢✩✖✪✖✧✫ ✗✖ ✬✖✢✩✖✛✭✘ ✮✖✥ ✗✯❢✫★ ❢✫✔✰✢✭ ✱✖✯✲✖✳ (1, 0),
(2, 2) ★✰✴ (3, 1) ✦❣✴✵
✑✶ ✕✖✙ ✘✯❢✫★ A (1, 0), B (2, 2) ★✰✴ C (3, 1) ❢✩✖✪✖✧✫
ABC ✰✢✭ ✱✖✯✲✖✳ ✦❣✴ ♦✷✖✰✸✭❢✥ ✹✺✻✹✼
✽ ABC ✗✖ ✬✖✢✩✖✛✭✘ = ✽ ABD ✗✖ ✬✖✢✩✖✛✭✘ + ✔✕✘✴✾
♣✥✧✪✖✧ ✳✫ BDEC ✗✖ ✬✖✢✩✖✛✭✘ – ✽ AEC ✗✖ ✬✖✢✩✖✛✭✘
✷✾ ✪✖✧✫✖★✈ AB, BC ★✰✴ CA ✰✢✭ ✔✕✯✗✤✿✖ ❀✕✱✖❁
y = 2 (x – 1), y = 4 – x, y =
❞❡❤❥❦qr st✉s
1
(x – 1) ✦❣✴✵
2
✷✥❁ ❂ ABC ✗✖ ✬✖✢✩✖✛✭✘
=
2
3
3
❅ 1 2 ( x ❃ 1) dx ❄ ❅ 2 (4 ❃ x) dx ❃ ❅ 1
2
3
x ❃1
dx
2
3
❆ x2
❇
❆
❇
x2 ❇
1 ❆ x2
x
x
2
4
❈
❉
❈
❈
❈ x❋
= ❊
❋
❊
❋
❊
2 ❍2 2 ● 2
● 2
❍1 ●
❍1
■❑ 2 2
▲ ❑1
32 ▲ ❑
2 2 ▲❏
▲ ❏ ■❑
▼ 2 ❙ ▼ ❘ ▼ 1❙ ◗ ◆ P❘ 4 ❖ 3 ▼
❙ ▼ ❘ 4❖ 2 ▼
❙◗
2❯ ❚
2 ❯❲
❯ ❚2
❯ ❲ ❱❚
❱❚ 2
= 2 P❘
–
♠✎✏✑✒✓✏ ❳❨ ♥✖✢ ✰✸❩✖✖✢✴
3
▲ ❑1
1 ■❑ 32
▲❏
▼ 3 ❙ ▼ ❘ ▼ 1 ❙◗ =
P❘
2
2 ❱❚ 2
❯ ❚2
❯❲
x2 + y2 = 4 ★✰✴ (x – 2)2 + y2 = 4 ✰✢✭
✕❬✜✰✥✯✳ ✬✖✢✩✖ ✗✖ ✬✖✢✩✖✛✭✘ ✮✖✥ ✗✯❢✫★✵
✑✶ ❢♥★ ✦✧★ ✰✸❩✖✖✢✴ ✰✢✭ ✔✕✯✗✤✿✖ ✦❣❁✴
✷✖❣✤
x2 + y2 = 4
... (1)
(x – 2)2 + y2 = 4
... (2)
✔✕✯✗✤✿✖ (1) ★✢✔✖ ✰✸❩✖ ✦❣ ❢✫✔✗✖ ✰✢✴✭♥❭ ✕❪✘ ❫✾♥✧ O ✛✤ ✦❣ ✷✖✢✤ ❢✫✔✗✯ ❢✩✖✐✜✖ ❴ ❵✗✖❵✳ ✦❣✵
✔✕✯✗✤✿✖ ♦❴✼ ★✗ ★✢✔✖ ✰✸❩✖ ✦❣ ❢✫✔✗✖ ✰✢✴✭♥❭ C(2, O) ✦❣ ✷✖❣✤ ❢✫✔✗✯ ❢✩✖✐✜✖ ❴ ❵✗✖❵✳ ✦❣ ✵
✔✕✯✗✤✿✖ ♦✻✼ ✷✖❣✤ ♦❴✼ ✗✖✢ ✦✘ ✗✤✙✢ ✛✤ ✦✕ ✛✖✥✢ ✦❣❁✴
(x –2)2 + y2 = x2 + y2
✷❛✖✰✖
✷❛✖✰✖
x2 – 4x + 4 + y2 = x2 + y2
x = 1 ❢✫✔✔✢ y = ❜ 3 ✛❭✖❝✥ ✦✖✢✥✖ ✦❣ ✵
388
① ✁✂✄
✈☎✆ ✝✞✟ ✠✡✟ ☛☞✌✍✍✎✏ ☛✎✑ ✒✓✝ ☎✔✕✎✞✖
A (1, – 3 )
✛
☛☞✌✍✍✎✏
☛✎✑
✠✙ ❣
=2[
ODCAO
ODAO
✮✍✎✯✍
✮✍✎✯✍✒✑④
= 2
1
✺
✶
✸
0
✶✺
✸
✛
❞✍
✮✍✎✯✍✒✑④
2
✵
✺
1
y dx
✻
+
=
✼
❅
❊
●
▲
◆
✾
✮✍✎✯✍
2
✵
✺
1
x 4 x2
2
■
✾
✾
3
❏
=
❏
4sin –1
❴❪
❛
=
3 4
✾
❅ ❃
❊ ❇
8
3
✾
❙
3
2
3
❨ ❀
❂
❩
❪
2 3
❫
6
2
1
2
◗ ■
❯
❲ ✿
❬
❁
❁
❂
❄ ❁
4
▼
❖
❈
❭
❩
❬
❵ ❫ ❴
❜
❛
2
2
✼
❩ ❪
❅
❊
–1 ✿
❃
4
3
+
❄❆
❈❋
0
1
P
❁
1
❀✽
2
x 2
2
✾
4sin 1 ( 1)
❨❳
?)
❍
■
❨
❂
2
2
3
✾
❩ ❭
❪
❵
❜
1
❀✽
▼
✼
❁
❄❆
❈❋
❍
❖
❆
❋
✾
❃
❱❱
❂
❉✪✍✎✏
❇
4sin
❘❘
❚❚ ■
(
✷
✹
✿
❑
2
✴
x 2
1
4sin –1
2
2
x
1
4sin –1
2
2
✾
◆ ❯
=
❞✍
✻
❇
■
DCAD
4 x 2 dx
1
✾
( x 2) 4 ( x 2)
▲ ❙
?)
❉✪✍✎✏
❝❡❢❤✐❥❦ ❧♠♥♣
✻
✾
● ◗
=
](
✴
1
( x 2) 4 ( x 2) 2
= 2
2
2
✠✙ ✬
✈✰✍✭✱✲
✮✍✎✯✍✒✑④
✼
❅
❊
❞✍
✫✪✍
✈✍✙✚
✷
✹
4 ( x 2) 2 dx
0
❞✍
✞★✍✍✩ ✪✍
3)
]
y dx
1
✳
✧✎✏
O ACA O
✮✍✎✯✍
=2[
✳
✣✤✥✦
✈✍☛☞✑✝☎
✧♦✪☛☎✭✩
✮✍✎✯✍
✮✍✎✯✍✒✑④
= 2
✜✙ ✢✍
A(1,
✗✘✞✡
0
●
❏
▲
◆
❅
❊
x 4 x
✾
4sin –1 1
3 4
✾
❨✽
❂
6
❆
❋
■
2
❁
4sin
3 4sin
■
–1
P
1
2
x
2
❆
❋
1
2
▼
❖
✽
❍
1
❧ ✁✂✄☎✁✆✝ ✞✆✟ ✠☎✡☛✌☞ ✁✆✍
389
✐✎✏✑✒✓✔✕ ✖✗✘
1. ✙✚✛✜✢ x2 = 4y ✈✣✤✚ ✛✥✦✣ 4x2 + 4y2 = 9 ✛♦✧ ★✩✢✛✪✫✬ ✭✣♦✮✣ ✯✣ ✭✣♦ ✮✣✙✧✜ ✰✣✪ ✯✫✱✲✳✴
2. ✛✵✣♦✶ (x – 1)2 + y2 = 1 ✳✛✶ x2 + y2 = 1 ✷♦ ✱✸✣✚♦ ✭✣♦ ✮✣ ✯✣ ✭✣♦✮✣✙✧✜ ✰✣✪ ✯✫✱✲✳✴
3. ✛✵✣♦✶ y = x2 + 2, y = x, x = 0 ✳✛✶ x = 3 ✷♦ ✱✸✣✚♦ ✭✣♦✮✣ ✯✣ ✭✣♦✮✣✙✧✜ ✰✣✪ ✯✫✱✲✳✴
4. ✷★✣✯✜✹ ✯✣ ✺✙✢✣♦✻ ✯✚✪♦ ✼✽✳ ✳✯ ✳♦✷♦ ✱✮✣✾✣✽✲ ✯✣ ✭✣♦✮✣✙✧✜ ✰✣✪ ✯✫✱✲✳ ✱✲✷✛♦✧ ✿✣✫❀✣✬
(– 1, 0), (1, 3) ✳✛✶ (3, 2) ✼✤✶✴
5. ✷★✣✯✜✹ ✯✣ ✺✙✢✣♦✻ ✯✚✪♦ ✼✽✳ ✳✯ ✳♦✷♦ ✱✮✣✯✣♦❁✣✫✢ ✭✣♦✮✣ ✯✣ ✭✣♦ ✮✣✙✧✜ ✰✣✪ ✯✫✱✲✳ ✱✲✷✯✫
✾✣✽✲✣✈✣♦ ✶ ✛♦✧ ✷★✫✯✚❁✣ y = 2x + 1, y = 3x + 1 ✳✛✶ x = 4 ✼✤✴✶
✙❂✿✹ ❃ ✳✛✶ ❄ ★♦✶ ✷✼✫ ✺✦✣✚ ✯✣ ❅✢✹ ✯✫✱✲✳❆
6. ✛✥✦✣ x2 + y2 = 4 ✳✛✶ ✚♦❇✣✣ x + y = 2 ✷♦ ✱✸✣✚♦ ❈✣♦❉♦ ✾✣✣✻ ✯✣ ✭✣♦✮✣✙✧✜ ✼✤❆
(A) 2 (❊ – 2)
(B) ❊ – 2
(C) 2❊ – 1
(D) 2 (❊ + 2)
7. ✛✵✣♦✶ y2 = 4x ✳✛✶ y = 2x ✛♦✧ ★✩✢✛✪✫✬ ✭✣♦✮✣ ✯✣ ✭✣♦ ✮✣✙✧✜ ✼✤❆
(A)
2
3
(B)
1
3
(C)
1
4
(D)
3
4
❢❋❢❋● ❍■❏❑▲▼❏
♠◆✒❖P◗✒ ❘❘ ✙✚✛✜✢ y2 = 4ax ✈✣✤✚ ✺✷✛♦✧ ✹✣✱✾✣✜✶❙ ✷♦ ✱✸✣✚♦ ✭✣♦✮✣ ✯✣ ✭✣♦✮✣✙✧✜ ✰✣✪ ✯✫✱✲✳✴
❖✔ ✈✣✛✥✧✱✪ 8.20 ✷♦❚ ✙✚✛✜✢ y2 = 4ax ✯✣ ✿✣✫❀✣✬ ★❞✜ ❯❙❱✽ ✙✚ ✼✤✴ ✹✣✱✾✣✜✶❙ ✲✫✛✣ LSL❲ ✯✣
✷★✫✯✚❁✣
x = a ✼✤✴
✱❱✢✣ ✼✽✈✣ ✙✚✛✜✢
x-✈✭✣
✛♦ ✧ ✙✱✚✪❆
✷★✱★✪ ✼✤✴
✭✣♦✮✣ OLL❳O ✯✣ ✈✾✣✫❀❉ ✭✣♦✮✣✙✧✜ = 2 (❨✭✣♦✮✣ OLSO
✯✣ ✭✣♦✮✣✙✧✜)
a
a
= 2❩ 0 ydx = 2 ❩ 0 4ax dx
a
= 2 ❬ 2 a ❩ 0 xdx
a
3
❝ ❡
8
8
2❭ 2❪
a ❣a 2 ❤ = a2
= 4 a ❫ ❴x ❵ =
3
3
3 ❛❴ ❜❵
❥❣ ❦❤
0
3
♥♣qrst✉ ✇①②③
390
① ✁✂✄
♠ ☎✆✝✞✟✆
x = –1
✝❣
✠✡
✎✏✑
✣t❧ ✌
y = 3x + 2, x-
❥ ☛☞✌✌
x=1
❧☛
✈✌✏✥✙✓✜
✓✒
y = 3x + 2, x-
✈✍✌
1,
x
✜r✌✌
2
3
✏✙
☛
✒✌
✦✧★✩
♦❧✒✌
✓ ✚✎
✎ ✏✑
✒✌ ☛ ✓ ✔✕✌ ☛ ✑
✍✌☛ ✗✌✘✙✚
✛✌✜
✒✢✓✣✎✤
✪☛✑
✫✬✌✌✭✕✌
2
3
x=
✒✌☛
✏☛✙ ✓✚✎
2
,1
3
x
✍✌☛✗✌
✓✖✌❥☛
✈✍✌
✘❥
✮✕✌
✯t
✈✌t❥
✏☛✙ ✉ ✢✱☛
✈✍✌
x-
✈✌ ✚☛☞✌
♦❧ ✒✌
❥☛ ☞✌✌
✓✪✚✜✢
x-
✈✌✚☛☞✌
✯t✰
✯t
❧☛
✈✍✌
➴✘❥ ✯t✤
✈✲✌✢✳✔
❄❅❆❇❈❉❊ ❋●❍■
=
✍✌☛✗✌✘✙✚
✍✌☛✗✌
ACBA
2
3 (3 x ✵
✶ 1
✴
✍✌☛✗✌✘✙✚
✒✌
✴
=
2)dx
1
✵
2
3
✶ ✴
2
3
+
✸
=
✻
✽
♠☎✆✝✞✟✆
✠✿
y = cos x
❧☛
✝❣
=
✈✌✏✥✙✓✜
✍✌☛✗✌
x=0
✍✌☛✗✌
✓✖✌❥☛
✦✧★★
❧☛✰
✒✌
2x
✸
✺
✼
✾✷
✒✌
✍✌☛✗✌✘✙✚
✻
✽
1
x=2
3x 2
2
❀ ✏☛✙
✈✲✌✢✳✔
✍✌☛✗✌✘✙✚
✍✌☛✗✌
✈✲✌✢✳✔
✺
✹
✎✏✑
OABO
+
DEFD
✍✌☛ ✗✌✘✙✚
♦❧✓✚✎
3x
2
✒✌
✍✌☛✗✌✘✙✚
✵
1
✺
2x
✹
✼
✾
✪❁✕
✛✌✜
✒✌
(3x 2) dx
✷
2
ADEA
✍✌☛ ✗✌
=
2
3
✷
1
6
25
6
13
3
✏❂
✒✢✓✣✎✤
✍✌☛✗✌✘✙✚
+
BCDB
✍✌☛✗✌
✒✌
✍✌☛✗✌✘✙✚
✍✌☛ ✗✌✘✙✚
❄❅❆❇❈❉❊ ❋●❍❍
3❃
2
❃
=
2
✶
0
cos x dx
= sin x
2
0
✵
✶ ❃
sin x
2
3
2
2
cos x dx
sin x
2❃
✵
✶
3❃
2
cos x dx
2
3
2
=1+2+1=4
❧ ✁✂✄☎✁✆✝ ✞✆✟ ✠☎✡☛✌☞ ✁✆✍
391
♠✎✏✑✒✓✏ ✔✕ ❢✖✗ ✘✙❢✚✛ ❢✘ ✜✢ y2 = 4x ✛✜✱ x2 = 4y, ❥✣✤✥✥✦✥✣✱
x = 0, x = 4, y = 4 ✛✜✱ y = 0 ✖✣ ❢✧✥❥✣ ✜★✩ ✜✣✪ ✫✥✣✬✥✭✪✮ ✘✥✣ ✯✙✰
❝❥✥❝❥ ✲✥✥★✥✣✱ ✳✣✱ ❢✜✲✥✥❢✚✯ ✘❥✯✣ ✴✵✱✶
✑❣ è✷✥✰ ✸✙❢✚✛ ❢✘ ✭❥✜✮✷✥✣✱ y2 = 4x ✛✜✱ x2 = 4y ✜✣✪ ✭♦❢✯✹✺✣✸
➥❝✸✻ (0,0) ✛✜✱ (4,4) ✴✵✱ ✚✵✖✥ ❢✘ ✦✥✜✼✪❢✯ ✽✾✿❀ ✳✣✱ ✸❁✥✥✩✷ ✥ ★✷✥ ✴✵✶
✦❝ ✜✢✥✣✱ y2 = 4x ✛✜✱
x2 = 4y ✖✣ ❢✧✥❥✣ ✫✥✣✬✥ OAQBO ✘✥
❍■❏❑▲▼◆ ❖P◗❘
✫✥✣✬✥✭✪✮
=
=
4
0
32
3
x2
dx = 2
4
2 x
16
3
3
2 2
x
3
x3
12
4
0
16
3
... (1)
✭✻✰✐ ✜✢✥✣✱ x2 = 4y, x = 0, x = 4 ✛✜✱ x-✦✫✥ ✖✣ ❢✧✥❥✣ ✫✥✣✬✥ OPQAO ✘✥ ✫✥✣✬✥✭✪✮
=
4
0
x2
dx
4
1 3
x
12
16
3
4
0
... (2)
❜✖✙ ✭♦✘✥❥ ✜✢ y2 = 4x, y-✦✫✥, y = 0 ✛✜✱ y = 4 ✖✣ ❢✧✥❥✣ ✫✥✣✬✥ OBQRO ✘✥ ✫✥✣✬✥✭✪✮
=
4
0
xdy
4
0
y2
dy
4
1 3
y
12
4
0
16
3
... (3)
✖✳✙✘❥❂✥✥✣✱ ❃❄❅❆ ❃✿❅ ✯❇✥✥ ❃❀❅ ✖✣ ✷✴ ❢✰❈✘❈✥✩ ❢✰✘✮✯✥ ✴✵ ❢✘
✫✥✣✬✥ OAQBO ✘✥ ✫✥✣ ✬✥✭✪✮ = ✫✥✣✬✥ OPQAO ✘✥ ✫✥✣ ✬✥✭✪✮ = ✫✥✣✬ ✥ OBQRO ✘✥ ✫✥✣✬ ✥✭✪✮
✦❇✥✥✩✯ ✈❆ ✭❥✜✮✷✥✣✱ y2 = 4x ✛✜✱ x2 = 4y ✖✣ ❢✧✥❥✥ ✫✥✣✬✥✭✪✮ ❢✸✛ ✴✻✛ ✜★✩ ✜✣✪ ✫✥✣✬✥✭✪✮ ✘✥✣ ✯✙✰ ❝❥✥❝❥ ✲✥✥★✥✣✱
✳✣✱ ❢✜✲✥✥❢✚✯ ✘❥✯✥ ✴✵✶
♠✎✏✑✒✓✏ ✔❉ ✫✥✣✬✥ {(x, y) : 0 ❊ y ❊ x2 + 1, 0 ❊ y ❊ x + 1, 0 ❊ x ❊ 2}✘✥ ✫✥✣✬✥✭✪✮ ❞✥✯ ✘✙❢✚✛✶
✑❣ ✦✥❜✛ ✖✜✩✭❇♦ ✥✳ ✴✳ ❋✖ ✫✥✣✬✥ ✘✥ ❥✣✤✥✥❢●✬✥ ✯✵✷ ✥❥ ✘❥✣✱ ❢✚✖✘✥ ✴✳✣✱ ✫✥✣✬✥✭✪✮ ❞✥✯ ✘❥✰✥ ✴✵✶ ✷✴
✫✥✣✬✥ ❢✰④✰❢✮❢✤✥✯ ✫✥✣✬✥✥✣✱ ✘✥ ✳è✷✜✯✙✩ ✫✥✣✬✥ ✴✵ ✐
A1 = {(x, y) : 0 ❊ y ❊ x2 + 1}
A2 = {(x, y) : 0 ❊ y ❊ x + 1}
✦✥✵❥
A3 = {(x, y) : 0 ❊ x ❊ 2}
392
① ✁✂✄
y = x2 + 1
♦✍✎✏✑
✤✬✎✭✮✯
✰✎✏ ✲✎✫
=
✘✎✳✎✑ ✕✴✖
1
=
✸
0
✹✻
=
✾❀
❄❂
❈❋
=
▲❏
P◆
♦✏ ✒
✴✎
( x 2 1) dx
✷
2
✷
x
✽
✼✺
✸
1
❃❅
1
1
3
❍
✹✻
✽ ✾❀
❁✿
●
❑ ■
❄❂
0
0
❖
❊
▼
◗
+
✰✎✏ ✲✎✓✒✶
1
x3
3
❍
❈
❆
▲
P
✕✙✱
❣✜✱
♦✍✎✏✑
✱♦✑
x
✽
✱♦✑
✓✔ ❥✎❡
❦✖✜ ❥✎✎❧♠✎
✦✧★✩
✪✏ ✫
✴✎
✰✎✏ ✲✎✓✒✶
?)
✼✺
❁✿
1
2 2
❍
❋
❇ ■
❏
◆
✪✏
♦✏ ✒
❡✏✑
❉✳✎✏ ✑
✤✎♦✥✒✕✖
❃❅
(i) y = x ; x = 1, x = 2
(ii) y = x4; x = 1, x = 5
y=x
y = x2
♦✍✎✏✑
(
❣✢✑✣
✰✎✏ ✲✎✓✒✶
TSRQT
✰✎✏ ✲✎
Q(1, 2)
✱♦✑
2
x2
2
❢✏❛✎✎✤✎✏✑
P(0, 1)
✚✛✙✜
✕✵✪✴✎
✷
2
2.
3.
❣✢
( x 1) dx
❘❙❚❯❚
1.
✓✔✕✖✗✘✏✙
✰✎✏ ✲✎
OTQPO
✰✎✏✲✎
y=x+1
OPQRSTO
✱♦✑
1
1
2
●❊
❍
❱
✕❜✎❢✏
✱♦✑
✱♦
❲❳
✱♦✑
✴✎
✈☎✆✝ ✞✟✠ ✡☛☞✌
❲❭❪❫❯❩❴❵
✰✎✏✲✎✓✒✶
❝✎✖
✴✭✕✵✱❞
✤✰✎
✤✰✎
✰✎✏ ✲✎
❡❤✳♦✖✭✐
✪✕♥❡✕✶✖
❖◗
❨❩❨❩❬
✰✎✏ ✲✎
xx-
23
6
=
❑▼
✴✎
✰✎✏ ✲✎✓✒✶
2
y = 4x , x = 0, y = 1
❝✎✖
✴✭✕✵✱✣
✖❥✎✎
y=4
✪✏
✕❜✎❢✏
✰✎✏ ✲✎
✴✎
✰✎✏ ✲✎✓✒✶
❝✎✖ ✴✭✕✵✱✣
0
4.
y= x 3
5. x = 0
6.
7.
✴✎
x=2
y = 4ax
4y = 3x2
✱♦✑
✓❢♦✶✳
✙✭❜✎✐ ♦✥✉✎
9.
✙✭❜✎✐ ♦✥✉✎
t ✖❥✎✎
♦✍
✱♦✑
❢✏ ❛✎✎
2
✓❢♦✶✳
8.
q
♣✔✎✓✒ ❛✎✭✑✕ ❦✱
✱♦✑
✱♦✑
✸
r
6
x 3 dx
✷
y = sin x
y = mx
2y = 3x + 12
✪✏
✪✏
✕❜✎❢✏
✕❜✎❢✏
x y
x2 y2
1
1
3 2
9
4
x2 y 2
x y
1
1
2
2
a b
a b
x2 = y,
y=x+2
x✇
②
✕ ❜✎❢✎
❣✢
✰✎✏ ✲✎✓✒✶
❝✎✖
✴✭✕✵✱✣
✴✎
✰✎✏ ✲✎✓✒✶
❝✎✖
✴✭✕✵✱✣
✱♦✑
❢✏ ❛✎✎
✪✏
✕❜✎❢✏
✶❜✎✜
✰✎✏ ✲✎
✴✎
✰✎✏ ✲✎✓✒✶
❝✎✖
✴✭✕✵✱✣
✱♦✑
✰✎✏✲✎ ✫
❢✏ ❛ ✎✎ ✤✎✏✑
✤✰✎
✪✏
✕❜✎❢✏
✰✎✏✲✎
✴✎
✰✎✏ ✲✎✓✒✶
❝✎✖
x y 1
x + y = 1, x– y = 1, – x + y = 1
✴✭✕✵✱✣
✪✏ ✕❜✎❢✏ ✰✎✏ ✲✎ ✴✎ ✰✎✏ ✲✎✓✒✶ ❝✎✖ ✴✭✕✵✱✣
✱♦✑
–x–y=1
]
{(x, y) : y
x2
y= x }
♦✍✎✏✑
13.
✪❡✎✴✶s ✕♦✕③ ✴✎ ④✓✳✎✏ ♣ ✴❢✖✏ ❣✜✱ ✱✴ ✱✏ ✪✏ ✕✲✎✬✎✜✵
♦✏✒
✴✎
✴✭✕✵✱✣
✰✎✏ ✲✎
12.
♠✎✭✮✎✎✏❧
✰✎✏✲✎
❝✎✖
✶❜✎✜
✪❡✎✴✶s ✕♦✕③ ✴✎ ④✓✳✎✏ ♣ ✴❢✖✏ ❣✜ ✱ ♦✍
✪✏
✕❜✎❢✏
✰✎✏✲✎✓✒✶
✴✭✕✵✱✣
❝✎✖
✕❜✎❢✏
11.
✤✎♦♠✳✴
✴✎
✰✎✏ ✲✎✓✒✶
✴✭✕✵✱✣
✪✏
❢✏❛✎✎
:
✴✎
❝✎✖
❢✏❛✎✎
✓❢♦✶✳
⑤⑥ ⑦⑨
⑧ ⑩
✪✏
❡✎s
✱♦✑
10.
[
✰✎✏ ✲✎
✰✎✏✲✎
❢✏❛✎✎
✴✎
❶
✕s✙✏ ✐♠✎✎✑✴
✖❥✎✎
✪✏
A(2, 0), B (4, 5)
✕❜✎❢✏
✱♦✑
✰✎✏ ✲✎
✴✎
C (6, 3)
✰✎✏ ✲✎✓✒✶
ABC,
❣✢✑✣
❝✎✖
✴✭✕✵✱✣
✴✎ ✰✎✏ ✲✎✓✒✶ ❝✎✖ ✴✭✕✵✱ ✕✵✪♦✏✒
❧ ✁✂✄☎✁✆✝ ✞✆✟ ✠☎✡☛✌☞ ✁✆✍
393
14. ✎✏✑✒✓✔ ✕✖✕✗ ✒✑ ✘✙✚✑✛✜ ✒❞✢✛ ✣✤✥✦ ❞✛❥✑✑✈✑✛✧ 2x + y = 4, 3x – 2y = 6 ✥✖✧ x – 3y + 5 = 0
✎✛ ✕★✑❞✛ ✩✑✛✪✑ ✒✑ ✩✑✛✪✑✙✫✓ ✬✑✢ ✒✭✕✮✥✯
15. ✩✑✛✪✑ {(x, y) : y2 ✰ 4x, 4x2 + 4y2 ✰ 9}✒✑ ✩✑✛✪✑✙✫✓ ✬✑✢ ✒✭✕✮✥✯
✶✱ ✎✛ ✲✳ ✢✒ ✙✴✵✔✑✛✧ ✏✛✧ ✎✣✭ ✘✷✑❞ ✒✑ ✸✚✔ ✒✭✕✮✥✹
16. ✖♦ y = x3, x-✈✩✑ ✥✖✧ ✒✑✛✕✺✚✑✛✧ x = – 2, x = 1 ✎✛ ✕★✑❞✛ ✩✑✛✪✑ ✒✑ ✩✑✛✪✑✙✫✓ ✣✻✹
✼15
15
17
(D)
4
4
4
17. ✖♦ y = x | x | , x-✈✩✑ ✥✖✧ ✒✑✛✕✺✚✑✛✧ x = – 1 ✢r✑✑ x = 1 ✎✛ ✕★✑❞✛ ✩✑✛✪✑ ✒✑ ✩✑✛✪✑✙✫✓ ✣✻✹
(A) – 9
(B)
(C)
2
4
(D)
3
3
[✽✾✿❀❁❂ : y = x2 ✚✕❃ x > 0 ✥✖✧ y = – x2 ✚✕❃ x < 0]
18. ✩✑✛✪✑ y2 ❄ 6x ✈✑✻❞ ✖❅✷✑ x2 + y2 = 16 ✏✛✧ ✎✕❡✏✕✓✢ ✩✑✛✪✑ ✒✑ ✩✑✛✪✑✙✫✓ ✣✻✹
(A) 0
(A)
19.
(B)
4
(4❆ ❇ 3) (B)
3
1
3
(C)
4
4
4
(4❆ ❈ 3) (C)
(8❆ ❇ 3) (D)
(8❆ ❈ 3)
3
3
3
y-✈✩✑, y = cos x ✥✖✧ y = sin x, 0 ❊ x ❊
(A) 2 ( 2 ❋ 1)
(B)
❉
✎✛ ✕★✑❞✛ ✩✑✛✪✑ ✒✑ ✩✑✛✪✑✙✫✓ ✣✻✹
2
(C)
2 ●1
2 ❍1
(D)
2
■❏❑❏▲▼❏
◆ ✖♦ y = f (x), x-✈✩✑ ✥✖✧ ❞✛❥✑✑✈✑✛✧ x = a ✢r✑✑ x = b (b > a) ✎✛ ✕★✑❞✛ ✩✑✛✪✑ ✖✛✫ ✩✑✛✪✑✙✫✓ ✒✑
b
✎❖✪✑ : ④P◗sP✐❘②
a
ydx
b
a
f ( x)dx ✣✻ ✯
◆ ✖♦ x = ❙ (y), y-✈✩✑ ✥✖✧ ❞✛❥✑✑✈✑✛✧ y = c, y = d ✎✛ ✕★✑❞✛ ✩✑✛✪✑ ✖✛✫ ✩✑✛✪✑✙✫✓ ✒✑ ✎❖✪✑ :
④P◗sP✐❘② =
d
xdy
c
d
c
( y )dy ✣✻ ✯
◆ ❃✑✛ ✖♦✑✛✧ y = f (x), y = g (x) ✥✖✧ ❞✛❥✑✑✥❚ x = a, x = b ✖✛✫ ✏❯✚ ✕★✑❞✛ ✩✑✛✪✑ ✒✑ ✩✑✛✪✑✙✫✓
✕✔❡✔✕✓✕❥✑✢ ✎❖ ✪✑ ❢✑❞✑ ❃✛✚ ✣✻ ?
④P◗sP✐❘② =
◆
b
f ( x) g ( x) dx , ✮✣✑❚ [a, b] ✏✛✧ f (x) ❄ g (x)
a
✚✕❃ [a, c] ✏✛✧ f (x) ❄ g (x) ✥✖✧ [c, b] ✏✛✧
f (x) ✰ g (x) , a < c < b, ✢✑✛ ✣✏ ✩✑✛✪✑✙✫✓
✒✑✛ ✕✔❡✔✕✓✕❥✑✢ ✙✴✒✑❞ ✎✛ ✕✓❥✑✢✛ ✣✻ ✧ ✹
✩✑✛✪✑✙✫✓ =
c
a
f ( x) g ( x) dx
b
c
g ( x) f ( x) dx
394
① ✁✂✄
✱☎✆✝✞✟✆✠✡
❧✒✓✔✕✖ ✗✘✙✓✚ ✔✓ ✛✜✓✢✣✤✓
✗✘✙✓✚✲✓✦✣
✳✓✢✓
✘✥✔✘❧✚
☛☞✌✍✎✟✏✆✑
✗✘✙✓✚ ✥✦✧ ✛✜✓ ✢✣✘✤✓✔ ✘✥✔✓❧
✘✖✴✵✓✦✶✓✚✓
✘✥✘✷
✛✢
✔✓✕ ❧✦ ★✩ ★✪✫✓ ★✬✭ ✮★ ✛✜ ✓✯✩✖ ✮✰✖✓✖✩
✫✓✷✓✘✢✚
★✬✭
✸❧
✘✥✘✷
✔✓
✛✜✓ ✢✣✤ ✓
❧✒✚✕✩✮
✫✓✥✈ ✧✘✚✮✓✦✣ ✥✦✧ ✹✓✦✺✓✛✧✕ ✫✓✬ ✢ ✻✓✦❧ ✥✼✚✪✫✓✦ ✣ ✥✦✧ ✫✓✮✚✖ ✔✩ ✗✙✓✖✓ ❧✦ ★✪✫✓✭ ✸❧ ✚✢★ ❧✦ ✘✖✴✵✓✦✶✓✚✓
✘✥✘✷❢
✔✓
❧✒✓✔✕✖
❧✥✓✦❞❁✔✈✶ ❂
✘✥✘✷
✘✥✔✓❧
(Archimedes (300
✔✕✖
✥✦✧
✔✩
✘❧❇✓✣✚
✛✜✓✢✣ ✘✤✓✔
✛✜✓ ✢✣✘✤✓✔
✸❞❜
✛✰❜
✔✓
)
✥✦✧
✘✼✽✓✘✚
✥✦✧
✾✛
✔✓✕
✒✦✣
✮✰❃✓✦❄❧
✔✓✮✓✦♦
❧✦
✛✜✓ ❆✚
❈✒❉❇
✘✥✔✓❧
✸❞❧✓
✸❧
✘✔✮✓✭
✘❧❇✓✣✚
❡✮✰❂✖
✖✦
✔✓
✛✜✮✓✦✗
❖✮✪❁❈✒
✥❈
✛✧✕✖
✥✦✧
✔✩
❧✒✿✩
❀✓
❧✔✚✩
(Eudoxus (440
★✪✫✓
✥✦✧
✸❞❜
✛✰❜
✘✔❧✩
▲❉■✪
✷✓✢✙✓✓
❧✦
P✫✘✖✘✵✯✚ ❧✒✓✔✕✖◗ ✮✓ ✼✛✘✵✓❞✮✓✦ ✣ ✔✩ ❖✮✪❁❈✒
★✬✭
)
✘✖✴✵✓✦✶✓✚✓
✫✓✬ ✢
✘✥✘✷
✫✓❅✔✘✒❃✩❀
★✬✭
✛✵✯✓✚❊
❋●✥✩✣ ✵✓✚✓❍■✩
(Theory of fluxion)
✒✦✣ ❡✮✰❂✖ ✖✦ ✔✕✖ ✛✢ ✫✛✖✓ ✔✓✮❞ ✛✜ ✥✓★✖ ✘❧❇✓✣✚
♠❡★✓✦✖
✣ ✦
✒✦✣
✛✢
✼✛✵✓✩❞
✛✘✢✯✮
✘✥✘✷
✫✓✬ ✢
✔✢✓✮✓
✒✦✣ ★✪✫✓✭
❧✖❊ ❋❏❏❑
✥✦✧ ✾✛ ✒✦✣ ✛✜✓✢✣✤✓ ✘✔✮✓✭
✥❈✚✓▼✘✺✓◆✮✓
✫✓✬ ✢
✸❧✔✓✦
✲✓✚
✔✢✖✦
✒✦ ✣
✛✜ ✘✚✫✥✔✕❀
(Inverse Method of tangents)
✔✓
✖✓✒✔✢✙✓ ✘✔✮✓✭
1684–86,
Eruditorum)
✖✓✒
✘■✮✓❢
✥✦ ✧
❉✩✯
✥✦ ✧
❄✮✓✦✣✘✔
❧✪✿✓✥
✔✓✦
✔✢
(Leibnitz)
✕✬ ✥✘✖❘✓
✖✦
✉✔
✛✜✛✺✓
✒✦✣ ✛✜✔✓✘✵✓✚ ✘✔✮✓ ✫✓✬✢ ✸❧✦ ✥✬✧✕❄✮✰✕❧ ❧❙✒✬❂✓✦✘✢✮❧
✮★
‘’
❯
✮✓✦✗✛✧✕ ✥✦✧ ✛✜✚✩✔
✛✘✢✥✘✚❞✚
✒✦ ✣
✒✓✖✔✢
✘■✮✓✭
✫✖✣✚
✳✓✢✓
❚✓✦❂✦
❖✮❄✚
✫✛✖✦
✮★
✹✓✦✺✓✛✧✕✓✦ ✣
✮✓✦✗✛✧✕
✘✔✮✓✭ ❧✖❊ ❋❏⑥❏
✛✜✛✺✓
❡✮✰❂✖
✥✦ ✧
✔✓✦
✳✓✢✓
✸❞❜
✔✬ ✕❄✮✰✕❧
✼✛✘✵✓❞✮✓✦✣
✔✩
❧✦
(Acta
(Calculous Summatorius)
✉✔❂✓
❧✣❉ ✘
✣ ✷✚
✸✢✓✦✘❃❂✓✦✘✢✮✒
✽✓✓❢
✥★✩✣
✛✢
♠❡★✓✦✖
✣ ✦
✸❧✦
(J.Bernoulli)
(Calculus Integrali)
✒✦✣ ♠❡★✓✦✖
✣ ✦ ❀✦❜ ❉✢✖✓✬ ✕✩
✸✣ ❂✗
✦ ✓
✜ ✕✩
❖✮✪ ❁❈✒
✖✓✒
✘✥✘✷
✥✦ ✧
❧✣ ✗✚
✒✦✣
✽✓✓✭
❡✮✰❂✖ ✫✓✬ ✢ ✕✬ ✥✘✖❘✓ ■✓✦✖✓✦✣ ✖✦ ✛✰✙✓❞✚✴ ✼✥✚✣✺✓ ✒✓✗❞ ✫✛✖✓✮✓ ❀✓✦ ✒✰✕✚✴ ✘✤✓❡✖ ✽✓✦✭ ✚✽✓✓✘✛ ♠✖ ■✓✦✖✓✦✣
✥✦ ✧
✘❧❇✓✣✚✓✦✣
✥✦✧
❧✣✗✚
✛✜ ✘✚✛✧✕
✚❁❧✒
✛✓✉
✗✉✭
✕✬ ✥✘✖❘✓
✖✦ ✘✖✘✵✯✚
❧✒✓✔✕✖
✔✩ ✷✓✢✙✓✓
✔✓
✛✜✮✓✦✗ ✘✔✮✓✭
✮★
✘✖✘✵✯✚
❧✣❉✷
✣
✔✓✦
★✬
✘✔
♠❡★✓✦✖
✣ ✦
★✩
✼✛✶❂✚✮✓
❧✢✓★✓✭
❧✥❞✛✜ ✽✓✒
✛✜ ✘✚✫✥✔✕❀
✫✓✬ ✢
✘✖✘✵✯✚
❧✒✓✔✕✖
✥✦✧
❉✩✯
✥✦ ✧
✘✖✶✔✶✓❞ ✮★ ★✬ ✘✔ ❧✒✓✔✕✖ ✗✘✙✓✚ ✥✦✧ ✫✓✷✓✢✤✓✰✚ ✷✓✢✙✓✓✫✓✦❢
✣
✘❧❇✓✣✚✓✦✣ ✚✽✓✓ ✫✥✔✕✖ ✗✘✙✓✚
❧✦ ✸❧✥✦✧
✵✓✚✓❍■✩
✛✜✓ ✢✣✘✤✓✔ ❧✣❉ ✷
✣
✓✦✣
✥✦✧
✫✣✚
✒✦ ✣
★✪ ✫✓✭
✔✓
✘✥✔✓❧
✚✽✓✓✘✛
✛✩❜❃✩❜
✸❧✔✓
✛✧✒✓❞❢
✫✓✬ ✘✯❁✮❢
❡✮✰❂✖❢
❧✩✒✓
✫✓✬ ✢
✔✩
✕✬ ✥✘✖❘✓
❧✣✔❱✛✖✓
✥✦✧
✥✦✧
✔✓✮✓✦♦
✫✓✷✓✢
✳✓✢✓
❋●✥✩✣
✛✢
❋⑥✥✩✣
(A.L.Cauchy)
(Lie Sophie)
"It may be said that the conceptions
of differential quotient and integral which in their origin certainly go back to
Archimedes were introduced in Science by the investigations of Kepler, Descartes,
Cavalieri, Fermat and Wallis... The discovery that differentiation and integration
are inverse operations belongs to Newton and Leibnitz".
✵✓✚✓❍■ ✩
✥✦ ✧
✛✜ ✓ ✢✣✤ ✓
✒✦ ✣
✉❜✉ ✕❜✔✓✦✵ ✓✩
✥✦ ✧
✔✓ ✘✖❙✖✘✕✘❲✓✚ ♠❇✢✙✓ ✥✙✓❞✖✩✮ ★✬✭
—
❳
—
✳✓ ✢✓
✘✔✮✓
✗✮✓✭
✫✣✚
✒✦ ✣
✕✩
❧✓ ✛
✦ ✧✩
❇❈❉❊❉
9
❋●■❏ ❑▲▼■◆❖P
Differential Equations
He who seeks for methods without having a definite problem in mind
seeks for the most part in vain – D. HILBERT
❍✁✂✄☎✆✁ ✭Introduction✮
❞✝✞✞ XI ✱✟✠ ✡☛ ☞✌✍✎❞ ✟✏✑ ✒✓✔✞✔ ✕ ✖✏✠ ✗✖✘✏ ✙✙✞✚ ❞✛ ✜✞✛✢ ✣❞
✱❞ ✍✟✎✠✤✞ ✙✥ ✟✏✑ ☛✞☞✏✝✞ ✣❞☛✛ ☞✑✦✘ f ❞✞ ✒✟❞✦✧ ✟★✑☛✏ ✩✞✎
✣❞✔✞ ✧✞✎✞ ✗★ ✒✜✞✞✚✎❢ ✣❞☛✛ ☞✑✦✘ f ❞✛ ☞✣✥✪✞✞✣✫✞✎ ☞✬✞✎✠ ✟✏✑ ☞✬✯✔✏❞
x ✟✏✑ ✣✦✱✢ f ✰(x) ✟★✑☛✏ ✩✞✎ ✣❞✔✞ ✧✞✎✞ ✗★♦ ✡☛✟✏✑ ✒✣✎✣✥✲✎
☛✖✞❞✦ ❧✣✳✞✎ ✟✏✑ ✒✓✔✞✔ ✖✏✠ ✗✖✘✏ ✙✙✞✚ ❞✛ ✜✞✛✢ ✣❞ ✔✣✴ ✣❞☛✛
☞✑✦✘ f ❞✞ ✒✟❞✦✧ ☞✑✦✘ g ✗★ ✎✞✏ ☞✑✦✘ f ✟★✑☛✏ ✩✞✎ ✣❞✔✞
✧✞✱♦ ✡☛❞✞✏ ✣✘t✘ ✵☞ ✖✏✠ ☛✶✤✞✷✸ ✣❞✔✞ ✧✞ ☛❞✎✞ ✗★✹
✣❞☛✛ ✣✴✱ ✗✌✱ ☞✑✦✘ g ✟✏✑ ✣✦✱ ☞✑✦✘ f ✩✞✎ ❞✛✣✧✱ ✎✞✣❞
dy
✺ g ( x) ✧✗✞✻ y = f (x)
... (1)
dx
☛✖✛❞✥✳✞ ✼✽✾ ✟✏✑ ✵☞ ✟✞✦✏ ☛✖✛❞✥✳✞ ❞✞✏ ✒✟❞✦ ☛✖✛❞✥✳✞
❞✗✎✏ ✗★✠♦ ✡☛❞✛ ✒✞★☞✙✞✣✥❞ ☞✣✥✪✞✞✫✞ ✷✞✴ ✖✏✠ ✴✛ ✧✞✱❧✛♦
✒✟❞✦ ☛✖✛❞✥✳✞✞✏✠ ❞✞ ✈☞✔✞✏❧ ✖✌✿✔ ✵☞ ☛✏ ✪✞✞★✣✎❞✛✢ ✥☛✞✔✘ ✣✟✩✞✘✢ ✧✛✟ ✣✟✩✞✘✢ ✖✞✘✟
✣✟✩✞✘✢ ✪✞✶✣✟✩✞✘✢ ✒✜✞✚❀✞✞✍✤✞ ✒✞✣✴ ✣✟✣✪✞❁✘ ✝✞✏✤✞✞✏✠ ✖✏✠ ✣❞✔✞ ✧✞✎✞ ✗★♦ ✒✎✹ ☛✪✞✛ ✒✯✔✞❂✌✣✘❞ ✟★✩✞✣✘❞
✒❁✟✏✫✞✳✞✞✏✠ ✟✏✑ ✣✦✱ ✒✟❞✦ ☛✖✛❞✥✳✞✞✏✠ ✟✏✑ ❧✗✘ ✒✓✔✔✘ ❞✛ ✒✯✔✠✎ ✒✞✟❀✔❞✎✞ ✗★♦ ✡☛ ✒✓✔✞✔ ✖✏✢✠
✗✖ ✒✟❞✦ ☛✖✛❞✥✳✞ ❞✛ ✟✌✑❣ ✒✞❂✞✥✪✞✶✎ ☛✠❞❃☞✘✞✒✞✏✢✠ ✒✟❞✦ ☛✖✛❞✥✳✞ ✟✏✑ ❄✔✞☞❞ ✱✟✠ ✣✟✣❀✞✫❅
✗✦✢ ✒✟❞✦ ☛✖✛❞✥✳✞ ❞✞ ✣✘✖✞✚✳✞✢ ☞✬✜✞✖ ❞✞✏✣❅ ✱✟✠ ☞✬✜✞✖ ❆✞✞✎ ✟✏✑ ✒✟❞✦ ☛✖✛❞✥✳✞ ❞✞✏ ✗✦ ❞✥✘✏
❞✛ ✟✌✑❣ ✣✟✣❂✔✞✻ ✒✞★✥ ✣✟✣✪✞❁✘ ✝✞✏✤✞ ✏✠ ✖✏✠ ✒✟❞✦ ☛✖✛❞✥✳✞✞✏✠ ✟✏✑ ✟✌✑❣ ✈☞✔✞✏❧✞✏✠ ✟✏✑ ✷✞✥✏ ✖✏✠ ✒✓✔✔✘ ❞✥✏✠❧✏♦
9.1
Henri Poincare
(1854-1912 )
396
① ✁✂✄
9.2
❣✒
✈☎✆☎✝✞☎✟✠
✓❣✔✕
✖✕
❣✗
(Basic Concepts)
✡☛ ☞✌✍✎☎✏✑
✘✙✚✙✘✔✘✛✜✢
✓✣✤✜✥
✦✕✧
✖✒✗✤✥★✜✜✕ ✩
✖✕
✓✘✥✘✪✢
❣✫✩
2
x – 3x + 3 = 0
sin x + cos x = 0
x+y=7
✬✜✭✮
✘✙✚✙✘✔✘✛✜✢
✖✒✗✤✥★✜
✓✥
✘✦✪✜✥
✤✥✕✩
dy
dx
x
... (1)
... (2)
... (3)
y= 0
... (4)
❣✒
✓✜✢✕ ❣✫✩ ✘✤ ✖✒✗✤✥★✜✜✕ ✩ ✯✰✱✲
✯✳✱ ✮✦✩ ✯✴✱ ✒✕✩ ✦✕✧✦✔ ✵✦✢✩✶✜ ✬✜✫✥✷✬✸✜✦✜
❀✜ ✬✘✺✤✱ ✻✜✜✘✒✔ ❣✫✩ ✼✽ ✘✤ ✖✒✗✤✥★✜ ✯✾✱ ✒✕ ✩ ✪✥ ✦✕✧ ✖✜✸✜✿✖✜✸✜ ✵✦✢✩✶✜ ✪✥
✪✥
✯
y)
✤✜
✬✦✤✔✼
✖✜✒✜❧❀✢❂
✖✘✚✒✘✔✢
❣✜✕✲
✩
✮✤
❞✜✗
✮✕ ✖✜
✬✦✤✔
✻✜✜✘✒✔
❣✫❁
✖✒✗✤✥★✜✲
✖✒✗✤✥★✜
✓✣✤✜✥
✭✖
✤✜
✘✼✖✒✕ ✩ ✵✦✢✩✶✜
✤❣✔✜✢✜
✪✥
✖✒✗✤✥★✜
✖✜✓✕♦✜
(x)
✬✜✘✹✢
✯✮✤
✦✕✧ ✖✜✓✕♦✜ ✬✜✘✹✢
✖✒✗✤✥★✜
✬✦✤✔
✯✪✥✜✕ ✩✱ ✦✕✧
✬✜✘✹✢ ✪✥
✪✥
✤❣✔✜✢✜
✦✕✧
❣✫❁
✬✦✤✔✼
❣✫❁
✮✤ ✮✕ ✖✜ ✬✦✤✔ ✖✒✗✤✥★✜✲ ✘✼✖✒✕ ✩ ✦✕✧✦✔ ✮✤ ✵✦✢✩✶✜ ✪✥ ✦✕✧ ✖✜✓✕♦✜✲ ✬✜✘✹✢ ✪✥ ✦✕✧ ✬✦✤✔✼
✖✘✚✒✘✔✢
❣✜✕✩✲
✖✜✒✜❧❀
✖✒✗✤✥★✜
✬✦✤✔
2
d y
dx 2
2
✮✤
✖✜✒✜❧❀
dy
dx
✖✒✗✤✥★✜
✬✦✤✔
✤❣✔✜✢✜
❣✫❁
❃❄✜❣✥★✜✢❀✜
3
=0
... (5)
❣✫❁
✘✙❂✖❧❄✕❣ ✮✕✖✕ ❞✜✗ ✬✦✤✔ ✖✒✗✤✥★✜ ❣✜✕✢✕ ❣✫✩ ✘✼✙✒✕ ✩ ✮✤ ✖✕ ✬✘✺✤ ✵✦✢✩✶✜ ✪✥✜✕✩ ✦✕✧ ✖✜✓✕♦✜ ✬✦✤✔✼
✻✜✜✘✒✔ ❣✜✕✢✕ ❣✫✲
✩
✭✖ ✓✣✤✜✥
✵✢✥
✬✜❅✕
✓✥
❣✒
❣✒
✬✓✙✕
✖✜✒✜❧❀
✬✜✓
✦✕✧
✤✜✕
✬✦✤✔
✖✒✗✤✥★✜ ✬✜✩✘✻✜✤ ✬✦✤✔ ✖✒✗✤✥★✜
✬✦✤✔
✦✕✧✦✔
✖✜✒✜❧❀ ✬✦✤✔
✖✒✗✤✥★✜
✦✕✧
✘✔✮
✖✒✗✤✥★✜✜✕✩ ✦✕✧
✬✦✤✔
✖✒✗✤✥★✜
✢✤
✬▲❀❀✙
✻✜❆❄
✤✜
✤❣✔✜✢✕ ❣✫❁
✩
❣✗
✖✗✘✒✢
❃✓❀✜✕❅
✔✕✘✤✙ ✭✖
✥✛✜✕❅
✩ ❁
✕
✭✖✖✕
✤✥✕❅✕
✩ ❁
❢❈❉❊❋●❍
❇
1.
❣✒
✬✦✤✔✼✜✕ ✩
✦✕✧
✘✔✮
✘✙✚✙✘✔✘✛✜✢
dy
dx
2.
❃♠✪
✤✜✕✘ ❑
✦✜✔✕
✬✦✤✔✼✜✕ ✩
❏
✦✕ ✧
✖✩✦✕ ✧✢✜✕ ✩
d2y
y , 2
dx
■
✘✔✮✲
✭✢✙✕
✒✕ ✩ ✓✣❀❡P✢ ✤✥✙✜ ✬✖❡✘✦✺✜✼✙✤ ❣✜✕❅✜ ✭✖✘✔✮
yn
✤✜
❃✓❀✜✕❅
✤✥✕✩❅✕❁
❏
✦✕ ✧
❃✓❀✜✕❅
d3y
y, 3
dx
■■
✬✘✺✤
n
❏
▼✫✻✜✜✕ ✩
✤✜✕
y
✦✥✗❀✢✜
✩ ✕
❄✕❅
■■■
dashes
✯
✦✕✩ ✤✜✕✘ ❑ ✦✜✔✕ ✬✦✤✔✼
✱✤✜✕
dny
dx n
❃♠✪
✓✣◆ ❀❀
✦✕✧
❖✓
✦✕✧ ✘✔✮ ❣✒ ✖✩✦✧
✕ ✢
✈ ✁✂ ✄☎✆✁✝✞✟
397
9.2.1 ✠✡☛☞ ✌✍✎☛✏✑✒ ☛✎ ☛✒✓✔✕ (Order of a differential equation)
❢✖✗✘ ✙✚✖✛ ✗✜✘✖✢✣✤ ✖✘ ✖✤✥❢✦ ✧✗ ✙✚✖✛ ✗✜✘✖✢✣✤ ✜★✥ ✗❢✩✜❢✛✪ ✫✚✪★✬✤ ✭✢ ✚✥✮ ✗✤✯✥✰✤ ✙✤❢✱✪
✭✢ ✚✥✮ ✧♣✭✪✜ ✖✤✥❢✦ ✚✥✮ ✙✚✖✛✲ ✖✘ ✖✤✥❢✦ ✳✤✢✤ ✯❢✢✴✤✤❢✵✤✪ ✶✤✥✪ ✘ ✶✷✸
❢✹✩✹❢✛❢✺✤✪ ✙✚✖✛ ✗✜✘✖✢✣✤✤✥★ ✯✢ ❢✚✭✤✢ ✖✘❢✲✻✼
dy
x
✽e
dx
... (6)
d2y
✾ y✿0
dx 2
... (7)
d3y
dx3
x
2
3
d2y
dx 2
0
...(8)
✗✜✘✖✢✣✤ ❧❀❁❂ ❧❃❁ ✻✚★ ❧❄❁ ✜✥★ ❅✜❆✤✼ ✯❇❈✤✜❂ ❢✳✪✘❉ ✻✚★ ✪❊✪✘❉ ✖✤✥❢✦ ✚✥✮ ✧♣✭✪✜ ✙✚✖✛✲
✧✯❢✫❈✤✪ ✶✷★ ♠✗❢✛✻ ♠✹ ✗✜✘✖✢✣✤✤✥★ ✖✘ ✖✤✥❢✦ ❅✜❆✤✼ ❋❂ ● ✻✚★ ❍ ✶✷✸
9.2.2 ✠✡☛☞ ✌✍✎☛✏✑✒ ☛✎ ■✒✒❏ (Degree of a differential equation)
❢✖✗✘ ✙✚✖✛ ✗✜✘✖✢✣✤ ✖✘ ❑✤✤✪ ✖✤ ✙▲❉❉✹ ✖✢✹✥ ✚✥✮ ❢✛✻ ✜▼✺❉ ◆❖P▼ ❉✶ ✶✷ ❢✖ ✚✶ ✙✚✖✛
✗✜✘✖✢✣✤❂ ✙✚✖✛✲✤✥★ y◗, y❘, y❘◗ ♠❜❉✤❢P ✜✥★ ❖✶▼✯P ✗✜✘✖✢✣✤ ✶✤✥✹✤ ✭✤❢✶✻✸ ❢✹✩✹❢✛❢✺✤✪ ✗✜✘✖✢✣✤✤✥★
✯✢ ❢✚✭✤✢ ✖✘❢✲✻✼
d3y
d2y
2
dx3
dx 2
dy
dx
2
2
dy
dx
dy
sin 2 y
dx
dy
dy
sin
dx
dx
y 0
0
0
... (9)
... (10)
... (11)
✶✜ ✯❇❢✥ ✰✤✪ ✖✢✪✥ ✶✷★ ❢✖ ✗✜✘✖✢✣✤ ❧❣❁ y❘◗❙ y❘ ✻✚★★ y◗ ✜✥★ ❖✶▼✯P ✗✜✘✖✢✣✤ ✶✷✸ ✗✜✘✖✢✣✤ ❧❋❡❁
y◗ ✜✥★ ❖✶▼✯P ✗✜✘✖✢✣✤ ✶✷ ❧❉❚❢✯ ❉✶ y ✜✥★ ❖✶▼✯P ✹✶✘★ ✶✷❁ ♠✗ ✯❇ ✖✤✢ ✚✥✮ ✙✚✖✛ ✗✜✘✖✢✣✤✤✥★ ✖✘
❑✤✤✪ ✖✤✥ ✯❢✢✴✤✤❢✵✤✪ ❢✖❉✤ ✲✤ ✗✖✪✤ ✶✷✸ ✯✢★✪▼ ✗✜✘✖✢✣✤ ❧❋❋❁ y◗ ✜✥★ ❖✶▼✯P ✗✜✘✖✢✣✤ ✹✶✘★ ✶✷ ✙✤✷✢
♠✗ ✯❇✖✤✢ ✚✥✮ ✙✚✖✛ ✗✜✘✖✢✣✤ ✖✘ ❑✤✤✪ ✖✤✥ ✯❢✢✴✤✤❢✵✤✪ ✹✶✘★ ❢✖❉✤ ✲✤ ✗✖✪✤ ✶✷✸
398
① ✁✂✄
❀☎✆ ✝✞ ✟✠✞✡ ☛☞✌✞✍✎✏ ✟✠✞✡✑✏✒✓ ✞✏ ✔✕✖✗ ✆ ☛☞✌✞✍✎✏ ✕✘ ✙✏✒ ✚☛ ✟✠✞✡ ☛☞✌✞✍✎✏ ✞✌
❄✏✏✙ ☛✒ ✕☞✏✍✏ ✙✏✛✗❀✜ ✕✘ ✚☛ ✟✠✞✡ ☛☞✌✞✍✎✏ ☞✒✓ ✚✗☎✢✣✏✙ ✚✤✥✙☞ ✞✏✒☎✦ ✠✒✧ ✟✠✞✡✑ ✞✌ ✚✤✥✙☞
❄✏✏✙
★✩✪✏✛☞✞
✚✗✍✏✒♠✙
✗✫ ✎✏✏✬✞✭
★✶✭ ☞✒✓ ☛✒ ✗✰✛❀✒✞
✞✌
❄✏✏✙
☞✒✓
✷ ✕✘✳
✕☞
✗✰ ☎
✒ ✱✏✙
✞✍
☛✞✙✒
✕✘ ✓
❄✏✏✙
☛☞✌✞✍✎✏ ★✷✸✭ ✞✌
☎✞
✹ ✕✘
☛☞✌✞✍✎✏✏✒ ✓
✑✔
☎✞
★✲✭✳
★✴✭✳
★✵✭
✝✠✓
✟✠✞✡ ☛☞✌✞✍✎✏ ★✷✷✭
✪✕✌✓ ✕✘❞
☎✞☛✌
✟✠✞✡
☛☞✌✞✍✎✏
✞✏✒☎✦
✞✌
❄✏✏✙
✝✠✓
★❀☎✆
✗☎✍✮✏✏☎✯✏✙
✕✏✒ ✭
✕☞✒❂✏✏
✕✏✒✙✒ ✕✘❞
✓
✩✪✏✛☞✞ ✗✫✎✏✏✬✞
❃❅✿❆❇✾✿ ❈
☛✓✆✮✏✜
❄✏✏✙
✞✌
✗☎✍✮✏✏☎✯✏✙
❢✻✼✽✾✿❁
✺
❑✏✙
✠✒✧
✗☎✍✮✏✏✯✏✏
☎✪❉✪☎✡☎❊✏✙ ✟✠✞✡ ☛☞✌✞✍✎✏✏✒ ✓ ☞✒✓ ☛✒ ✗✰✛❀✒✞ ✞✌ ✞✏✒☎✦ ✝✠✓ ❄✏✏✙ ★❀☎✆ ✗☎✍✮✏✏☎✯✏✙ ✕✏✒✭
✞✌☎✑✝❋
dy
cos x 0
dx
(i)
●
(iii)
y
(ii)
❍
y2
❚❚❚ ▲
▲
ey
❙
◆
d2y
xy 2
dx
▲
dy
x
dx
2
■
❏
❖
P
◗
❘
▼
dy
dx
y
0
◆
0
❆❣
(i)
❜☛
✟✠✞✡
✷ ✕✘❞ ❀✕
☛☞✌✞✍✎✏
(ii)
☛☞✌✞✍✎✏
☞✒✓
✚✗☎✢✣✏✙
y
❯ ☞✒✓ ✔✕✖✗ ✆ ☛☞✌✞✍✎✏ ✕✘ ✝✠✓
✞✌
❄✏✏✙
✷
dy
dx
✷
✕✘✳
❜☛☎✡✝
❜☛
d2y
dx 2
☛☞✌✞✍✎✏ ✞✏ ✔✏❀✏❳ ✗✱✏
✝✠✓
✟✠✞✡
❜☛ ✟✠✞✡ ☛☞✌✞✍✎✏ ☞✒✓ ✚✗☎✢✣✏✙
❜☛
dy
dx
✞✌☎✑✝❞
1.
✙✞
d4y
dx 4
✠✒✧
❤
✗✰ ❂✪✏✒ ✓ ☞✒✓ ✗✰✛❀✒✞
sin( y ) 0
❡❡❡
❥
✕✘❞
❜☛☎✡✝
d2y
dx 2
❜☛✞✌
✞✏✒ ☎✦
☛☞✌✞✍✎✏
✞✌
❄✏✏✙
✟✠✞✡✑✏✒✓ ☞✒✓ ✔✕✖✗✆ ✪✕✌✓ ✕✘
☛☞✌✞✍✎✏
2. y + 5y = 0
❯
✷
✚✤✥✙☞ ✞✏✒☎✦ ✟✠✞✡✑
✟✠✞✡
✕✘ ❞ ❜☛☎✡✝ ❜☛✞✌ ✞✏✒☎✦
d2y
dx 2
☞✒✓ ✔✕✖✗ ✆ ☛☞✌✞✍✎✏ ✕✘ ✟✏✘ ✍
✐❨❩❬❭❪❫❴
☛✒ ✷✸
dy
dx
✞✌ ✟☎✩✞✙☞ ❄✏✏✙✏✓✞ ✷ ✕✘✳ ❜☛☎✡✝ ❜☛ ✟✠✞✡
❜☛ ✟✠✞✡ ☛☞✌✞✍✎✏ ☞✒✓ ✚✗☎✢✣✏✙ ✚✤✥✙☞ ✞✏✒☎✦ ✟✠✞✡✑
❄✏✏✙✏✓✞
✷
✟✠✞✡✑
✕✘❞
✹ ✕✘❞ ❀✕ ✟✠✞✡ ☛☞✌✞✍✎✏
(iii)
✞✏✒☎✦
✚✤✥✙☞
✞✌ ✟☎✩✞✙☞
✕✘❞
y
❱❱❱ ✕✘❞ ❜☛☎✡✝ ❜☛✞✌ ✞✏✒☎✦ ❲ ✕✘❞
❜☛☎✡✝
❜☛✞✌
❄✏✏✙ ✗☎✍✮✏✏☎✯✏✙ ✪✕✌✓ ✕✘❞
❵❛❝
✞✌
✞✏✒☎✦ ✝✠✓
3.
■
❖
◗
❄✏✏✙
ds
dt
★❀☎✆ ✗☎✍✮✏✏☎✯✏✙
4
❏
P
❘
▲
3s
d 2s
dt 2
◆
0
✕✏✒✭
❑✏✙
✈ ✁✂ ✄☎✆✁✝✞✟
2
4.
✠ d2y ✡
✠ dy ✡
☛ cos ✌
✌
✍☞0
2 ✍
✎ dx ✏
✎ dx ✏
5.
6. ( y✓✓✓) 2 + (y✔)3 + (y✕)4 + y5 = 0
399
d2y
✑ cos3 x ✒ sin 3 x
dx 2
7. y ✖✖✖ + 2y✔ + y✕ = 0
9. y✔ + (y✕)2 + 2y = 0 10. y✔ + 2y✕ + sin y = 0
8. y✕ + y = ex
11. ✗✘✙✚ ✛✜✢✙✣✤✥
3
2
✠ d2y ✡
✠ dy ✡
✠ dy ✡
☛✌
✌
✍ ☛ sin ✌
✍ ☛ 1 ☞ 0 ✙✢ ❞✥✥✦ ✧★✩
2 ✍
✎ dx ✏
✎ dx ✏
✎ dx ✏
(A) 3
(B) 2
12. ✗✘✙✚ ✛✜✢✙✣✤✥ 2 x 2
(A) 2
(D) ✐✪✣✫✥✥✪✬✥✦ ✭✧✢✮ ✧★
(C) 1
d2y
dy
✯3
✰ y ✱ 0 ✙✢ ✙✥✲✪✳ ✧★ ✩
2
dx
dx
(B) 1
(D) ✐✪✣✫✥✥✪✬✥✦ ✭✧✢✮ ✧★
(C) 0
9.3. ✴✵✶✷ ✸✹✺✶✻✼✽ ✶✽ ✾✿✽❀✶ ❁✵❂
Solutions of a Differential Equation)
❃✵❃❄✽❅❆ ❇✷
(General and Particular
✪✐❢✚✢ ✙❈✥✥✗✥✲✮ ✜✲✮ ✧✜✭✲ ✪✭❉✭✪✚✪❊✥✦ ✐❋✙✥✣ ✘✲● ✛✜✢✙✣✤✥✥✲✮ ✙✥✲ ✧✚ ✪✙❍✥ ✧★✩
x2 + 1 = 0
... (1)
sin2 x – cos x = 0
... (2)
✛✜✢✙✣✤✥✥✲✮ ❧■❏ ✦❑✥✥ ❧▲❏ ✙✥ ✧✚ ▼✙ ▼✲✛✢ ✘✥◆✦✪✘✙ ✗❑✥✘✥ ✛✪❉✜❖ ✛✮❊❍✥ ✧★ P✥✲ ✪◗▼ ✧❘▼
✛✜✢✙✣✤✥ ✙✥✲ ✛✮✦❘✬✳ ✙✣✦✢ ✧★ ✗❑✥✥❙✦❚ P❯ ❱✛ ✛✮❊❍✥ ✙✥✲ ✛✜✢✙✣✤✥ ✜✲✮ ✗❲✥✦ x ✘✲● ◆❑✥✥✭ ✐✣
✐❋ ✪✦◆❑✥✥✪✐✦ ✙✣ ✪◗❍✥ P✥✦✥ ✧★ ✦✥✲ ◗✥❍✥❳ ✐❈✥ ✗✥★✣ ❯✥❍✥❳ ✐❈✥ ✗✥✐✛ ✜✲✮ ❯✣✥❯✣ ✧✥✲ P✥✦✲ ✧✮★❨
✗❯ ✗✘✙✚ ✛✜✢✙✣✤✥
d2y
dx 2
y=0
... (3)
✐✣ ✪✘❩✥✣ ✙✣✦✲ ✧★✮❨
✐❋❑✥✜ ◗✥✲ ✛✜✢✙✣✤✥✥✲✮ ✘✲● ✪✘✐✣✢✦ ❱✛ ✗✘✙✚ ✛✜✢✙✣✤✥ ✙✥ ✧✚ ▼✙ ▼✲✛✥ ✐●✚✭ ❬ ✧★ P✥✲ ❱✛
✛✜✢✙✣✤✥ ✙✥✲ ✛✮✦✬❘ ✳ ✙✣✲❭✥ ✗❑✥✥❙✦❚ P❯ ❱✛ ✐●✚✭ ❬ ✙✥✲ ✗✘✙✚ ✛✜✢✙✣✤✥ ✜✲✮ ✗❲✥✦ y ❧✗✥✪❖✦ ❩✣❏
✘✲● ◆❑✥✥✭ ✐✣ ✐❋✪✦◆❑✥✥✪✐✦ ✙✣ ✪◗❍✥ P✥✦✥ ✧★ ✦✥✲ ❯✥❍✥❳ ✐❈✥ ✗✥★✣ ◗✥❍✥❳ ✐❈✥ ❯✣✥❯✣ ✧✥✲ P✥✦✲ ✧★ ❨✮
✘♦ y = ❬ (x) ✗✘✙✚ ✛✜✢✙✣✤✥ ✙✥ ✧✚ ✘♦ ❧✛✜✥✙✚✭ ✘♦❏ ✙✧✚✥✦✥ ✧★❨ ✪✭❉✭✪✚✪❊✥✦
✐●✚✭ ✐✣ ✪✘❩✥✣ ✙✢✪P▼
y = ❬ (x) = a sin (x + b)
... (4)
400
① ✁✂✄
a, b
t☎✆✝
✞
R.
❀✟✠
✡☛
☞✌✍✎
✏✆✑✒
✡☛✓✔ ✌
✏✓✕✍t✆✔ ✖
✕✆✔
☛✗✘✕✒✙✆
✚✛✜
✗✔ ✖
☞✢✟✣✤✥✆✆✟☞✣
✕✒
✟✠❀✆ t✆❢ ✣✆✔ ✦✆❀✆✝ ☞✧✆ ✏✆✑✒ ✠✆❀✆✝ ☞✧✆ ✦✒✆✦✒ ☎✆✔ t✆✣✔ ☎✑★
✖
✡☛✟✍❢ ❀☎ ☞✌✍✎ ✏✓✕✍ ☛✗✘✕✒✙✆ ✚✛✜
✕✆
☎✍
☎✑★
✗✆✎
✍✘✟t❢
✟✎✯✎✟✍✟✰✆✣
a
✟✕
✏✆✑✒
☞✢✆ ✱✣
☞✌✍✎
b
✓❞ ✌✩
✕✆✔
✟✓✟✪✆✫✬
a=2
✗✆✎
b
❢✓✖
✭
✮
✠✔
4
✟✠❢
t✆✣✔
☎✑✖
✣✆✔
☎✗✔ ✖
☎✑✲
☎✆✔✣✆
y=
✳
✴ ✶
✵
1
(x) = 2sin x
✷
✸
✺
4
... (5)
✹
✻
❀✟✠ ✡☛ ☞✌✍✎ ✏✆✑✒ ✡☛✓✔ ✌ ✏✓✕✍t✆✔ ✖ ✕✆✔ ☛✗✘✕✒✙✆ ✚✛✜ ✗✔ ✖ ☞✢✟✣✤✥✆✆✟☞✣ ✕✒ ✟✠❀✆ t✆❢ ✣✆✔ ☞❞✎ ✲
✦✆❀✆✝
☞✧✆
✏✆✑✒
✳
☞✌✍✎
☛✗✘✕✒✙✆
☎✑
✟✓✟✪✆✫✬
✗✔✖
☞✢✆✼✍✆✔✖
☎✍
✤✓✔ ❡✩
✠✆✔
a
✕☎✆
✦✒✆✦✒
☞✧✆
❧❀✆☞✕
✕✆
✍✔✟✕✎
✠✆❀✆✝
☎✍
✣✥✆✆
✏✼✒
b
✓✔ ✌
✟t☛✗✔ ✖ ✤✓✔❡✩
❢✔☛✆
☎✍✿
t✆✔ ✤✓✔❡✩ ✏✼✒✆✖✔ ☛✔
♠❃❄❅❆❇❄
❢✕
❅❣
✕✒✣✔
✏✓✕✍
❈
✡☛✟✍❢ ✳
☎✖★
✑
1
✚☞✢✆✼✍✜
☎✑★
✟✓✟✪✆✫✬
a, b
t✦✟✕
✗✆✎
❍✆✘
☛✗✘✕✒✙✆
☛✟✯✗✟✍✣
☞✌✍✎
♦☞✟✤✥✆✣
✽
1
✗✔ ✖
☎✑ ✖ ✏✆✑✒
☎✑✖
✚✛✜
✣✥✆✆
✕✆✔✡✾
❀☎
❍✆✘
✡☛✟✍❢
❢✕
✕✆
☞✌✍✎
✤✓✔ ❡✩
✡☛✕✆✔
☎✍
☎❞ ❢
✟✠❢
✏✼✒
☎✑★
✏✓✕✍
☛✟✯✗✟✍✣
✏✓✕✍
☛✗✘✕✒✙✆
✏✼✒
♦☞✟✤✥✆✣
☛✗✘✕✒✙✆
☛❉❀✆✟☞✣
☎✆✔
✏✓✕✍
☛✗✘✕✒✙✆
✗❞ ❁✣ ☎✑ ✏✥✆✆✾ ✣❂ ❧❀✆☞✕
✕✆
✕✘✟t❢
✟✓✟✪✆✫✬
✟✕
☎✍
☞✌✍✎
y=e ,
❧❀✆☞✕
☎✍ ✕☎✍✆✣✆
☎✍ ✗✔ ✖ ✤✓✔ ❡✩ ✏✼✒✆✔ ✖ ✕✆✔
✕☎✍✆✣✆
– 3x
✕✆
✕✆
☎✑★
✟✓✟✪✆✫✬
✗✆✎
✠✔✎✔
☞✒
0
✕✆
☎✑★
✏✓✕✍
d2y
dx 2
☛✗✘✕✒✙✆
❊
dy
dx
❋
6y
●
☎✑★
☎✍
✟✠❀✆
☎❞✏✆
☞✌✍✎
y = e– 3x
☎✑★ ✡☛✓✔✌ ✠✆✔ ✎✆✔ ✖ ☞✧✆✆✔ ✖ ✕✆
x
✓✔ ✌ ☛✆☞✔✧ ✆
✏✓✕✍✎ ✕✒✎✔ ☞✒
☎✗ ☞✢✆✱✣
☎✑✲
dy
= 3e –3x
dx
✏✦
✎☎✘✖
☎✑★
t✆✣✆
☎✍✿
☎✍✿
t✆✣✔
✕☎✍✆✣✆
❢✔☛✆
☞✢✆✱✣
☎✆✔
☛✗✘✕✒✙✆
✚✈✜
✕✆
x
✓✔ ✌
☛✆☞✔✧✆
☞❞ ✎✲
✏✓✕✍✎
... (1)
✕✒✎✔
☞✒
☎✗
✠✔ ✰✆✣✔
☎✑✖
✟✕
d2y
= 9e –3x
dx 2
d 2 y dy
,
dx 2 dx
✦✆❀✆✝
☞✧✆
✡☛✟✍❢
✏✆✑✒
y
✕✆
✗✆✎✿
✟✠❢
■❢
✏✓✕✍
☛✗✘✕✒✙✆
✗✔ ✖
☞✢✟✣✤✥✆✆✟☞✣
= 9e– 3x + (–3e– 3x) – 6.e– 3x = 9 e– 3x – 9 e– 3x = 0 =
✟✠❀✆
☎❞ ✏✆
☞✌✍✎
✟✠❢
☎❞ ❢
✏✓✕✍
☛✗✘✕✒✙✆
✕✆
❢✕
✕✒✎✔
✠✆❀✆✝
☎✍
☎✑★
☞✧✆
☞✒
☞✒
☎✗
☞✆✣✔
☎✑✖
✟✕
✈ ✁✂ ✄☎✆✁✝✞✟
401
♠✠✡☛✡☞✌✡ ✍ ❧✎✏✑✒✓✔ ✕✖✒✗✘ ✒✕ ✓✙✚✛ y = a cos x + b sin x, ✒✗❧❢✜✢ a, b ✣ R, ✤✥✕✚
d2y
★ y ✩ 0 ✕✑ ❞✚ ❞✪✫
❧❢✖✕✦✧✑
dx 2
☛❣ ✒✬✏✑ ❞✭✤✑ ✓✙✚✛ ❞✪
y = a cos x + b sin x
... (1)
❧❢✖✕✦✧✑ ✮✯✰ ✥✜✙ ✬✑✜✛✑✜✢ ✓✱✑✑✜✢ ✕✑ x, ✥✜✙ ❧✑✓✜✱✑ ♦✲✑✦✑✜✲✑✦ ✤✥✕✚✛ ✕✦✛✜ ✓✦ ❞❢ ✬✜✳✑✔✜ ❞✢✪✴
dy
= –a sin x + b cos x
dx
d2y
= – a cos x – b sin x
dx 2
d2y
✘✥✢ y ✕✑ ❢✑✛ ✒✬✘ ❞✭✘ ✤✥✕✚ ❧❢✖✕✦✧✑ ❢✜✢ ✓✵✒✔✶✷✑✑✒✓✔ ✕✦✛✜ ✓✦ ✓✵✑✸✔ ✕✦✔✜ ❞✪✴✢
dx 2
❝✑✏✑✹ ✓✱✑ = (– a cos x – b sin x) + (a cos x + b sin x) = 0 = ✬✑✏✑✹ ✓✱✑
❜❧✒✚✘ ✒✬✏✑ ❞✭✤✑ ✓✙✚✛✺ ✒✬✘ ❞✭✘ ✤✥✕✚ ❧❢✖✕✦✧✑ ✕✑ ❞✚ ❞✪✫
✐✻✼✽✾✿❀❁ ❂❃❄
✯ ❧✜ ✯❅ ✔✕ ✓✵✎✏✜✕ ✓✵❆✛ ❢✜✢ ❧✎✏✑✒✓✔ ✕✖✒✗✘ ✒✕ ✒✬✏✑ ❞✭✤✑ ✓✙✚✛ ✮✶✓❇❈ ✤✷✑✥✑ ✤✶✓❇❈✰ ❧✢❉✔
✤✥✕✚ ❧❢✖✕✦✧✑ ✕✑ ❞✚ ❞✪✴
1. y = ex + 1
2. y = x2 + 2x + C
3. y = cos x + C
:
:
:
y❊ – y❋ = 0
y❋ – 2x – 2 = 0
y❋ + sin x = 0
4. y =
:
y❋ =
5. y = Ax
:
xy❋ = y (x ■ 0)
6. y = x sin x
:
xy❋ = y + x
7. xy = log y + C
:
y2
y❋ =
(xy ■ 1)
1 ❑ xy
8. y – cos y = x
:
(y sin y + cos y + x) y❋ = y
1● x
2
xy
1 ❍ x2
x 2 ❏ y 2 (x ■ 0 ✤✑✪✦ x > y ✤✷✑✥✑ x < – y)
402
① ✁✂✄
9. x + y = tan–1y
10. y =
11.
a2
✆
y2 y + y2 + 1 = 0
:
x2 x
☎
dy
= 0 (y
dx
(– a, a) : x + y
✝
✞
0)
♣✟✠ ✡✟☛☞✌ ✍✟✎☛ ☞✡✏✑ ✒✍✡✎ ✏✓✑✡✠✔✟ ✍☛✕ ✖✗✟✘✡ ✙✎ ✓☛✚ ✛✘☞✜✢✟✣ ✜✍☛✤✥ ✒♣✠✟☛✚ ✡✑ ✏✚✦✗✟ ✙✧★
(A) 0
12.
✣✑r
(B) 2
✡✟☛ ☞✌
✏✚✦ ✗✟
✍✟✎☛
☞✡✏✑
(C) 3
✒✍✡ ✎
✏✓✑✡✠✔ ✟
✍☛ ✕
(D) 4
☞✍☞✩✟✪✌
(B) 2
(C) 1
❣✟r✣☛
✭✮✬
✙✧✚
✛✘☞✜✢✟✣
✜✍☛✤ ✥
✒♣✠ ✟☛ ✚
✯✰✱✲✳
☞✡
✭✴
✵✱✴✶
✷✵✳✴
✸✹✺✳✻✼✱
✳✱
❢✽✹✱✾ ✼✱
(Formation of a
✏✓✑✡✠✔✟
x2 + y2 + 2x – 4y + 4 = 0
✿✡
✿☛✏ ☛
✍❀ ✣
✏✓✑✡✠✔✟
✡✑
(D) 0
9.4.
Differential Equation whose Solution is Given)
✙✓
✓☛✚
✙✧★
(A) 3
❢✫✬
✙✎
✡✟☛
❄❆❉
☞r❁☞✘✣
✡✟
x,
✍☛ ✕
✡✠✣✟
✏✟✘☛♦✟
✙✧
☞❣✏✡✟
✒✍✡✎r
✍☛✚ ✕❂❃
✡✠r☛
dy x 1
, (y
dx 2 y
✗✙ ✿✡ ✒✍✡✎ ✏✓✑✡✠✔✟ ✙✧■ ✒✟✘ ❑✟❂
❄❅❆❇
✘✠
... (1)
❈❉
✘❃✟❏✣
✙✧
✒✟✧✠
✡✠✣☛
☞❊✟❋✗✟
❆
●✡✟●❍
✙✧■
✙✧✚
2)
✞
... (2)
✓☛✚ ❂☛✦✟☛▲
✚ ☛ ☞✡
❄✒r✈▼ ✟✟▲
◆❖P❖❆ ✡✟ ✛❂✟✙✠✔✟ ◆
❂☛ ☞✦✟✿❉ ☞✡
✗✙ ✏✓✑✡✠✔✟ ✍❀◗✟✟☛✚ ✍☛✕ ✿✡ ✍✈✕✎ ✡✟☛ ☞r❁☞✘✣ ✡✠✣✟ ✙✧ ✒✟✧✠ ✛✏ ✍✈ ✕✎ ✡✟ ✿✡ ✏❂✜✗ ✏✓✑✡✠✔✟ ❄❆❉
✓☛✚
☞❂✗✟
✙✈✒ ✟
✍❀✣
✙✧■
✒✟●✿
☞r❡r☞✎☞✦✟✣
✏✓✑✡✠✔✟
✘✠
☞✍♣✟✠
✡✠☛★
✚
x2 + y2 = r2
r,
✡ ✟☛ ☞✍☞▼✟❞r
✙✟☛✣☛
✙✧✚
●❜✗✟☞❂
✓✟r ❂ ☛r ☛ ✘✠
✛❂ ✟✙✠✔ ✟✣★
❄✒✟✍❀✕☞✣
✿✡
✿☛✏ ☛ ✏✚✍☛✚ ✕❂❃✑
✍☛ ✚✕❂❃
✓❙ ✎
❚❑❂✈
✙✓✟✠✑
◆❖❆
2
✍☛ ✕
2
❂☛☞✦✟✿❉■
✍☛✕
✒✟✧✠
●✏
✙✓☛ ✚ ✍✈ ✕ ✎
☞▼✟❞r
✏❂✜ ✗ ✘❃ ✟❏ ✣
x + y = 1, x + y = 4, x2 + y2 = 9
✍❀◗ ✟✟☛✚
✙✧
❱☞♣
2
... (3)
●✏
2
✘❃✡ ✟✠
✏✓✑✡✠✔✟
✍✈✕✎ ✡✟☛ ☞r❁☞✘✣ ✡✠✣✟
☞❣r✡✑
✍✈ ✕✎ ✍☛✕
☞❊✟❋✗✟✿❯
✘❃❜✗☛✡
☞▼✟❞r
❄❘❉
✙✧ ☞❣r✡✟
✙✧■
✚
✏❂✜✗ ❲✟✠✟ ✏✚✣✪
✈ ✌
☞✡✿
❣✟r☛ ✍✟✎✟ ✒✍✡✎ ✏✓✑✡✠✔✟ t✟✣ ✡✠r☛ ✓☛✚ ✙✧ ■
✚
✗✙ ✏✓✑✡✠✔✟
r
✏☛
☞✎✿
✓✈❧✣
r
✡✟
✙✟☛r ✟
✓✟r
♣✟☞✙✿
☞▼✟❞r
❧✗✟☛☞
✚ ✡
✙✧■
✍✈✕✎
✏✓✑✡✠✔✟
✍☛✕
☞✍☞▼✟❞r
❄❘❉
✡✟
x
✏❂✜✗✟☛✚
✍☛ ✕
✍☛✕
✏✟✘☛♦✟
❳❨❩❬❭❪❫ ❴❵❛
✈ ✁✂ ✄☎✆✁✝✞✟
403
✠✡☛☞✌ ☛✍✌✎ ✏✍ ✑✒ ✓✔✕☛✍✖✗ ✏✘✗✙✚ ✛☛✑✗ ✜✗✚✗ ✒✢✣ ✠✤✗✗✥✚✦
2x + 2y
dy
dx
✠✤✗✡✗
0
x+y
dy
dx
0
... (4)
✑✒ ✠✡☛☞ ✓✔✕☛✍✖✗❀ ✓✔✕☛✍✖✗ ✧★✩ ✪✗✍✗ ✛✌✫✛✏✚ ✓✡✎✭✬ ✮✘✕ ✡✯✰✗✗✎✬ ✡✎✭ ✡✱✭☞ ☛✗✎ ✛✌✫✛✏✚ ☛✍✚✗ ✒✢✣
✠✗✲✳ ✛✏✭✍ ✓✎ ✛✌✴✌✛☞✛✵✗✚ ✓✔✕☛✍✖✗ ✏✍ ✛✡✶✗✍ ☛✍✎✷✬
y = mx + c
... (5)
✏✘ ✗✶☞✗✎✬ m ✚✤✗✗ c, ✡✎✭ ✛✡✛♦✗✸✌ ✔✗✌✗✎✬ ✓✎ ✒✔✎✬ ✡✱✭☞ ✡✎✭ ✛✡✛♦✗✸✌ ✓✮✹✑ ✏✘ ✗✙✚ ✒✗✎✚✎ ✒✢✬ ✺✮✗✒✍✖✗✚✑✗
y=x
y=
(m = 1, c = 0)
(m =
3x
3 , c = 0)
y=x+1
(m = 1, c = 1)
y=–x
(m = – 1, c = 0)
y=–x–1
(m = – 1, c = – 1)
✲❜✑✗✛✮ ✧✠✗✡✯✭✛✚ ✻✼✽ ✮✎✛✵✗✳✩✣
✲✓ ✏✘☛✗✍ ✓✔✕☛✍✖✗ ✧✾✩ ✓✍☞ ✍✎✵✗✗✠✗✎✬ ✡✎✭ ✡✱✭☞ ☛✗✎ ✛✌✫✛✏✚
☛✍✚✗ ✒✢ ✛✜✓✔✎✬ m, c ✏✘✗✶☞ ✒✢✣
✠✿ ✒✔✗✍✕ ❁✛✶ ✲✓ ✡✱✭☞ ✡✎✭ ✏✘❜✑✎☛ ✓✮✹✑ ✪✗✍✗ ✓✬✚❂✱ ❃ ✛☛✳
✜✗✌✎ ✡✗☞✗ ✠✡☛☞ ✓✔✕☛✍✖✗ t✗✚ ☛✍✌✎ ✔✎✬ ✒✢✣ ✲✓✡✎✭ ✠✛✚✛✍❄✚
✡✒ ✓✔✕☛✍✖✗ m ✚✤✗✗ c ✓✎ ✔✱❄✚ ✒✗✎✌✗ ✶✗✛✒✳ ❄✑✗✎✛✬ ☛ ✡✱✭☞ ✡✎✭
✛✡✛♦✗✸✌ ✓✮✹✑✗✎✬ ✡✎✭ ✛☞✳ m ✚✤✗✗ c ☛✗ ✔✗✌ ✛♦✗✸✌ ✒✢✣ ✑✒ ✠✡☛☞
✓✔✕☛✍✖✗❀ ✓✔✕☛✍✖✗ ✧✾✩ ☛✗ x ✡✎✭ ✓✗✏✎❅✗ ❆✔✗✌✱✓✗✍ ✮✗✎ ✿✗✍
✠✡☛☞✌ ☛✍✌✎ ✏✍ ✏✘ ✗✙✚ ✒✗✎✚✗ ✒✢ ✠✤✗✗✥✚✦
❴❵❛❝❞❡❣ ❤❥❦
2
d y
dy
❇ m ✚✤✗✗
❈0
dx
dx 2
... (6)
✓✔✕☛✍✖✗ ✧❧✩❀ ✓✔✕☛✍✖✗ ✧✾✩ ✪✗✍✗ ✛✮✳ ✒✱✳ ✓✍☞ ✍✎✵✗✗✠✗✎✬ ✡✎✭ ✡✱✭☞ ☛✗✎ ✛✌✫✛✏✚ ☛✍✚✗ ✒✢✣
✛❃✙✏✖✗✕ ✓✔✕☛✍✖✗ ✧★✩ ✚✤✗✗ ✧✾✩ ❆✔❢✗✷ ✓✔✕☛✍✖✗ ✧❉✩ ✳✡✬ ✧❧✩ ✡✎✭ ❊✑✗✏☛ ✒☞ ✒✢✣✬
9.4.1
❋●❍ ■❏❍ ❑▲▼◆❖ ❑❖P ❑❏P◗ ❘▼❖ ❋❙❚❋❯❱ ❘❲❙❖ ❑▼◗❖ ❳❑❘◗ ❨❩❬❘❲❭▼ ❑❖P ❋❙❩▼❪❭▼ ❘❬
❯✐❋ ▲❫▼ (Procedure to form a Differential Equation that will represent a given
Family of curves)
(a) ✑✛✮ ✛✮✳ ✒✱✳ ✡❆✗✎✬ ☛✗ ✡✱✭☞ F1 ✡✎✭✡☞ ✳☛ ✏✘✗✶☞ ✏✍ ✛✌♦✗✥✍ ☛✍✚✗ ✒✢ ✚✗✎ ✲✓✎ ✛✌✴✌✛☞✛✵✗✚
✫✏ ✡✗☞✎ ✓✔✕☛✍✖✗ ✪✗✍✗ ✛✌✫✛✏✚ ✛☛✑✗ ✜✗✚✗ ✒✢✷
F1 (x, y, a) = 0
... (1)
404
① ✁✂✄
♠☎✆✝✞✟✆✠✡☛
❢✘♦❢☞✠
❢❞✎✆
✱❞
✌✆✍✆
✙✆
❧✜✢
✔✕✖❞✞✟✆
y2 = ax
☞✞✌✍✎✆✏✑
x
✔✆☞✏✣✆
✌✏✓
f(x, y, a) : y2 = ax
✌✒✓✍
✌✏✓
♦☞
✌✆✍✏
✔✕✖❞✞✟✆
✤✌❞✍✘
✔✕✖❞✞✟✆
❢✘✦✘❢✍❢✧✆✠
❧✜✢
g (x, y, y , a) = 0
a
♦☞
✕✏✑
❞✞✘✏ ☞✞
☞★✆✩✠
✝✆✏✠✆
✝✕✏✑
y , y, x,
✥
✱✌✑
a
❞✆✏
✔❢✦✕❢✍✠
❢✘✦✘❢✍❢✧✆✠
✠✪✆✆
♦☞
✕✏ ✑
❧✫✢
☞★✆✩✠
✔✏
... (2)
❢✌✍✒✩✠
❞✆✏
✝✆✏✠✆
❞✞✘✏
☞✞
✤✆✌✬✎❞
✝✕✏✑
✤✌❞✍
✔✕✖❞✞✟✆
✝✚ ✡
F (x, y, y ) = 0
F2
a,
... (3)
✥
(b)
❢☎✱
✎❢☎
✌✆✍✏
✝✒✱
✌❀✆✏✑ ❞✆
✔✕✖❞✞✟✆
☞★✆✐✍✆✏ ✑
✌✒✓✍
❢✘♦❢☞✠
✗✆✞✆
❢❞✎✆
b
✠✪✆✆
✙✆✠✆
❢✘✭✆✮✞
☞✞
❞✞✠✆
✠✆✏ ✯✔✏ ❢✘✦✘❢✍❢✧✆✠
✝✚
(4)
✱❞
x
❞✆
✌✏✓ ✔✆☞✏✣✆
❢✘✦✘❢✍❢✧✆✠
✔✕✖❞✞✟✆
... (4)
✤✌❞✍✘ ❞✞✘✏ ☞✞
♦☞
✕✏✑
☞★✆✩✠
✝✆✏✠✆
✝✕✏ ✑
y x, y, a, b
✥✰
❞✆✏ ✔❢✦✕❢✍✠ ❞✞✘✏ ✌✆✍✆
✝✚ ✡
g (x, y, y , a, b) = 0
... (5)
✥
☞✞✑✠ ✒
☎✆✏
✔✕✖❞✞✟✆✆✏✑
✱❞
✠✖✔✞✏
✤✌❞✍✘
❞✖
✔✕✖❞✞✟✆
❞✞✘✏
✔✝✆✎✠✆
✔✏
✤✆✌✬✎❞✠✆
❞✖
❢✘✦✘❢✍❢✧✆✠
☞✞
☞★✆✐✍✆✏✑
☎✆✏
♦☞
✝✚✛
✕✏✑
✎✝
☞★✆✩✠
❢✌✍✒✩✠
❞✆✏
✔✕✖❞✞✟✆
❧✴✢☛
❧✲✢
✔✕✖❞✞✟✆
❢✘✦✘❢✍❢✧✆✠
✱✌✑
❧✵✢
♦☞
✕✏✑
❞✞✘✆
✔✕✖❞✞✟✆☛
❢❞✎✆
h (x, y, y , y , a, b) = 0
a
b
✥
✔✦✭✆✌
❧✲✢
✔✕✖❞✞✟✆
✙✆✠✆
✘✝✖✑
✝✚
❞✆
✯✔❢✍✱
x
✠✪✆✆
☞★✆✩✠
✝✆✏✠✆
✝✚ ✡
... (6)
❢✌✍✒✩✠
❞✆✏
❞✞✘✏
☞✞
✝✕✏✑
✤✆✌✬✎❞
❢❞✔✖
✷✸✹✺✻✼✽
✶
✝✆✏✠ ✖
❢✙✠✘✏
✝✚
❄❅✼❆❇✻✼
✙t❢❞
m
❈
✌❀✆✏✑
✱❞
❢☎✎✆
❆❣
✔✕✖❞✞✟✆
♠✔
❧✜✢
✌❀
✌✏✓
❁✌✏❂ ❃
✝✒✤✆
✝✚
✌✏✓
✌❀
✌✒✓✍
✌✒✓✍
✔✑✿✠
y = mx
✌✒ ✓✍
✤✐✞
✌✏✓
... (7)
✳
❢✘♦❢☞✠
❞✆✏
❞✆✏
✤✌❞✍
✝✚ ✡
F (x, y, y , y ) = 0
✥
✝✕✏✑
✔✆☞✏✣✆
✌✏✓
✳
✔✏
♦☞
✝✚ ✡
F2 (x, y, a, b) = 0
✔✕✖❞✞✟✆
❞✞✘✏
✝✚✡
✥
✔✕✖❞✞✟✆
✗✆✞✆
✝✚✛
✔❞✠✆
❞✆
❞✆
❞✞✘✏
✔✕✖❞✞✟✆
❢✘♦❢☞✠
✌✆✍✏
✕✏✑
✤✌❞✍
❁✌✏❂ ❃
❞✞✘✏
✌✆✍✏
✔✕✖❞✞✟✆
✤✐✞
✝✆✏✠✏
✤✌❞✍
❞✖
❞✆✏❢✾
♠✠✘✖
✝✖
✝✚✑✛
✔✕✖❞✞✟✆
❞✆✏
❉✆✠
❞✖❢✙✱
✝✚✛
❢❞
☎✆✏✘✆✏
y = mx
x
dy
m
dx
☞✣✆✆✏✑
❞✆
✌✏✓
... (1)
✔✆☞✏ ✣✆
✤✌❞✍✘
❞✞✘✏
☞✞
✝✕
☞★✆✩✠
❞✞✠✏
✝✚✛
✑
❊
m
❞✆
✝✆✏✠✆
✕✆✘
✝✚✛
✔✕✖❞✞✟✆
✎✝
☞★ ✆✐✍
❧✜✢
m
✔✏
✕✏✑ ☞★❢✠❁✪✆✆❢☞✠
✕✒●✠
✝✚
✤✆✚✞
❞✞✘✏ ☞✞
✯✔❢✍✱
✎✝
✝✕✏ ✑
y
✤✭✆✖❍✾
dy
x
dx
✤✌❞✍
✤✪✆✌✆
x
✔✕✖❞✞✟✆
dy
dx
✝✚✛
❋
y
❊
0
☞★✆✩✠
✈ ✁✂ ✄☎✆✁✝✞✟
405
♠✠✡☛☞✌✡ ✍ ♦✎✏✑✒ ♦✑✓ ♦✔✓✕ y = a sin (x + b), ❢✖✗✘✑✒ a, b ▲♦✑✙✚ ✛✜✢ ✣✤✥✒ ✦✏✑ ❢✧★❢✩✪ ✦✢✧✑ ♦✏✕✑
✛♦✦✕ ✗✘✫✦✢✬✏ ✦✏✑ ✭✏✪ ✦✫❢✖✮✯
☛❣
❢✰✱✏ ✣✔✛✏ ✣✤ ❢✦ y = a sin (x + b)
... (1)
✗✘✫✦✢✬✏ ❧✲✳ ♦✑✓ ✰✏✑✧✏✑✒ ✩✴✏✏✑✒ ✦✏ x ♦✑✓ ✗✏✩✑✴✏ ✵✶✏✢✏✑✶✏✢ ✛♦✦✕✧ ✦✢✧✑ ✩✢ ✣✘ ✩✷✏✸✪ ✦✢✪✑ ✣✤✹
dy
✺ a cos( x ✻ b)
dx
... (2)
d2y
✼ – a sin ( x ✽ b)
dx 2
✗✘✫✦✢✬✏ ❧✲✳✥ ❧✾✳ ✪✿✏✏ ❧❀✳ ✗✑ a ✪✿✏✏ b ✦✏✑ ❢♦✕✔✸✪ ✦✢✧✑ ✩✢ ✣✘ ✩✷✏✸✪ ✦✢✪✑ ✣✤✯✒
d2y
✽ y✼0
dx 2
... (3)
... (4)
✗✘✫✦✢✬✏ ❧❁✳ ▲♦✑✙✚ ✛✜✢✏✑✒ a ✪✿✏✏ b ✗✑ ✘✔❂✪ ✣✤ ✛✏✤✢ ❃✗❢✕✮ ✱✣ ✛❄✏✫❅❆ ✛♦✦✕ ✗✘✫✦✢✬✏ ✣✤✯
♠✠✡☛☞✌✡ ❇ ✮✑✗✑ ✰✫❈✏❉♦❊✶✏✏✑✒ ♦✑✓ ♦✔✓✕ ✦✏✑ ❢✧★❢✩✪ ✦✢✧✑ ♦✏✕✏ ✛♦✦✕ ✗✘✫✦✢✬✏ ✭✏✪ ✦✫❢✖✮ ❢✖✧✦✫
✧✏❢❄✏✱✏✉ x-✛✴✏ ✩✢ ✣✤✒ ✪✿✏✏ ❢✖✧✦✏ ♦✑✓✒ ✰✷ ✘❋✕ ●❍✰✔ ✣✤✯
☛❣
✣✘ ✖✏✧✪✑ ✣✤ ✒ ❢✦ ✦❢✿✏✪ ✰✫❈✏❉♦❊✶✏✏✑✒ ♦✑✓ ♦✔✓✕ ✦✏ ✗✘✫✦✢✬✏ ❢✧■✧❢✕❢❏✏✪ ✩✷ ✦✏✢ ✦✏ ✣✏✑✪✏ ✣✤
❧✛✏♦❊✓❢✪ ❑▼❀ ✰✑❢❏✏✮✳
x2 y 2
✽
✼1
... (1)
a 2 b2
✗✘✫✦✢✬✏ ❧✲✳ ✦✏ x ♦✑✓ ✗✏✩✑✴✏ ✛♦✦✕✧ ✦✢✧✑ ✩✢ ✣✘✑✒
2 x 2 y dy
◆
❖ 0 ✩✷✏✸✪ ✣✏✑✪✏ ✣✤✯
a 2 b 2 dx
✛✿✏♦✏
y ◗ dy ❘ Pb 2
❙
❚✼
x ❯ dx ❱ a 2
... (2)
❤✐❥❦♥♣q rst
✗✘✫✦✢✬✏ ❧✾✳ ♦✑✓ ✰✏✑✧✏✑✒ ✩✴✏✏✑✒ ✦✏ x ♦✑✓ ✗✏✩✑✴ ✏ ✛♦✦✕✧ ✦✢✧✑ ✩✢ ✣✘✑✒ ✩✷✏✸✪ ✣✏✑✪✏ ✣✤✹
❲ dy
❳
x ❨ y❬
dy
❲ y ❳ ❲ d y ❳ ❩ dx
❭
❪0
❬
❩ ❬❩
2 ❬ ❩
2
❫ x ❴ ❫ dx ❴ ❫
x
❴ dx
2
2
✛✿✏♦✏
d2y
dy
❵ dy ❛
xy 2 ◆ x ❜ ❝ – y
❖0
dx
❞ dx ❡
dx
✗✘✫✦✢✬✏ ❧❀✳ ✛❄✏✫❅❆ ✛♦✦✕ ✗✘✫✦✢✬✏ ✣✤✯
... (3)
406
① ✁✂✄
x
♠☎✆✝✞✟✆ ✠
✫✡☛☞ ✌☞✍ ✎✏✑ ✒✓✔✕ ✖✗ ✘✖✙☞✚ ✌✗✛✍ ✜☞✑✍ ✜✢✣☞☞✍✤ ✜✍✥ ✜✕✥ ✑
✌☞ ✡✜✌✑ ❞✎✦✌✗✧☞ ★☞r
✝❣
✜✍✥
✎☞✛ ✑✦✩✪✬❡
✜✕✥ ✑
C
✌☞✍
❞✍
✌✦✩✪✬✭
x
✫✡☛☞ ✌☞✍ ✎✏ ✑ ✒✓✔✕ ✖✗ ✘✖✙☞✚ ✌✗✛✍ ✜☞✑✍ ✜✢ ✣☞☞✍✤
✩✛✩✔✚❧✮
✩✌✯☞
✪☞r☞
(0, a)
✰✱✭
✲❞
✜✕✥ ✑
✜✍✥
✩✌❞✦ ❞✔✘✯ ✜✍✥ ✜✍✥
✤ ✔❢ ✒✓✔✕ ✜✍✥ ✩✛✔✍✚✙☞☞✤✌ ✰✱✤ ✳✡☞✜✢✥ ✩r ✴✵✶ ✔✍✩✷☞✬✸✭
❜❞✩✑✬
C
✜✕✥ ✑
❞✎✦✌✗✧☞
✌☞
x2 + (y – a)2 = a2
a
✩✪❞✎✍✤
✌☞
x
x2 + y2 = 2ay
✡✈☞✜☞
✡✜✌✑✛
✌✗✛✍
✖✗
✖❢☞♦r
✌✗r✍
✰✱ ✤✹
✿
✾
✡✈☞✜☞
✡✈☞✜☞
❞✎✦✌✗✧☞
✳❀✸
❞✍
a
✌☞
✿
❞✎✦✌✗✧☞
✎☞✛
✳✽✸
✎✍ ✤
x2
y2 2 y
✡✈☞✜☞
dy 2
(x
dx
✡✈☞✜☞
dy
dx
✩✔✬
✰✕ ✬
✜✢ ✣☞☞✍✤
✝❣
✲❞
✎☞✛
✜ ✕✥ ✑
✜✕✥ ✑
✌☞
✖✗
✖❢ ☞♦r
✌✗r✍
✰✱✤ ✹
dy
dx
dy
dx
y 2 ) 2 xy 2 y 2
dy
dx
2 xy
x – y2
2
✡❁☞✦❧✮
❞✎✦✌✗✧☞
✡✜✌✑
✰✱✭
✬✍❞✍ ✖✗✜✑✯☞✍✤ ✜✍✥ ✜✕✥ ✑ ✌☞✍ ✩✛❃✩✖r ✌✗✛✍ ✜☞✑☞ ✡✜✌✑ ❞✎✦✌✗✧☞ ★☞r ✌✦✩✪✬ ✩✪✛✌☞
♠☎✆✝✞✟✆ ❂
✎✏ ✑
✜✍✥
... (2)
✗✷☞✛✍
x y
✙☞✦❧☞✚
❇❈❉❊ ❋●❍ ■❏❑
dy
dy
2x 2 y
2a
dx
dx
dy
dy
x y
a
dx
dx
dy
x y
dx
a
dy
dx
✾
✯✰
... (1)
✬✌ ✘✜✍✺✻ ✡✼✗ ✰✱✭ ❞✎✦✌✗✧☞ ✳✽✸ ✜✍✥ ✔☞✍✛☞✍ ✤ ✖☛☞☞✍ ✤
❞☞✖✍☛☞
✜✍✥
✰✱✹
✒✓✔✕
✖✗
✑✦ ✩✪✬
✜ ✍✥
✰✱
r✈☞☞
✩✌
✩✌❞✦
✩✪✛✌☞
✲✖ ✗☞ ❆
✍ r
❞✔✘✯
✌✦
✡☛☞
✼✩✼✚ r
✛☞ ✩❁☞
❄✛☞❅✎✌
x-
✖✗✜ ✑✯☞ ✍ ✤
(a, 0)
✖✗
✡☛☞
✜ ✍✥
✰✱
✌✦
✜ ✕✥ ✑
✩✔✙☞☞
✌☞ ✍
✩✪❞✎✍✤
P
a
✰✱✭
✎✍ ✤
❞✍
✩✛ ✩✔✚ ❧ ✮
✬✌
✩✌✯☞
❄✛☞ ❅✎✌
✪☞r☞
✘✜ ✍✺✻
✰✱
✡☞ ✱ ✗
✡✼✗
✰✱
✈ ✁✂ ✄☎✆✁✝✞✟
407
✭✠✡☛☞✌✍✎ ✏✑✺ ♥✒✍✓✡✔✕✖ ✗✘✍✙✔ ☛✚✌✙ P ❞✡ ✘✛✜❞✢✣✡ ✤✥✦
y2 = 4ax
... (1)
✘✛✜❞✢✣✡ ✭❧✕ ☛✒✌ ♥✡✒✧✡✒★ ✩✪✡✡✒★ ❞✡ x ☛✒✌ ✘✡✩✒✪✡ ✠☛❞✙✧
❞✢✧✒ ✩✢ ✤✛ ✩✡✎✒ ✤✥★✦
2y
dy
✫ 4a
dx
... (2)
✘✛✜❞✢✣✡ ✭✬✕ ✘✒ ✮a ❞✡ ✛✡✧ ✘✛✜❞✢✣✡ ✭❧✕ ✛✒★ ✢✓✡✧✒ ✩✢
✤✛ ✩✡✎✒ ✤✥✦★
y2
2y
y 2 ✰ 2 xy
✠✯✡☛✡
❳❨❩❬❭❪❫ ❴❵❛
dy
( x)
dx
dy
✫0
dx
... (3)
✘✛✜❞✢✣✡ ✭✱✕ ✍♥✔ ✤✚✔ ✩✢☛✙✲✡✒★ ☛✒✌ ☛✚✌✙ ❞✡ ✠☛❞✙ ✘✛✜❞✢✣✡ ✤✥✖
✐✳✴✵✶✷✸✹ ✻✼✽
❧ ✘✒ ✺ ✎❞ ✩✾✿✲✒❞ ✩✾❀✧ ✛✒❁★ ❂☛✒❃❄ ✠❅✢✡✒★ a ✎✯✡✡ b ❞✡✒ ✍☛✙✚❆✎ ❞✢✎✒ ✤✚✔ ✍♥✔ ✤✚✔ ☛❇✡✒★ ☛✒✌ ☛✚✌✙
❞✡✒ ✍✧❈✍✩✎ ❞✢✧✒ ☛✡✙✡ ✠☛❞✙ ✘✛✜❞✢✣✡ ❉✡✎ ❞✜✍❊✔✖
x y
❋ ●1
2. y2 = a (b2 – x2)
3. y = a e3x + b e– 2x
a b
4. y = e2x (a + bx)
5. y = ex (a cos x + b sin x)
6. y-✠✪✡ ❞✡✒ ✛❍✙ ■❏♥✚ ✩✢ ❂✩❀✡❑ ❞✢✧✒ ☛✡✙✒ ☛☞▲✡✡✒★ ☛✒✌ ☛✚✌✙ ❞✡ ✠☛❞✙ ✘✛✜❞✢✣✡ ❉✡✎ ❞✜✍❊✔✖
7. ✔✒✘✒ ✩✢☛✙✲✡✒★ ☛✒✌ ☛✚✌✙ ❞✡ ✠☛❞✙ ✘✛✜❞✢✣✡ ✍✧✍✛❑✎ ❞✜✍❊✔ ✍❊✧❞✡ ❀✡✜▼✡❑ ✛❍✙ ■❏♥✚ ✩✢ ✤✥
✠✡✥✢ ✍❊✧❞✡ ✠✪✡ ◆✧✡✿✛❞ y-✠✪✡ ❞✜ ✍♥❀✡✡ ✛✒★ ✤✥✖
8. ✔✒✘✒ ♥✜❖✡❑☛☞▲✡✡✒★ ☛✒✌ ☛✚✌✙ ❞✡ ✠☛❞✙ ✘✛✜❞✢✣✡ ❉✡✎ ❞✜✍❊✔ ✍❊✧❞✜ ✧✡✍P✡✲✡◗ y-✠✪✡ ✩✢ ✤✥★ ✎✯✡✡
1.
✍❊✧❞✡ ☛✒✌★ ♥✾ ✛❍✙ ■❏♥✚ ✤✥✖
9. ✔✒✘✒ ✠✍✎✩✢☛✙✲✡✒★ ☛✒✌ ☛✚✌✙ ❞✡ ✠☛❞✙ ✘✛✜❞✢✣✡ ❉✡✎ ❞✜✍❊✔ ✍❊✧❞✜ ✧✡✍P✡✲✡◗ x-✠✪✡ ✩✢ ✤✥★
✎✯✡✡ ✍❊✧❞✡ ☛✒★✌♥✾ ✛❍✙ ■❏♥✚ ✤✥✖
10. ✔✒✘✒ ☛☞▲✡✡✒★ ☛✒✌ ☛✚✌✙ ❞✡ ✠☛❞✙ ✘✛✜❞✢✣✡ ❉✡✎ ❞✜✍❊✔ ✍❊✧❞✡ ☛✒✌★ ♥✾ y-✠✪✡ ✩✢ ✤✥ ✠✡✥ ✢ ✍❊✧❞✜
✍❢✡❘✲✡ ✱ ✗❞✡✗❑ ✤✥✖
11. ✍✧❙✧✍✙✍✓✡✎ ✠☛❞✙ ✘✛✜❞✢✣✡✡✒★ ✛✒★ ✘✒ ✍❞✘ ✘✛✜❞✢✣✡ ❞✡ ❚✲✡✩❞ ✤✙ y = c1 ex + c2 e–x ✤✥❣
(A)
d2y
❯ y ❱ 0 (B)
dx 2
d2y
❲ y ❱ 0 (C)
dx 2
d2y
❯ 1 ❱ 0 (D)
dx 2
d2y
❲1 ❱ 0
dx 2
408
① ✁✂✄
12.
❢☎✆☎❢✝❢✞✟✠
✡☛☞✌✍✎✟✟✏✑
☛✏✑
✡✏
❢✌✡
(A)
d2y
dx 2
(C)
d 2 y 2 dy
x
xy 0
dx
dx 2
dy
x
dx
2
✙
✚
xy
✡☛☞✌✍✎✟
x
✛
✌✟
✒✌
❢✓❢✔✟✕✖
y=x
✗✝
(B)
d2y
dx 2
✚
x
dy
dx
✚
xy
(D)
d2y
dx 2
✜
x
dy
dx
✜
xy 0
✛
✗❣✘
x
✢
9.5.
(Methods of Solving First order, First Degree Differential Equations)
✥✣ ✤✦✧
❜✡
✽❢✍✾✿✏❀
❢✓❢❆❇✟✏✑
9.5.1
★✦✩✪✫
☛✏✑
✌☞
✬✭✮
✽❁ ❂✟☛
✗☛
❈❈✟❉
✥✣ ✤✦✧
✯✦✦✰
✌✟✏ ❢✖
✭✩✱
✲✭★✳
✽❁ ❂✟☛
✒✓✑
❃✟✟✠
✓✏ ❄
✴✧✵★✶✷✦✦✩✮
❅✓✌✝
★✦✩
✸✳
✡☛☞✌✍✎✟✟✏✑
★✶✹✩
★✵
✗✝
✌✍☎✏
✌✟✏
✪✭✪✺✻✦✼
✌☞
✠☞☎
✌✍✏✑ ❊✏❋
✐●❍■❏❑▲▼■◆❖
P▲
◗■❘❙
❚◗❑❘
❯❱◆❑▲▼■
(Differential equations with variables
separable)
✽❁❂✟☛
✌✟✏ ❢✖
✒✓✑
✽❁❂✟☛
❃✟✟✠
dy
dx
❇❢❀
F (x, y)
✽❄✝☎
✗❣
✌✗✝✟✠✟
✌✟✏
❅✟❣✍
✗❣❋
❊❞ ✎✟☎✽❄✝
h(y), y
✒✏✡ ✟
❨
❅✓✌✝
✡☛☞✌✍✎✟
❢☎✆☎❢✝❢✞✟✠
❲✽
✌✟
✗✟✏ ✠✟
✗❣❳
F ( x, y )
... (1)
g (x), h(y)
✌✟
✽✍
✗✟✏☎✏
✌✟
✒✌
✓✏ ❄
✽❄✝☎
✡☛☞✌✍✎✟
❲✽
✗❣
❫❴❵
✠✟✏
✌✟✏
☛✏✑
❅❢♦✟❩❇❬✠
✡☛☞✌✍✎✟
❢✌❇✟
❫❴❵
❢☎✆☎❢✝❢✞✟✠
❲✽
❭✟
✡✌✠✟
✽❛ ❂✟❬✌✍✎✟☞❇
☛✏✑
❢✝✞✟✟
❭✟
✗❣
❈✍
g x), x
❭✗✟❪
❫
✓✟✝✟
✡✌✠✟
✡☛☞✌✍✎✟
✗❣❳
dy
= h (y) . g (x)
dx
❇❢❀
h (y)
❝
0,
✠✟✏
❈✍✟✏ ✑
✌✟✏
✽❛❂✟✓r ❄
✌✍✠✏
... (2)
✗❞ ✒
✡☛☞✌✍✎✟
❫❡❵
✌✟✏
1
dy = g (x) dx
h( y )
... (3)
✓✏❄ ❲✽ ☛✏✑ ❢✝✞✟✟ ❭✟ ✡✌✠✟ ✗❣❋ ✡☛☞✌✍✎✟ ❫❤❵ ✓✏❄ ❀✟✏☎✟✏✑ ✽❥✟✟✏✑ ✌✟ ✡☛✟✌✝☎ ✌✍☎✏ ✽✍ ✗☛
1
dy
h( y )
❜✡
✽❁✌✟✍
✡☛☞✌✍✎✟
❫❧❵♠
❢❀✒
✗❞✒
✽❁✟❦✠ ✌✍✠✏ ✗❣✑ ❳
g ( x) dx
... (4)
❅✓✌✝ ✡☛☞✌✍✎✟
✌✟
✗✝
❢☎✆☎❢✝❢✞✟✠
❲✽
☛✏✑
✽❁❀✟☎
✌✍✠✟
H (y) = G (x) + C
❇✗✟❪
H (y)
✒✓✑
G (x)
Ø☛✔✟❳
1
h( y )
✒✓✑
✌✟
✗❣❳
... (5)
g (x)
✓✏❄
✽❁ ❢✠❅✓✌✝❭
✗❣✑
❅✟❣✍
C
♥✓✏✾ ✿
❅❈✍
✗❣❋
✈ ✁✂ ✄☎✆✁✝✞✟
♠✠✡☛☞✌✡ ✍
☛❣
✎✏✑✒ ✓✔✕✑✖✗✘
409
dy x ✙ 1
, (y ✜ 2) ✑✘ ❞✢✘✣✑ ✤✒ ✥✘✦ ✑✕✧★✩✪
✚
dx 2 ✛ y
✧❢✢✘ ✫✢✘ ✤✬ ✧✑
dy
x ✭1
(y ✜ 2)
✮
dx
2✯ y
... (1)
✓✔✕✑✖✗✘ ❧✰✱ ✔✲✳ ✴✖✘✲✳ ✑✘✲ ✣✵✶✘✏✷✸ ✑✖✹✲ ✣✖ ✤✔ ✣③✘✺✦ ✑✖✦✲ ✤✬✳ ✻
(2 – y) dy = (x + 1) dx
... (2)
✓✔✕✑✖✗✘ ❧✼✱ ✏✲✸ ❢✘✲✹✘✲✳ ✣✽✘✘✲✳ ✑✘ ✓✔✘✑✒✹ ✑✖✹✲ ✣✖ ✤✔ ✣③✘✺✦ ✑✖✦✲ ✤✬✳ ✻
❁ (2 ✾ y ) dy ✿ ❁ ( x ❀ 1) dx
y2 x2
❃
❄ x ❄ C1
2
2
x2 + y2 + 2x – 4y + 2 C1 = 0
x2 + y2 + 2x – 4y + C = 0
C = 2C1
2y ❂
✎✶✘✏✘
✎✶✘✏✘
✎✶✘✏✘
★✤✘t
... (3)
✓✔✕✑✖✗✘ ❧❅✱ ✎✏✑✒ ✓✔✕✑✖✗✘ ❧✰✱ ✑✘ ❞✢✘✣✑ ✤✒ ✤✬✪
♠✠✡☛☞✌✡ ❆❇
✎✏✑✒ ✓✔✕✑✖✗✘
dy 1 ✙ y 2
✑✘ ❞✢✘✣✑ ✤✒ ✥✘✦ ✑✕✧★✩✪
✚
dx 1 ✙ x 2
☛❣ ✴♣t✧✑ 1 + y2 ✜ 0, ❜✓✧✒✩ ✴✖✘✲✳ ✑✘✲ ✣✵✶✘✏✷✸ ✑✖✦✲ ✤❈✩ ✧❢✢✘ ✤❈✎ ✘ ✎✏✑✒ ✓✔✕✑✖✗✘ ✧✹❉✹✧✒✧❊✘✦
❋✣ ✔✲✳ ✧✒❊✘✘ ★✘ ✓✑✦✘ ✤✬✻
dy
dx
✚
1 ✙ y 2 1 ✙ x2
... (1)
✓✔✕✑✖✗✘ ❧✰✱ ✏✲✸ ❢✘✲✹✘✲✳ ✣✽✘✘✲ ✳ ✑✘ ✓✔✘✑✒✹ ✑✖✦✲ ✤❈✩ ✤✔ ✣✘✦✲ ✤✬ ✻✳
dy
dx
✮●
2
1✭ y
1 ✭ x2
tan–1 y = tan–1x + C
●
✎✶✘✏✘
✢✤ ✓✔✕✑✖✗✘ ❧✰✱ ✑✘ ❞✢✘✣✑ ✤✒ ✤✬✪
♠✠✡☛☞✌✡ ❆❆
✎✏✑✒ ✓✔✕✑✖✗✘
dy
dx
4 xy 2 ✑✘ ✧✏✧❍✘■❏ ✤✒ ✥✘✦ ✑✕✧★✩❑ ✢✧❢ y = 1 ★▲
x = 0 ✤✘✲
☛❣
✢✧❢ y ✜ 0, ✧❢✢✘ ✤❈✎✘ ✎✏✑✒ ✓✔✕✑✖✗✘ ✧✹❉✹✧✒✧❊✘✦ ❋✣ ✔✲✳ ✧✒❊✘✘ ★✘ ✓✑✦✘ ✤✬✻
dy
✮ ✯ 4 x dx
y2
... (1)
410
① ✁✂✄
❧☎✆✝✞✟✠
✡☛☞
✌✍ ✎
✏✠✍ ✑✠✍ ✒
✓✔✠✠✍ ✒
dy
y2
✜
✝✠
❧☎✠✝✕✑
1
= – 2x2 + C
y
✈✢✠✌✠
y
✤
C
y
✝✠
☎✠✑
☎✍✒
✓✠✗✍
✖✘✙
✒
1
y=1
❧☎✆✝✞✟✠
✖☎
✜
✣
✡✦☞
✓✞
4 x dx
✚ ✛
✈✢✠✌✠
❧☎✆✝✞✟✠
✝✞✑✍
2x
2
... (2)
C
x=0
✈✠✘✞
✡✦☞
✥
☎✍ ✒
✓✐✧✗★✢✠✠✧✓✗
✓✐✧✗★✢✠✠✧✓✗
✝✞✑✍
✝✞✑✍
✓✞
✓✞
✧✏❞
C=–1
✖☎✍✒
✈✌✝✕
✖✫❞
✓✐✠✩✗
✖✠✍ ✗✠
❧☎✆✝✞✟✠
✖✘✪
✝✠
✧✌✧✬✠✭✮
✖✕
1
✤
2x
2
♠✰✱✲✳✴✱
✯
✓✐✠✩✗
✵✶
❧☎✆✝✞✟✠
✲❣
1
✖✠✍ ✗✠
➥✷✏✫ ✡☛✸
✖✘✪
☛☞
❧✍
✹✫✺ ✞✑✍
x *dy = (2x2 + 1) *dx (x
✧✏❞
✈✌✝✕
✖✫❞
❧☎✆✝✞✟✠
dy
❄
✤ ❆
❈
dy
✈✢✠✌✠
❊
● ■
✌✠✕✍
✼
✝✠✍
0)
✡☛☞
✌✍ ✎
✏✠✍ ✑✠✍ ✒
✓✔✠✠✍✒
✝✠
✝✠
❧☎✆✝✞✟✠
✝✆✧✺❞
✧✺❧✝✠
✈✌✝✕
✖✘✪
✿✓
☎✍
✈✧❀✠❁❂❃✗
✧✝❂✠
✺✠
❧✝✗✠
✖✘✙
❅
✯
❇
❉
2x
1
dx
x
❋
❍
... (1)
❏
▲
❧☎✠✝✕✑
✝✞✑✍
✓✞
✖☎
✓✐✠✩✗
✝✞✗✍
✖✘✙
✒
1
dx
x
2x
y = x2 + log | x | + C
✈✢✠✌✠
❧☎✆✝✞✟✠ ✡✦☞
▼❧
✌✻
2 x2 1
dx
x
dy
✖☎
❞✍❧✍
✧✑✽✑✧✕✧✾✠✗
❑
❧☎✆✝✞✟✠
❞✝
✌✫✎✕
✌✍ ✎
✧✏❞
❞✝
✈✌✝✕ ❧☎✆✝✞✟✠
✖✫❞
❞✍❧✍
✧✌✧✬✠✭✮
❧✏★❂
✝✠
✌✍ ✎
... (2)
✖✕
❧☎✆✝✞✟✠
✌✻✠✍ ✒ ✌✍ ✎
◆✠✗
✌✫✎✕
✝✞✑✠
✝✠✍
❖✠✖✗✍
✧✑✿✧✓✗
✖✘✒
✺✠✍
✝✞✗✠
➥✷✏✫
✖✘
✡☛✸
✓✞✒ ✗✫
☛☞
❧✍
✹✫✺ ✞✗✠ ✖✠✍✪
dy
*
②P◗❘❙❚
❯❱❲❱
❳❨❩ ❬❱
❤P ❥ ❚❤❱③
❤✉
❜❱P ❲
dy
dx
❜❝❡❪❛
②❢❙②❱
❤P ❥
❛❦❱❱
◗❤♥ ❛
❭❙
♦♣❱❘❱
q❫❪
❜❱P ❳ ❢❱r❲s
t❳❱❪❛❲♣❱❱❴ ❪
✉❴ ❪
❳❨❡ ✇
♥ ❛
dx ❜❱P❲ dy s❱❴ ❳⑧❦❱❫⑨❵⑩❳⑧❦❱❫⑨❵
❭❬❱❱ ✉❱❘s❲ ❤✉ ◗❤♥ ❛ ❭❙ ♦♣❱❘❱❜❱❴ ❪ s❙ ❭♥❶❳❷❱❸ ⑥❡❱⑤❡❱ s❲ ❭s❛❴ ❤P⑦
❪
❭❪❩ ❹❱❺ ❻ Introduction to calculus and
Analysis, volume-I page 172, By Richard Courant, Fritz John Spinger — Verlog New York.
❤❱❴❛❱
dx
❭❪❫❴❵❛
s❱❴
❭❱④❱❲♣❱ ❭❪⑤❡❱❜❱❴ ❪
s❙
❛❲❤
⑥❡❫❤❱❲
✉❴ ❪ ②❱❛❴
❤P❪⑦
✈ ✁✂ ✄☎✆✁✝✞✟
411
❜✠✡☛☞ ✠✌✍✎✏✑✒ ✓✔✕ ✌✖✗ x = 1, y = 1 ✐✘✡✙✚✛✒✒✡✐✙ ✎✏✜✖ ✐✏ ✢✌✖✗ C = 0 ✐✘✒✣✙ ✢✒✖✙✒ ✢✤✥ C ✎✒
✌✒✜ ✠✌✍✎✏✑✒ ✓✔✕ ✌✖✗ ✐✘✡✙✚✛✒✒✡✐✙ ✎✏✜✖ ✐✏ ✢✌✖✗ ❡✦✒✍✧★ ✩✪ ✎✒ ✠✌✍✎✏✑✒ y = x2 + log | x | ✩✖♦
✿✐ ✌✖✗ ✐✘✒✣✙ ✢✒✖✙✒ ✢✤✥
♠✫✬✭✮✯✬ ✰✱
➥✲✳✴ (–2, 3), ✠✖ ❧✴✵✏✜✖ ✩✒☛✖ ☞✖✠✖ ✩✪ ✎✒ ✠✌✍✎✏✑✒ ✶✒✙ ✎✍✡✵☞ ✡✵✠✩✖♦ ✡✎✠✍
➥✲✳✴ (x, y) ✐✏ ✚✐✷✒✸ ✏✖✹✒✒ ✎✍ ✐✘✩✑✒✙✒
✭❣
2x
✢✤✥
y2
✢✌ ✵✒✜✙✖ ✢✤✗ ✡✎ ✡✎✠✍ ✩✪ ✎✍ ✚✐✷✒✸ ✏✖✹✒✒ ✎✍ ✐✘✩✑✒✙✒
dy
✩✖♦ ✲✏✒✲✏ ✢✒✖✙ ✍ ✢✤✥ ❜✠✡☛☞
dx
dy 2 x
✺
dx y 2
... (1)
♣✏✒✖✗ ✎✒✖ ✐✻✛✒✩✼♦ ✎✏✙✖ ✢✴☞ ✠✌✍✎✏✑✒ ✓✽✕ ✎✒✖ ✡✜✾✜✡☛✡✹✒✙ ✿✐ ✌✖✗ ✡☛✹✒✒ ✵✒ ✠✎✙✒ ✢✤ ❀
y2 dy = 2x dx
... (2)
✠✌✍✎✏✑✒ ✓✔✕ ✩✖♦ ✳✒✖✜✒✖✗ ✐❁✒✒✖✗ ✎✒ ✠✌✒✎☛✜ ✎✏✜✖ ✐✏ ✢✌ ✐✘✒✣✙ ✎✏✙✖ ✢✤✗ ❀
2
❃ y dy ❂ ❃ 2 x dx
y3
x2 C
3
✠✌✍✎✏✑✒ ✓❄✕ ✌✖✗ x = –2, y = 3 ✐✘✡✙✚✛✒✒✡✐✙ ✎✏✜✖ ✐✏ ✢✌✖✗ C = 5 ✐✘✒✣✙ ✢✒✖✙✒ ✢✤✥
C ✎✒ ✌✒✜ ✠✌✍✎✏✑✒ ✓❄✕ ✌✖✗ ✐✘✡✙✚✛✒✒✡✐✙ ✎✏✜✖ ✐✏ ✢✌✖✗ ❡✦✒✍✧★ ✩✪ ✎✒ ✠✌✍✎✏✑✒
❡✛✒✩✒
... (3)
1
y3
2
2
❅ x ❆ 5 ❡✛✒✩✒ y ❇ (3x ❈ 15) 3
3
✩✖♦ ✿✐ ✌✖✗ ✐✘✒✣✙ ✢✒✖✙✒ ✢✤✥
♠✫✬✭✮✯✬ ✰❉ ✡✎✠✍ ✲✤✗ ✎ ✌✖✗ ✌❢ ☛❊✜ ✎✍ ✩✻✡ ❋ 5% ✩✒✡✧✒✸ ✎ ✎✍ ✳✏ ✠✖ ✢✒✖✙ ✍ ✢✤ ✥ ✡✎✙✜✖ ✩✧✒✒✖● ✌✖✗
Rs 1000 ✎✍ ✏✒✡✷✒ ✳✴❧✴✜✍ ✢✒✖ ✵✒☞❧✍❍
✭❣ ✌✒✜ ☛✍✡✵☞ ✡✎✠✍ ✠✌■ t ✐✏ ✌❢☛❊✜ P ✢✤ ✥ ✳✍ ✢✴❜✸ ✠✌✚■✒ ✩✖♦ ❡✜✴✠✒✏
dP
dt
❡✛✒✩✒
dP
dt
5
100
P
P
20
... (1)
✠✌✍✎✏✑✒ ✓✽✕ ✌✖✗ ♣✏✒✖✗ ✎✒✖ ✐✻✛✒✩✼♦ ✎✏✜✖ ✐✏❏ ✢✌ ✐✘✒✣✙ ✎✏✙✖ ✢✤✗ ❀
dP
P
dt
20
... (2)
412
① ✁✂✄
❧☎✆✝✞✟✠
✡☛☞
✌✍ ✎
✏✠✍ ✑✠✍ ✒
✓✔✠✠✍ ✒
✝✠
P C
✈✜✠✌✠
✢
P
✈✠✚✞
t
❜❧✧✕★
✝✠
☎✠✑
❧☎✆✝✞✟✠
❧☎✆✝✞✟✠
✡❞☞
❧✍
✖☎
✓✗ ✠✘✙
✝✞✙✍
✖✚ ✒ ✪
ec1
t
20
e
(
t✖✠✣
P = 1000,
✈✥
✓✞
✛
t
20
e
P
✝✞✑✍
t
C1
20
log P =
✈✜✠✌✠
❧☎✠✝✕✑
✡❞☞
✖☎
✞✦✠✑✍
✓✗✠✘✙
✤
C)
... (3)
t=0
t✥
☎✍ ✒
e C1
✓✞
✝✞✙✍
✖☎
C = 1000
✓✗ ✠✘✙
✝✞✙✍
✖✚✐
✒
✖✚✒ ✪
t
P = 1000 e 20
☎✠✑
✕✆✧t★
t
✌♦✠✠✍ ✩
☎✍✒
☎✫ ✕✬✑
✏✭✮✭ ✑✠
t✠✙✠
✖✠✍
✖✚✯
t
2000 = 1000 e 20
✰
✙✥
t = 20 loge2
✱✲✳✴✵✶✷✸
✼
❧✍
✼✽
1.
3.
✙✝
dy
dx
❄
dy
dx
✛
✓✗✾ ✑✠✍ ✒
✌✍ ✎
☎✍✯
✒
✓✗ ✿❀✍ ✝
✈✌✝✕
✹✺✻
❧☎✆✝✞✟✠
✝✠
❁❀✠✓✝
✖✕
2.
y 1 ( y 1)
4. sec2 x tan y dx + sec2 y tan x dy = 0
❃
❅
❇
❈
dy
dx
❄
4 y2 ( 2
❃
❃
❆
y 2)
❆
dy
dx
6.
7. y log y dx – x dy = 0
5
8. x
✼✼
❧✍
✌✠✕✠
dy
dx
✼❊
❇
3
11. ( x
sin 1 x
❉
✙✝
✧✌✧✾✠♦❋
❅
✝✆✧t★✐
1 cos x
1 cos x
5. (ex + e–x) dy – (ex – e–x) dx = 0
9.
❂✠✙
✌✍ ✎
✖✕
x2
❅
❄
dy
dx
(1 x 2 ) (1 y 2 )
❅
❄ ❃
❅
y5
10. ex tan y dx + (1 – ex) sec2 y dy = 0
✓✗✾ ✑✠✍ ✒
❂✠✙
☎✍✯
✒
✈✌✝✕
❧☎✆✝✞✟✠
✌✍ ✎
✧✕★
✝✆✧t★✐
x 1)
❅
✓✗✿ ❀✍ ✝
dy
= 2x2 + x; y = 1
dx
❀✧✏
x=0
✧✏★
✖✭★
✓✗✧✙✥✒ ✬
✝✠✍
❧✒✙♦
✭ ❋
✝✞✑✍
✈ ✁✂ ✄☎✆✁✝✞✟
12.
x ( x 2 ✠ 1)
413
dy
✡ 1 ; y = 0 ❀☛☞ x = 2
dx
✌ dy ✍
✑ ✎ a (a ✔ R); y = 1 ❀☛☞ x = 0
✒ dx ✓
13. cos ✏
14.
dy
✡ y tan x ; y = 2 ❀☛☞ x = 0
dx
15. ➥✕☞✖ ✗✘✙ ✘✚ ✛✜ ✢✖✣✤✥✜ ✦✧★✜ ✩✪ ✩✜✛✜ ✦✫ ✪✧ ✛✬✭✪✤✮✧ ✯✧✰ ✪✭☛✣✩ ☛✣✛✪✧ ✱✦✪★
✛✬✭✪✤✮✧ y✲ = ex sin x ❣✳✴
16. ✱✦✪★ ✛✬✭✪✤✮✧ xy
dy
dx
( x 2) ( y 2) ✦✜♦ ☛★✩ ➥✕☞✖ ✗✵✙ ✶✵✚ ✛✜ ✢✖✣✤✥✜ ✦✧★✧ ✦✫
✯✧✰ ✪✭☛✣✩✴
17. ➥✕☞✖ (0, –2) ✛✜ ✢✖✣✤✥✜ ✦✧★✜ ✩✪ ✩✜✛✜ ✦✫ ✪✧ ✛✬✭✪✤✮✧ ✯✧✰ ✪✭☛✣✩ ☛✣✛✦✜♦ ☛✪✛✭ ➥✕☞✖
(x, y) ✐✤ ✷✐✸✧✹ ✤✜✺✧✧ ✪✭ ✐✻✦✮✧✰✧ ✱✧✳✤ ✼✛ ➥✕☞✖ ✦✜ ♦ y ☛✥☞✜✸✹ ✧✧❢✪ ✪✧ ✢✖✮✧✥✐♦★ ✼✛ ➥✕☞✖ ✦✜♦ x
☛✥☞✜✸✹ ✧✧❢✪ ✦✜ ♦ ✕✤✧✕✤ ❣✳✴
18. ✩✪ ✦✫ ✦✜♦ ☛✪✛✭ ➥✕☞✖ (x, y) ✐✤ ✷✐✸✧✹ ✤✜✺✧✧ ✪✭ ✐✻ ✦✮✧✰✧✙ ✷✐✸✧✹ ➥✕☞✖ ✪✧✜✙ ➥✕☞✖ (– 4, –3).
✛✜ ☛✬★✧✥✜ ✦✧★✜ ✤✜✺✧✧✺✧❢❧ ✪✭ ✐✻✦✮✧✰✧ ✪✭ ☞✖✢✥
✖ ✭ ❣✳✴ ❀☛☞ ❀❣ ✦✫ ➥✕☞✖ (–2, 1) ✛✜ ✢✖✽✧✤✰✧ ❣✧✜
✰✧✜ r✛ ✦✫ ✪✧ ✛✬✭✪✤✮✧ ✯✧✰ ✪✭☛✣✩✴
19. ✩✪ ✢✧✜★✧✪✧✤ ✢✖✾✕✧✤✜ ✪✧ ✱✧❀✰✥✙ ☛✣✛✜ ❣✦✧ ✿✧✤✪✤ ✐✖♦★✧❀✧ ✣✧ ✤❣✧ ❣✳✙ ☛✷❁✧✤ ✢☛✰ ✛✜ ✕☞★
✤❣✧ ❣✳ ❀☛☞ ✱✧✤❢✿✧ ✬✜❢ r✛ ✢✖✾✕✧✤✜ ✪✭ ☛❥✧❂❀✧ ❃ r✹✪✧r✹ ❣✳ ✱✧✳✤ ❃ ✛✜✦✜❢♦❧ ✕✧☞ ❄ r✹✪✧r✹ ❣✳✙ ✰✧✜
t ✛✜✦✜❢♦❧ ✕✧☞ ✼✛ ✢✖✾✕✧✤✜ ✪✭ ☛❥✧❂❀✧ ✯✧✰ ✪✭☛✣✩✴
20. ☛✪✛✭ ✕✳❢✪ ✬✜❢ ✬❅★❆✥ ✪✭ ✦❇☛❈ r % ✦✧☛❉✧✹✪ ✪✭ ☞✤ ✛✜ ❣✧✜✰✭ ❣✳✴ ❀☛☞ ✵✘✘ ❊✐❀✜ ✵✘ ✦❉✧✧✜❋ ✬✜❢
☞✖✢✥
✖ ✜ ❣✧✜ ✣✧✰✜ ❣✳❢✙ ✰✧✜ r ✪✧ ✬✧✥ ✯✧✰ ✪✭☛✣✩✴ (loge2 = 0.6931).
21. ☛✪✛✭ ✕✳❢✪ ✬✜❢ ✬❅★❆✥ ✪✭ ✦❇☛❈ 5% ✦✧☛❉✧✹✪ ✪✭ ☞✤ ✛✜ ❣✧✜✰✭ ❣✳✴ r✛ ✕✳❢✪ ✬✜❢ Rs 1000 ✣✬✧
✪✤✧✩ ✣✧✰✜ ❣✳❢✴ ✯✧✰ ✪✭☛✣✩ ☛✪ ✵✘ ✦❉✧✹ ✕✧☞ ❀❣ ✤✧☛✸✧ ☛✪✰✥✭ ❣✧✜ ✣✧✩✢✭❞ (e0 5 = 1.648)
22. ☛✪✛✭ ✣✭✦✧✮✧✖ ✛✬❅❣ ✬✜❢ ✣✭✦✧✮✧✖✱✧✜❢ ✪✭ ✛❢✺❀✧ ✵✙ ✘✘✙ ✘✘✘ ❣✳✴ ● ❍✧❢■✧✜ ✬✜❢ r✥✪✭ ✛❢✺❀✧ ✬✜❢ 10%
✪✭ ✦❇☛❈ ❣✧✜✰✭ ❣✳✴ ☛✪✰✥✜ ❍✧❢■✧✜❢ ✬✜❢ ✣✭✦✧✮✧✖✱✧✜❢ ✪✭ ✛❢✺❀✧ ●✙ ✘✘✙ ✘✘✘ ❣✧✜ ✣✧✩✢✭✙ ❀☛☞ ✣✭✦✧✮✧✖✱✧✜❢
✦✜♦ ✦❇☛❈ ✪✭ ☞✤ ✼✥✦✜ ♦ ✼✐☛✷❁✧✰ ✛❢✺❀✧ ✦✜ ♦ ✛✬✧✥✖✐✧✰✭ ❣✳✴
dy
x❏ y
✡e
✪✧ ❑❀✧✐✪ ❣★ ❣✳▲
dx
(A) ex + e–y = C
(B) ex + ey = C
23. ✱✦✪★ ✛✬✭✪✤✮✧
(C) e–x + ey = C
(D) e–x + e–y = C
414
① ✁✂✄
9.5.2
x
✱✏✑
❧☎✆✝✝✞✟✠
y
✏♦✒
✡☛☞✌
✓✔✕✔✓✖✓✗✘✙
✚✒✖✔✘♦✑
✚✛
✓✏✜✘✛
2
F1 (x, y) = y + 2xy,
F3 (x, y) = cos
❀✓✫
✬✚✛✘♦✭✙
✚✐✓✙✷✸✘✘✓✚✙
✢✛
✥
✧
✩
✓✫❀✘
✦
F1 ( x, y) =
F2 ( x, y) =
✻
✻
✻
✻
✻
✻
F4 (x, y) = sin x + cos y
★
✪
✮♦✑
✤✘✱
2
✢✣✓✤✱
F2 (x, y) = 2x – 3y,
y
,
x
✚✒✖✔✘♦✑
(Homogenous differential equations)
❧☎✟☞✍✎✝
x
✹✮
✙✘♦
y
✈✘✯ ✛
✢✘♦
✚✐✘✺✙
2
✓✢❞✣
✰✘✲✳❀♦✙ ✛
✈✜✛
✻
✹✮
✾
✽
❁
❂ ✿
❁
❂
❃ ✼
❄
❃
❄
✻
✹✯✑
F4
F(x, y), n
F ( x, y) = n F(x, y)
✻
✹✮
●✘✘✙
✻
y
❞♦
✓✢
✾
✻
❅ ✻
✓✢❞✣
F1, F2, F3
✚✒✖✔✘♦ ✑
❢✘✣
n
✏♦✒
F( x, y) =
✻
✢✘♦
✻
✓✖✱
n
✻
F (x, y)
✏♦✒
❈✚
✮♦✑
✓✖✗✘✘
✢✘♦ ❉❞ ❈✚ ✮♦✑ ✔✹✣✑ ✓✖✗✘✘ ✤✘ ❞✢✙✘ ✹✯❊ ❉❞❞♦ ✹✮ ✓✔✕✔✓✖✓✗✘✙ ✚✓✛❢✘✘❋✘✘
❀✹
✏✘✖✘
❞✮●✘✘✙✣❀ ✚✒✖✔ ✢✹✖✘✙✘
F4
❢✘✣
❞✮●✘✘✙✣❀
✚✐✓
♦ ❇✘✙
✢✛✙♦
✚✒✖✔
✹✯✑
✔✹✣✑
✵✮✰✘✶
✹✯ ❊
▲
x
2
❏
◆
P
F1 ( x, y )
y2
x2
❘
❚
y2 1
❱
❯
❳
F2 ( x, y )
✽
✿
x1 2
❁
❩
❃
✈✸✘✏✘
F1, F2, F3
✓✢
F1 ( x, y )
✈✸✘✏✘
✹✯ ❊ ❀✓✫ ✓✢❞✣
✰✘✲✳❀♦✙ ✛ ✈✜✛ ✴ ✏♦✒ ✓✖✱
✻
✔✘♦❣ ✢✛✙♦ ✹✯✑ ✓✢ ✬✚✛✘♦✭✙ ✬✫✘✹✛❍✘✘♦✑ ✮♦✑
✤■✓✢
✹✮
✴
✢✛✙♦ ✹✯✑❊
✚✒✖✔
✹✯✑
✱✏✑
✹✯✶
✑
✢✛✙♦
✽ ✼
✢✛✙♦
✤✘ ❞✢✙✘ ✹✯ ✚✛✑✙t ✚✒✖✔
✚✐✘✺✙
x
✻
✻
✚✐✓
♦ ❇✘✙
✴
2
y
y
cos
= 0 F3 (x, y)
x
x
n
F4 (x, y),
F4 ( x, y) = sin x + cos y
❀✹✘❆
✵✮✰✘✶
✻
✻
✻
✓✖✱
(y + 2xy) = F1 (x, y)
(2x – 3y) = F2 (x, y)
F3 ( x, y) = cos
✻
✴ ✏♦✒
F2 ( x, y )
❘
❚
y1 2
❱
❳
F3 ( x, y )
x cos
x
y
▼
2y
x
2x
y
3y
x
❬
❑
❖ ▲
❚
x1h3
❄
3
❚
❨
y
x
❲
y
x
✽
❁
y1h4
❘
❱
❳
x h5
◗
❨
❃
❙
❲
❑
❖
❙
❱
❳
❂ ✿
y
x
x
,
y
❘
y 2 h2
❨
✾
❏
◆
P
◗
❙
❲
x 2 h1
✾
❂
❄
x
,
y
y
x
❙
❲
❨
2, 1, 0
●✘✘✙ ✏✘✖♦ ❞✮●✘✘✙✣❀ ✚✒✖✔
✈ ✁✂ ✄☎✆✁✝✞✟
415
✠ y✡
F4 ( x, y ) ☛ x n h6 ☞ ✌ , n ✏ N ♦✑✒ ✓✔✕✖ ✗✘✖ ✙✘✚ ♦✑✒ ✓✛✜
✍ x✎
✥ x✦
n
F4 (x, y) ✤ y h7 ✧ ★ , n ✫ N
✢✣✘♦✘
✩ y✪
❜✕✓✛✜ ✜✔ ✬✒✛✚ F (x, y), n ❄✘✘✭ ♦✘✛✘ ✕✙❄✘✘✭✖✮ ✬✒✛✚ ✔✯✛✘✭✘ ✯✰ ✮✓✱
n
F (x, y) = x g
y
x
✲✳✴✵✴
ynh
x
y
dy
= F (x, y) ♦✑✒ ✶✬ ♦✘✛✘ ✢♦✔✛ ✕✙✖✔✷✸✘ ✕✙❄✘✘✭✖✮ ✔✯✛✘✭✘ ✯✰ ✮✓✱ F(x, y) ✹✘✺✻✮ ❄✘✘✭ ♦✘✛✘
dx
✕✙❄✘✘✭✖✮ ✬✒✛✚ ✯✰❧
dy
✠ y✡
✾ F ✼ x, y ✽ ✾ g ☞ ✌
dx
✍ x✎
♦✑ ✒ ✶✬ ♦✘✛✑ ✕✙❄✘✘✭✖✮ ✢♦✔✛ ✕✙✖✔✷✸✘ ✔✘✑ ✯✛ ✔✷✚✑ ♦✑ ✒ ✓✛✜ ✯✙
y=vx
... (1)
y
= v ✢✣✘✘✿✭❀
x
... (2)
✬✐✓✭❁✣✘✘✓✬✭ ✔✷✭✑ ✯✰❂
✕✙✖✔✷✸✘ ❃❅❆ ✔✘ x ♦✑ ✒ ✕✘✬✑❇✘ ✢♦✔✛✚ ✔✷✚✑ ✬✷ ✯✙ ✬✐✘❈✭ ✔✷✭✑ ✯✰❂ ❉
dy
dv
❊v❋ x
dx
dx
✕✙✖✔✷✸✘ ❃●❆ ✕✑
✢✣✘✘✿✭❀
... (3)
dy
✔✘ ✙✘✚ ✕✙✖✔✷✸✘ ❃❞❆ ✙✑❂ ✬✐✓✭❁✣✘✘✓✬✭ ✔✷✚✑ ✬✷ ✯✙ ✬✐✘❈✭ ✔✷✭✑ ✯✰❂ ❉
dx
dv
v ❍ x ■ g (v )
dx
dv
x
❊ g (v ) ❏ v
... (4)
dx
✕✙✖✔✷✸✘ ❃❑❆ ✙✑❂ ▲✷✘✑❂ ✔✘✑ ✬▼✣✘♦❀✒ ✔✷✚✑ ✬✷ ✯✙ ✬✐✘❈✭ ✔✷✭✑ ✯✰❂ ❉
dv
dx
◆
g (v ) ❖ v x
... (5)
✕✙✖✔✷✸✘ ❃P❆ ♦✑ ✒ ✱✘✑ ✚✘✑❂ ✬❇✘✘✑❂ ✔✘ ✕✙✘✔✛✚ ✔✷✚✑ ✬✷ ✯✙✑❂ ✬✐✘❈✭ ✯✘✑✭✘ ✯✰❉
❘
✮✓✱ v ✔✘✑
1
dv
◆ ❘ dx ◗ C
g (v) ❖ v
x
... (6)
y
✕✑ ✬✐✓✭❁✣✘✘✓✬✭ ✔✷ ✓✱✮✘ ❙✘✜ ✭✘✑ ✕✙✖✔✷✸✘ ❃❚❆❯ ✢♦✔✛ ✕✙✖✔✷✸✘ ❃❞❆ ✔✘
x
❱✮✘✬✔ ✯✛ ✬✐✱✘✚ ✔✷✭✘ ✯✰❧
416
① ✁✂✄
☎
❀☛☞
❢✆✝✞✟✠✡
F (x, y)
✬✏★ ✩❀
✎✏✏✑
✔✏✖✏
✣✤✢ ✓✏✤ ✗ ☛✜✚✗ ❣✜✗✏♦ ✯✑
❑✏✑
✕✗✪♦
♠✹✠✺✻✟✠
✔♦ ✚
✼✽
☛✖✴
✌✍✎✏✏✑✒❀
✌✍✎✏✏✑✒❀
✰✰✏✫
☞✬✏✏✫♥✴ ☛✕
✌✍✒✕ ✗✘ ✏
✜✚✖✪
✣✤
✑✏♦
✣✍
dx
dy
x
y
✙
✙
F( x , y )
v
✓✈✏✏✫✑ ✭
dx
x
F( x, y ) h
dy
y
✔♦✚ ✓✪✱✌ ✏✗
✶✷✸✑♦
✓✏✵♦
✓✔✕✖
✔♦✚
✛✜
x = vy
✔♦ ✚ ✛✜
✍♦✢
✣✤ ✥
✦✣✏✧
✜✐☛✑✮✈✏✏☛✜✑
✕✗✑♦
✍♦✢ ☛✖✲✏✕✗ ✳❀✏✜✕
✣✖
✣✤✥
✢
✓✔✕✖
(x – y)
✌✍✒✕✗✘✏
dy
= x + 2y
dx
✌✍✎✏✏✑✒❀
✣✤
✓✏✤ ✗
♥✌✕✏
✣✖
❑✏✑ ✕✒☛✦✴✥
✺✾
☛☞✴
✵✴
✓✔✕✖
✌✍✒✕✗✘✏
dy
dx
✍✏✪
✕✏♦
♥✌☛✖✴
✓✑❂
F(x, y)
☛☞❀✏
✣✱✓ ✏
✬✏★ ✩❀
✓☛❁✏✳❀✯✑
☛✕❀✏
✦✏
✌✕✑✏
✣✤ ❂
... (1)
❅
✎✏✏✑
✓✔✕✖
✍♦ ✢
x 2y
x y
x 2y
x y
( x 2 y)
( x y)
F( x, y )
✓✶
✛✜
❃
❄
F (x, y) =
✖✒☛✦✴
☛✪✿✪☛✖☛✲✏✑
✔✏✖✏
✌✍✎✏✏✑✒❀
✴✕
✌✍✒✕✗✘✏
❆
✜✚✖✪
F( x, y )
✣✤✥
✌✍✎✏✏✑✒❀
✓✔✕✖
✌✍✒✕✗✘✏
✣✤✥
❇❈❉❊❋●❍
■
dy
dx
▼
❖ ▼
▼
P
2y
x
y
1
x
1
▲
❘
✌✍✒✕✗✘✏ ❧❙❚ ✕✏ ☞✏❀✏✧ ✜❯✏
✜✚✖✪
✣✤✥
♥✌✕✏♦
✣✖
♥✌☛✖✴
✕✗✪♦
✌✍✒✕✗✘✏
✔♦✚
☛✖✴
❧❱❚
✣✍
g
❏
◆
◆ ❖
g
■
▼
P
◆
y
x
❏
◆
... (2)
◗
◗
y
x
✴✕
✔♦ ✚ ✛✜ ✍♦ ✢ ✣✤ ♥✌☛✖✴ ❀✣ ✬✏★ ✩ ❀ ✎✏✏✑ ✔✏✖✏ ✴✕ ✌✍✎✏✏✑✒❀
✌✍✎✏✏✑✒❀
✜✐☛✑✮✈✏✏✜✪
✕✗✑♦
y = vx
✓✔✕✖
✌✍✒✕✗✘✏
✣✤✥
✣✤❂✢
... (3)
✈ ✁✂ ✄☎✆✁✝✞✟
417
❧✠✡☛☞✌✍ ✎✏✑ ☛✍ x ♦✒✓ ❧✍✔✒✕✍ ✖♦☛✗✘ ☛☞✘✒ ✔☞ ✙✠ ✔✚✍✛✜ ☛☞✜✒ ✙✢✣ ✪
dy
dv
✤v ✥x
dx
dx
... (4)
dy
☛✍ ✠✍✘ ✔✚❞✜✧★✍✍❞✔✜ ☛☞✘✒ ✔☞ ✙✠ ✔✚✍✛✜ ☛☞✜✒ ✙✢✣ ✪
dx
❧✠✡☛☞✌✍ ✎✦✑ ✠✒✣ y ✱♦✣
v✩x
dv 1 ✩ 2v
✫
dx 1 ✬ v
✖★✍✍✭✜✮
x
dv 1 ✩ 2v
✫
✬v
dx 1 ✬ v
✖★✍✍✭✜✮
x
dv v 2 ✯ v ✯ 1
✰
1✲ v
dx
✖★✍✍✭✜✮
v ✳1
✳ dx
dv
✴
x
v2 ✵ v ✵ 1
... (5)
❧✠✡☛☞✌✍ ✎✶✑ ♦✒✓ ✷✍✒✘✍✒✣ ✔✕✍✍✒✣ ☛✍ ❧✠✍☛✗✘ ☛☞✘✒ ✔☞ ✙✠ ✔✚✍✛✜ ☛☞✜✒ ✙✢✣ ✪
✸
v ✳1
dx
dv ✴ ✳ ✸
x
v ✵ v ✵1
2
✖★✍♦✍
1 2v ✵ 1 ✳ 3
dv ✴ ✳ log x ✵ C
2 ✸ v2 ✵ v ✵ 1
✖★✍♦✍
1
2v ✵ 1
3
1
dv ✳ ✸ 2
dv ✴ ✳ log x ✵ C
2
✸
2 v ✵ v ✵1
2 v ✵ v ✵1
✖★✍♦✍
1
3
1
dv ✴ ✳ log x ✵ C
log v 2 ✵ v ✵ 1 ✳ ✸ 2
2
2 v ✵ v ✵1
✖★✍♦✍
1
log v 2
2
✖★✍♦✍
v 1
3
2
1
1
v
2
2
3
2
2
dv
log x
1
3 2
✺ 2v ✵ 1 ✻
log v 2 ✵ v ✵ 1 ✳ .
tan ✹1 ✼
✽ ✴ ✳ log x ✵ C
2
2 3
3 ✿
✾
C
418
① ✁✂✄
1
log v 2
2
✈☎✆✝✆
v
✥✆✏
y
,
x
❧✏
✑✒✓✔✕☎✆✆✓✑✔
✥✖✗✏
✟
✣
y
1
x
✣
✮
✲
log ( y 2
log ( x 2
✈☎✆✝✆
✟
✟
✣
✥✖✔✏
✰
✳
xy
✟
✟
x2 )
y2 )
☛
☛
✈✝✥✶
♠✾✿❁❂❃✿
✹✺✻
❧✙✷✥✖✸✆
❄❅
♥❆✆✆❇❈❉
✼❀✆✑✥
✥✆
✘✶
✓✥ ✈✝✥✶ ❧✙✷✥✖✸✆
3 tan
✱
❁❣
✘✶
✓♥❀✆
❜✆✔
✘❢✈✆
✈✝✥✶
dy
F( x, y )
dx
❙
✝✏♦
y cos
❀✘✆❘
F (x, y) =
✟
C
✎
2y
1✤
★
✭
1✮
2y
✲
2 3 tan
2 3 tan
✞
2y
✍
x
3x
1✠
x 2y
1✠
☞
✞
x
3x
✰
☞
✟
✟
3x
✯
✰
✳
✦
✩
✣
C
✬
C1
✵
✡
✌
x
3x
✣
✟
2C1
✟
C
✎
✡
✌
✎
✘✛✽
x cos
❊
■
y dy
x dx
❋
❏
●
y cos
▲
❊
■
❑
y
x
❋
❏ ❍
x
❧✙▼✆✆✔✷❀ ✘✛ ✈✆✛✖
▲
✥✷✓◆❉✽
❧✙✷✥✖✸✆
y cos
dy
dx
❀✘✆❘
✢
✴
❑
❈❧✥✆
✡
✌
3
✫
✍
❀✘
✟
☞
3 tan
✧
✵
xy
2v 1
1✠
✍
1
log x 2
2
y
1 x2
x
✞
3 tan
☛
✘✛✜ ✪
✯
✰
✴
✈☎✆✝✆
1
log x 2
2
✑✒✆ ✚✔
✘✙
y2
1
log 2
2
x
✈☎✆✝✆
✟
✑✖
1
y2
log 2
2
x
✈☎✆✝✆
v 1
✟
❲
❨
y
x
◗✑
y
x
✥✆
❯
❳ ❱
❩
y
x cos
x
✈✝✥✶
x
✘✛✽
❚
❯
❲
❳
❨
❩
✙✏✜
✓✶P✆✆
◆✆
❧✥✔✆
✘✛ ✪
x
... (1)
y
x cos
x
◗✑
❚
✓✗❖✗✓✶✓P✆✔
❧✙✷✥✖✸✆
✘✛✽
✈ ✁✂ ✄☎✆✁✝✞✟
419
x ❞✠✡ ☛x ❧✡ ☞✌✍ y ❞✠✡ ☛y ❧✡ ✎✏✑✒✓✔✠✠✑✎✒ ❞✕✖✡ ✎✕ ✗✘ ✎✏✠✙✒ ❞✕✒✡ ✗✚✍✛
✢ y✣
✪ [ y cos ✥ ✦ ✤ x ]
✧ x★
0
✩ ✪ [F( x, y )]
F (✜x, ✜y) =
y✣
✢
✪ ✥ x cos ✦
x★
✧
F (x, y) ✬✠✫✭✮ ✯✠✠✒ ✌✠✰✠ ❧✘✯✠✠✒✱✮ ✎✲✰✖ ✗✚✳ ✴❧✑✰☞ ✑✵✮✠ ✗✶✷✠ ✷✌❞✰ ❧✘✱❞✕✸✠ ☞❞ ❧✘✯✠✠✒✱✮
✷✌❞✰ ❧✘✱❞✕✸✠ ✗✚✹ ✴❧❞✠✡ ✗✰ ❞✕✖✡ ✌✡✲ ✑✰☞ ✗✘ ✎✏✑✒✓✔✠✠✎✖ ❞✕✒✡ ✗✚✍✛
y = vx
... (2)
❧✘✱❞✕✸✠ ✺✻✼ ❞✠ x ✌✡✲ ❧✠✎✡♦✠ ✷✌❞✰✖ ❞✕✖✡ ✎✕ ✗✘ ✎✏✠✙✒ ❞✕✒✡ ✗✚✍ ✛
dy
dv
✽v✾ x
dx
dx
❧✘✱❞✕✸✠ ✺✿✼ ✘✡✍ y ☞✌✍
... (3)
dy
❞✠ ✘✠✖ ✎✏✑✒✓✔✠✠✑✎✒ ❞✕✖✡ ✎✕ ✗✘ ✎✏✠✙✒ ❞✕✒✡ ✗✚✍ ✛
dx
v✾ x
dv v cos v ✾ 1
✽
dx
cos v
✷✔✠✌✠
x
dv v cos v ✾ 1
✽
❀v
cos v
dx
✷✔✠✌✠
x
dv
1
❁
dx cos v
✷✔✠✌✠
cos v dv ✽
✴❧✑✰☞
dx
x
1
❂ cos v dv ✽ ❂ x dx
✷✔✠✌✠
sin v = log | x | + log | C|
✷✔✠✌✠
sin v = log | Cx |
v ❞✠✡
y
✎✏✑✒✓✔✠✠✑✎✒ ❞✕✖✡ ✎✕ ✗✘ ✎✏✠✙✒ ❞✕✒✡ ✗✚✍✹
x
❃ y❄
sin ❆ ❇ ❅ log Cx
❈ x❉
✮✗ ✷✌❞✰ ❧✘✱❞✕✸✠ ✺✿✼ ❞✠ ❊✮✠✎❞ ✗✰ ✗✚✹
420
① ✁✂✄
♠☎✆✝✞✟✆ ✠✡
x=0
✜✏✥❀
✝❣
✏✥✜☞
✥☛☞☞✌ ✍✎ ✏✑ ✒✓✑✔ ✕✖✗✑✘✙☞
✤✦
✢❢✒☞
y=1
✏✥✜☞
✒✓✑✔
✢❢ ✒☞
✢☞✧
✕✖✗✑✘✙☞
2x e
dx
dy
F(x, y)
✔✗✏✤✎
x
y
F (x, y)
✍✕✏✔✎❀
✍✕✑☞
✏✥✜☞
✢✔
✕✖✗✑✘✙☞
✕✖✗✑✘✙☞
☛☞✶✷ ✜
✚☞☞✛
✢❢ ✒ ☞
✒✓✑✔
✪☞✛
❧✻✼
❧✿✼
✑✘✬✧
2 xe
x
y
✑☞
✖✧ ✱
y
x
✓✧✸
✕☞✰✧♦☞
dx
dy
✾
v
❅
y
dv
dy
y
✰✸✔✬
✎✑
y
dv
dy
❈
✒✺☞✓☞
y
dv
dy
✾ ❉
✒✺☞✓☞
2ev dv
✵
✑✗✏✤✎✫
✢✣✲
x
y
✵
✒✓✑✔
✰✐ ✏✛✹✺☞☞✰✬
✑✘✬✧
y
[F ( x, y )]
x
y
✢✣ ✫
✕✖✚☞☞✛✗✜
✰✘
✑✘✛✧
✕✖✗✑✘✙☞
✢✣ ✫
✢✣ ✫
✱
✢✖
✰✐ ☞✽✛
✑✘✬✧
✰✘
✑✘✛✧
✢✣ ✱ ✲
dv
dy
✖☞✬
✰✐ ✏✛✹✺☞☞✏✰✛
2v e v 1
2e v
2v e v 1
v
2ev
✒✺☞✓☞
✕✑✛☞
2 ye
❄
❆
✤☞
✪☞✛
F( x, y)
✛✦
x
y
✒✓✑✔✬
✑☞
✏✔✮☞☞
✢✔
... (1)
x = vy
✢✖
v+ y
dx
dy
❁❂❃
✳
✕✖✗✑✘✙☞
✏✔✎❀
✖✧✱
✏✓✏☛☞★✩
2 xe
✕✖✚☞☞✛✗✜
✓✧✸
✯✰
✑☞
✕✖✚☞☞✛✗✜ ✢✣ ✒☞✣ ✘
y
✴
✓☞✔☞
✕✖✗✑✘✙☞
0
x
y
2 ye
✒✛✲
✍✕
x
y
2 y e dx ( y 2 x e ) dy
✏✬✭✬✏✔✏✮☞✛
2y e
✖☞✬
✛☞✧
x
y
❇
❇
1
2ev
dy
y
✢✖
✰✐☞✽✛
✑✘✛✧
✢✣✱ ✲
✈ ✁✂ ✄☎✆✁✝✞✟
v
421
dy
✠✡☛☞☛
✎ 2e . dv ✌ ✍ ✎ y
✠✡☛☞☛
2 ev = – log | y | + C
x
v ❞☛✏ y ❧✏ ✑✒✓✔✕✡☛☛✓✑✔ ❞✖✗✏ ✑✖ ✘✙ ✑✒☛✚✔ ❞✖✔✏ ✘✛❛ ✪
x
y
2 e + log | y | = C
❧✙✜❞✖✢☛ ✣✤✥ ✙✏❛✦ x = 0 ✱☞❛ y = 1 ✑✒✓✔✕✡☛☛✓✑✔ ❞✖✗✏ ✑✖ ✘✙ ✑✒☛✚✔ ❞✖✔✏ ✘✛❛ ✪
... (3)
2 e0 + log | 1 | = C ✧ C = 2
C ❞☛ ✙☛✗ ❧✙✜❞✖✢☛ ✣✤✥ ✙✏❛ ✑✒✓✔✕✡☛☛✓✑✔ ❞✖✗✏ ✑✖ ✘✙ ✑✒☛✚✔ ❞✖✔✏ ✘✛❛ ✪
x
y
2 e + log | y | = 2
❀✘ ✓★✱ ✘✩✱ ✠☞❞✫ ❧✙✜❞✖✢☛ ❞☛ ✱❞ ✓☞✓✬☛✭✮ ✘✫ ✘✛✯
♠✰✲✳✴✵✲ ✶✷
★✬☛☛♥✸✱ ✓❞ ☞✹☛✏❛ ❞☛ ☞✩✺✫✦ ✓✻✗☞✏✺ ✓❞❧✜ ✼✽★✩ (x, y) ✑✖ ✕✑✬☛♥ ✖✏✐☛☛ ❞✜ ✑✒☞✢☛✔☛
x2 ✾ y2
✘✛✦ x2 – y2 = cx ⑥☛✖☛ ✑✒★✿☛ ✘✛ ✯
2 xy
✳❣ ✘✙ ✻☛✗✔✏ ✘✛❛ ✓❞ ✱❞ ☞✹ ☞✏✺ ✓❞❧✜ ✼✽★✩ ✑✖ ✕✑✬☛♥ ✖✏✐☛☛ ❞✜ ✑✒☞✢☛✔☛
✸❧✓✫✱
dy
☞✏✺ ✽✖☛✽✖ ✘☛✏✔✜ ✘✛✯
dx
y2
1❂ 2
dy x 2 ✾ y 2
dy
x
❁
❃
❀☛
2
y
dx
2 xy
dx
x
✕✑✭✮✔✪ ❧✙✜❞✖✢☛ ✣▲✥ ❧✙❄☛☛✔✜❀ ✠☞❞✫ ❧✙✜❞✖✢☛ ✘✛✯
✸❧❞☛✏ ✘✫ ❞✖✗✏ ☞✏✺ ✓✫✱ ✘✙ y = vx ✑✒✓✔✕✡☛☛✑✗ ❞✖✔✏ ✘❛✛ ✯
y = vx ☞✺☛ x ☞✏✺ ❧☛✑✏♦☛ ✠☞❞✫✗ ❞✖✗✏ ✑✖ ✘✙ ✑☛✔✏ ✘✛✪❛
dy
dv
❅v❆ x
dx
dx
✠✔✪
❀☛ v ❇ x
dv 1 ❇ v 2
❈
2v
dx
2v
dx
2v
dx
dv 1 ❉ v 2
dv ❊
dv ❊ ❋
x ❈
❀☛
❀☛ 2
2
x
x
v ❋1
1❋ v
2v
dx
... (1)
422
① ✁✂✄
2v
❜☎✆✝✞
✈☛☞✌☞
✈☛☞✌☞
y
x
❞☞✍
1
dx
x
dv
✟ ✠
❧✒
✔✕
❧✎ ☞✖✏
✢ ✣
C1
✡
v 1
log | v2 – 1 | = – log | x | + log | C1 |
log | (v2 – 1) (x) | = log |C1|
(v2 – 1) x = ± C1
✈☛☞✌☞
v
2
✡
☎✍
❧✎ ✆✏✑☛☞☞✆❧✏
✚
✤
✦
✠
❞✒✓✍
y2
x2
✛
✜
1 x
✥
✔✗✙
✘
✧
(y2 – x2) = ± C1 x
✈☛☞✌☞
❞✒✏✍
❀☞
x2 – y2 = Cx
✐★✩✪✫✬✭✮
✶
☎✍
☎✍
✶✲
✏❞
❧✎✴ ❀✍ ❞
✌✍ ✳
❞☞✍
❧✎ ✴ ❀✍❞ ❧✎ ✵ ✓
✔✝
✕✍✘ ✷✵☞☞✸❜✞
✆❞
✆✷❀☞
❆
7.
❉
dy
dx
❂
x cos
10.
x
✤
✦
❋
8.
✚
dy
dx
❏
y
x2
y
x
❃
✛
✥ ❈
2 y2
1 e
☎✕✺❞✒✻☞
☎✕✼☞☞✏✺❀
✔✗ ✈☞✗✒
✧
❑
✦
x sin
dx e
✚
✤
❍
▼
x
y
y
x
1
y
x
✾
y
x
4. (x – y ) dx + 2xy dy = 0
2
x2
6. x dy – y dx =
✛❇
✥❊
y dx
✧●
■
◆ ▲
x
❆
✢ ❉
y sin
✚
✤
✦
❋
0
9.
y
x
✛
✥
✜
x cos
✧
y dx
P
✚
✤
✦
❑
y
x
✛❇
✥❊
❅
y 2 dx
x dy
✧●
x log
❍
▼
❖
y
dy 2 x dy
x
■
◆
❏
▲
0
P
x
dy 0
y
✶✶ ☎✍ ✶◗ ✏❞ ✌✍✳ ❧✎✵✓☞✍ ✘ ✕✍ ✘ ❧✎✴ ❀✍ ❞ ✈✌❞✝ ☎✕✺❞✒✻☞ ✌✍✳ ✆✝✞ ✆✷✞ ✔✹✞ ❧✎ ✆✏❘✘❙ ❞☞✍ ☎✘✏❚
✹ ❯ ❞✒✓✍
✆✌✆✵☞❚❯
✔✝
❢☞✏
❜✓✕✍ ✘
✿ ❁
2
xy
❄
y sin
❖
x
y
✈✌❞✝
y
2.
3. (x – y) dy – (x + y) dx = 0
x2
✔✹✈ ☞
❞✺✆✽✞✙
1. (x2 + xy) dy = (x2 + y2) dx
5.
✯✰✱
❞✺✆✽✞❱
11. (x + y) dy + (x – y) dx = 0; y = 1
x=1
2
2
x=1
12. x dy + (xy + y ) dx = 0; y = 1
❀✆✷
❀✆✷
✌☞✝☞
✈ ✁✂ ✄☎✆✁✝✞✟
423
✠
✡
☛
2☞ y✌
13. ✓ x sin ✑ ✒ ✍ y ✔ dx ✎ x dy ✏ 0; y ✏ ❀✙✚ x = 1
4
✕ x✖
✗
✘
dy y
✛ y✜
✢ ✣ cosec ✥ ✦ ✤ 0 ; y = 0 ❀✙✚ x = 1
14.
dx x
✧ x★
2
2 dy
✫ 0 ; y = 2 ❀✙✚ x = 1
15. 2 xy ✩ y ✪ 2 x
dx
dx
☞x✌
✏ h ✑ ✒ ♦✬✭ ✮✯ ♦✰✱✬ ✲✳✴✰✰✵✶❀ ✷♦✸✱ ✲✳✶✸✹✺✰ ✸✰✬ ✻✱ ✸✹✼✬ ♦✬✭ ✙✱✽ ✙✼✾✼✙✱✙✿✰✵
16.
dy
✕ y✖
✳✬❡ ✲✬ ✸✰❁✼ ✲✰ ✯❂✙✵❃❄✰✰✯✼ ✙✸❀✰ ❅✰✵✰ ✻❁❆
(A) y = vx
(B) v = yx
(C) x = vy
(D) x = v
17. ✙✼✾✼✙✱✙✿✰✵ ✳✬❡ ✲✬ ✸✰❁✼ ✲✰ ✲✳✴✰✰✵✶❀ ✷♦✸✱ ✲✳✶✸✹✺✰ ✻❁❢
(A) (4x + 6y + 5) dy – (3y + 2x + 4) dx = 0
(B) (xy) dx – (x3 + y3) dy = 0
(C) (x3 + 2y2) dx + 2xy dy = 0
(D) y2 dx + (x2 – xy – y2) dy = 0
9.5.3 ❥❇❈❉❊❋ ●❍❋■ ❏❑▲❋❥▼❊ (Linear differential equations)
dy
◆ Py ❖ Q ,
dx
♦✬✭ ✮✯ ♦✰✱✰ ✷♦✸✱ ✲✳✶✸✹✺✰P ✙❅✲✳✬❡ P ✽♦❡ Q ✷◗✹ ✷❄✰♦✰ ♦✬✭♦✱ x ♦✬✭ ✯✭✱✼ ✻❁❡P ✯❂ ❄✰✳ ✸✰✬✙❘
✸✰ ✹❁✙✿✰✸ ✷♦✸✱ ✲✳✶✸✹✺✰ ✸✻✱✰✵✰ ✻❁❞ ✯❂❄✰✳ ✸✰✬ ✙❘ ♦✬✭ ✹❁✙✿✰✸ ✷♦✸✱ ✲✳✶✸✹✺✰ ♦✬✭ ♦❙✭❚
♠✚✰✻✹✺✰ ❯✲ ✯❂✸✰✹ ✻❁❡❆
dy
◆ y ❖ sin x
dx
dy ❱ 1 ❲
x
❳❩ ❬y ❨e
dx ❭ x ❪
dy ☞ y ✌ 1
✎
✏
dx ✑✕ x log x ✒✖ x
✯❂❄✰✳ ✸✰✬✙❘ ♦✬✭ ✹❁✙✿✰✸ ✷♦✸✱ ✲✳✶✸✹✺✰ ✸✰ ✚✐✲✹✰ ✮✯ ✲✬♦❡✭✬ ❫
dx
❴ P1 x ❵ Q1 ✻❁P ✙❅✲✳✬❡ P1
dy
✷✰❁✹ Q1 ✷◗✹ ✷❄✰♦✰ ♦✬✭♦✱ y ♦✬✭ ✯✭✱✼ ✻❁❡❞ ❯✲ ✯❂✸✰✹ ♦✬✭ ✷♦✸✱ ✲✳✶✸✹✺✰ ♦✬✭ ♦❙✭❚ ♠✚✰✻✹✺✰
✙✼✾✼✙✱✙✿✰✵ ✻❁❡❆
dx
❴ x ❵ cos y
dy
424
① ✁✂✄
dx
dy
✐✟✠✡☛
☞✡✌✍✎
✏✌✑
✒✓✍✔✡☞
❞✖
☞✒✜✌
✏✌✑
✍✖✢
2x
y
✕✏☞✖
dy
dx
☞✡✌
✆
✝
y 2e
✞
☎
y
✗☛✘☞✒✙✡
... (1)
Py Q
✚
✛
✗☛✘☞✒✙✡
✏✌✑
✣✡✌✜✡✌✤ ✐✥✡✡✌✤
x
☞✡✌
✏✌✑
✐✑✖✜
g (x)
✗✌
✦✧ ✙✡✡
☞✒✜✌
✐✒
❞☛
✐✟✡★✩
☞✒✩✌ ❞✓✤ ✪
dy
+ P. g (x) y = Q . g (x)
dx
g (x)
g (x)
☞✡
✫✬✜
✭✗
✐✟☞✡✒
☞✘✍✮✢
✗☛✘☞✒✙✡
☞✡
✯✡✬✡✰
✐✥✡
y . g (x)
☞✡
✕✏☞✖✮
✯✜
✮✡✢ ✪
dy
d
+ P. g (x) y =
[y . g (x)]
dx
dx
g (x)
✕✠✡✡✈✩✱
✩✡✍☞
... (2)
dy
dy
+ P. g (x) y = g (x)
+ y g (x)
dx
dx
P. g (x) = g (x)
g ( x)
P=
g ( x)
g (x)
✕✠✡✏✡
✳
✲
✲
✴
✕✠✡✏✡
✣✡✌✜✡✌✤
✐✥✡✡✌✤
x
☞✡
✏✌✑
✗✡✐✌✥✡
✵
✕✠✡✏✡
✸
✗☛✘☞✒✙✡
✐✑✖✜
✾✿❀❁❂❃
✗☛✘☞✒✙✡
✐✒
❞☛
✐✟✡★✩
☞✒✩✌
❞✓✤ ✪
g ( x)
dx
g ( x)
✴
✵
P. dx = log g ( x)
✶
g (x) = e P dx
g(x) = e pdx
❧✺✻
☞✡
✕✏☞✖✮
❄❅❆❀❁
❧❇✻
✼
☞✡✌
☛✌✤
(I.F.)
g (x)
e
✕✠✡✏✡
Pdx =
☞✒✜✌
✷
✹
✕✠✡✏✡
✍☞✗✘
✗☛✡☞✖✜
❉
✯✜
✮✡✩✡
☞❞✖✡✩✡
☞✡
pdx
☛✡✜
dy
dx
d
ye
dx
●
■
❊
✗✌
✦✧ ✙✡✡
☞✒✜✌
❞✓ ❢
✬❞
✐✑✖✜
✐✒
✽✗
✗☛✘☞✒✙✡
g(x) = e pdx
✼
✍✣✢
☞✡
❞✧✢
❞✓ ❢
✐✟✍✩❈✠✡✡✍✐✩
Pe
❉
P dx
❍ ✛
pdx
☞✒✜✌
y Q. e
❋
Qe
■
P dx
❉
✐✒
pdx
❞☛
✐✟✡ ★✩
☞✒✩✌
❞✓✤ ✪
✯✡✬✡✰
✐✥✡
✕✏☞✖
x
✩✠✡✡
y
✗☛✘☞✒✙✡
✏✌✑
☞✡
✈ ✁✂ ✄☎✆✁✝✞✟
425
♥✠✡☛✠✡☞ ✌✍✠✠✡☞ ✎✠ x, ♦✡✏ ✑✠✌✡✍ ✠ ✑✒✠✎✓☛ ✎✔☛✡ ✌✔ ✕✒ ✌✖✠✗✘ ✎✔✘✡ ✕✙✚☞
y e
✛✜✠♦✠
P dx
Q e
P dx
y e
P dx
Q e
P dx
dx
dx C
❀✕ ✛♦✎✓ ✑✒✢✎✔✣✠ ✎✠ ✤❀✠✌✎ ✕✓ ✕✙✥
✐✦✧★✩ ✪★✫✬✭ ✮✫✯ ✰✱✬✲★✪ ✳✮✪✴ ✵✩✶✪✰✷★ ✪★✫ ✸✴ ✪✰✹✫ ✮✫✯ ✬✴✺ ✵✬✻✩✬✴✼ ✽✰✷★✾
(i) ❢♥✿ ✕❁✿ ✛♦✎✓ ✑✒✢✎✔✣✠ ✎✠✡
dy
❂ Py ❃ Q ♦✡✏ ❄✌ ✒✡☞ ❢✓❢❅✠✿ ❢❆✑✒✡☞ P, Q ✛❇✔ ✛✜✠♦✠
dx
♦✡✏♦✓ x ♦✡✏ ✌✏✓☛ ✕✙✥☞
(ii) ✑✒✠✎✓☛ ❧❁✣✠✎ (I.F.) = e❈ P dx ❑✠✘ ✎✢❢❆✿✥
(iii) ❢♥✿ ✕❁✿ ✛♦✎✓ ✑✒✢✎✔✣✠ ✎✠ ✕✓ ❢☛❉☛❢✓❢❅✠✘ ❄✌ ✒✡☞ ❢✓❢❅✠✿✚
y . (I.F.) =
Q × I.F. dx C
❀❢♥ ✌✖✜✠✒ ✎✠✡❢❊ ✎✠ ✔✙❢❅✠✎ ✛♦✎✓ ✑✒✢✎✔✣✠
dx
❋ P1 x ● Q1 ♦✡✏ ❄✌ ✒✡☞ ✕✙ ❢❆✑✒✡☞ P ✛✠✙✔ Q
1
1
dy
✛❇✔ ✛✜✠♦✠ ♦✡✏♦✓ y ♦✡✏ ✌✏✓☛ ✕✙❍☞ ✘■ I.F. = e ❏ P1 dy ✛✠✙✔
x . (I.F.) =
♠▲★✸✰✷★ ▼◆
✸✴
✛♦✎✓ ✑✒✢✎✔✣✠
Q1 × I.F. dy C ✛♦✎✓ ✑✒✢✎✔✣✠ ✎✠ ✕✓ ✕✙✥
dy
❖ y P cos x ✎✠ ✤❀✠✌✎ ✕✓ ❑✠✘ ✎✢❢❆✿✥
dx
❢♥❀✠ ✕❁✛✠ ✛♦✎✓ ✑✒✢✎✔✣✠
dy
◗ Py P Q ✕✙, ❆✕✠t P = –1 ✛✠✙✔ Q = cos x
dx
❜✑❢✓✿ I.F.
e
1 dx
e
x
✑✒✢✎✔✣✠ ♦✡✏ ♥✠✡☛✠✡☞ ✌✍✠✠✡☞ ✎✠✡ I.F. ✑✡ ❧❁✣✠✠ ✎✔☛✡ ✌✔ ✕✒ ✌✖✠✗✘ ✎✔✘✡ ✕✙✚☞
e❘ x
✛✜✠♦✠
dy ❘ x
❘x
❖ e y P e cos x
dx
d
❯x
❯x
❙ y e ❚ ❃ e cos x
dx
426
♥☎✆✝☎✆✞
① ✁✂✄
✟✠☎☎✆✞
✡☎
x
♦✆☛
☞☎✟✆✠ ☎
ye
☞✌☎✡✍✝
x
✖
✗
✡✎✝✆
✟✎
x
✖
✏✌
✟✑ ☎✒✓
✡✎✓✆
✏✔ ✕
✞
e cos x dx C
✙
... (1)
✘
x
I = e cos x dx
✖
✌☎✝
✍❡✚✛✜
✚✡
✙
= cos x
e
✢
✖
✥
✧ ✤
✈✰☎♦☎
☞✌❡✡✎❧☎
✤
cos x e
✖
=
✬
cos x e
✩
=
✤
cos x e
✖
✳✴✵
✌✆✞
I
✡☎
x
✤
x
✙
✬✪
✭
x
( sin x) ( e x ) dx
✤
✖
x
dx
sin x(– e x )
✩
sin x e
✘
✖
✤
sin x e
(sin x cos x) e
2
✲
✌☎✝
ye
✟✑ ✚✓❞✰☎☎✚✟✓
✖
x
✤
✙
✬
✯
cos x ( e x ) dx
✬
cos x e
✖
x
✩
✫
✮
dx
✚♥✜
✏✾✜
♠❂❃❄❅❆❃
✈♦✡✍
❇❈
✚♥❀☎
☞✌❡✡✎❧☎
✈♦✡✍
✏✾✈ ☎
✲
x
☞✌❡✡✎❧☎
✳✴✵
♦✆☛
♥☎✆✝☎✆ ✞
❜☞✚✍✜
❑
Py Q ,
▲
♦✆☛
❉
✟✠☎☎✆✞
dy
dx
dy
dx
✷
❉
▼✟
✡☎
✏✍
dy
dx
2y
x
x
✹
✡✎✓✆
✏✔ ✕
✞
C
x2
✡☎✆
x
☞✆
2
y
x
✡☎
❊
❉
✏✔❁
❊
x 2 ( x 0)
❋
✡☎
✿❀☎✟✡
✏✍
●☎✓
✡❡✚✛✜❁
✏✔ ✕
❊
P
✱
✟✑ ☎✒✓
C ex
✿❀☎✟✡
2y
I.F. = e
✏✌
✽
☞✌❡✡✎❧☎
dy
dx
✟✎
✻
☞✌❡✡✎❧☎
✈♦✡✍
x
✡✎✝✆
sin x cos x
2
y
✈✰☎♦☎
✱
sin x cos x
e
2
✶
✸ ✺
x
✱
✼
❀✏❏
✦ ✤ ✙
★
1
=
I=
✈✰☎♦☎
❄❣
✣
I = – e–x cos x + sin x e–x – I
2I = (sin x – cos x) e–x
✈✰☎♦☎
❀✏
x
... (1)
❍☎☎■
♥✆✝✆
✟✎
✏✌
✟✑☎✒✓
✡✎✓✆
✏✔✞✕
x
✎✔ ✚◆☎✡
2
dx
x
✈♦✡✍
☞✌❡✡✎❧☎
log x
= e2 log x = e
2
◗
x2
✏✔❁
❀✏☎❖
P
[
▲
2
x
✜♦✞
t❙❘❦ ❢❚
Q=x
elog f ( x )
✏✔❁
f ( x)]
✈ ✁✂ ✄☎✆✁✝✞✟
427
❜✠✡☛☞ ✡✌☞ ✍✎☞ ✠✏✑✒✓✔✕ ✒✕ ✍☛ ✍✖✗
2
3
y . x2 = ✙ ( x) ( x ) dx ✘ C = ✙ x dx ✘ C
x2
✢2
y✣
✤Cx
4
✚✛✕✜✕
❀✍ ✡✌☞ ✍✎☞ ✚✜✒☛ ✠✏✑✒✓✔✕ ✒✕ ✥❀✕✦✒ ✍☛ ✍✖✧
♠★✩✪✫✬✩ ✭✮
✪❣
✚✜✒☛ ✠✏✑✒✓✔✕ y dx – (x + 2y2) dy = 0 ✒✕ ✥❀✕✦✒ ✍☛ ❞✕✯ ✒✑✡✰☞✧
✡✌❀✕ ✍✎✚✕ ✚✜✒☛ ✠✏✑✒✓✔✕ ✡❢✱❢✡☛✡✲✕✯ ✳✦ ✏✴✵ ✡☛✲✕✕ ✰✕ ✠✒✯✕ ✍✖✗
dx x
✶ ✷ 2y
dy y
❀✍✸
1
dx
✹ P1 x ✺ Q1 , ✜✴♦ ✳✦ ✜✕☛✕ ✓✖ ✡✲✕✒ ✚✜✒☛ ✠✏✑✒✓✔✕ ✍✖ ✧ ❀✍✕✻ P1 ✺ ✼
☞✜✵
y
dy
1
Q1 = 2y ✍✖✧ ❜✠✡☛☞ I.F ❁ e
✿ ✾ y dy
❁e
✾ log y
❁e
log ( y )✽1
❁
1
y
✚✯✗ ✡✌☞ ✍✎☞ ✚✜✒☛ ✠✏✑✒✓✔✕ ✒✕ ✍☛ ✍✖✗
x
✚✛✕✜✕
✚✛✕✜✕
✚✛✕✜✕
1
❂1❃
❄ ✙ (2 y ) ❅ ❆ dy ✘ C
y
❇ y❈
x
2dy C
y
x
✷ 2y ❉ C
y
x = 2y2 + Cy
❀✍ ✡✌☞ ✍✎☞ ✚✜✒☛ ✠✏✑✒✓✔✕ ✒✕ ✥❀✕✦✒ ✍☛ ✍✖✧
♠★✩✪✫✬✩ ✭✭ ✚✜✒☛ ✠✏✑✒✓✔✕
dx
✹ y cot x = 2x + x2 cot x (x ❊ 0)
dy
✒✕ ✡✜✡❋✕●❍ ✍☛ ❞✕✯ ✒✑✡✰☞✸ ✡✌❀✕ ✍✎✚✕ ✍✖ ✡✒ y = 0 ❀✡✌ x ❏
✪❣ ✡✌❀✕ ✍✎✚✕ ✚✜✒☛ ✠✏✑✒✓✔✕
■
2
dy
❑ Py ▲ Q , ✜✴♦ ✳✦ ✒✕ ✓✖✡✲✕✒ ✚✜✒☛ ✠✏✑✒✓✔✕ ✍✖✧ ❀✍✕✻
dx
428
① ✁✂✄
P = cot x
✈☎✆ ✝
Q = 2x + x2 cot x
I.F = e
✈✎✏
✈✑✒☛
✠✓✔✒✝✕☎
✒☎
❣✆ ✞
✟✠✡☛☞
cot x dx
✌
✍
❣☛
y . sin x
elog sin x
❣✆ ✏
(2x + x2 cot x) sin x dx + C
❂ ✖
2x sin x dx + x2 cos x dx + C
✈✗☎✑☎
y sin x
✈✗☎✑☎
y sin x = sin x
❂ ✖
✖
✘
✜
✣
y sin x
✈✗☎✑☎
2 x2
2
x 2 sin x
✦
sin x
✍
✚
✙
✢ ✚
✤
✥
✥
cos x
✘
✜
✣
x 2 cos x dx
2 x2
dx
2
✙
✢
✛
✤
✛
✥
✥
x 2 cos x dx C
✛
x 2 cos x dx C
✛
y sin x = x2 sin x + C
✈✗☎✑☎
✠✓✔✒✝✕☎
❧✧★
y=0
✓✩ ✪
✫
x
☞✑✪
... (1)
✬
✐✭ ✡✎✮✗☎☎✡✐✎
2
✒✝✯✩
✐✝
❣✓
✐✭☎✰✎
✒✝✎✩
❣✆ ✪ ✏
2
0
C=
✈✗☎✑☎
✠✓✔✒✝✕☎
❧✧★
C
✓✩ ✪
sin
2
✒☎
✱✲
2
4
✓☎✯
✐✭ ✡✎✮✗☎☎✡✐✎
y sin x = x 2 sin x
y = x2
✈✗☎✑☎
❀❣
✡✶☞
♠✻✼✽✾✿✼
✡✒✠✔
y
❣✷☞
❁❃
●❢✶✷
✡✯✶✩ ❍✸ ☎☎✪✒
✽▲
❣✓
✈✗☎✑☎
✈✑✒☛
✡❢❄✶✷
(x, y)
✐✝
❧✒☎✩ ✡✺★
❈☎✯✎✩
❣✆✪
✳
✴
✠✓✔✒✝✕☎
❧❅❆
✧★
✮✐✸☎❍
✑✩❋
✡✒
✠✩
✒☎
✲
✱
✐✝
❣✓
✒✔
✒✔
(sin x
✐✭ ✑✕☎✎☎❆
✑✩ ❋
✬
x xy
◆
xy
✒✝✎✩
❣✆ ✏
✪
4
✡✑✡✸☎✹✺
✮✐✸☎❍
✐✭ ☎✰✎
2
✵
❣☛
0)
❣✆✞
❇✷ ❈✝✯✩ ✑☎☛✩ ☞✒ ✑❉
✝✩ ■☎☎
✑❉
✒✝✯✩
2
4 sin x
❇✷ ✕☎✯✐❋☛
dy
dx
dy
dx
C
2
❀☎✩ ❇
✝✩ ■☎☎
❏✠
✑✩❋
✒✔
✒☎
●❢✶✷
✑✩❋
❢✝☎❢✝
❣✆✞
✐✭✑ ✕☎✎☎
✠✓✔✒✝✕☎ ❊☎✎
x
dy
dx
✡✯✶✩❍✸ ☎☎✪✒
✑✩❋
❢✝☎❢✝
✒✔✡❈☞❆
❧❑☎✷ ❈★
❣☎✩ ✎✔
✎✗☎☎
❣✆✞
❀✡✶
x
✟✠
✑❉ ✑✩ ❋
✡✯✶✩❍✸ ☎☎✪✒
✈☎✆✝
✟✠✡☛☞
▼
❖
x
... (1)
✈ ✁✂ ✄☎✆✁✝✞✟
❧✠✡☛☞✌✍ ✎✏✑✒
429
dy
✓ Py ✔ Q ♦✕✖ ✗✘ ☛✍ ☞✙✚✛✍☛ ✜♦☛✢ ❧✠✡☛☞✌✍ ✣✙✤ ✥✣✍✦ P = – x ✧♦★
dx
Q = x ✣✙✤ ❣❧✚✢✧
x dx
I.F. = e
e
x2
2
✜✩✪ ✚✫✧ ✣✬✧ ❧✠✡☛☞✌✍ ☛✍ ✣✢ ✣✙✪
✯ x2
✯ x2
2
✭
y .e
✰ ✲ ( x ) e 2 ✮ dx ✱ C
✠✍❡ ✢✡✚✳✧
✠✍❡ ✢✡✚✳✧
❣❧✚✢✧
I ✵ ✶ ( x) e
✷x
2
... (2)
✴ x2
2
dx
2
✰ t , ✩r – x dx = dt ✥✍ x dx = – dt
t
t
I ✵ ✸ ✶ e dt ✵ ✸ e ✵ – e
✴ x2
2
❧✠✡☛☞✌✍ (2) ✠✕★ I ☛✍ ✠✍❡ ✘✹✚✩✺✻✍✍✚✘✩ ☛☞❡✕ ✘☞✒ ✣✠ ✘✍✩✕ ✣✙★✪
✼ x2
✼ x2
y e 2 ✽ ✾e 2 + C
y ✵ ✸1 ✿ C
✜✻✍♦✍
x2
e2
... (3)
❧✠✡☛☞✌✍ ✎❀✑ ♦❁✍✕★ ♦✕✖ ♦✬✖✢ ☛✍ ❧✠✡☛☞✌✍ ✣✙ ✘☞★✩✬ ✣✠ ❣❧ ♦✬✖✢ ♦✕✖ ✧✕❧✕ ❧✫✺✥ ☛✍ ❧✠✡☛☞✌✍ ❂✍✩
☛☞❡✍ ❞✍✣✩✕ ✣✙★ ✳✍✕ ❃r✫✬ ✎❄✒ ✏✑ ❧✕ ❅✬✳☞✩✍ ✣✍✕✤ ❧✠✡☛☞✌✍ ✎❀✑ ✠✕ ★ x = 0 ✧♦★ y = 1 ✘✹✚✩✺✻✍✍✚✘✩ ☛☞❡✕
✘☞ ✣✠ ✘✍✩✕ ✣✙★✪
1 = – 1 + C . eo ✜✻✍♦✍ C = 2
❧✠✡☛☞✌✍ ✎❀✑ ✠✕★ C ☛✍ ✠✍❡ ✘✹✚✩✺✻✍✍✚✘✩ ☛☞❡✕ ✘☞ ✣✠ ✘✹✍❆✩ ☛☞✩✕ ✣✙★✪
y ✵ ✸1 ✿
x2
2e2
✥✣ ♦❁ ☛✍ ✜❇✍✡❈❉ ❧✠✡☛☞✌✍ ✣✙✤
✐❊❋●❍■❏❑ ▲▼◆
✏ ❧✕ ✏❖ ✩☛ ♦✕ ✖ ✘✹P❡✍✕ ★ ✠✕✒★ ✘✹◗✥✕☛ ✜♦☛✢ ❧✠✡☛☞✌✍ ☛✍ ❘✥✍✘☛ ✣✢ ❂✍✩ ☛✡✚✳✧✪
1.
dy
❙ 2 y ❚ sin x
dx
4.
dy
❱❳
❲
❨ (sec x ) y ❩ tan x ❪ 0 ❬ x ❭ ❫
2❵
dx
❴
2.
dy
❯2 x
❙ 3y ❚ e
3.
dx
2
5. cos x
dy y
2
❙ ❚x
dx x
dy
❲
❙ y ❚ tan x ❪ 0 ❬ x ❭
❴
dx
❱❳
❫
2❵
430
① ✁✂✄
x
6.
dy
dx
☎
2y
✆
x 2 log x
8. (1 + x2) dy + 2xy dx = cot x dx (x
x
9.
dy
dx
x log x
7.
✝
✟
✞
✠
✡
y)
✞
2
log x
x
✆
dy
1
dx
✠
dy
dx
2
12. ( x 3 y )
11. y dx + (x – y2) dy = 0
y
☎
0)
y x xy cot x 0 ( x 0) 10. ( x
✞
dy
dx
☎
y ( y 0) .
✆
☛
✶☞ ✌✍ ✶✎ ✏✑ ✒✍✓ ✔✕✖✗✘✍✙ ✚✍✙ ✔✕ ✛✜✍✑ ✢✒✑✣ ✌✚✤✑✥✦✘ ✒✍✓ ✧✣★ ✧✩★ ✪✫★ ✔✕ ✧✏✬✙✭ ✑✘✍ ✌✙✏✮
✫ ✯ ✑✥✗✍ ✒✘✣✘
✧✒✧✖✘✮✯
13.
✪✣
dy
dx
2 y tan x sin x; y 0
dy
dx
2
14. (1 x )
15.
16.
✚❡✣
✺✬✩✫
✒✍✓
✔✥
✻✫✼ ✘✥✗✍
✐✔✖✘✿
➥❅❆
❇❈
✧✗✩✍✖
✿ ✘✘✙✑✘✍✙
✢✧✭✑
18.
✌✍
✺✬✩✫
✥✍❁✘✘
✌✍
✑✘
✜✘✍✻
✔✕✒✦✘✏✘
✒✘✣✍
❂✌
x
✑✘
❂✌
✒✽
2
✌✚✤✑✥✦✘
✺✬✩✫
✑✘
✔✥
✺✬✩✫
x 1
✳ ✴✵
✷ ✸✹
✒✽
3
✒✍✓
✧✗✩✍✖
✿ ✘✘✙✑✘✍ ✙
✌✚✤✑✥✦✘
❁✘✤✙♦✤
✑✤✧✰★
❢✘✏
✻✾✿
✒✍✓
✜✘✍ ✻
✑✤✧✰★
❢✘✏
✐✔✖✘✿
✥✍❁ ✘✘
✜✧✩
✒✍✓
✜✧✩
✑✤
✒✽
✒✍✓
✬✥✘✬✥
✪❃❄
✒✽
✒✍✓
✾✌
✾✌
✔✕ ✒✦✘✏✘
✒✍✓
✧✑✌✤
✺✬✩✫
✧✑✌✤
✺✬✩✫
✔✧✥✚✘✦✘
✪❃ ❄
✢✒✑✣
✢✒✑✣
✌✚✤✑✥✦✘
x
dy
dx
❉
y
✆
2x2
✑✘
(B) e –y
✌✚✤✑✥✦✘
(1 y 2 )
❊
y2 1
dx
dy
(B)
✌✚✘✑✣✗
❋
✻✫✦ ✘✑
(D) x
yx = ay ( 1 y 1)
●
y2 1
■
✪❃✱
1
x
(C)
1
1
(A)
0
2
★✑
✑✤
✻✫✰✥✗✍
(A) e –x
19.
✒✘✣✍
x
❀ ✲♥
1
;y
1 x2
2 xy
dy
3 y cot x sin 2 x; y
dx
(x, y)
17.
✑✤✧✰★✱
❢✘✏
❍
❍
1
(C)
1 y2
❊
✑✘
✌✚✘✑✣✗
✻✫✦ ✘✑
✪❃ ✱
1
(D)
1 y2
■
✌✍
✎
✈ ✁✂ ✄☎✆✁✝✞✟
431
❢✠❢✠✡ ☛☞✌✍✎✏✌
♠✑✒✓✔✕✒ ✖✗
❧✘✙✚✛✜✢ ✣✤✛✥✦ ✛✣ ✜✧★✩ y = c1 eax cos bx + c2 eax sin bx, ✥t✚✪ c1, c2 ▲✫✬✭✮
✯✰✱ t✲✳ ✯✫✣★ ❧✴✤✣✱✵✚
d2y
dy ✶ 2
2
✸ 2a
✹ a ✹ b ✷ y ✺ 0 ✣✚ t★ t✲❞
2
dx
dx
✓❣
✛✻✙✚ t✼✯✚ ✜✧★✩ t✲✽
y ❀ e ax ✾ c1 cos bx ❁ c2 sin bx ✿
... (1)
❧✴✤✣✱✵✚ ❂❃❄ ✫✬ ✧ ✻✚✬✩✚✬❅ ✜❆✚✚✬❅ ✣✚ x ✫✬ ✧ ❧✚✜✬ ❆✚ ✯✫✣★✩ ✣✱✩✬ ✜✱ t✴ ✜✚✢✬ t✲❅ ✛✣
dy
ax
ax
❉ e ❇ – bc1 sin bx ❊ b c2 cos bx ❈ ❊ ❇ c1 cos bx ❊ c2 sin bx ❈ e . a
dx
dy
ax
❉ e [(bc2 ❊ a c1 )cos bx ❊ (a c2 ● bc1 )sin bx]
dx
❧✴✤✣✱✵✚ ❂❍❄ ✫✬✧ ✻✚✬✩✚✬❅ ✜❆✚✚✬❅ ✣✚ x, ✫✬✧ ❧✚✜✬❆✚ ✯✫✣★✩ ✣✱✩✬ ✜✱ t✴ ✜✚✢✬ t✲❅ ✛✣
✯❋✚✫✚
... (2)
d2y
ax
■ e [(b c2 ❏ a c1 ) (❑ sin bx . b) ❏ ( a c2 ❑ b c1 ) (cos bx . b)]
2
dx
+ [(b c2 ▼ a c1 ) cos bx ▼ ( a c2 ◆ b c1 ) sin bx] e ax . a
= e ax [(a 2 c2 ◆ 2ab c1 ◆ b 2 c2 ) sin bx ▼ ( a 2 c1 ▼ 2ab c2 ◆ b 2 c1 ) cos bx]
✛✻✦ ❖✦ ✯✫✣★ ❧✴✤✣✱✵✚ ✴✬❅
❝✚✙✚✪ ✜❆✚
d 2 y dy
,
✦✫❅ y ✣✚ ✴✚✩ ✜P✛✢▲❋✚✚✛✜✢ ✣✱✩✬ ✜✱ t✴ ✜✚✢✬ t✲❅✽
dx 2 dx
e ax [a 2 c2 2abc1 b 2 c2 )sin bx (a 2 c1 2abc2 b 2 c1 ) cos bx]
2ae ax [(bc2 ac1 ) cos bx (ac2 bc1 )sin bx]
(a 2 b 2 ) e ax [c1 cos bx c2 sin bx]
e
ax
a 2 c2 2abc1 b 2 c2 2a 2 c2 2abc1 a 2 c2 b 2 c2 sin bx
(a 2 c1 2abc2 b 2 c1 2abc2 2a 2 c1 a 2 c1 b 2 c1 )cos bx
= e ax [0 ◗ sin bx ▼ 0cos bx] = eax × 0 = 0 = ✻✚✙✚✪ ✜❆✚
❜❧✛★✦ ✛✻✙✚ t✼✯ ✚ ✜✧★✩ ✛✻✦ t✼✦ ✯✫✣★ ❧✴✤✣✱✵✚ ✣✚ t★ t✲❞
432
① ✁✂✄
♠✌✍✎✏✑✍ ✒✓ ❢✔✕✖✗ ✘✕✙✚✛✛✜✢✛ ✣✤✥ ✦✤ ✧✤ ★✩ ✪✛✛✤✥ ★✤✫ ★✙✫✬ ✭✛ ✮★✭✬ ✧✣✖✭✯✰✛ ✱✛✕ ✭✖❢✲✦ ✲✛✤ ❢✳✴✤ ✵ ✢✛✛✥✭
✮✶✛✛✤✥ ✭✛
✷✸✢✛✵ ✭✯✕✤
✎❣
✬✖❢✲✦❡
C
(
✣✛✳
✔✛✯✛
❢✳❢✴✵⑥✽
★✙✫✬
C
✭✛✤
❢✳✴✤✵✢✛✛✥✭
❢✭✗✛
✹✺ ✻
✲✛✕✛
✾✧
❢✳❞❢✸✕
❧❆❇
✭✯✳✤
★✙✫✬
★✛✬✛
✭✛
x
✮✚✛★✛
❢✭✧✖
✮✛✺✯
✧✴✷✗
❢✔✕✖✗
★✤ ✫
✘✕✙ ✚✛✛✜ ✢✛
★✤✫
✥ ✴✿
❀✼✴✙
✣✤✥
★✤✫
✼✳✛
★✩ ✪✛✛✤✥
❢✳✴✤✜✢✛✛✥✭
✭✛
★✙✫✬
(–a, a)
✹✺✥
x
◗
■
2a 2a
❈
a
❉
❋
❑
▼
yy
y 1
P
✸✯
✭✯✳✤
dy
1
dx
❏
dy
dx
... (1)
... (2)
✹✣
✸✛✕✤
✹✥❅
✺
0
❊
●
▲
◆
❖
❖ ❘
✧✣✖✭✯✰✛
❧❙❇
✣✤✥
a
✭✛
❱
❩
x
✣✛✳
x
❯
✸✿❢ ✕✷✚✛✛❢✸✕
yy
y 1
❯
xy
❵❛
❫
✮✚✛★✛
(x + y)2 y
✮✚✛★✛
(x + y)2 [ y
❜
✹✙✦
★✩ ✪✛✛✤✥
♠✌✍✎✏✑✍ ✒♥
★✤ ✫
★✙✫✬
✭✛✤
❯
❬
❪
❤
2
❢✴✗✛
✗❢✴
✹✙✮✛
❳
❚ ❳
2
❵❴
yy
y 1
❯
❜❫
yy
❵❛
✸✛✕✤
2
❚❲
❬
✈☎✆✝✞✟✠ ✡☛☞
✹✺ ✥❅
❱
❨
❪
❩
❭
x
yy
y 1
❯
❚ ❳
y x yy
❛
❛
2
❚❲
❬
❪
2
❵❴
❝❫
x
❜
2
dy
dx
log
✧✣✖✭✯✰✛
❊
e(3 x
♣
2
❵❴
+ 1] = [x + y y ]2
✐
❵❦
❋
❑
✭✯✳✤
dy
dx
★✛✬✛
●
▲ ■
✮★✭✬
3x 4 y
❍
✧✣✖✭✯✰✛
✹✺✻
✭✛ ❢★❢✢✛⑥✽ ✹✬ ✱✛✕ ✭✖❢✲✦✻
◆
x=0
✮★✭✬
yy
✐
▼
y=0
y
✹✣
+ [x + y]2 = [x + y y ]2
❢✳❞❢✸✕
✮★✭✬ ✧✣✖✭✯✰✛
❩
✸✯
x
❭
❜
❥
❢✴✦
❱
x x yy
✮✚✛★✛
✭✯✳✤
2
❚❲
❚ ❳
❭
✎❣
★✛✬✛
✹✺❅
✮★✭✬✳
dy
dx
dy
dx
y
❍
a
✮✚✛★✛
❢✭
★✤✫
✧✣✖✭✯✰✛
✧✛✸✤✶✛
★✤✫
❈
✹✺
✭✯✳✤
(x + a)2 + (y – a)2 = a2
x2 + y2 + 2ax – 2ay + a2 = 0
x
2x 2 y
✲✛✤
✷✸✢✛✵
✭✛✤
)
✮✚✛★✛
✧✣✖✭✯✰✛
✮✶✛✛✤✥
✴✤❢ ❄✛✦ ✻
❁❂❃
✮✛★✩✫❢✕
✹✺✥ ✻
❢✳♦✳❢✬❢❄✛✕
4 y)
❞✸
✣✤✥
❢✬❄✛✛
✲✛
✧✭✕✛
✹✺ ❅
❢✴✗✛ ✹✙✮ ✛
✈ ✁✂ ✄☎✆✁✝✞✟
dy
dx
✠✡☛☞☛
e3 x e 4 y
433
... (1)
♣✌☛✍✎ ✏☛✍ ✑✒✡☛☞✓✔ ✏✌✕✍ ✑✌ ✖✗ ✑☛✘✍ ✖✙✎✚
dy
3x
✛ e dx
e4 y
❜✜✢✣✤
✦e
✥4y
dy ✛ ✦ e3 x dx
e ✧ 4 y e3 x
★
✩C
3
✪4
4 e3x + 3 e– 4y + 12 C = 0
✠✡☛☞☛
✜✗❧✏✌✫☛ ✬✭✮ ✗✍✎ x = 0 ✤☞✎ y = 0 ✑✐✢✘✯✡☛☛✢✑✘ ✏✌✕✍ ✑✌ ✖✗ ✑☛✘✍ ✖✙✎✰
✠✡☛☞☛
4 + 3 + 12 C = 0 ✠✡☛☞☛ C =
... (2)
✱7
12
✜✗❧✏✌✫☛ ✬✭✮ ✗✍✎ C ✏☛ ✗☛✕ ✑✐✢✘✯✡☛☛✢✑✘ ✏✌✕✍ ✑✌ ✖✗✚
4 e3x + 3 e– 4y – 7 = 0, ✑✐☛✲✘ ✏✌✘✍ ✖✙✎
❀✖ ✢✳✤ ✖✴✤ ✠☞✏✣ ✜✗❧✏✌✫☛ ✏☛ ✤✏ ✢☞✢✵☛✶✷ ✖✣ ✖✙✸
♠✹✺✻✼✽✺ ✾✿
✠☞✏✣ ✜✗❧✏✌✫☛
❁ y❂
❁ y❂
(x dy – y dx) y sin ❃ ❄ = (y dx + x dy) x cos ❃ ❄ ✏☛✍ ✖✣ ✏❧✢❞✤✸
❅x❆
❅x❆
✻❣
✢✳❀☛ ✖✴✠☛ ✠☞✏✣ ✜✗❧✏✌✫☛ ✢✕❢✕✢✣✢❇☛✘ ❈✑ ✗✍✎ ✢✣❇☛☛ ❞☛ ✜✏✘☛ ✖✙✸
❉
❉
❋ y●
❋ y ●❊
❋ y●
❋ y ●❊
2
2
▲ x y sin ❏ ❑ ❍ x cos ❏ ❑ ▼ dy ✛ ▲ xy cos ❏ ❑ ■ y sin ❏ ❑ ▼ dx
◆ x❖
◆ x ❖◗
◆ x❖
◆ x ❖◗
P
P
✠✡☛☞☛
❘ y❙
❘ y❙
xy cos ❯ ❱ ❚ y 2 sin ❯ ❱
dy
❲ x❳
❲ x❳
❨
y
dx
❘ ❙
❘ y❙
xy sin ❯ ❱ ❩ x 2 cos ❯ ❱
❲ x❳
❲ x❳
✳☛❀✍✎ ✑♥☛ ✑✌ ✠✎✵☛ ✤☞✎ ✖✌ ✳☛✍✕☛✍✎ ✏☛✍ x2 ✜✍ ❬☛☛❭ ✳✍✕✍ ✑✌ ✖✗ ✑☛✘✍ ✖✙✎✰
2
y
❪ y❫ ❪ y ❫
❪ y❫
cos ❵ ❛ ❴ ❵ 2 ❛ sin ❵ ❛
dy x
❝ x❡ ❝ x ❡
❝x❡
❤
y ❪ y❫
dx
❪ y❫
sin ❵ ❛ ❥ cos ❵ ❛
x ❝x❡
❝x❡
... (1)
434
① ✁✂✄
▲☎✆✝✞✟
✩✠
✠✡☛☞✌✍✎
✠✡☛☞✌✍✎
☞✎✛
✧✦
dy
dx
✏✑✒✓
☞✌❜✛
g
✖
✔
✗
✙
♦✛✜
✪✦✫
y
x
✕
♦✛✜
✘
✢☎
☞✎
✠✡✣✎✎✞☛✤
✥♦☞✦
dy
dx
✥✬✎♦✎
v
✯
x
dv
dx
x
✥✬✎♦✎
dv
dx
dv
dx
v cos v v 2 sin v
[
v sin v cos v
✯
✰
✠✡☛☞✌✍✎ ✏✑✒ ✥✎★✌ ✏✈✒☞✎ ☎✐✤✎✛❞ ☞✌❜✛ ☎✌
✿
2v cos v
v sin v cos v
✴
✡✛✭
v
tan v dv
☞✎✛
✽
❆
❈
y
x
❄
❆
✥♦☞✦
✾
log | v | 2log | x | log | C1 |
❁
❂
log C1
... (3)
☎✐✪✞▲✬✎✎✪☎✞
☞✌❜✛
☎✌
y
x
✧✡
❅
❇
❉
❊
C
✓
❪✧✎❋
C = ± C1
❉
✗
1
dx
x
✿
❇
✔
✼
1
1
dv 2
dx
v
x
❅
✙
✧❍✫
✠✛
y
x
✿
✰
y
( x2 )
x
sec
✥✬✎♦✎
2
C1
✲ ❃
❈
❄
❀
sec v
v x2
sec
✷
✻
sec v
v x2
✥✬✎♦✎
✶
✹
✺
log
✥✬✎♦✎
2 dx
x
dv
v sin v cos v
dv
v cos v
log sec v
✥✬✎♦✎
]
✱
✳
✸
✼
✥✬✎♦✎
✪●✫
v x
✲
✵
✩✠✪✦✫
✤✧
... (2)
v sin v cos v
v cos v
✥✬✎♦✎
✏❧✒
✩✠✪✦✫
✧★ ✭✮
✥✬✎♦✎
✠✡☛☞✌✍✎
✧★✓
✧✡
y = vx
☎✐ ✪✞▲✬✎✎✪☎✞ ☞✌✞✛
✠✡☛☞✌✍✎
✚
✕
✘ ✖
C xy
✚
✠✡☛☞✌✍✎
☞✎
■✤✎☎☞
✧✦
✧★ ✮
☎✎✞✛
✧★✭
✪☞
✈ ✁✂ ✄☎✆✁✝✞✟
♠✠✡☛☞✌✡ ✍✎
435
✏✑✒✓ ✔✕✖✒✗✘✙
(tan–1y – x) dy = (1 + y2) dx ✒✙ ❞✓ ✚✙✛ ✒✖✜✢✣✤
☛❣
✜❢✥✙ ❞✦✏✙ ✏✑✒✓ ✔✕✖✒✗✘✙ ✜✧★✧✜✓✜✩✙✛ ✪✫ ✕✬✭ ✜✓✩✙✙ ✢✙ ✔✒✛✙ ❞✮✯
dx
x
tan ✰1 y
✱
✲
dy 1 ✱ y 2 1 ✱ y 2
✔✕✖✒✗✘✙ ❧✳✴✵
... (1)
dx
✶ P1 x = Q1, ✑✬♦ ✪✫ ✒✙ ✗✮✜✩✙✒ ✏✑✒✓ ✔✕✖✒✗✘✙ ❞✮✤ ✥❞✙✷
dy
1
tan ✰1 y
P1 ✸
✣✑✭ Q1 ✲
❞✮✤ ❜✔✜✓✣
1 ✱ y2
1 ✹ y2
I.F. e
1
dy
1 y2
e tan
1
y
❜✔✜✓✣ ✜❢✣ ❞✦✣ ✏✑✒✓ ✔✕✖✒✗✘✙ ✒✙ ❞✓ ❞✮✯
xe
✕✙✧ ✓✖✜✢✣
tan ✺1 y
✼ tan✻1 y ✽ tan ✺1 y
e
dy ✿ C
✾❄ ❀
2 ❁
❂ 1✿ y ❃
✼ tan ✻1 y ✽ tan ✺1 y
e
dy
2 ❁
✿
1
y
❂
❃
I✾❄ ❀
❇ 1 ❈
dy ✲ dt
tan–1 y = t ✫✐✜✛❅❆✙✙✜✫✛ ✒✗✧✬ ✫✗ ❞✕ ✫✙✛✬ ❞✮✭ ✜✒ ❉
2 ❊
❋ 1✱ y ●
✏✛✯
t
I = ❄ t e dt , I = t et – ❍1 . et et, I = t et – et = et (t – 1)
✏❆✙✑✙
I = e tan■ y (tan–1y –1)
1
✔✕✖✒✗✘✙ ❧❏✴ ✕✬✭ I ✒✙ ✕✙✧ ✫✐✜✛❅❆✙✙✜✫✛ ✒✗✧✬ ✫✗ ❞✕
x . e tan
✏❆✙✑✙
❑1 y
▼e
tan ❑ 1 y
(tan ▲1 y ◆ 1) ❖ C ✫✙✛✬ ❞✮✭
P1 y
x = (tan ◗1y ❘ 1) ❙ C e◗ tan
✥❞ ✜❢✣ ❞✦✣ ✏✑✒✓ ✔✕✖✒✗✘✙ ✒✙ ❚✥✙✫✒ ❞✓ ❞✮✤
... (2)
436
1.
① ✁✂✄
❢✒✓✒❢✔❢✕✖✗
✘✙✚✔
✈☎✆✝✆
✞
✛✜✢✚✣✤✖✖✥✦
✜✥✦
✟✠
✛✥
✡☛✡☛☞
✧★✩✪✥✚
✟✌✍✎✝☛✏✑
✚✢
✚✖✥❢✫
✬✙✦
✭✖✖✗
✮✪❢✯
✧❢✣✰✖✖❢✱✖✗
✲✖✥✳
✴✖✗
7y
sin x
✚✢❢❞✬✵
2.
d2y
(i)
dx 2
✸
d4y
(iii)
dx 4
❁
5x
dy
dx
✶
✻
✽
sin
2
✷
✼
❃
❅
6y
✛✦▲✗
✘✙✚✔
❀
✚✖
✲✔
✼
✾
dy
dx
✶
✻
✽
2
✷
✼
✸
✺
✾
✲●✵
: x
2
d2y
dx 2
d2y
dx 2
(ii) y = ex (a cos x + b sin x)
:
(iii) y = x sin 3x
d2y
:
dx 2
(iv) x2 = 2y2 log y
2
: (x
✬✚
❊✙✥❙❚
✘❯✣
✧★❋ ✖✜
✚✢❢❞✬
❯✗❉❋ ✖✖✐ ❇✖
✘✙✚✔
✯❇✖✖◗♥ ✬
❡
❤
❍
x2
❍
■
2 0
❏
2y 0
❏
9 y 6cos3 x 0
y2 )
dy
dx
▼
xy 0
◆
✬✥✛✥
✙❨❩ ✖✖✥ ✦
✙✥❈
✙❉❈✔
c
❞✲✖❘
✬✚
✧★✖❯✔
✲●❲
✚✖
❳✪✖✧✚
✲✔
✚✖
✘✙✚✔
✛✜✢✚✣✤✖
✴✖✗
x 1
❳✪✖✧✚
✘✙✚✔
✛✜✢✚✣✤✖
✲●✵
✚✢❢❞✬
❞✖✥
❢✒✯✥◗✦ ❇✖✦✖✚
✘❬✖✖✥✦
✚✣✗✥ ✲●✵
✦
✛✜✢✚✣✤✖
❢✚
dy
dx
✘✙✚✔
1 y2
1 x2
❭
❪
❫
0,
❦
✛✜✢✚✣✤✖
0
❲
❞❴❢✚
✚✖
❵
✲✔
✴✖✗
✚✢❢❞✬✵
❭
✛✜✢✚✣✤✖
dy
dx
❛
(x + y + 1) = A (1 – x – y – 2xy)
➥❴✯❉
xy
■
✲●✵
❢✚
✜✥✦
dy
dx
dy
dx
2
■
❑
2
❍
⑥✖✣✖ ❢✒❖❢✧✗ ✙P✖✥✦ ✙✥❈ ✙❉ ❈✔ ✚✖ ✘✙✚✔ ✛✜✢✚✣✤✖ ❢✒❢✜◗ ✗ ✚✢❢❞✬ ❞✲✖❘
x2 – y2 = c (x2 + y2)2
(x3 – 3x y2) dx = (y3 – 3x2y) dy
❢✛❱
✚✖✥ ❊✧❇✖◗
8.
✻
4
✹
❆
–x
3. (x – a)2 + 2y2 = a2,
a
7.
(ii)
3
✷
0
❄ ❂
(i) y = a e + b e + x
6.
dy
dx
✶
✽
✛✜✢✚✣✤✖
x
5.
log x
❢✒✓✒❢✔❢✕✖✗ ✧★❇✒✖✥✦ ✜✥✦ ✧★✩✪✥✚ ✙✥❈ ❢✔✬ ✛✩✪✖❢✧✗ ✚✢❢❞✬ ❢✚ ❢✯✪✖ ✲❉ ✘✖ ✧❈✔✒ ✮✘❊✧✱✫ ✘❋✖✙✖
❊✧✱✫✳
4.
✺
✾
d3y
dx3
✿
✹
y2
x2
✲●❲
❛
y 1
0
x 1
❛
❜
❛
✚✖
✲✔
❳✪✖✧✚
❛
❢❞✛✜✥✦
A
✬✚
✧★✖❯✔
✲●✵
❝ ❣
❥
✛✥
▲❉ ❞✣✒✥
✙✖✔✥
✬✚
✬✥✛✥
✙P
✚✖
4
sin x cos y dx + cos x sin y dy = 0
✛✜✢✚✣✤✖
❧
✲●✵
✴✖✗
✚✢❢❞✬
❢❞✛✚✖
✘✙✚✔
✈ ✁✂ ✄☎✆✁✝✞✟
437
9. ✠✡☛☞ ✌✍✎☛✏✑✒ (1 + e2x) dy + (1 + y2) ex dx = 0 ☛✒ ❞☛ ✓✡✓✔✒✕✖ ✗☞ ✘✒✙ ☛✎✓✚❞✛
✓❢✜✒ ✗✢✠✒ ✗✣ ✓☛ y = 1 ✜✓❢ x = 0.
✤
x
✥
x
10. ✠✡☛☞ ✌✍✎☛✏✑✒ y e y dx ★ ✦ x e y ✩ y 2 ✧ dy ( y ✪ 0) ☛✒ ✗☞ ✘✒✙ ☛✎✓✚❞✭
✫
✬
11. ✠✡☛☞ ✌✍✎☛✏✑✒ (x – y) (dx + dy) = dx – dy ☛✒ ❞☛ ✓✡✓✔✒✕✖ ✗☞ ✘✒✙ ☛✎✓✚❞✛ ✓❢✜✒
✗✢✠✒ ✗✣ ✓☛ y = –1, ✜✓❢ x = 0 (✌❧✡✯✮ ✙ : x – y = t ✏❥✒✮❧)✭
✱ e ✰2 x
y ✲ dx
12. ✠✡☛☞ ✌✍✎☛✏✑✒ ✶
✳
✴ 1 ( x ✵ 0) ☛✒ ✗☞ ✘✒✙ ☛✎✓✚❞✭
✷
x
x ✹ dy
✸
13. ✠✡☛☞ ✌✍✎☛✏✑✒
dy
✩ y cot x = 4x cosec x (x ✺ 0) ☛✒ ❞☛ ✓✡✓✔✒✕✖ ✗☞ ✘✒✙ ☛✎✓✚❞✛
dx
✓❢✜✒ ✗✢✠✒ ✗✣ ✓☛ y = 0 ✜✓❢ x ✼
14. ✠✡☛☞ ✌✍✎☛✏✑✒ (x + 1)
✻
2
.
dy
= 2 e–y – 1 ☛✒ ❞☛ ✓✡✓✔✒✕✖ ✗☞ ✘✒✙ ☛✎✓✚❞✛ ✓❢✜✒ ✗✢✠✒
dx
✗✣ ✓☛ y = 0 ✜✓❢ x = 0.
15. ✓☛✌✎ ✽✒✾✡ ☛✎ ✚✿✌❧❥✜✒ ☛✎ ✡❀✓❁ ☛✎ ❢✏ ✓☛✌✎ ❂✒✎ ✌✍✜ ❃✌ ✽✒✾✡ ✡✮✯ ✓✿✡✒✓✌✜✒✮❧ ☛✎ ✌❧ ❥✜✒
✡✮✯ ✌✍✒✿✢♦✒✙✎ ✗✣✭ ✜✓❢ ✌✿❄ ❅❆❆❆ ✍✮❧ ✽✒✾✡ ☛✎ ✚✿✌❧❥✜✒ ❇❈✛❈❈❈ ❉✒✎ ✠✒✣✏ ✌✿❄ ❇❈❈❊ ✍✮❧ ❇❋✛❈❈❈
❉✒✎✛ ✙✒✮ ✘✒✙ ☛✎✓✚❞ ✓☛ ✌✿❄ ❇❈❈❆ ✍✮❧ ✽✒✾✡ ☛✎ ✚✿✌❧❥✜✒ ●✜✒ ✗✒✮✽✎❍
16. ✠✡☛☞ ✌✍✎☛✏✑✒
(A) xy = C
17.
y dx ■ x dy
❏ 0 ☛✒ ❑✜✒♦☛ ✗☞ ✗✣▲
y
(B) x = Cy2
(C) y = Cx
(D) y = Cx2
dx
▼ P1 x ◆ Q1 ✡✮✯ ❖♦ ✡✒☞✮ ✠✡☛☞ ✌✍✎☛✏✑✒ ☛✒ ❑✜✒♦☛ ✗☞ ✗✣▲
dy
(A) y e ❘
P1 dy
(B) y . e❳
❙ ❯ P Q1e ❘
P1 dx
P1 dy
❨ ❬ ❱ Q1e❳
◗ dy ❚ C
P1 dx
❲ dx ❩ C
(C) x e❳
P1 dy
❨ ❬ ❭ Q1e❳
P1 dy
❪ dy ❩ C
(D) x e ❘
P1 dx
❙ ❯ ❫ Q1e ❘
P1 dx
❴ dx ❚ C
18. ✠✡☛☞ ✌✍✎☛✏✑✒ ex dy + (y ex + 2x) dx = 0 ☛✒ ❑✜✒♦☛ ✗☞ ✗✣▲
(A) x ey + x2 = C (B) x ey + y2 = C (C) y ex + x2 = C (D) y ey + x2 = C
438
① ✁✂✄
❧☎✆☎✝✞☎
✟
✱✠
✱✡☛☞
☛✌✍✠✎✏☞
✙✣✕✠✥✒☞✡ ✓✚
✟
✑✠☛✍
✟
✠✍
✦☞✡ ✓✧
☛✌✍✠✎✏☞
✠☞✡ ✑✫
✔✕✖✓✗☞
✘✎
✣✕✠✥
✌✡ ✓
✙✘✎☞✡ ✚
✓
☛✌✍✠✎✏☞
☛✑✭✌✑✥✖
✕✡✛
☛☞✜✡✢☞
✠✦✥☞✖☞
❢✪✘✖✌
✣☞✑✤✖
✘✎
✕✡ ✛
✣✕✠✥✒
✦★✩
✣✕✠✥✒
✠✍
✠☞✡ ✑✫✧
❢☛
✣✕✠✥
✦★✩
✠✦✥☞✖✍
❀✑✬ ✠☞✡ ✮✯ ✣✕✠✥ ☛✌✍✠✎✏☞ ✣✕✠✥✒☞✡ ✓ ✌✡ ✓ ✰✦✲✜✬ ☛✌✍✠✎✏☞ ✦★✓ ✖☞✡ ❢☛ ✣✕✠✥ ☛✌✍✠✎✏☞
❞☞☞✖
✠✍
✟
☛✑✭✌✑✥✖
✣✕✠✥
☛✌✍✠✎✏☞
✑✒☛✌✡ ✓
✑✠☛✍
✦☞✡✖✍
✜✑✎✳☞☞✑✴☞✖
✣✕✠✥
✦★✩
☛✌✍✠✎✏☞
❞☞☞✖
✠✍
✙❀✑✬
✜✑✎✳☞☞✑✴☞✖
✦☞✡✚
❢☛
✣✕✠✥
☛✌✍✠✎✏☞
✌✡ ✓
☛✑✭✌✑✥✖ ❢✪✘✖✌ ✠☞✡ ✑✫ ✣✕✠✥✒ ✠✍ ❢✪✘✖✌ ❞☞☞✖ ✙✕✡✛✕✥ ✵✶☞✷✌✠ ✜✸ ✏☞☞✹✠ ✚ ✦☞✡✖✍ ✦★✩
✟
✱✠ ✑✬✱ ✦✲✱ ✣✕✠✥ ☛✌✍✠✎✏☞ ✠☞✡ ☛✓✖ ✲✴✫ ✠✎✶✡ ✕☞✥☞ ✜✛✥✶ ❢☛ ✣✕✠✥ ☛✌✍✠✎✏☞ ✠☞
✦✥
✠✦✥☞✖☞
✦★✩
✱✠
✱✡☛☞
✦✥
✦★✧ ✺❀☞✜✠
☛✌✍✠✎✏☞ ✠✍ ✠☞✡ ✑✫
✑✒☛✌✡ ✓
✦✍
❢✖✶✡
✔✕✡✪ ❣
✣✘✎
✦☞✡✓✧
✑✒✖✶✍
❢☛
✦✥ ✠✦✥☞✖☞ ✦★ ✣☞★✎ ✔✕✡✪ ❣ ✣✘✎☞✡✓ ☛✡ ✌✲✻✖ ✦✥
✣✕✠✥
✑✕✑✼☞✴✫
✦✥ ✠✦✥☞✖☞ ✦★ ✩
✟
✑✠☛✍
✦✲✱
✑✬✱
✜✛✥✶
☛✡
✣✕✠✥
☛✌✍✠✎✏☞
✰✶☞✶✡
✕✡ ✛
✑✥✱
✦✌
❢☛
✜✛✥✶
✠☞
❢✽☞✎☞✡ ✽☞✎
❢✖✶✍ ✦✍ ✰☞✎ ✣✕✠✥✶ ✠✎✖✡ ✦★✓ ✑✒✖✶✡ ❢☛ ✜✛✥✶ ✌✡✓ ✔✕✡✪ ❣ ✣✘✎ ✦☞✡ ✖✡ ✦★✓ ✣☞★✎ ✖✰ ✔✕✡✪ ❣
✣✘✎☞✡ ✓ ✠☞✡
✟
✑✕✥✲✈✖
✘✎ ✜♣✾☞✻✠✎✏☞✍❀
✦★✓ ✩
✠✎✖✡
✑✕✑✵ ✱✡☛✡ ☛✌✍✠✎✏☞
✠☞✡ ✦✥ ✠✎✶✡ ✕✡ ✛
✘✎☞✡ ✓ ✠☞✡ ✜✸ ✎✍ ✖✎✦ ☛✡ ✜♣✾☞✕❁✛ ❁ ✑✠❀☞ ✒☞ ☛✠✖☞ ✦★ ✣✾☞☞✯✖❁
✣☞★✎
✟
x
✕☞✥✡
✜✬
dx
✕✡ ✛
☛☞✾☞
✎✦✶✡
dy
f ( x, y )
dx
f (x, y)
g(x, y)
❂❃❄♦❄
✱✠ ✱✡☛☞ ✣✕✠✥ ☛✌✍✠✎✏☞✧ ✑✒☛✠☞✡
✦★✧
✓
☛✌❞☞☞✖✍❀
dy
+ Py
dx
x
✕✡✛
✣✕✠✥
Q,
✜✛✥✶
✦★✧
✓
✕✡ ✛
☛✌✍✠✎✏☞
❅✜
✜❈✾☞✌
✕☞✥☞
✠☞✡✑✫
✕☞✥✡ ✜✬
dy
✒☞✖✍ ✦★ ✑✒☛✌✡ ✓
✕✡✛ ☛☞✾☞ ✎✦✶✡ ✘☞✑✦✱
✘☞✑✦✱✩
✣✑✳☞✺❀✻✖ ✑✠❀☞ ✒☞ ☛✠✖☞ ✦★✧ ✒✦☞❆
✟
y
❢✜❀☞✡ ✿ ✠✍
✑✥✱
✱✕✓
✠✦✥☞✖☞
✣✕✠✥
✎★✑❉☞✠
dx
dy
g ( x, y )
✕✡✛ ❅✜ ✌✡ ✓
✼☞✸ ❇❀ ❞☞☞✖ ✕☞✥✡ ☛✌❞☞☞✖✍❀ ✜✛✥✶
✦★✩
☛✌✍✠✎✏☞✧
✣✕✠✥
❊❋●❍■❏●❑▲
✑✒☛✌✡ ✓
☛✌✍✠✎✏☞
P
✖✾☞☞
✠✦✥☞✖☞
Q
✣✘✎
✣✾☞✕☞
✕✡✛✕✥
✦★✩
▼◆❖P◗❏❘●❙
✣✕✠✥ ☛✌✍✠✎✏☞ ✑✕❚☞✶ ✠✍ ✜❈✌❉
✲ ☞ ✳☞☞✴☞☞✣☞✡✓ ✌✡ ✓ ☛✡ ✱✠ ✦★✩ ✎☞✡✘✠ ✖✾❀ ❀✦ ✦★ ✑✠ ✣✕✠✥
☛✌✍✠ ✎✏☞☞✡ ✓
✠☞
(1646-1716)
✣✑✔✖✷✕
✶✡ ☛✕✯✜❈✾☞✌
✶✕ ✓✰✎
❯❯✧
☛✕✯☛ ✑✌✠☞✧
Gottfried Wilthelm Freiherr Leibnitz
1 2
y dy
y ,
2
❯❱❲❳
❩
❨
✠☞✡
✑✥✑❉☞✖
❅✜ ✌✡ ✓ ✜❈✔✖✲✖
✑✠❀☞
✖✾☞☞
✈ ✁✂ ✄☎✆✁✝✞✟
439
♠✠✡☛ ☞✌☛✠✌☛✍ ✎✏✑✒✓✌☛✍ ✔ ✕✌✖✗ dy ✡☛ ✎❧✗❧✘✑ ✓✗✌✙✌✚ ✛✜✑✢✑✣ Leibnitz ✱☛✡✒ ✛✤ ✓✌☛ ✥✌✑ ✓✗✠☛ ✓✒
✡✦✜✙✌ ✦☛✍ ✦✧✠ ★✌☛ ❧✩✡✓✒ ✜✎✪✌✫ ✗☛✬✌✌ ❧✠❧☞✫✭✮ ✯✌☛✍✰ ✲✡ ✡✦✜✙✌ ✠☛ ✡✠✳ ✴✵✶✴ ✦☛✍ ♠✷✯☛✍ ✸✘✗✌☛✍ ✛☛✹
✎✐★✌✺✓✗✻✌✒✙ ❧✛❧✼✽ ✛☛✹ ✕✷✛☛✭✌✻✌ ✓✌ ✦✌✾✫☞✪✌✫✠ ✓✗✌✙✌✚ ✱✓ ✛✭✌✫ ✎✪✘✌✑✳ ♠✷✯✌☛✍✠☛ ✸✎✏★✌✦ ✓✌☛❧✮ ✛☛✹
✡✦✿✌✌✑✒✙ ✡✦✒✓✗✻✌✌☛✍ ✛☛✹ ✯❀ ✓✗✠☛ ✓✒ ❧✛❧✼✽ ✓✌ ✡❁❂✌✒✓✗✻✌ ❧✓✙✌✚ ✛☛ ✕✌✾☛ ❃❄❅☛ ✕✌✖✗ ✕❆✎
✡✦✙ ✦☛✍ ♠✷✯✌☛✍✠☛ ✸✎✏★✌✦ ✓✌☛❧✮ ✛☛✹ ✗✖ ❧✬✌✓ ✕✛✓❀ ✡✦✒✓✗✻✌✌☛✍ ✓✌☛ ✯❀ ✓✗✠☛ ✓✒ ❧✛❧✼✽ ✓✌
✕✷✛☛✭✌✻✌ ❧✓✙✌✚ ❧✓✑✠✌ ✕✌✪✘✙✫✩✠✓ ✯✖ ❧✓ ♠✎✙✢✺
✫ ✑ ✡❇✌✒ ❧✛❧✼✙✌☛✍ ✓✒ ✬✌✌☛✩ ✕✛☛✹❀☛ ✱✓ ❈✙❧✺✑
⑥✌✗✌ ✕✛✓❀ ✡✦✒✓✗✻✌✌☛✍ ✛☛✹ ✩✷✦ ✛☛✹ ✎❉✘✒✡ ✛✭✌✌☛❊ ✛☛✹ ✕❆✎✌✛❧✼ ✛☛✹ ✕✍✑ ✾✫✑ ✡✍✎✷✠ ✯✢✲✫✚
✎✏✌✗✍❇✌ ✦☛✍ ✛☛✹✛❀ ✡✦✒✓✗✻✌✌☛✍ ✛☛✹ ✸✯❀✽ ✓✗✠☛ ✓✒ ✎✏❧✛❧✼ ✓✌☛ ✕✛✓❀ ✡✦✒✓✗✻✌✌☛✍ ✛☛✹
✸✡✦✌✓❀✠✽ ✛☛✹ ❫✎ ✦☛✍ ❧✠❧☞✫✪✮ ❧✓✙✌ ✾✙✌ ★✌✌✚ ✙✯ ✪✌❋☞ ✡✠✳ ✴✵✶● ✦☛✍ ✎✏★✌✦✑✣ James
Bernoulli, ❍✴✵■❏ ❑ ✴▲●■▼ ⑥✌✗✌ ✎✏✘❀✠ ✦☛✍ ❀✌✙✌ ✾✙✌✚ ✪✌❋☞ ✸✯❀✽ ✓✌ ✡✛✫✎✏★✌✦ ✎✏✙✌☛✾
Joseph Louis Lagrange ❍✴▲◆✵❑✴❖✴◆▼✰ ⑥✌✗✌ ✡✠✳ ✴▲▲❏ ✦☛✍ ❧✓✙✌ ✾✙✌✚ ✙✯ ✿✌✮✠✌
✕✛✓❀ ✡✦✒✓✗✻✌✌☛✍ ✛☛✹ ✩✷✦ ✡☛ ❀✾❇✌✾ ✴●● ✛✭✌✌☛❊ ❃✌☞ ✿✌❧✮✑ ✯✢✲✚✫ ✙☛ Jules Henri Poincare
❍✴❖■❏ ❑ ✴✶✴P▼✰ ★✌☛✰ ❧✩✷✯✌☛✍✠☛ ✪✌❋☞ ✸✯❀✽ ✛☛✹ ✎✏✙✌☛✾ ✛☛✹ ❧❀✱ ✕✓✌✮✳✙ ✑✛✫✹ ✎✏✜✑✢✑ ❧✓✙✌✰
✎✹❀✑✣ ✕✌✼✢❧✠✓ ✪✌❋☞✌✛❀✒ ✦☛✍ ✪✌❋☞ ✯❀ ✓✌☛ ✕✎✠✌ ♠❧✘✑ ✜★✌✌✠ ✎✏✌◗✑ ✯✢✕✌✚ ✸✘✗✌☛✍ ✛☛✹
✎✐★✺✓✗✻✌✒✙ ❧✛❧✼ ✓✌ ✠✌✦✓✗✻✌ John Bernoulli ❍✴✵✵▲❑✴▲❏❖▼✰ James Bernoulli ✛☛✹
✕✠✢✩ ⑥✌✗✌ ❧✓✙✌ ✾✙✌✚ ✦✲✫ P●✰ ✴▲✴■ ✓✌☛ Leibnitz ✓✌☛ ❧❀✬✌☛ ✕✎✠☛ ✎❂✌ ✦☛✍✰ ♠✷✯✌☛✠☛ ❧✠❞✠❧❀❧✬✌✑
✕✛✓❀ ✡✦✒✓✗✻✌ ✛☛✹ ✯❀ ✓✒ ✬✌✌☛✩ ❧✓✱
x2 y❘ = 2y
✛☛✹ ✯❀ ✑✒✠ ✎✏✓✌✗ ✓✒ ✛✤✌☛✍ ✠✌✦✑✣ ✎✗✛❀✙✰ ✕❧✑✎✗✛❀✙ ✕✌✖✗ ✿✌✠✒✙ ✛✤✌☛✍ ✛☛✹ ✱✓ ✡✦❁✯ ✓✌
✦✌✾✫☞✪✌✫✠ ✓✗✌✑☛ ✯✖✍✚ ✙✯ ☞✪✌✌✫✑ ✌ ✯✖ ❧✓ ✱☛✡☛ ✡✗❀ ❧☞✬✌✌✲✫ ✎❡❅✠☛ ✛✌❀☛ ✕✛✓❀ ✡✦✒✓✗✻✌✌☛✍ ✛☛✹ ✯❀
✛✖✹✡☛ ✠✌✠✌ ❫✎ ✼✌✗✻✌ ✓✗✑☛ ✯✖✍✚ P●✛✒✍ ✪✌✑✌❋☞✒ ✛☛✹ ♠✑✗✌✼✫ ✦☛✍ ✸✕✛✓❀ ✡✦✒✓✗✻✌✌☛✍ ✛☛✹ ✾✢✻✌✌♦✦✓
❧✛✪❀☛✭✌✻✌✽ ✪✌✒✭✌✫✓ ✛☛✹ ✕✍✑✾✫✑ ✕✛✓❀ ✡✦✒✓✗✻✌✌☛✍ ✛☛✹ ✯❀✌☛✍ ✓✒ ✩❧✮❀ ✎✏✛✐✹❧✑ ✛☛✹ ✕✌❧✛✭✓✌✗ ✯☛✑✢
è✙✌✠ ✕✌✓❧✭✌✫✑ ❧✓✙✌ ✾✙✌✚ ✕✌✩✓❀ ✲✡✠☛ ❀✾❇✌✾ ✡❇✌✒ ✕❧✛✭✓✌✗✌☛✍ ✯☛✑ ✢ ✕♦✙✍✑ ✎✏❧✛❧✼ ✛☛✹ ❫✎
✦☛✍ ✎✏✦✢✬✌ ✜★✌✌✠ ✎✏✌◗✑ ✓✗ ❧❀✙✌ ✯✖✚
—❙—
❙❚❯❱❯ 10
❲❳❨❩❬ ❭❪❫❴❳❵❬❛ (Vector Algebra)
In most sciences one generation tears down what another has built and what
one has established another undoes. In Mathematics alone each generation
builds a new story to the old structure. – HERMAN HANKEL
❍✁✂✄☎✆✁ (Introduction)
✈✝✞✟ ✠✡☛✞☞ ✌✍✎✞ ✏✟✑ ✒✏✟✑ ✈✞✟☞ ✝✓✔✞ ☛✏✕✖✟ ✒✡✑ ✌✡✗✟ ☛☞ ✈✘✝☞✍
➴✙✚✘✛✜ ✢✣✘ ✒✡✤ ✥☞ ✝✦✧★✩✘✕ ✎✟✧ ☛✪✘✕✘✫✬✍ ☞✘✟ ✈✝✞✍ ✒✍ ★✍✏ ✎✟✧ ✠✭✗✮✟
☛✪✘✕✘✫✬✍ ✎✟✧ ✝✘✗ ❢✟✑✠ ✝✒✦✙✚✘✞✟ ✎✟✧ ☛✕✥ ❢✟✑✠ ✝✮ ☛☞✗ ✝✓☞✘✮ ✝✓✒✘✮ ☞✮✞✘
✚✘☛✒✥✤ ✈✎✕✘✟☞✞ ☞✍☛✌✥ ☛☞ ✝✓♣✘✏ ✝✓✔✞ ☞✘ ✗✑✯✘✘☛✎✖ ✰✱✘✮ ✲✳✴
✏✍★✮ ✒✘✟ ✗☞✖✘ ✒✡❡ ✣✒ ✥☞ ✥✟✗✍ ✮✘☛✔✘ ✒✡ ☛✌✗✏✟✑ ✎✟✧✎✕ ✥☞ ✏✘✞
✝☛✮✏✘✐✘ ✌✘✟ ✥☞ ✎✘✵✖☛✎☞ ✗✑✪✣✘ ✒✡✶ ✗☛✷✏☛✕✖ ✒✡❡ ✥✟✗✍ ✮✘☛✔✘✣✘✙
✈☛✠✔✘ ☞✒✕✘✖✍ ✒✡❡ ✖♣✘✘☛✝ ✠✭✗✮✟ ✝✓✔✞ ☞✘ ✰✱✘✮ ✥☞ ✥✟✗✍ ✮✘☛✔✘ ✒✡
✸☛✌✗✟ ✩✕ ☞✒✖✟ ✒✡✑✹ ☛✌✗✏✟✑ ✏✘✑✗✝✟☛✔✘✣✘✟✑ ☞✍ ✔✘☛✢✖ ✝☛✮✏✘✐✘ ✎✟✧
✗✘♣✘❧✗✘♣✘ ☛✠✔✘✘ ✸☛✌✗✏✟✑ ✠✭✗✮✘ ☛✪✘✕✘✫✍✬ ☛✵♣✘✖ ✒✡✹ ✯✘✍ ✗☛✷✏☛✕✖ ✒✡❡
✥✟✗✍ ✮✘☛✔✘✣✘✑✙ ✗☛✠✔✘ ☞✒✕✘✖✍ ✒✡❡ ❢☛✐✘✖✶ ✯✘✘✡☛✖☞✍ ✥✎✑ ✈☛✯✘✣✘✑☛✺✘☞✍ ✏✟✑
✣✟ ✠✘✟✞✘✟✑ ✝✓☞✘✮ ☞✍ ✮✘☛✔✘✣✘✙ ✞✘✏✖❀ ✈☛✠✔✘ ✮✘☛✔✘✣✘✙✶ ✌✡✗✟ ☛☞ ✕✑✩✘✛✶✜
✠✓♥✣✏✘✞✶ ✗✏✣✶ ✠✭✮✍✶ ❢☛✖✶ ✻✘✟✺✘✝✧✕✶ ✈✘✣✖✞✶ ✖✘✝✏✘✞✶ ☞✘✣✜✶ ✼✞✶
✎✘✟♦★✖✘✶ ✽✘✞✾✎✶ ✝✓☛✖✮✘✟✼☞ ✛✾✣✘☛✠ ✥✎✑ ✗☛✠✔✘ ✮✘☛✔✘✣✘✙ ✌✡✗✟ ☛☞ ☛✎✵♣✘✘✝✞✶ ✎✟❢✶ ✾✎✮✐✘✶ ✩✕✶ ✯✘✘✮✶ ✗✑✎❢✟ ✶
☛✎✿✦✖ ✻✘✟✺✘ ☞✍ ✖✍✎✓✖✘ ✛✾✣✘☛✠ ✩✒✦✼✘ ☛✏✕✖✍ ✒✑❡✡
✛✗ ✈❜✣✘✣ ✏✟✑ ✒✏ ✗☛✠✔✘✘✟✑ ☞✍ ✎✦✧❁ ✈✘✼✘✮✯✘✭✖ ✗✑☞♦✝✞✘✥✙✶ ✗☛✠✔✘✘✟✑ ☞✍ ☛✎☛✯✘❂✞ ✗✑☛❃✣✘✥✙ ✈✘✮✡
✛✞✎✟✧ ✩✍✌✍✣ ✥✎✑ ❄✣✘☛✏✖✍✣ ❢✦✐✘✼✏✘✟❅ ☞✘ ✈❜✣✣✞ ☞✮✟✑❢❡✟ ✛✞ ✠✘✟✞✘✟✑ ✝✓☞✘✮ ✎✟✧ ❢✦✐✘✼✏✘✟❅ ☞✘ ✗☛✷✏☛✕✖
❆✝ ✗☛✠✔✘✘✟✑ ☞✍ ✗✑☞♦✝✞✘ ☞✘ ✝✭✐✘✜ ✈✞✦✯✘✭☛✖ ✠✟✖✘ ✒✡ ✈✘✡✮ ✰✝✣✦✢✜ ✖ ✚❇✚✖ ✻✘✟✺✘✘✑✟ ✏✟✑ ✛✞☞✍ ☛✎✔✘✘✕
✰✝✣✘✟☛❢✖✘ ☞✍ ✈✘✟✮ ✝✓✟☛✮✖ ☞✮✖✘ ✒✡❡
10.2 ❈❉❊❋ ●✁■✁❏❍✁✂❑ ▲▼✆◆❖P✁◗❘ (Some Basic Concepts)
✏✘✞ ✕✍☛✌✥ ☛☞ ☛☞✗✍ ✖✕ ✈♣✘✎✘ ☛✺✘❧☛✎✏✍✣ ✈✑✖☛✮✻✘ ✏✟✑ l ☞✘✟✛✜ ✗✮✕ ✮✟✪✘✘ ✒✡❡ ✖✍✮ ✎✟✧ ☛✞✔✘✘✞✘✟✑ ☞✍
✗✒✘✣✖✘ ✗✟ ✛✗ ✮✟✪✘✘ ☞✘✟ ✠✘✟ ☛✠✔✘✘✥✙ ✝✓✠✘✞ ☞✍ ✌✘ ✗☞✖✍ ✒✡❡✑ ✛✞ ✠✘✟✞✘✑✟ ✏✟✑ ✗✟ ☛✞☛✔✚✖ ☛✠✔✘✘ ✎✘✕✍ ☞✘✟✛✜
10.1
W.R. Hamilton
(1805-1865)
441
❧ ✁✂✄ ☎✆✝✞ ✟✄✠
❍✡☛
☞✌
✍✎✏✡✡
✑✒✓✔
✍✎✏✡✡
✌✕✖✡✗☛
[
✕✘
✈✡✙✚ ✛✑✗
10.1 (i), (ii)]
❆
❪❫❴❵❛❜❝
✢✣✎✑ ✤✡✗
✈✜
✌☛✑ ✥☞
✒✡ ✎ ♥ ✡ ✎❞
✧✎ ✩✎ ✑✌✩☛
✑✒✓✔
✍✎ ✏✡✡ ✏✡❞★
✒✡ ✎ ♥ ✡ ✎❞
✕✡ ✎✗✎
✑✌
☞✌ ✑ ✒✪✡✡
✢✣✡ ❢✗ ✕✡ ✎✗✡
✕✘
✦✑ ✒
✕✧
✙✡✖☛
(
✍✎✏✡✡
✈✡ ✙✚✛✑ ✗
‘l’
✍✎ ✏✡✡
‘l’
✢✍
❡❣❤❡
✌✡ ✎
✍✎ ✏✡✡ ✏✡❞★
✢✑ ✍✧✡ ✐✡
10.1(iii))
AB
✑♥r✡✫ ✑ ✍✗ ✕✡✎
❆ ✈✗✭
✢✣✑ ✗✜❞ ✑ r ✗
✗✌
✥✡ ✗✡
✕✘ ❆ ✬✩
✍✎✏✡✡ ✏✡❞ ★ ✧✎❞
☞✌ ✑ ✒✓✔
✌✍
✕✘❞
✒✎ ✗ ✎
✢✣ ✌✡ ✍ ✕ ✧✎❞
✢✑ ✍✧✡✐✡
☞✙❞
✗✜
☞✌
✑✒✪✡✡
✕✘❞ ❆
1
✮✯✰✱✲✲✳✲✲
è✦✡♥
☞✌
☞✎✩☛
✍✡✑✪✡
✑✥✩✧✎❞
✑✌
☞✌
✑✒✓✔
✒☛✑✥☞
✍✎ ✏✡✡✏✡❞ ★
☞✙❞
✒✡✎♥✡✎❞
✑✒✪✡✡
✩✑✒✪✡
✕✡✎ ✗✡
✕✡✎✗✎
(
✕✘
✕✘❞✴
✩✑✒✪✡
✌✕✖✡✗☛
✙✎✛
♦✢
✧✎❞
✑♥✑✒✫ ✓✔
✕✘❞
✌✍✗✎
✬✩✎
✈✡✘✍
✹
✩✑✒✪✡
AB
✕✘❆
✵✵✵
✶
10.1(iii)),
✈✡✙✚✛✑✗
AB
✑✥✩✎
✵✵✵
✶
✸
a,
✩✡r✡✍✐✡✗✭
✢✑✍✧✡✐✡
✈✷✡✙✡
✻
✺ ✈✷✡✙✡
✹
✩✑✒✪✡
a
✺
✙✎✛
♦✢
✧✎❞
✢✼✽ ✗✎ ✕✘❞ ❆
✾✜✒✿
✙✕
A
✵✵✵
✶
✥✕✡t ✩✎ ✩✑✒✪✡
AB
✢✣✡ ✍❞❍ ✡
✕✡✎✗✡
✕✘✴
✢✣✡ ✍❞✑❍✡✌
✾✜✒✿ ✌✕✖✡✗✡
✕✘ ✈✡✘✍
✙✕
✾✜✒✿
B
✥✕✡t
✵✵✵
✶
AB ,
✢✍ ✩✑✒✪✡
✩✧✡❢✗ ✕✡✎✗✡ ✕✘ ✈❞ ✑✗✧ ✾✜✒✿ ✌✕✖✡✗✡ ✕✘❆ ✑✌✩☛ ✩✑✒✪✡ ✙✎✛ ✢✣✡✍❞✑❍✡✌ ☞✙❞ ✈❞✑✗✧ ✾✜✒✿✈✡✎❞
❄❄❄
❅
✙✎ ✛
♦✢
✜☛❀
✧✎❞
✌☛
✒❁ ✍☛
✑♥✑✒✫ ✓✔
✩✑✒✪✡
✑✌✦✡
✌✡
✥✡✗✡
✢✑✍✧✡✐✡
✕✘❆
✗☛✍
❂✈✷✡✙✡
✌✡
✑♥✪✡✡♥
✖❞ ✜✡✬✫ ❃
✩✑✒✪✡
✌✕✖✡✗✡
✌☛
✑✒✪✡✡
✕✘
✬✩✎
✈✡✘ ✍
✌✡✎
✑♥✑✒✫ ✓✔
❇
| AB |
✌✍✗✡
✈✷✡✙✡
|a |
✙✎ ✛
✕✘❆
▲
❈
✈✷✡✫
✯▼◆✲✯❖
✌✤✡ ✡
✯❉❊✮❋✲●
♥✕☛❞
❂✈✡✙✚ ✛✑✗
✑♥✒✎✫✪✡✡❞ ✌
|a |<0
✌✡ ✌✡✎✬✫
✕✘❆
P✯◗❘✲
XI
■✦✡✎❞✑✌ ✖❞✜✡✬✫ ✌❍✡☛ ❍✡☛ ❏✐✡✡❑✧✌ ♥✕☛❞ ✕✡✎ ✗☛ ✕✘ ✬✩✑✖☞ ✩❞✙✎ ✛✗♥
✩ ✎✴
(Position Vector)
✑ ❙ ✡ ❚✑ ✙ ✧☛ ✦
i
❲❳❨❩ ❂ ❃❃❆
(x, y, z)
✒✑ ✤✡ ✐✡✡ ✙✗☛✫
✈❞ ✗✑✍✤✡
✧✎❞
✧❁✖
✾✜✒✿
✩ ✧✌✡ ✎ ✑ ✐✡ ✌
O (0, 0, 0)
❬❬❬
❭
✕✘❆
✗✜
✩✑✒✪✡
OP
✑✥✩✧✎❞
O
✈✡✘✍
P
✑ ♥ ✒ ✎ ✫ ✪✡ ✡❞ ✌
✙✎✛
✩✡✢✎ ✤✡
Ø✧✪✡✭
✢ ❯✑ ✗
☞✌
☞✎✩✡
✢✣✡ ✍❞✑❍✡✌
☞✙❞
✌✡ ✎
✾✜✒✿
❱✧✍ ✐✡
P
✈❞ ✑✗✧
✌☛ ✑ ✥☞
✖☛✑✥☞
✾✜✒✿
✑✥✩✙✎ ✛
✕✘❞✴
O
✙✎ ✛
442
① ✁✂✄
▼◆P◗❘❙❚
❧☎✆✝✞☎
✟✠✡☛
P
❞☎
☞✌✍☎☞✎
❧☞✡✏☎
❞✑✒☎✎☎
❯❱❲❳
✑✓✔
✡✕✖✗
✧✝✭
✆✮☎✯✎
❧✕✘☎
✙❞✞☎☎
XI )
❧✝
✣✣✣
✤
❞☎
✚✆✛☎✝✜
❞✖✎✝
✑☛✢
OP
✦
(
✈✍☎✥☎
r)
❞☎
✆☞✖✧☎★☎
☞✩✪✩☞✒☞✫☎✎
✬✆
✑☎✝✎☎
✱✱✱
✦
| OP | = x 2
❖✛✥✑☎✖
❧✝ ☞✩☞✡✷✸✹
✧✝✭
☞❞✢
✧✕✒
✺☎✎✝
✟✠✡☛
✜✛☎
✑✓✔
✑✭ ✓
P(x, y, z)
❧☎✆✝✞☎✴
[
✟✠✡☛ ✈☎✝✭
✲
A, B, C
z2
✦
❜✵✛☎☞✡
✥✝✳
☞✌✍☎☞✎
❧☞✡✏☎
✶✧✏☎✰
✦
✦
a, b , c
10.2(ii)]
✈☎✥✻✳☞✎
✔
❇❇❇
❈
❞☎
❑
❧☞✡✏☎
✥✝✳
y2
(Direction Cosines)
❢✼✽✾✿❀❁❂❃❄ ❂❅❆
✢❞
O
✟✠✡☛
✲
✑✓✰
r
⑥☎ ✖☎
☞✌✍☎☞✎
x, y
❧☞✡ ✏☎
✢✥✭
z-
OP
✈✞☎
❞✗
▼◆P◗❘❙❚
❈
❉❊❋●❋
r
▲✩☎✵✧❞
❯❱❲❨
✒✗☞ ✺✢
✺✓❧ ☎
☞✡ ✏☎☎✈☎✝✭
✥✝✳
☞❞
❧☎✍☎
✈☎✥✻✳ ☞✎
✠✩☎✢
②❍■❏
✜✢
✧✝✭
✡✏☎☎ ✛
✷ ☎
✶✧✏☎✰
❞☎✝★☎
443
❧ ✁✂✄ ☎✆✝✞ ✟✄✠
✡
, ,
★
r
☛
☞✓✜
❢✎☞✧✜✫✒✑✓✢ ✑✚✛
10.3,
✤✑☞✈ ✜❢✗
x
r
r
cos
OBP
cos , cos
l, m
n
✱☞✌ ✍ ❢✎✏✑✑ ✒✑✓✔✑ ✒✕✖✑✗✓ ✕✘✌✙ ✚✛ ✒✑✓✔ ✑✑✓✌ ☞✓✜ ✒✑✓✢ ✑✚✛ ✣✑✛ ✤✥✑✑✦✗✧
✭✣✏✑✬
✚✛✒✑✓
OAP
☛ ✱☞✌
✢✓ ❢✛❢✎✦✮✯
✱☞✌
✱✒ ✢✣✒✑✓✔ ✑ ❢✳✑✴✑✵✰
✕✘ ✤✑✘❞
cos
✍ ✢❢✎✏✑
✰✑✗✑
❢✒✪✑
✕✘✙
✚✢ ❢✳✑✴✑✵✰
✢✓ ✕✣
✢✣✒✑✓✔ ✑
❢✳✑✴✑✵✰ ✑✓✌
✶
|r |
♦✸◗ ✹②✺ ✐③❀ ❦✻
✸
y
r
cos
✢✓ ✕✣
(lr, mr, nr)
mr
nr
✒✑✓
✱☞✌
✢✑✣✑✩✪✗✬
✢✓ ✕✣ ✎✓✲✑✗✓ ✕✘✌ ❢✒ ❢✳✑✴✑✵✰
✷❦✸
OCP
✱☞✌
✕✘✌ ✤✑✘❞
✒✕✖✑✗✓
✡
☞✓✜ ❃✼
✹✷❀❦
✻❀ ❦
z
r
cos
✺♦❛
✼✽ ✑✾✗
❣❙
✒❞✗✓
✕✘✌✙
✚✢✿
❢✖✲✑ ✢✒✗✓ ✕✘✌✙ ✚✢ ✼✽ ✒✑❞ ❁❂✎✵
r
P
☞✓ ✜ ❢✛✎✓✦ ✏✑✑✌ ✒✑✓✌
lr,
✣✓✌ ✤❢✴✑❄✪❅✗ ❢✒✪✑ ✰✑ ✢✒✗✑ ✕✘ ✙ ❢✎☞✧✜✫✒✑✓✢✑✚✛ ☞✓✜ ✢✣✑✛✵✼ ✑✗✿ ✢✌ ✲✪✑✱❆
❇
✢❢✎✏✑
✼✽ ✒✑❞
❢✎☞✧ ✜✫✤✛✵✼ ✑✗
☞✓✜
✒✕✖✑✗✓
✕✘✌
✤✑✘❞
a, b
✚✛✒✑✓ ✭✣✏✑✬
✗✥✑✑
c
✢✓ ❢✛❢✎✦✮ ✯
❢✒✪✑
✰✑✗✑ ✕✘ ✙
❉❊❋●❍■❏
❈
10.3
❬■❭❪❫
✛✑✓✯
✕✣
▲▼◆❖PP❘❚
❯❘ ❱
✒❞
✢✒✗✓
✕✘✌
l 2 + m 2 + n2 = 1
✼❞✌✗✵
✢✑✣✑✩✪✗✬
a2 + b2 + c 2
1
❑
(Types of Vectors)
❲❳❨P❩
[Zero (null) Vector]
❴❉❵❬■
❢✒
✱✒
✢❢✎✏✑
❢✰✢☞✓✜
✼✽✑❞✌ ❢✴✑✒
✱☞✌
✤✌❢✗✣
❁❂✎✵
✢✌ ✼✑✗✿
✕✑✓✗✓
✕✘✌❜
❡
✏✑❝ ✩✪ ✢❢✎✏✑ ✒✕✖✑✗✑ ✕✘ ✤✑✘❞ ✚✢✓
❢✎✏✑✑
✼✽✎ ✑✛
✛✕✿✌
✰✑
✒✿
0
☞✓ ✜ ❃✼ ✣✓✌ ❢✛❢✎✦✮ ✯ ❢✒✪✑ ✰✑✗✑ ✕✘✙ ✏✑❝✩✪ ✢❢✎✏✑ ✒✑✓ ✒✑✓✚✦ ❢✛❢✏❤✗
❅✪✑✓❢
✌ ✒
✢✒✗✿
✚✢✒✑
✼❢❞✣✑✔✑
✏✑❝✩✪
♥♥♥♣
✴✑✿
❢✎✏✑✑
♠✑❞✔✑
❢✒✱
✕✵ ✱
✣✑✛✑
(Unit Vector)
q■r■s ❴❉❵❬■
✰✑
✢✒✗✑
✕✘ ✙
✕✑✓✗✑
AA, BB
✢❢✎✏✑
✕✘ ✤✥✑☞✑
✏✑❝ ✩✪
✢❢✎✏✑
✕✘✙
❢✒✢✿
❢✎✱
❴④⑤⑥■❉❵q ❴❉❵❬■
✕✘❜
✢✕
✤✑❢✎✣
✢✌❞ ✓✲✑
✢❢✎✏✑
✒✕✖✑✗✓
✒✕✖✑✗✓
➼❍■■❽qs
✕✘✌ ✙
❴❉❵❬■
✒✑✓
❢✛❃❢✼✗
✒❞✗✓
✕✘✌✙
❢✎✏✑✑
✣✓✌
✣✑✳✑✒
✢❢✎✏✑
✒✑✓
â ✢✓
❢✛❢✎✦✮ ✯
❢✒✪✑
✰✑✗✑
✕✘✙
✎✑✓ ✤✥✑☞✑ ✤❢♠✒ ✢❢✎✏✑ ❢✰✛✒✑ ✱✒ ✕✿ ✼✽ ✑❞✌❢✴✑✒ ❁❂✎✵
✎✑✓ ✤✥✑☞✑
✤❢♠✒ ✢❢✎✏✑
✪❢✎
✱✒
✕✿
❞✓✲✑✑
☞✓ ✜ ✢✣✑✌ ✗❞
✕✘ ✗✑✓
✕✘✌ ✙
(Equal Vectors)
✢✣✑✛
✒✑✓ ✚✦
✕✘✌✙
✚✛✒✑✓
❷
❷
✎✑✓ ✢❢✎✏✑
a
❸❹❺❺
b
✢✣✑✛ ✢❢✎✏✑ ✒✕✖✑✗✓ ✕✘✌ ✪❢✎ ❻✛☞✓✜ ✼❢❞✣✑✔✑
❼
❼
❢✎✏✑✑
✒✿
(Co-initial Vectors)
✢❢✎✏✑
✢❢✎✏✑
❴q■❶ ❴❉❵❬■
✱☞✌
a
(Collinear Vectors)
❴⑦⑧⑨⑩ ■ ❴❉❵❬■
☞✓
✕✵ ✱
✚✢✒✑✓
✱✒ ✢❢✎✏✑ ❢✰✢✒✑ ✼❢❞✣✑✔✑ ✱✒ t✤✥✑☞✑ ✉ ✚✒✑✚✦✇ ✕✘ ✣✑✳✑✒ ✢❢✎✏✑
①
✒✕✖✑✗✑
❢☞✒❥✼✗✬
♥♥♥
♣
a =b
☞✓✜
❃✼
✣✓✌
❢✖✲✑✑
(Negative of a Vector)
✱✒
✰✑✗✑
✢❢✎✏✑
✕✘✙
❢✰✢✒✑
✼❢❞✣✑✔✑
❢✎✱
✕✵ ✱
✢❢✎✏✑
(
✣✑✛
❾❾❾
❼
✖✿❢✰✱
AB )
☞✓✜ ✢✣✑✛ ✕✘ ✼❞✌✗✵ ❢✰✢✒✿ ❢✎✏✑✑ ❢✎✱ ✕✵ ✱ ✢❢✎✏✑ ✒✿ ❢✎✏✑✑ ☞✓✜ ❢☞✼❞✿✗ ✕✘❜ ❢✎✱ ✕✵ ✱ ✢❢✎✏✑
❾❾❾❼
✒✑
❿✔✑✑➀✣✒
❾❾❾❼
BA
✒✕✖✑✗✑
✕✘✙
❻✎✑✕❞✔✑✗✬
❾❾❾
❼
➁ ➂
AB
☞✓ ✜
❃✼
✣✓✌
❢✖✲✑✑
✰✑✗✑
✕✘✙
✢❢✎✏✑
BA ,
❾❾❾
❼
✢❢✎✏✑
AB
✒✑
❿✔✑✑➀✣✒
✕✘
✤✑✘❞
✚✢✓
444
① ✁✂✄
❢✎✏✑✒✓✔
✥✕✖✗✘✙✚ ✕✛✜✢✣✣✛✤✣✚ ✦✛✧★✣ ✩✦ ✕✪✫✣✜ ✬✭ ✛✫ ✥✮✯✰✱ ✦✰ ✛✫✦✲ ✫✣✰ ✢✣✲ ✥✦✳✰✴ ✕✛✜✯✣✵✣ ✶✳✱ ✛✧★✣✣
✫✣✰ ✕✛✜✳✛✚✘✚ ✛✫✶ ✛❞✮✣ ✷✳✖✱ ✳✰✴ ✦✯✣✱ ✚✜ ✛✳✷✸✣✣✛✕✚ ✛✫✖✣ ✹✣ ✦✫✚✣ ✬✭✺ ✩✦ ✕✪✫✣✜ ✳✰ ✴ ✦✛✧★✣ ✷✳✚✱ ✻✣
✦✛✧★✣
✫✬❧✣✚✰
♠❁❂❃❄❅❂
✫✣
1
✬✭✱✺
✩✦
✧✛♥✣✵✣
✽✣❧✰❈✣✲✖
✕✼✜✰
30°
✦✰
✛✮❉✕✵✣
✽✾✖✣✖
✕✛★✿✯
✯✰✱
✯✰✱✐
✬✯
❆❇
✷✳✚✱✻✣
km
✦✛✧★✣✣✰✱
✳✰✴
✫✲
✬✲
✿✿✣✘
✫✜✰✱❀✺
✰
✛✳✷✸✣✣✕✮
✫✲✛✹✶✺
❊❊❊
❋
❃❣
✦✛✧★✣
OP
(
■❇❏❆
♠❁❂❃❄❅❂
2
✽✣✳❍✴✛✚
✽✢✣✲✤●
✛✮❑✮✛❧✛❈✣✚
✯✰✱ ❡✰✵✣✲❞▲
✛✳✷✸✣✣✕✮
✫✣✰
✛✮❉✛✕✚
✫✜✚✣
✬✭
)
✧✰✛❈✣✶ ✺
✯✣✕✣✰✱ ✫✣✰ ✽✛✧★✣
✶✳✱ ✦✛✧★✣
✳✰✴ ❉✕
✫✲✛✹✶✺
(ii) 1000 cm3
(v) 10 g/cm3
(i) 5 s
(iv) 30 km/h
(vi) 20 m/s
✥▼✣✜
✫✲
(iii) 10 N
✽✣✰✜
✈☎✆✝✞✟✠
❃❣
(i)
(iv)
❀✛✚◆✽✛✧★✣
♠❁❂❃❄❅❂
(i)
(ii)
(iii)
(ii)
(v)
✦✯✖◆✽✛✧★✣
3
✽✣✳❍ ✴✛✚
✦✱ ✜✰❈✣
■❇❏◗
✯✰✱
(iii)
(vi)
✽✣✖✚✮◆✽✛✧★✣
❖✣✮P✳◆✽✛✧★✣
✫✣✭✮
✦✰
❞❧◆✦✛✧★✣
✳✰❀◆✦✛✧★✣
✦✛✧★✣
✬✭✱
✦✯✣✮ ✬✭✱
✦✬◆✽✣✛✧✯ ✬✭✱
❃❣
(i)
(ii)
(iii)
❘
✦✱✜✰❈✣
✦✛✧★✣
❘
❘
: a, c
d
:a
c
r❙❚❚
❯
✦✯✣✮
✦✛✧★✣
❯
r❙❚❚
❘
✦✬◆✽✣✛✧✯
✦✛✧★✣
: b, c
✈☎✆✝✞✟✠
❘
❯
r❙❚❚
d
❱❲ ❳❨❩❬❭❪
1.
✥▼✣✜
2.
✛✮❑✮✛❧✛❈✣✚
✦✰
30°
✕✼✳✘
✯✰✱
✯✣✕✣✱ ✰
❆❇
✫✣✰
km
✽✛✧★✣
(i) 10 kg
(iv) 40
3.
(i)
(iv)
✦✯✖
✳✰❀
✫✣✰
✛✳✷✸✣✣✕✮
✶✳✱
✦✛✧★✣
(ii) 2
✽✛✧★✣
✫✣❧✣✱ ★✣
✶✳✱
✦✛✧★✣
✧✼✜✲
(v)
✫✣✖✘
✽✣❧✰❈✣✲✖
❉✕
✯✰✱
✥▼✣✜◆✕✛★✿✯
✳✰✴
❉✕
✛✮❉✕✵✣
❡✰✵✣✲❞▲
✫✲✛✹✶✺
✫✲✛✹✶✺
(iii) 40°
(vi) 20 m/s2
✳✼✴❧✱ ❞
✜✣✛★✣✖✣✰✱
(ii)
✫✣
✳✰✴
✯✲●✜
(v) 10–19
✳✣●
✛✮❑✮✛❧✛❈✣✚
✳✰✴
❫❴❵❫
✯✰✱
❡✰✵✣✲❞▲
✫✲✛✹✶✺
(iii)
❞❧
✡☛☞✌
✡☛☞✍
❧ ✁✂✄ ☎✆✝✞ ✟✄✠
4.
✈✡☛☞✌✍✎
✏✑✒✓
✔✕✖
☛✗✘ ✙
✚✛✜
✍✢✣✢✍✤✍✥✡✎
✦✍✧★✡✡✛ ✜
445
✖✡✛
✐✩✪✡✍✢✕✫
(i)
(iii)
5.
(ii)
✦✩✬✈✡✍✧✚
✦✜ ✭✛✥✡
✐✭✜ ✎✮
✦✚✡✢
✈✦✚✡✢
✍✢✣✢✍✤✍✥✡✎ ✖✡ ❢✯✡✭ ✦✰✱ ✈✲✡☛✡ ✈✦✰✱ ☛✛✌ ✳✐ ✚✛✜ ✧✴✍✵✕✫
✸
✶
(i) a
(ii)
10.4
✎✲✡✡
✧✡✛
✷
✦✜ ✭✛✥✡
a
✩✹✜✫
✦✜ ✭✥
✛ ✡
✦✍✧★✡✡✛ ✜
✐✍✭✚✡♥✡
✖✡
✦✧✹ ☛
(iii)
✦✚✡✢
✐✍✭✚✡♥✡
☛✡✤✛
✧✡✛
✦✍✧★✡
(iv)
✦✚✡✢
✐✍✭✚✡♥✡
☛✡✤✛
✧✡✛
✦✜✭✛✥✡
✺✻✼✽✾✾✿❀
❁✾
✩✡✛ ✎✡
✦✚✡✢
✩✡✛ ✎✛
✦✜ ✭✥
✛ ✡
✦✍✧★✡
✩✹✫
❲❳❨❩❬❭❪
❫❴❵❝
✩✹✜✫
✩✡✛ ✎✛
✦✚✡✢
✩✹ ✜✫
(Addition of Vectors)
❂✾✿❃❄❅❆
❇❇❇
❈
✦✍✧★✡
AB
✦✛
✎✖
✍☛r✲✡✡✐✢✫
✕✖
✤❍■ ✖✴
B
C
❋●✧✮
❋●✧✮
✦✛
✈●
A
❋●✧✮
C
❋●✧✮
✎✖
✩✚✡✭✡
✦✡❉✡✭♥✡✎❊
✕✖
✦✛
✎✖
✤❍■ ✖✴
✕✛ ✦✴
❋●✧✮
B
✪✤✎✴
❏✡✭✡
✎✡✰✐✱✘
✍r✲✡✍✎
✎✖
(
✩✹
✍✖✱✡
✦✛
✐▼✡ ◆✎
✈✍❡✡◗✱❘✎
✱✩
✦✍✧★✡
✩✡✛✎✡
✍✖✱✡
✱✡✛ ✗
✖✡
✩✹
✈✡✹✭
✵✡✎✡
❖✦✛
✩✹
A
✦✛
✈✡✹ ✭
❋●✧✮
✍☛r✲✡✡✐✢
❑❑❑
▲
B
✍✵✦✚✛ ✜
❢✦☛✛ ✌
✫
☛✮✌✤
❋●✧✮
✖✴✍✵✕
10.7)
✈✡☛☞ ✌✍✎
❑❑❑
▲
AC
❋●✧✮
✪✪✡✘
✪✤✎✴
✗✱✡
❑❑❑
▲
✚✛✜
✖✴
✩✹
●✡✧
A
✦✛
✦✍✧★✡
,
❑❑❑
▲
AC = AB BC
P
☛✛ ✌
✳✐
❲❳❨❩❬❭❪
✩✹✫
✍❙✡❡✡✮✵
✍✢✱✚
✖✩✤✡✎✡
✩✹✫
▲
✶
✦✡✚✡❚✱✎❊❯ ✱✍✧ ✩✚✡✭✛ ✐✡✦ ✧✡✛ ✦✍✧★✡
a
❫❴❵❛
✎✲✡✡
b
✩✹✜
[
✈✡☛☞✌✍✎
10.8 (i)],
✎✡✛ ❢✢✖✡ ✱✡✛ ✗ ❱✡✎ ✖✭✢✛
☛✛ ✌ ✍✤✕ ❢❚✩✛✜ ❖✦ ✍r✲✡✍✎ ✚✛✜ ✤✡✱✡ ✵✡✎✡ ✩✹ ❯ ✎✡✍✖ ✕✖ ✖✡ ✐▼✡ ✭✜ ✍❡✡✖ ❋●✧✮ ✧♦ ✦✭✛ ☛✛ ✌ ✈✜ ✍✎✚ ❋●✧✮ ☛✛ ✌ ✦✜ ✐✡✎✴
✩✡✛ ✵✡✕
[
✈✡☛☞✌✍✎
10.8(ii)]
✫
❲❳❨❩ ❬❭❪
❫❴❵❜
446
① ✁✂✄
10.8 (ii)
♠☎✆✝✞✟✆✠✡ ☛✆☞✌✍✎✠
✖
b
❡✏✒
✑ ✝❡✓✏ ✔✎☎✕✆
☞✏✍ ♦✎✞❡✆✟✆ ✗☞✑ ✎☎✕✆✆ ✘✆✏ ♦✎✞☞✎✠✙✠ ✎✘✗ ✎✚✓✆
★
a
♦✐✘✆✞ ✜✢✆✆✓✆✑✠✎✞✠ ✎✘✣✆ ✝✤ ✠✆✎✘ ✛✔✘✆ ♦✐✆✞✑✎✥✆✘ ✦✚☎✧✒
✘✩
✮
✮
AC
✥✆✧✫✆
✠✩✔✞✩
⑥✆✞✆
✎✓✬✎♦✠
✝❡✏✑
✭
a
✔✎☎✕✆✆✏✑
✳✳✳
✮
♦✐☎✆✓
✘✞✠✆
✝✤✒
☛✢✆✆✙✠✲
✶✶✶
✷
☛✚
❡✏ ✑
✝❡
♦✆✠✏
✝✤✑
✎✘
✳✳✳
✮
b
✣✆✏❞
✘✆
✰☛✢✆☞✆
♦✎✞✟✆✆❡✩✱
✳✳✳
✮
AB BC = AC [
✭
☛✆☞✌ ✍✎✠
CA ,
✴ ✵
♠♦✣✧✙ ✈✠
✛✔✎❜✗
✳✳✳
✮
✔❡✩✘✞✟✆
✳✳✳
✮
✔✏
✭
✠✆✺♦✣✙ ✣✝
✝✤
✎✘
✎✘✔✩
10.8 (ii)]
❆
♦✆✠✏
✝✤✑
✎✘
✖
✭
✘✩ ✥✆✧✫✆☛✆✏ ✑ ✘✆✏
✎✪✆✥✆✧✫
✝❡
✹✹✹✖
✳✳✳
✮
AB BC CA = AA
✛✔✘✆
✠✢✆✆
✶✶✶✷
AC
✈✣✆✏ ✑✎✘
♦✧✓✡
ABC
✎✪✆✥✆✧✫
0
✸
✣✎☎
✗✘ ✻❡
✫✆✗
❡✏ ✑ ✎❜✣✆
♦✎✞✟✆✆❡✩ ✘✩ ☛✆✏✞ ♦✐✏✎✞✠ ✘✞✠✆ ✝✤ ✈✣✆✏✎
✑ ✘ ♦✐✆✞✑✎✥✆✘ ✗☞✑ ☛✑✎✠❡ ✦✚☎✧ ✔✑♦✆✠✩ ✝✆✏ ✫✆✠✏ ✝✤✑
✣✝
✠✆✏
[
✗✘
❆
✳✳✳
✮
BC
✾
✔✎☎✕✆
✕✆✼✽✣
10.8(iii)]
☛✆☞✌ ✍✎✠
✳✳✳✳
✮
☛✚
ABC
✖
✯
a b
✔✎☎✕✆
☞✏✍ ☛✑✎✠❡ ✦✚☎✧ ☞✏ ✍ ✔✑♦✆✠✩ ✝✤ ✠✚ ✎✪✆✥✆✧✫
✛✔
✘✩
✞✿✓✆
✛✔
♦✐✘✆✞
✘✩✎✫✗
✠✆✎✘
✛✔✘✆
♦✎✞❡✆✟✆
BC ,
✔✎☎✕✆
☞✏✍
✹✹✹
✖
♦✎✞❡✆✟✆
☞✏ ✍
✔❡✆✓
♦✞✑✠✧
✝✆✏✒
✛✔✘✩
BC =
❄
✵
BC
✎✪✆✥✆✧✫
✠✚
✎✓✣❡
✶✶✶
✷
AC
AB BC = AB ( BC)
❄
✎✘✔✩
✗✘
✓✆☞
✓✆☞
✘✆✏
✓✆☞
☞✆✜✠☞
✎☎✣✆
♦✎✞✟✆✆❡✩
❉
r❋●●
b
✓☎✩
☞✏✍
✘✩
❞✣✆
☞✏ ✍
✎☞♦✞✩✠
✝✆✏
☛✆☞✌ ✍✎✠
✘✞✠✏
[
✝✧✗
☛✆☞✌ ✍✎✠
❀❁❂❃
(iii)]
iii
❀❁❂❃✰
✱ ☛✢✆✆✙ ✠✲
✘✆✏
✎✘✓✆✞✏
✘✞✠✏
☞✏❞ ☛✆✤✞
☞✏ ✍
✝ ❡✆✞ ✏
✝✤❆
✑
☎✼ ✔✞✆
✎✓✬✎♦✠
✔✏
✠✚
☎✼✔✞✏
✛✔
✓☎✩
✿❜✓✆
✣✢✆✆✢✆✙
✎☞✿✆✞
✚✆✞✏
❡✏ ✑
✘✞✠✆
✰☛✆ ☞✌ ✍ ✎✠
✥✆✧✫✆☛✆✏ ✑
♦✆✔
✗✘
✔✏
✎☎ ✕✆ ✆
✔❡✆ ✑✠✞
✎ ✓✬✎♦ ✠
✔✎✝ ✠✱ ☎✆✏
❀❁❂❏✱
☞✏ ✍
✎✘✓✆✞✏
✓✆☞
✕✆✧✬
♦✞
✘✞✠✩
❜✆✓✏
☞✏ ✍
✿ ✠✧ ✥ ✆ ✧ ✙ ✫
✎✘✗
✠✚
✔✎☎ ✕✆
✫✆ ✓✏
♠✥✆✣✎✓❑▲
✦✚☎✧
✠✘
☎✆✏
✝✤❆
✎❜✗
♦✆✓✩
☞✏ ❞
✔✎☎✕✆
✘✆
✛✔
☞✏ ✍
☞✏ ❞❆
✓✆☞
✝❡✆✞✏
✚✝✆☞
✘✆✣✙
✎☎✕✆✆
✘✞
✛✓ ☎✆✏
✘✩
♦✆✔
✘✩
✝✤✑
✎✘
✷
✣✆✏❞♦✍❜
✷
✎☎✕✆✆
✝✤❆ ✣✝
✘✆
✔❡✆✑✠✞
✿✠✧✥✆✧✙✫
✎✓✣❡
✘✞✠✆
✘✝❜✆✠✆
b
✛✓
☞✆❜✆
✷
✗☞✑
✝✤❆
✘✆✏ ♦✎✞❡✆✟✆
✔✎☎✕✆
✣✆✏ ❞
▼◆❖P◗❘❙
✝✤✒
✑
❜✑✚ ☞✠✲✲
✫✆✓✏
✛✑✫ ✓
⑥✆✞✆
✗✘
❞✎✠ ✗☞✑ ✎☎✕✆✆
☞✆ ❜✏
✘✩
☞✏ ✍
☞✏ ❞✆✏ ✑ ☞✏ ✍ ✣✧❞♦✠ ♦✐✥✆✆☞
♦✐✥✆✆☞✩
✔✎☎✕✆
✞✝✏
✘✩
r❋●●
❞✧✫✞✓✏
✔✏
a +b
✎✓✬✎♦✠
a
✿✠✧✥✆✧✙✫
✔❡✆✑✠✞
✎☞✘✟✆✙ ✛✓ ☎✆✏ ✓✆✏ ✑ ✔✎☎✕✆✆✏ ✑ ☞✏ ✍ ✣✆✏ ❞
✔✎✝✠
♦✆✠✏
✝✤❆
☞✏ ✍ ♦✆✓✩ ☞✏ ✍ ✚✝✆☞
✎✥✆✽✓ ☞✏ ❞ ✔✏
✥ ✆ ✧ ✫ ✆☛ ✆✏ ✑
✗☞ ✑
✝❡
❊
✷
✰♦ ✎✞ ❡✆✟ ✆
✔✏
a b
✝✤❆
✔✑❜♥ ✓
☎✆✏ ✓✆✏ ✑
☞✏ ✍
✮
✮
❈
✵
☛✑✠✞
✗✘
✿✿✆✙
❡✏ ✑ ✗✘
☞✏ ❞✱
✣ ✎☎
✝✤
✎☎✕✆✆
✷
✷
AC , a
☛✚
☎✆ ✏
✾
✭
✹✹✹✹
✖
✔✎☎✕✆
✎✓✣❡
✘✩
✐ ✆✏ ❞
☛✓✧♦✣
✘✆
✳✳✳
✮
✾ ❈
✳✳✳✳
✮
✶✶✶
✷
✳✳✳✳
✮
☞✆❜✩
BC
✎☎✕✆✆
❅❅❅
❇
✹✹✹✹
✖
❚❯❱❲
✘✆
✔✏
✰☛✢✆✆✙ ✠✲
✎✓❍✓✎❜✎■✆✠
447
❧ ✁✂✄ ☎✆✝✞ ✟✄✠
✑✒✓✔✓✕ ✖
❢☛☞✌✍✎✏
✡
✵✵✵✶
✵✵✵
✶
✵✵✵
✶
✿❢❀❁✎
★✥✺
✦✙
❂✎❃❄✌❅❆
✵✵✵
✶
❄❈✍✎❊❋●
▲
✦✳ ✲
❍
✵✵✵
✶
✑✚
✬✭✮✯
✵✵✵
✶
AC
✰✑✻✼✓
✘✓✢✣
✩✢ ✫
✻✓✢
✰✢
✦✙
✗✓✢✱
✩✢✫
✷
✑✗✘✙
OB )
✧✚
✖✓✢
✑✚
✻✽✰✤✢
✩✢✫
(
a
Ø✙✑✩✑✗✙✘✥✓
ABC
✑✒✓✔✓✕ ✖
✦✕✧
✦✙
✜✓✥✢
★❚❳
✰✙✓✗
ABCD
ADC
❲
✰✙✓✳✥✤
t✥✕✔ ✓✕ ✸ ✖
✰✙✥✕ ✾✘
✦✲✳✹
✚✓
✛✜✘✓✢ ✣
❖❖❖
P
P
AB a
✈◗❘
❙
P
BC b ,
✚✤✥✢
❲
✦✲❳
❭✰✑♦✧
❪❪❪
■
❫❑❴
❵
✚❞ ✰❨✙✕ ❩✓
t✥✕ ✔✓✕✸ ✖
★✓✩✪ ✫✑✥
❪❪❪
■
✙✢✳ ✑✒✓✔✓✕✖ ✑✗✘✙ ✩✢✫ ✜❡✘✓✢✣ ✰✢
✔✓✕ ✖✓✧❬
✬✭✮✬✭
✙✢✳
■
DC = AB = a
✦✲✹ ✜✕✗✺ ✑✒✓✔✓✕✖
❪❪❪■
❪❪❪❪
■
✶
✑✚
AC = a + b
■
AD = BC = b
✑✗✘✙
❯❯❯
❱
✰✙✓✳ ✥✤
✰✙✓✳✥✤
❪❪❪
■
✚✓✢ ♦❞✑✖✧ ❍★✓✩✪ ✫✑✥ ✬✭✮✬✭◆ ✙✓✗ ♦❞✑✖✧
✑✒✓✔✓✕ ✖
✑✚
❉✘✓✢✳✑✚
✧✩✳
❪❪❪■
✦✲✳
✙✢✳
✦✲✳
)
❖❖❖
P
✥❚
✰✚✥✢
✑♦✧
■
b = b
♠✌✌❢▼✎ ✰✙✓✳✥✤ t✥✕✔✓✕✸✖
✚✤
✵✵✵
✶
Properties of vector addition)
b
r❏❑❑
■
■
■
a
★✓✩✪ ✫✑✥
■
a
✰✑✻✼✓✓✢✳
✦✕ ✧
❉✘✓✢✳✑✚
✰✚✥✢
■
✻✓✢
①
❄❈✍✎❊❋●
✚✤✥✢
✴
✚✦
❇❃❅
✛✜✘✓✢✣
OA OB = OC (
✘✓
✴
✦✲✹
✚✓
✵✵✵✶
OA AC = OC
✑✗✘✙
✑✗✘✙
❪❪❪
■
AC = AD + DC
✶
=b +a
❱
❱
❱
a b = b
★✥✺
❛
2
❄❈✍ ✎❊❋●
■
■
a
■
✥ ❞✗
■
✰✑✻✼✓✓ ✢ ✳
❲
■
(a b ) c = a (b
▲
❱
❛
▲
❜
■
❜
c)
■
✣✘✓
✦✲
✙✓✗
✖✲✰✓
♦❞✑✖✧❳
✑✚
✰✑✻✼✓✓✢✳
★✓✩✪ ✫✑✥
❫❑❵❴
c
(
✰✓✦t✘✸
■
♠✌✌❢▼✎
❦♥♣qs✉✇
■
a, b
■
a, b
10.11(i)
✩✢✫
✣✕ ❝✓
)
❪❪❪
■
■
r❏❑❑
★✓✲✤
c
(ii)
②③④②③
✑♦✧
✚✓✢
Ø✙✼✓✺
✙✢✳ ✻✼✓✓✸ ✘✓
❦♥♣qs✉✇
✣✘✓
②③④②②
❪❪❪■
PQ, QR
✦✲✹
❪❪❪
■
❣ ❤✐
RS
✰✢
✑✗❥✑✜✥
✑✚✘✓
448
① ✁✂✄
✏
✏
✑✑✑
✒
✑✑✑✒
✑✑✑
✒
✑✑✑✒
✑✑✑
✒
✑✑✑
✒
r✍
a b = PQ + QR = PR
✓✔✕✖
b
✎
✏
✒
✒
(a
❜✗✘✙✚
✏
c = QR + RS = QS
✎
b)
✛
✒
✑✑✑
✒
✑✑✒
✑✑✑
✒
✑✑✑
✒
✑✑✒
c = PR + RS = PS
✛
c ) = PQ + QS = PS
✒
✒
✑✑✑
✒
✛
✒
✓✔✕✖
a (b
✓r✜
(a b ) c = a (b
✛
✒
✒
✒
✛
✒
✒
✛
✢
✒
✢
c)
✒
✗✘❧✩✔
❢✣✤✥✦✧★
✪✔✫✬✭✮✙
✯✫✮
✗✔✰✱✪✲
✬✳✴✔✵✶✲
✒
✒
✪✔✫✬✭✮✙
✷✔✫❀✽✷✔✫✺
✷✔
✾✭✪✔✫✬
✘✷✚
a
✘✍✹✔
✗✰✔✪r✔
✷✸
✗✫
r✸✹
✰✶
✗✘❧✩✔✔✫✺
✒
a, b
✒
✻❋✼✼
c
✷✔
✒
b
c
✯✫✮
♦✭
✘✙✿✔r✫
✰✕❁
✺
✷✰✙✔r✔
✰✕❁
✶✫✺
❃
✹✔✫✉
✷✸✘❂✚
✘✷
✘✷✗✸
✗✘❧✩✔
a
✘✙✚
✯✫✮
✰✶
✏
✏
✰✕✜
✺
✭✔r✫
✏
✏
✏
a 0= 0 a
✎
❄
✎
a
❈
✪✰✔❅
✩✔❆❇ ✪
10.5
0
✗✘❧✩✔
❊●
✗✘❧✩✔
❍■❏❑▲
▼◆
✪✔✫✬✭✮✙
▼■❏❑▲
✘✙✚
✯✫✮
●▲
✪✔✫❉✪
✗✯✲✗✘✶✷✔
(Multiplication of a Vector by a Scalar)
❖P ◗▲❘
❙
✶✔✹
✙✸✘❂✚
a
✘✷
❙
✚✷
✘❧✪✔
✰✳✓ ✔
✗✘❧✩✔
✓✔✕✖
✰✕
❚ ✚✷
✓✘❧✩✔
✘❂✗✫ ❚
a
✯✫✮
♦✭
✘✹✘❧✲❀✉
✶✫✺
a
✷✔
| |
❚
✬✳✴✔✔
✰✔✫r✔
a
a
❲✔✸ ✗✘❧✩✔
❯
✰✔✫✹✫ ✯✫✮ ✓✹✳✗✔✖ ❚
✘✷✪✔
✰✕❱
❂✔r✔
a
✗✘❧✩✔
✓✘❧✩✔
✷✔
a
✷✔
✓✘❧✩✔
❚
,
❚ ✗✫ ✬✳ ✴✔✹
✷✰✙✔r✔
✯✫✮ ✗✺ ✖✿
✫ ✔ ✚✷ ✗✘❧✩✔ ✰✕❁ ❚ ✯✫✮ ✶✔✹ ✵✹✔❳✶✷ ✓❨✔✯✔ ❩✴✔✔❳✶✷
❯
✷✸ ✘❧✩✔✔❱
✰✕❱
✗✘❧✩✔
❃
❙
✰✕❁ ✹✔✫✉ ✷✸✘❂✚ ✘✷ ❚
r✍
❃
❯
✗✫ ✬✳✴✔✹✭✮✙
✰✕❁
❯
a
✯✫✮ ✗✶✔✹ ✓❨✔✯✔ ✘✯✭✖✸r ✰✔✫r ✸ ✰✕❁ ❚
a
❯
✷✔ ✭✘✖✶✔✴✔
a
✯✫✮ ✭✘✖✶✔✴✔
✓❨✔✔✲r❬
✒
❫
| a | = | || a |
❭
✚✷ ✓✘❧✩✔
✓✔✯❛✮ ✘r
❝❞❡❝❣
✗✫ ✗✘❧✩✔
✯✫✮ ✬✳ ✴✔✹
✶✫✺
✰✕❁
❧✸
✬❜✲
❪
❉✪✔✘✶r✸✪ ✱✔❴✔✳❀✔✸✷✖✴✔
✷✔
✈☎✆✝✞✟✠
❂✍ ❚
= – 1,
✒
r✍
❭
a
❤ ✐
a
✯✫✮
✘✯✭✖✸r
✒
✰✕ ✓✔✕✖
✰✶
✰✶✫ ✩✔✔
✰✕❁
✒
✗✘❧✩✔
–a
✒
a (– a ) = (– a )
✛
✷✸
✷❵✭✹✔
(visualisation)]
✡☛☞✡✌
❙
❂✔✫ ✚✷ ✚✫✗✔
✗✘❧✩✔
❙
✘❧✩✔✔
♦✭
✒
❙
a ✷✸
[
a
✯✫✮
✗✶✔✹ ✰✕ ✓✔✕✖ ✘❧✩✔✔
❙
✗✘❧✩✔
a
a
✷✔
✒
✒
✛
✰✕ ✘❂✗✷✔ ✭✘✖✶✔✴✔
❤
0
✭✔r✫
✰✕❁
✺
❩✴✔✔❳✶✷
❥✓❨✔✯✔
✪✔✫❉ ✪
✭❦ ✘r✙✔✫✶♠✷✰✙✔r✔
449
❧ ✁✂✄ ☎✆✝✞ ✟✄✠
✈✡☛☞
✌✍✎ ✏
1
,
|a|
=
✔
✍✎✌✡
✑
❢✒✈ ✡
❢☛
✦
✔
0,
a
✍✓
✫
★✩✓✡☞
✪
✓✢
✍✎✘✡✡
✍✓✛✢
✵✶✷✸✹✺✻
✴
✈✡❜❊✱
❞✬✣
❞✡✭✡✓
❀❁❂❃❄
❅❆❇
k
✈✍✎✘✡
✼✡✢
✈❏✡
✱♦✣
A(1, 0, 0), B(0, 1, 0)
z✲✬ ✤✬
✈❏✡ ★☞
❢☛✣✯
✤✥
◆◆◆❖
✈✡☛☞
✈✥
①✌✡
◆◆◆
❖
✛✍✎✘✡
✓✡✬
❢☛ ✯
❍❞✘✡■
✱✓
❞✡✜
●✥✎✒
✈✡☛☞
1
a
|a|
✽
♦✬✰
❢❞ ✎✬✳ ✡✤✬ ❢☛✣ ✍✓
♥♥♥♣
★✩✓✡☞
OZ
✈❏✡✡✬✣
kˆ
P◗❘❥
P (x, y, z)
✍✓
✈✡☛☞
✍❙✜❞✬ ✣
♦✬✰
C(0,0, 1)
♦✬✰
♦✬ ✰
,
x-
✈❏✡
| OC | 1
P◗❘❥
✛✬
✍✜✍✎❊❑▲
★✩❚✌✬✓
✓✡
✛✍✎✘✡
❞✡✭✡✓
✍✓✌✡
❱
❢☛ ✣
✓❢✲✡✤✬
❢☛ ✣
★✍☞❞✡❯✡
❢☛
❙✡✤✡
✓✡
✍✐❫✡✍✤
✛✬
✛✍✎✘✡
XOY
✤✲
OP
❲✈✡♦❳✰ ✍✤
★☞
✲✢✍❙✱
✳✡✢✣ ❵✬
iˆ, ˆj
❙☛ ✛✡
①✱
✲✣✥
kˆ
❝❡❣
♦✬✰
✍✜✎✬ ❊✘✡✡✣ ✓✡✬✣ ✓✢
▼▼▼✽
✈✡☛☞
OQ
❤
★✍☞✼✡✡❑✡✡ ♦✬✰ ✈✜✒ ✛✡☞
xiˆ .
❜✛
★✩✓✡☞
❢❞
⑤⑦⑧⑨⑩❶❷
✛✡★✬ ❏✡
P
❸❹❺❸❻
❱❨❬❱❭❪✯
✍✓
✓✡
✈✡♦❳✰ ✍✤
★✡✎
x, y
❍❞✘✡■
❢❞
❢☛✣
★✡✤✬
▼▼▼✽
▼▼▼✽
▼▼▼✽
rrr
s
▼▼▼✽
▼▼▼▼
✽
OP = OP1 + P1P
O
❢☛ ✣✯
❴❴❴
✦
✈✡☛☞
★✩✓✡☞
✍✲✳✡✤✬
✓✡✬ ❍❞✘✡■
OP1 = OQ + QP1
❜✛
❞✬✣
✮★
❱❨❬❱②
●✥✎✒
P1
✍✐❫✡✍✤
✛✍✎✘✡
xiˆ
q
yjˆ
❤
xiˆ
q
yjˆ
q
Pt◗❡◗
❜✛
★✩✓✡☞
✈❏✡ ♦✬✰ ✈✜✒ ✍✎✘✡
▼▼▼✽
P1P
❤
❤
zkˆ .
✮★
❞✬✣
OR
❜✛✢
zkˆ
✽
OP (
✎✘✡✡❊✌ ✡
✍✓
❤
▼▼▼
✽
✓✡
★✡✤✬ ❢☛✣ ✍✓
❢☛✯
❞✬ ✣
z-
✱♦✣
▼▼▼
✽
QP1 OS yjˆ
♠
❜✛✬
◆◆◆
❖
✈✜✒ ✍✎✘✡
⑥✡☞✡
P1
●✥✎✒
♥♥♥
♣
♠
❢❞
✽
✈❏✡ ♦✬✰ ✛❞✡✣ ✤☞ ❢☛✯ ❛✌✡✬✣ ✍✓
P
❢☛ ✯
k0 = 0
✍✲✱
P1 P, z-
❞✡✭✡✓ ✛✍✎✘✡ ❢☛ ✈✡☛☞
✓☞✤✡
✧
✧
▼▼▼
✽
OC
P◗❥
❘
iˆ, ˆj
✲✢✍❙✱
✍✜✮✍★✤
◆◆◆
❖
OA, OB
OX, OY
❜✜✓✡✬
✤✥
✧
| OA | 1, | OB | = 1
❍❞✘✡■
❢☛
✐★❑▲✤■
▼▼▼✽
✛✍✎✘✡
✜❢✢✣
(Components of a vector)
❈❄❉✿
●✥✎✒✈ ✡✬ ✣
y-
✛✍✎✘✡
✫
a, a
✾✿
✘✡✙✚✌
✧
â =
10.5.1
✱✓
1
|a| 1
|a|
✦
| a | | || a | =
❜✛
a
✕❋❦❦✖
❩
✗
r ) = xiˆ
q
yjˆ
q
zkˆ
♦✬✰
★✩✡✉✤
❢✡✬ ✤✡ ❢☛✯
✍✓✛✢
✓❢✲✡✤✬
❢☛✣
✓✼✡✢④✓✼✡✢
✼✡✢
✛✍✎✘✡
✓✡
xiˆ, yjˆ
x, y
z
✈✡☛☞
✱♦✣
✓✡✬
✌❢
❝❡❣
✮★
zkˆ
✇✡▲✓
✮★
❍❞✡①✤
✛❞✓✡✬ ✍❯✡✓
✇✡▲✓
✓❢✲✡✤✡
✈❏✡✡✬✣
✼✡✢
♦✬✰
✓❢✡
❢☛✯
✈✜✒ ✍✎✘✡
❙✡✤✡
x, y
✌❢✡③
❢☛✯
✽
r
♦✬ ✰
✽
✱♦✣
z, r
✛✍✎✘✡
♦✬✰
✇✡▲✓
✈✍✎✘✡
✇✡▲✓
✓❢✲✡✤✬
❢☛ ✣✯
450
① ✁✂✄
❉❊❋●❍■❏
☞
r
❢☎✆✝ ✆❢✞✟✠
✣✠
☎✝
✆☎✛✝
✤✥✦
xiˆ
✡
✤✗
yjˆ zkˆ ,
☛
☛
✧✠✔★
✤✥✌
☎✕✛✔
❑▲▼❑◆
☎✝ ❞✌✍ ✠✎✏ ✑✠✎✒✠✠✓✠✔✕✆
✆✗☎✠✔✩ ✠
❢☎
✭✭✭✭
☞
✆✗☎✠✔✩ ✠
OP1P,
❢✪✠✫✠✜✣
✗✔✌
✤✗
✑✠✛✔
✤✥✌
✰✰✰✱
✳
✙✔ ✚
♦✑
✗✔✌
✆❢✞✟✠
✑✖✠✴✛
✤✠✔ ✛✝
r
✓✺
xiˆ
x2
✮
| P1P |2
✯
y2
( x2
✳
yjˆ + zkˆ
☛
10.14)
✈✠✙✬✚❢✛
☎✝
❞✌ ✍✠✎✏
| r | = | xiˆ
✯
y2 )
yjˆ zkˆ | = x 2
✯
✵
a
✶✷✸
❙
b
a1iˆ a2 ˆj + a3 kˆ
❄✠★☎ ♦✑ ✗✔✌ ✹✗✟✠✲
✯
✈✠✥✕
✯
✯
✻
✻
✆❢✞✟✠✠✔✌
a
b
✼✽✾
✿
☎✠✔ ✘✠✔✓
✵
a b = (a1 b1 )iˆ (a2
✵
✯
❀
(ii)
a
✯
b
✼✽✾
✿
☎✠
✵
✻
(iv)
❢☎✆✝
✯
♦✑
✙✔ ✚
✯
❂
b2 ) ˆj (a3 b3 )kˆ
✙✔✚
✤✥✌
✘❢✞
✯
❂
♦✑
✻
a
b
a1 = b1, a2 = b2
✆❢✞✟✠
✯
❂
❁
(iii)
b2 ) ˆj (a3 b3 )kˆ
✗✔✌
✑✖✠✴✛
✤✠✔✛✠
✤✥✦
✈✌✛✕
✵
a b = (a1 b1 )iˆ (a2
❆
✯
✻
✻
✆❢✞✟✠
✼✽✾
✿
✈❢✞✟✠
✯
y2
b1iˆ b2 ˆj b3 kˆ
✤✥ ✌ ✛✠✔
(i)
z2
✯
z2
✤✥✦
✵
✘❢✞ ✞✠✔ ✆❢✞✟✠
✡
(
✰✰✰
✱
| OP1 |2
| OP | =
❢☎✆✝
✗✔ ✌
✭✭✭☞
❢☎
✭✭✭☞
✈✛✲
OQP1
| OQ |2 +|QP1|2
| OP 1 | =
✈✠✥✕
❢✪✠✫✠✜✣
✭✭✭✭☞
✑✖✗✘
✔
☎✠ ✞✠✔ ✍✠✕ ✑✖✘✠✔✓ ☎✕✙✔✚ ✛✜ ✕✛
✌
✢✠✛
✆✗✠✧
❃ ✆✔
✤✠✔✛✔
✈✠✥✕
✘❢✞
✙✔✚✙❞
a3 = b3
a
❅
✆❢✞✟✠
✈✠✥✕
☎✠
✓✜✩ ✠✧
❇
a = ( a1 )iˆ ( a2 ) ˆj ( a3 ) kˆ ⑥✠✕✠
✑✖✞❈✠
✤✥ ✦
✗✔✌
✑✖ ✠✴✛
✤✠✔✛✠
✤✥ ✦
⑥✠✕✠ ❢✞✺
451
❧ ✁✂✄ ☎✆✝✞ ✟✄✠
✡☛☞✌✍✍✎✏
✑✍
☛❢✥✙✜✍★☛✢✒✤
✒✍✎✓✔✕✖
✡✎
☛✤✖✥✍
✖✚☛❡✪
a
☛✑
✱
✯
b
✈✬✭
❙
✱
(i) ka ma
☛✑✡✚
✗☛☞✌✍
✡✎
✡☛☞✌✍
✑✍
✓✛✜✍✢
✡☛✣✤☛✖✥
✦✔
✤✎✏
☛✢✣✢☛✖☛✧✍✥
✫
✫
✤✍✢
✗✍✘✙
✩✘
☞✍✎
✯
✩✘✏
✡☛☞✌✍
k
✗✍✘✙
m
✪❢✏
✳
✱
(k
✰
✑✍✎❞✮
☞✍✎
✥♥
✱
✴
(ii) k (ma ) (km)a
m) a
✩✘✏
✗☛☞✌✍
✳
✱
✴
(iii) k ( a b ) ka kb
✲
✵✶✷✸✹✺✻
❁
❀
1.
✔✼ ✎☛✽✍✥ ✑✙
✗✍✔
✡✑✥✎ ✩✘ ✏
✾ ❢✎ ✕
☛✑
✩✘ ❂
❢✍❃✥❢
✤✎✏
☞✍✎
a
✡☛☞✌✍
b
❅❆❇
❈
✡✏✙ ✎ ✧✍
❄
a
✩✛✪ ✩✘✏ ❣ ✗❑✍✍✮ ✥▲
✒☛☞
▼
❢✎ ✕❢✖
✗✍✘ ✙
● ❍
◆
✩✍✎✥✎
✥♦✍✚
✿
a
✩✘✏
✒☛☞
✩✤✎✌ ✍✍
b
✗✍✘✙
▼
❢✎✕❢✖
✗✍✘✙
a
✡☛☞✌✍
❢✎ ✕
a
✩✍✎❂ ✒☛☞ ✡☛☞✌✍
b1iˆ b2 ˆj b3 kˆ ,
◆
◆
✒☛☞
✪✑
✪✎✡ ✎
✫
✫
a
✴
a1iˆ a2 ˆj a3 kˆ
◆
☛✖✪ ✡☛☞✌✍
❄
b
✌✍❉❊✒✎✥✙ ✗☛☞✌✍ ✾ ✑✍ ✗☛❃✥❋❢ ✩✘✏ ✥✍☛✑
✴
❢✎✕
✤✍✢
❄
❄
✡✏ ✙✎ ✧✍
♦✍✚
☛✑✡✚
✈✬✭
❙
b
■✍❏✑ ✦✔ ✤✎✏ ☛☞✪
✥♥ ☞✍✎ ✡☛☞✌✍ ✡✏✙ ✎ ✧✍ ✩✍✎ ✥✎ ✩✘✏
✒☛☞
b1iˆ b2 ˆj b3 kˆ = (a1iˆ a2 ˆj a3 kˆ)
◆
◆
b1iˆ b2 ˆj b3 kˆ = ( a1 )iˆ ( a2 ) ˆj ( a3 )kˆ
◆
P
P
b1
P
b1
a1
◆
◗ ❘
a1 , b2
◗ ❘
a2 , b3
◗ ❘
◆
❖
❖
a3
b3
a3
b2
a2
❚
◆
❖
a = a1iˆ a2 ˆj a3 kˆ
a1, a2, a3
l, m, n
liˆ mjˆ nkˆ = (cos )iˆ (cos ) ˆj (cos )kˆ
❯
2.
3.
✒☛☞
✥♥
✒☛☞
☛☞✪
y
◆
❖
◆
✪❢✏
z
☛✑✡✚
✩✛✪
✗✽✍
♠❫❴❵❛❜❴
✡☛☞✌✍
❢✎✕
4 x, y
✑✚
✗✍✘✙
☛☞✌✍✍
♥✢✍✪
✡✍❑✍
z
❢✎✕
✡☛☞✌✍
✤✎✏
✓✪
☛☞❢▲ ✕★✑✍✎✡✍❞✢ ✩✘✏
✤✍❱✍✑
✑✍✎✜✍
❀
a
✡☛☞✌✍
✩✘
✡☛☞✌✍
❢✎✕
☛☞❢▲ ✕★✗✢✛ ✔✍✥
✩✘✏ ❂
✑✩✖✍✥✎
✥♥
❡✩✍❲
❳
,
❨ ✪❢✏ ❩ ☛☞✪
✩✛✪
✡☛☞✌✍
❬✍✙✍
❭✤✌✍❪
x,
✩✘ ✏ ❂
✴
❢✎✕ ✤✍✢ ❝✍✥ ✑✚☛❡✪ ✥✍☛✑ ✡☛☞✌✍
a
▼
✴
xiˆ 2 ˆj zkˆ
◆
✗✍✘✙
◆
b
▼
2iˆ
◆
yjˆ kˆ
◆
✡✤✍✢ ✩✘ ✏ ❂
❵❤
è✒✍✢
☞✚☛❡✪
☛✑
☞✍✎
✡☛☞✌✍
✴
✗✥❪
♠❫❴❵❛❜❴
✡✤✍✢
✩✛✪
☛☞✪
5
✡☛☞✌✍
❥❦♣
q
b
✡✤✍✢
✩✍✎✥✎
a iˆ 2 ˆj
r
s
✩✍✎✏ ✓✎
✴
t
✤✍✢ ✖✚☛❡✪
✩✘✏
✒☛☞
✗✍✘✙
❢✎✕❢✖
✒☛☞
✗✍✘✙
b
✰
✒☛☞
2iˆ
✗✍✘✙
✯
❢✎✕❢✖
✒☛☞
✥♥ ✉✒✍
✡✏ ✓✥
■✍❏✑
✡✤✍✢
✒✩✍❲
|a|
✴
|a | |b | ?
▼
✩✘
12
②
22
③
①
5
✗✍✘✙
|b |
①
22 12
②
①
|a| |b |
✰
✔✙✏✥✛
☛☞✪
✩✛ ✪
✡☛☞✌✍
✡✤✍✢
✢✩✚✏
✩✘✏
a
5
✉✒✍✎✏☛✑
❞✢❢✎ ✕
✡✏ ✓✥
❄
❄
✉✒✍ ✡☛☞✌✍
✱
✱
❞✡☛✖✪
①
✩✘ ❂
x = 2, y = 2, z = 1
✴
ˆj
✐✢❢✎✕
✩✘✏ ✇
③
❵❤
a
✡✤✍✢
✴
■✍❏✑
☛♦✍❊✢
✩✘✏ ❂
❅❆❇
❈
b
452
① ✁✂✄
✝❣
2iˆ 3 ˆj kˆ
✎
6
♠☎✆✝✞✟✆
a
❧✠✡☛☞
✣
a
❧✠✡☛☞
♦✏✑
✌
✍
✒✓✔✠✡☛☞
♦✏✑
✍
✕☞✖☞✗
✒✓✔✠✡☛☞
✤
✎
|a| =
✒✈
aˆ
❜❧✠✮✜
7
♠☎✆✝✞✟✆
✯
✎
❧✠✡☛☞
a
✌
✲
✰
♦✏✑
✒✓✔✠✡☛☞
✠✡✜
✪✔✜
a
❧✠✡☛☞
♦✏✑
✒✓✔✠✡☛☞
✜✗
✕☞✖☞✗
✬
2 ˆ
i
14
aˆ
❧✠✡☛☞
a
✬
♦✏✑
✒✓✔✠✡☛☞
✒☞✫✦
✴
♦☞✮☞
✧✠✦✕☞✳☞
✗✚✠✛✜✢
❧✠✡☛☞
✪☞✏✙☞
✪✫✢
14
✭
3 ˆ
j
14
✱
1 ˆ
k
14
✱
✘☞✙ ✗✚✠✛✜ ✠✛❧✗☞
1
1 ˆ
a =
(i 2 ˆj )
|a|
5
✸
✷
7a = 7
✻
1
✽
2
✺
5
ˆj 3kˆ
i
8
❧✠✡☛☞
✝❣
✘☞✙
✠✡✜
✌
✪✔✜
❧✠✡☛☞☞✏❃
✗☞
✌
2iˆ
✍
♦✏✑
✍
✾
5
❀
❄☞✏❅ ✧✑✮
♦✏✑
✸
14 ˆ
j
5
✒✓✔✠✡☛☞
✪✫✢
✝❁❂
✕☞✖☞✗
❄☞✏❅ ✧✑✮
✎
✎
c,
✎
42
|c | =
✒☞✫✦
✒❉☞✚❊❋
✕☞✖☞✗
cˆ
9
b
7 ˆ
i
5
j =
2 ˆ
j
5
✸
✗✚✠✛✜✢
✎
♠☎✆✝✞✟✆
✒☞✫✦
✍
a b
✒✙❈
✎
2iˆ 2 ˆj – 5kˆ
✎
a
❧✠✡☛☞☞✏❃
1 ˆ
i
5
✯
✺ ✼
✲
✿
♠☎✆✝✞✟✆
✧✠✦✕☞✳☞ ✴ ❜✗☞❜✵ ✪✫✢
✷
✯
✺
✹
❜❧✠✮✜
32 12
✜✏❧☞ ❧✠✡☛☞
✶
✝❣
✘☞✙
✧★☞ ✩✙
⑥☞✦☞
✥
22
1
(2iˆ 3 ˆj kˆ) =
14
iˆ 2 ˆj
✰
1
a
|a|
❧✠✡☛☞
✥
aˆ
❧✠✡☛☞
✕☞✖☞✗
1
c
|c |
1
✯
✷
✎
❧✠✡☛☞
✍
32
✍
( 2) 2
✲
✪✫✢
29
✌
❧✠✡☛☞
✷
✯
✎
c = 4iˆ 3 ˆj 2kˆ
t❆❇→
a
✌
✰
29
iˆ
(4iˆ 3 ˆj 2kˆ)
✍
ˆj 2kˆ
✲
✸
4 ˆ
i
29
✯
✰
3 ˆ
j
29
2
✸
29
kˆ
✪✫✢
♦✏✑ ✠✡♦●✑❍✒✓✔✧☞✙ ✠✮✠■☞✜ ✒☞✫✦ ❜❧✗✚ ❧✪☞❄✙☞ ❧✏ ✠✡♦● ✑❍✗☞✏❧☞❜✓
✘☞✙ ✗✚✠✛✜✢
✎
✝❣ è❄☞✓ ✡✚✠✛✜ ✠✗ ❧✠✡☛☞
x, y, z
l, m
❃
✪☞✏✙✏ ✪✫✢
❄✠✡
✒☞✫✦
l
✌
xiˆ
✍
yjˆ
❜❧✠✮✜ ✠✡✜ ✪✔✜ ❧✠✡☛☞
n
✠✡✜
✪✔✜
✷
✠✡♦●✑❍✗☞✏❧☞❜✓
❧✠✡☛☞
1
,
6
a
|r |
✻
✒✙❈
r
✽
✿
1
6
,
♦✏✑
m
1
6
,–
♦✏✑
✍
zkˆ
✠✮✜ ✪✕
✠✡♦●✑❍✗☞✏❧☞❜✓
1
, n
6
b
|r |
✷
2
6
✼
✾
❀
✪✫❃✢
a, b, c
a = 1, b = 1
♦✏✑ ✠✡♦●✑❍✒✓✔ ✧☞✙
✧☞✙✏ ✪✫ ❃ ✠✗
✪✫ ❃
❧✠✡☛☞ ♦✏✑ ❏ ❑✕☞❅✙ ▲☞❋✗
✒☞✫ ✦
c = –2
✙☞✏❈
c
|r |
✷
2
(
6
✷
▼◆ ❦❢
s❖ ❞
|r |
6)
✪✫✢
✧✔✓❈
453
❧ ✁✂✄ ☎✆✝✞ ✟✄✠
10.5.2
❀✙✚
P1
♥✡☛
☞✌♥✍✎✡☛✏
P1(x1, y1, z1)
✈✛✜✢
✪✣
✙✼✛✽✛✦✾
OP1P2
✖✡✔✡
P2(x2, y2, z2)
✈✛✜✢
✙✫✬✛✭✣
P1
✶✴✵✶✷✸✹
✒✓✔✡✕☛
(Vector joining two points)
✗✒♥✘✡
✚✛✣
✤✥✚✦
✧✜★
✩✥
✰✰✰✰
✱
P2
❞✛✣
✑✡☛
✮✛✬✛
P2
✈✛✜✢
✙✭❀✫
❞✛✣
✫✺✬
✰✰✰✱
✻✛✩✣
✪✣
✧✜★
❣✈✛✮✲✳✙✩
✙✫✬✛✭✣
✧✫
✻✢
✙✼✛✽✛✦ ✾
✰✰✰✰
✱
❂
❁✦ ❃✛❄✫✛✣❅
✮✣✳
✰✰✰✰
✱
●✻
❞✢✩✣ ✧✦❇
✪✣ ✙✬❋✛✛ ✾✛✩✛ ✧✜ ✹
➄➅➆➇➈➉➊
✰✰✰✱
❍
OP1
❂
y2 ˆj
✰✰✰✰
✱
P1P2 = ( x2iˆ
= ( x2
❆✻❀✛✣❁
❞✛
✰✰✰✰
✱
P1P2 = OP2
❍
❂
z2 kˆ) ( x1iˆ
❍
x1 )iˆ ( y2
❂
❍
y1 ˆj
❂
y1 ) ˆj ( z2
❂
❂
❍
➋➌➍➋➎
z1kˆ)
z1 )kˆ
❑❑❑❑❑
▲
✰✰✰✰
✱
P1P2
✪✙✚✯✛
✪✣
✻✢
❞✢✭✣
✰✰✰✰
✱
❆✻❀✦♠❈✩ ✪✫❉❞✢❃✛ ✙✭❊✭✙✬✙❋✛✩
✈■✛✛♠✩❏
✧✜
OP1 P1P2 = OP2
✙❞
❀✛✣ ❁✻✳✬
✪✙✚✯✛
O
✤✥✚✦
✻✿❀✛✣❁
❞✛
P1P2
✪✙✚✯✛
P1P2 = ( x2
✻✙✢✫✛❃✛
❞✛
❍
x1 ) 2
❂
( y2
❍
y1 ) 2
❂
( z2
❍
z1 ) 2
✮✣✳
✫✣★
✻✿✛ ♦✩
✩✢✻✳
✙✚❯❱
●✻
✧✛✣✩✛ ✧✜✹
▼◆❖P◗❘❖❙❚
❲✛✩
✪✙✚✯✛
P❳
P
Q(– 1, – 2, – 4)
❇✮★
❞✛✣ ✙✫✬✛✭✣ ✮✛✬✛
❇✮★
P
Q
❞❉
✧✜
✈✛✜✢
Q
✈★✙✩✫
●✻
✫✣★
✻✿✛ ♦✩
✪✣
❞❉✙✾❇✹
❈❀✛✣★✙ ❞
❜✪✙✬❇
P(2, 3, 0)
✤✥✚✦ ✈✛✣★
P
✪✙✚✯✛
Q
✈✛✜✢
✪✣
Q
❞❉
✩✢✻✳
✧✜❨
✙✚❯❱
P
❩✻❯❱✩❬
✻✿ ✛✢★✙✽✛❞
✤✥✚✦
✤✥✚✦
✧✜❨
✰✰✰
✱
❞✛✣
✙✫✬✛✭✣
✮✛✬✛
✈✽✛❉❯❱
PQ ,
✪✙✚✯✛
✙✭❊✭✙✬✙❋✛✩
✧✛✣✩✛
✧✜✹
✰✰✰
✱
PQ = ( 1 2)iˆ ( 2 3) ˆj ( 4 0)kˆ
PQ = 3iˆ 5 ˆj 4kˆ
❭
❭
❪
❭
❭
❪
❭
❭
✰✰✰
✱
✈■✛✛♠✩❏
10.5.3
❫✡✏❴
✗❵❛ ✡
✫✛✭ ✬❉✙✾❇ ✫✺ ✬
(Section Formula)
O
P
Q
✤✥✚✦
✮✣✳
✰✰✰
✱
✙ ❩■✛✙✩
✪✙✚ ✯✛
✤✥✚✦✈✛✣ ★
P
R
⑥✛✢✛
❲✛✩
Q
❞✛✣
✤✥✚✦
✧✜✹
✧✫
OQ
❡❢❤
✐
✙✫✬✛✭✣
✪✣
✶✴✵✶❦✸
✫✺✬
❞✢✭✛
OP
✻✿❞✛✢
✚✛ ✣
❣✈✛ ✮✲✳✙✩
❆✚❏✚ ✣✯❀
❇✮★
✪✛✻ ❝
✣ ✛
✪✣
✮✛✬✛
✙✮✽✛✛✙✾✩
❇✮★
O
✈✛✜✢
✚✛✣ ✤✥✚✦ ✧✜ ★ ✙✾✭❞✛✣
✰✰✰✱
✮✣✳
✥✛♣
✪✛✻✣❝✛
✚✛✣✭✛✣★
✙ ✭●✙✻✩
✢✣❋✛✛
✙ ❞❀✛
❣✈✛✮✲✳✙✩
✤✥✚✦
✙❩■✛✙✩❀✛✣★
R
❞✛✣
❋✛★❥
✾✛
✙ ❞❀✛
✙ ❞✪❉
✩❉ ✪✢ ✣
✤✥✚✦
✧✜ ✹
✈★✩ ❬
✪❞✩✛
✶✴✵✶q✸✹
✧✜ ✹
❁❀✛
❀✧✛r
✧✫✛✢✛
ssst
❞✛
✙❩■✛✙✩
❇❞✉❇❞
✪✙✚✯✛
❞✢✮✣✳
OR
✬✣✩ ✣
➄➅➆➇➈➉➊
➋➌➍➋➏
✧✜★✹
➁➁➁
➂
✇①②❖✇③
❙
④⑤
R, PQ
⑦❖⑧
⑨⑩③ ❶
✇❷❸❖❖✇④③
✰✰✰✱
✻✿❞✛✢
✙✮✽✛✛✙✾✩
❞✢✩✛
✧✜
✙❞
⑦◗③❖
➁➁➁
➂
m RQ = n PR ,
✾✧✛r
P❹
❺⑨❖❷❻❼✇③
m
✈✛✜✢
n
❙❚❽❙❾❿➀
❀✙✚
R, PQ
❞✛✣
❜✪
❄✭✛➃✫❞ ✈✙✚✯✛ ✧✜ ★ ✩✛✣ ✧✫ ❞✧✩✣ ✧✜★
454
① ✁✂✄
✠✠✠
✡
❢☎
R, PQ
✆✝✞✟
m:n
☎❞☛
♦☛☞
✌✍✟✎❞✏
✑☛✒
✌✒✏✓
❢♦✔❞❞❢✕✏
✛✛✛
✜
✛✛✛✜
✛✛✛
✜
✦✦✦✧
☎✖✏❞
✛✛✛✜
RQ = OQ OR
✢
✤
✜
m (b
❜❧❢★✱
✜
✜
a)
✩
✰
✌✏✓
R
✆✝✞✟
✕❞☛
P
❢☎
✌❞✘✖
Q
☎❞☛
m:n
♦☛☞
❈❉✼❃ ❊ ❋ ✹✽
✱☎
R, PQ
✎✷✳✍
✫✗
●❍■●❏❑▲
♦☛☞
✶✎
✑☛✒
❀✼ ❁
✿✼ ❂
❧▼✫❞✎✍
✌✍✟✎❞✏
❖❞☛P◗ ✏☛
✗✘✒
❢☎
✦✦✦✧
☎❞
❢✲✈❞❢✏
❭❪❫❴❵❛❝
OR =
❧❢✞✳❞
✫❢✞
R, PQ
☎❞
❣❣❣❤
❢✲✈❞❢✏
11
♠❥✼❆❅❦✼
✗✘✒✙
✱☎
✌✍✟✎❞✏
❧❢✞✳❞
✞❞☛
✆✝✞✟
R
✱☛ ❧☛ ✆✝✞✟
✑☛✒
(i)
OR =
✌✒✏✓
P
☎❞
(ii)
✎✖
)
✑☛ ✒
✌✒✏✓
❢♦✔❞❞❢✕✏
☎✖✏❞
✗✘
☎❞
❢✲✈❞❢✏
❧❢✞✳❞
♦☛☞
✶✎
✑☛✒
✎✷❞ ✸✏
✗❞☛✏❞
✗✘✙
❀❅✽✼
✎❞◆☎
PQ
☎❞☛
♦☛☞
❲
❢★✱
m:n
PR
QR
R i.e.,
❆❇
❱
♦☛☞
m
n
❯
❳
❩
✵
mb na
m n
❬
♦☛☞
✶✎
✑☛✒ ✎✷❞ ✸✏
✗❞☛✏❞
t✉✇③④⑤⑥
✗✘✙
⑦⑧⑨⑦⑩
❬
✑❡✫
✰
☎❞
☎✖✍☛
✴
❨
✵
❧✖★
✴
❚
✌✍✟✎❞✏ ✑☛✒ ✝❞❙ ❢♦✔❞❞❢✕✏ ☎✖✍☛ ♦❞★☛ ✆✝✞✟
✭✪✫❞☛✬
✒ ✮
(
mb na
m n
✗✑
✖☛❘❞❞❘❞✒P
a
✵
✹❃❄✼✼ ✹✾ ✽
☎✖✍❞
❧☛
✯
✦✦✦✧
✾✿
OPR
✧
✤
✯
OR =
II
✱♦✒
✰
✵
✹✺✻✼✹✽
r
mb na
m n
✜
r =
✌✈❞♦❞
ORQ
✜
r ) = n (r
✩
❢✚❞✔❞✟✕❞☛✒
✢
✧
✥
✌✝
✜
b r
✣
✦✦✦
✧
PR = OR OP
✌❞✘✖
✗✘✙
✜
✆✝✞✟
✰
a b
2
Q
✗✘
m=n
✏❞☛
✌❞✘✖
❜❧❢★✱
❢✲✈❞❢✏
I
✛✛✛
✜
❧☛
PQ
♦☛☞
✑❡✫
✆✝✞✟
R
✯
✌❞✘✖
♦☛☞
✶✎
✗❞☛✐❞✙
✦✦✦
✧
★②❢✕✱
❢✲✈❞❢✏
✑☛✒
❧❢✞✳❞
❢✕✍♦☛☞
♣❞✏
✝❞❙ ❢♦✔❞❞❢✕✏
❢✲✈❞❢✏
☎②❢✕✱
☎✖✏❞
✕❞☛
OP
❧❢✞✳❞
P
Q
✱♦✒
☎❞☛
3a
✠✠✠✡
✧
✧
✥
✤
2b
✌❞✘✖
❢✑★❞✍☛ ♦❞★②
OQ
✖☛❘❞❞
✡
✡
a b
♥
✣
2:1
♦☛☞
✆✝✞✟
R
☎❞☛
✗✘✙
❆q
(i) P
✌❞✘✖
☎❞
Q
☎❞☛
❢✲✈❞❢✏
❢✑★❞✍☛
❧❢✞✳❞
♦❞★②
✖☛❘❞❞
☎❞☛
✗✘✓
2:1
♦☛☞
✌✍✟✎❞✏
✰
✰
✑☛ ✒
✰
✰
2( a b ) (3a 2b )
OR =
3
✦✦✦✧
(ii) P
✌❞✘✖
❢✲✈❞❢✏
Q
☎❞☛ ❢✑★❞✍☛ ♦❞★② ✖☛❘❞❞
❧❢✞✳❞
✗✘✓
✯
☎❞☛
2:1
✯
♦☛☞
✰
✰
✯
r
☎✖✍☛
♦❞★☛
s
5a
3
✌✍✟✎❞✏ ✑☛✒ ✝❞❙ ❢♦✔❞❞❢✕✏ ☎✖✍☛ ♦❞★☛ ✆✝✞✟
✰
✰
r
❢♦✔❞❞❢✕✏
✰
r
2(a b ) (3a 2b )
OR =
2 1
❣❣❣❤
✌✒✏✓
✰
r
s
4b
✰
r
a
R
☎❞
455
❧ ✁✂✄ ☎✆✝✞ ✟✄✠
12
♠✡☛☞✌✍☛
✔❢✏✜✏✘✢
☞❣
✣✚ ✤
✧✙
♥✎✏✏✑✒✓
✎✏✥✦✏✑
✫✏✬✚
✔✕
ˆj kˆ), B(iˆ 3 ˆj 5kˆ), C(3iˆ 4 j 4kˆ)
A(2iˆ
✖✗♥✘
✧★✩
✔✕
AB = (1 2)iˆ ( 3 1) ˆj ( 5 1)kˆ
BC = (3 1)iˆ ( 4 3) ˆj ( 4 5)kˆ
✯
✰
✯
✰
✰
✯
✰
✯
✰
✰
✯
iˆ 2 ˆj 6kˆ
✯
✲✲✲
✳
✯
✰
✴
2iˆ
✲✲✲✳
CA = (2 3)iˆ ( 1 4) ˆj (1 4)kˆ
✈✏★✷
✯
✹✺✏✻
✈✔✬✔✷❜✬
✰
♥✥✔✢✓
✯
✰
✴
✧✘✈ ✏
✰
✶
| CA |2
✭✭✭
✮
| AB |2 = 41 6 35 | BC |2
✔♥✺✏
✰
✔❢✏✜✏✘✢
✓✕
✶
✴
✔✻❈✻✔❉✔❊✏✬
✱✔♥✎✏✏✚✩
■
a
2.
3.
iˆ
❋
✱✙✕✏✚✛ ✏
✱✙✏✻
✫✔✷✙✏✛✏
✱✙✏✻
✔♥✎✏✏
4. x
5.
✈✏★ ✷
y
✔❢✏✜✏✘✢
✫✔✷✙✏✛✏
✣✏❉✚
♥✏✚
♥✏✚
✔✣✔✜✏❏✻
♦✏✬
✙✏✻
6.
7.
8.
9.
❳✏❨✕
✱✔♥✎✏
✵
✮
a
✱✔♥✎✏
✴
iˆ
✶
ˆj 2kˆ
✶
✰
✧★ ✪
❄❅❆❇
✫✔✷✕❉✻
✕✥✔✢✓✼
2iˆ 7 ˆj 3kˆ; c
❍
❍
1 ˆ
i
3
✱✔♥✎✏
✬✏✔✕
❋
●
PQ,
✶
✈✏★✷
❖❫◗
✔♥✓
✧✘✓
✣✚✤
✈✻✘✔♥✎✏
❴❚
✧★✩ ✪
✱✔♥✎✏✏✚✩
a
❱◗
❬
✈✻✘ ✔♥✎✏
10.
✱✔♥✎✏
11.
♥✎✏✏✑✒✓
12.
✱✔♥✎✏
✙✏❢✏✕
✔❉✔❊✏✓✪
2iˆ 3 ˆj
✱✔♥✎✏
2iˆ 4 ˆj 5kˆ
✴ ✵
✣✚✤
✶
✶
✈✻✘✔♥✎✏
5iˆ
✵
✔✕
ˆj 2kˆ
✶
✱✔♥✎✏
✶
♦✏✬
ˆj
❑▲▼
❙
xiˆ
yjˆ
✱✙✏✻
✧✏✚ ✩ ✪
✓✕
c iˆ 6 ˆj – 7 kˆ
✮
✈✏★✷
✙✏❢✏✕
✴
✵
✱✔♥✎✏
♦✏✬
✱✔♥✎✏
2kˆ
✕✥✔✢✓
❬
❑▲❙▼
✣✚ ✤ ✈✻✘ ✔♥✎✏ ✓✕
✕✥
♦✏✬
b
iˆ
✢✧✏❭
ˆj
✕✥✔✢✓✪
✖✗♥✘
kˆ ,
✕✏ ✺✏✚❞✫✤❉ ♦✏✬ ✕✥✔✢✓✪
P
✈✏★ ✷
Q
Ø✙✎✏✼
❖❘◗
✔❉✓◗
✱✔♥✎✏
❪❚
✳
✳
✣✚✤
P◗
a b
❵
✣✚✤
✕✥✔✢✓✪
2iˆ 3 ˆj
iˆ 2 ˆj 3kˆ
✶
✙✏❢✏✕
2iˆ
✱✔♥✎✏
1 ˆ
k
3
❍
✔❉✔❊✏✓✪
❩❩❩
❬
✱✔♥✎✏
1 ˆ
j
3
✕✥✔✢✓✪
✮
✴
✰
✖✗♥✘ ❖P◗ ❘❚ ✧★ ✈✏★ ✷ ✈✩ ✔✬✙ ✖✗♥✘ ❖❯❱◗ ❲❚ ✧★ ✪ ✒✱ ✱✔♥✎✏ ✣✚✤ ✈✔♥✎✏
iˆ 2 ˆj kˆ, b
✮
a
✱✔♥✎✏
✕✏
✱✔♥✎✏
✕✥✔✢✓
♦✏✬
iˆ 3 ˆj 5kˆ
✸ ✯
■
❋
✔✣✔✜✏❏✻
✓✕ ✱✔♥✎✏ ✕✏ ✫◆✏✷✩ ✔✜✏✕
✓✣✩
■
b
●
✣✏❉✚
✣✚✤
✣✚✤
ˆj kˆ;
●
✶
✭✭✭✮
✐✽ ✾✿❀❁❂❃
1.
ˆj kˆ
✵
✔✕
✭✭✭
✮
✈✬✼
✱✙✕✏✚✛ ✏
✧★✩ ✪
✭✭✭
✮
✒✱✣✚✤
✓✕
✔♥✣❝✤
4kˆ
✓✚✱ ✏ ✱✔♥✎✏ ♦✏✬ ✕✥✔✢✓
❑▲❙▼
cosine
4iˆ 6 ˆj 8kˆ
♦✏✬
✕✥✔✢✓✪
✱✩ ✷❊
✚ ✏
✔✢✱✕✏
✧★✩ ✪
✫✔✷✙✏✛✏ ❛ ✒✕✏✒✑
✧★ ✪
456
① ✁✂✄
13.
A (1, 2, –3)
cosine
➥✍✎✏✑✒✓✔
❞✚
✧✒✛
✖✎✕✦ ◗
14.
✎✤✒✒♥✪✱
✖❞
15.
➥✍✎✏✑✒✓✔
P ( iˆ
(i)
✗✓✔ ✓
16.
17.
✎✒✓
A, B
✎✤✒✒♥✪✱ ✖❞ ➥✍✎✏
c
✑✒✯✜
18.
P(2, 3, 4)
➥✍✎✏✑✒✓✔
❃
❆
●●●●
Q – iˆ
✕✒✘✓
❞✜✙✓
kˆ)
✕✒✘✓
➥✍✎✏
❞✒✓
✖✗✘✒✙✓
Q(4, 1, –2)
✱✕✔
A
B
❧✓
❞✚
✰
✛✜✢◗ ✖✎✥✣
❞✒✓
✕✓◗
❧✒♦✒
✖✗✘✒✙✓
R
❞✒
✍✜✒✍✜
✕✒✘✚
✖✼♦✒✖✛
✕✒✘✓
✭✏ ❞✒
✜✓✶✒✒
❞✒✓
❧✖✎✤✒
✧✒✛
❧✖✎✤✒
❧✖✎✤✒
❞✒
✗✽✾
❧✗❞✒✓❣✒
✕✓◗
✖❇✒✻✒✏★
✖✘✱
✕✓◗
✤✒✚✥✒✒✓❈ ❞✒
✖✙❊✙✖✘✖✶✒✛
✗✓✔
❧✓
❁
❁
❃
❞✚✖★✱✩
❄
ˆj kˆ
✙✮✚✔
✮✯✩
2iˆ
❅
❞✜✛✓ ✮✯✔✩
✖✙✗✒♥❣✒
❞✒✯✙
✑✙✏✢✒✛
✧✒✛
➥✍✎✏
❆
❀
✕✓◗
❞✚✖★✱✩
a 3iˆ 4 ˆj 4kˆ, b
✖★✙✕✓◗ ✖✼♦✒✖✛ ❧✖✎✤✒ ✿✗✤✒✸
✮✯✩
✮✏✑✒
✷✸✹
❂
C,
❆
●●●
OZ
✱✕✔
ˆj
✵
10.18),
✑✒✕❉ ◗✖✛
❆
●●●
✲✳✴❥
✮✯✔ ✺ ✱❞
ABC (
✖❇✒✻✒✏★
✑✬✒✒✓✔
✖✕✻✒✒✖★✛
✑✒✯✜
❄
✖✗✘✒✙✓
OX, OY
✫
✑✒✯✜
iˆ 3 ˆj 5kˆ
❄
ˆj kˆ
kˆ)
✍✒❝✺
❞✒✓
❞✚✖★✱✩
✫
2 ˆj
(ii)
✑✔✛✸
❆
iˆ
❧✖✎✤✒
B(–1, –2, 1)
✱✕✔
❧✒
❞♦✒✙
❧❋✾
❆
(A) AB + BC + CA = 0
❑❑❑
▲
❑❑❑
▲
❑❑❑
▲
(B) AB BC AC
❍
PPP
❘
■
PPP
❘
▼
◆
❑❑❑
▲
■
❍
b
✲✳❥
✴
❆
✎✒✓
a,
0
▲
❬ ❭
❧✔✜✓✶✒
❧✖✎✤✒
✮✯✔
✛✒✓
✖✙❊✙✖✘✖✶✒✛
✗✓✔
❧✓
❞✒✯✙
❧✒
❢❙❚❤ ✲❢❯❱✳
❲s❳
❞♦✒✙
❧✮✚
✙✮✚✔
✮✯✸
❢❨❩
❂
✲✳❥
✴
b
✕✓◗
a
❧✖✎✤✒✒✓✔
❦♠❜♣❡❡✐q
t❡
❧✗✒✙✏✢✒✛✚
✮✯✔✩
✙✮✚✔
❂
❂
✎✒✓✙✒✓✔
❴✒✣❞
✿✗✒❫✛
r❵✳✳
b
❞✚
✉✇②❡③④⑤⑥
✖✎✤✒✒
❧✗✒✙
✮✯
✢✜✔✛ ✏
✢✖✜✗✒❣✒
✖✕✖✻✒❛✙
❫✏❣✒✙✢◗✘ ✙✒✗❞ ✱❞
✮✯✔✩
(Product of Two Vectors)
✑✻✒✚ ✛❞ ✮✗✙✓ ❧✖✎✤✒✒✓✔ ✕✓◗ ✾✒✓❫ ✢◗✘ ✱✕✔ ⑦✾✕❞✘✙ ✕✓◗ ✍✒✜✓ ✗✓✔ ✑✽✾✾✙
❧✖✎✤✒✒✓✔ ❞✒
✡☛☞✡✌
b
❂
(C) a
(D)
✈☎✆✝✞✟✠
❪
❪
(B) a
❜❡✐
❏
❘
❂
(A) b
10.6
0
❂
❂
a
✾✖✎
❖
❑❑❑▲
(D) AB CB CA
19.
0
PPP❘
(C) AB BC CA
❑❑❑
▲
▲
❏
✎⑨❧✜✚
✍✚★✚✾
❧✔ ✖✿✾✒
❞✚ ⑩⑩✒♥
❞✜✙✒
✖❞✾✒ ✮✯✩ ✑✍ ✮✗✒✜✒ ⑧✎✦✎✤
✓ ✾
✮✯✩ ✮✗
✼✗✜❣✒
❞✜
❧❞✛✓ ✮✯✔
✖❞ ✎✒✓ ❧✔ ✶✾✒✑✒✓✔ ❞✒ ❫✏❣✒✙✢◗✘ ✱❞ ❧✔ ✶✾✒ ✮✒✓✛ ✚ ✮✯✺ ✎✒✓ ✑✒⑦✾⑨✮✒✓✔ ❞✒ ❫✏❣✒✙✢◗✘ ✱❞ ✑✒⑦✾⑨✮ ✮✒✓✛✒ ✮✯ ✢✜✔✛ ✏
✢◗✘✙✒✓✔
✱✕✔
❞✚
✖✼♦✒✖✛
✎✒✓ ✢◗✘✙✒✓✔
✙✒✗✛✸
✑✖✎✤✒
❧✖✎✤✒
✮✒✓✛✒
❞✒
✗✓✔
✮✗
❧✔✾✒✓★✙✩
❫✏❣✒✙✢◗✘
✮✯✩
⑧❛✮✓✔
❧✖✎✤✒✒✓✔
✢❶ ❞✒✜
✪❧✚ ✢❶❞✒✜
★✮✒❷
✕✓ ◗
✎✒✓
✢✖✜❣✒✒✗
✪✙
✎✒✓
❧✓
❫✏❣✒✒
❞✜
❧✖✎✤✒✒✓✔ ❞✒
✱❞
✢❶ ❞✒✜
✑✖✎✤✒
✕✓◗
❧❞✛✓
❫✏❣✒✙
✮✒✓✛✒
✻✒✚
✮✯
❫✏❣✒✙✢◗✘✒✓✔
✮✯✔
✙✒✗✛✸
✎✒✓
✢◗✘✙✒✓✔
✎✒✓ ✛✜✚✕✓◗
❧✓
✢✖✜✻✒✒✖✥✒✛
✑✒✯✜
✕✓◗
❧✖✎✤✒
✑✒❸✒✜
❫✏❣✒✙✢◗✘
✢✜
❞✒
➥✍✎✏✕✒✜
✖❞✾✒
★✮✒❷
❹✾✒✖✗✛✚✺
❫✏❣✒✙
★✒✛✒
✮✯✩
✢✖✜❣✒✒✗
✱❞
✾✒✔ ✖❇✒❞✚
✱✕✔
❶ ✒✓❫ ✮✯✔✩ ✪❧ ✢✖✜❺❻✓✎ ✗✓✔ ✮✗ ✪✙ ✎✒✓ ✢❶❞✒✜ ✕✓◗ ❫✏❣✒✙✢◗✘✒✓✔ ❞✚ ⑩⑩✒♥ ❞✜✓✔❫✓✩
✑✖✻✒✾✒✔ ✖❇✒❞✚ ✗✓✔ ✪✙✕✓◗ ✖✕✖✻✒❛✙ ✑✙✏✢✾
457
❧ ✁✂✄ ☎✆✝✞ ✟✄✠
10.6.1
♥✡☛
☞✌♥✍✡✡☛✎
✏✡
✑✌♥✍✡
✐✙✚✛✜✜✢✜✜
✣
✤✥✦
✰
✰
✳✥❁✬
✰
♦✦✹
✻
b,
✈✱❙✲
❄✸
❅✦
✳✮✤✧✥
a b
✻
✴✵ ✶✥✷✸✹✺
✸✮✬❆✥✥✮✽✥✫
✮❞✪✥
✰
0
a
0
❊◗
s ❝❤♣ ❋✱ ❋✱●
s ✱ ❣❙ ✈✱❙✲
✰
✰
✪✮✤
❞✥
⑥✥✬✥ ✮✷✮✤✼✽✾
✮❞✪✥
✿✥✫✥
❀❁
✿✥✫✥
❀❁❇
✰
a
❈❉
✰
✰
b
✈✱✲
❙
❃
✰
✰
0,
b
✈❖✱❊✱
❍✳✥♦■✹✮✫
❏❑▲❏▼◆❇
✰
✫✥✦
❈ ✸✮✬❆✥✥✮✽✥✫
✷❀P✯
❀❁
✳✥❁✬
❂✭
✮❘❚✥✮✫
❅✯✦
❯
❯
❀❅
a
✭✮✤✧✥✥✦✯
a b = | a | | b | cos
❂✭✦
✿❀✥t
✧✥★✩✪✦✫✬
[Scalar (or dot) product of two vectors]
✒✓✔✡✕✖✗✘
✰
0
a b
✸✮✬❆✥✥✮✽✥✫
❞✬✫✦ ❀❁❇
✯
➞➟➠➡➢➤➥
➦➧➨➦➩
✐ ❱ ❲ ❳ ✜❨✜
✰
✰
1. a b
♦✥❘✫✮♦❞
❩❞
✻
✭✯❬✪✥
2.
❅✥✷
✺P✮✿❩
a
b
✈✱✲
❙
✪✮✤ ❈
✸✬❘✸✬
✪✮✤
❈
✰
✻
♠
❀❁✯
①
✫❭
❡✪✥✦✮
✯ ❞
❃
❴
❵ q
❃
❴
0
♠ ♦✦✹
❭✬✥❭✬
✪✮✤
✳✥❁✬
♦✦✹ ♦✺
✪✮✤
❃
❃
a
❛
❪
✻
b
❜
♦✦✹
✮❘❚✥✮✫
❅✦ ✯
❈ ❥ ❦ ❀❁❇
❃
r
✭✯✤❆✥✼
❅✦✯
✿❁✭ ✥
✸✬❘✸✬
✮❞
❂✭
✺✯❭ ♦✫❫
✮❘❚✥✮✫
❅✥③✥❞
❅✦✯ ❈
,
iˆ, ˆj
✭✮✤✧✥✥✦ ✯
④ ❊⑤
❀❁❇
kˆ,
♦✦✹
✮✺❩
❀❅
✸✥✫✦
✮❞
iˆ
iˆ
iˆ
ˆj
✻
✻
✤✥✦
✧✥★ ✩✪✦ ✫✬
a
✭✮✤✧✥✥✦✯
✴✵ ✶✥✷✸✹✺
❪
b
✈✱✲
❙
ˆj
ˆj
✻
♦✦✹
⑧
⑧
❻❅
✻
ˆj = kˆ kˆ 1
kˆ = kˆ iˆ 0
❭P⑦
❞✥
✮♦✮✷❅✦✪
✳❚✥♦✥
❶
❿✜❲
,
❪
cos
–1 ⑨
❷
❹
❀❁
✰
a.b
| a || b |
✰
✰
⑩
❸
⑥✥✬✥
✮✤✪✥
✿✥✫✥
❀❁❇
❺
✳❚✥✥✼ ✫❫
✰
❽
❽
a b= b a
( ?)
(Two important properties of scalar
✻
➆❲➄
❈
✰
✰
➁➂ ❨✜➃✐➄➅
❞✥✦✶✥
⑧
a.b
| a || b |
cos
✳✮✤✧✥
❪
✰
✰
❾✙❿➀✜
❵
0
a b
|a ||b |
| a |2 ,
⑧
7.
✫❭
❃
a b
❂✭
❃
a b
r
②
❀❁✯
❃
❃
a ( a)
❩♦✯
✳❚✥✥✼✫❫❫
✭✮✤✧✥
✰
✫❚✥✥
6.
✧✥★✩✪✦ ✫✬
❵
❪
r
✸✦✉✇ ✥✶✥
❀❁✯
❃
❴
a a | a |2 ,
= ,
✮♦✮✧✥✽✾✫❢
5.
✤✥✦
a b |a ||b |
✫❭
✰
4.
❃
❃
= 0,
✮♦✮✧✥✽✾✫❢
✺✯❭ ♦✫❫ ❫
✰
✰
b
✈✱✲
❙
✰
✰
3.
a
✮❞
❀❁❇
✰
✰
➇➈➉➆✐➊ ❨✜➋
❼
❡✪✥✦✯
➁➂❨✜➌➇➋
product)
➁➂❨✜➌➇➋
1
➍
❍✳✮✤✧✥
❃
❃
✭✮✤✧✥
❀❁✯
➁➂ ❨ ✜➌➇➋
✫❭
2
✴✵✶✥✷✸✹ ✺
a (b
❴
❯
c) = a b
➓
✺P✮✿❩
a
➔
✸✬
✮ ♦✫✬ ✶✥
✮✷✪❅◆ ❅✥✷
✺P✮✿❩
❯
a c
➓
❯
❯
❅✥✷
✪✥✦✴ ✸✹ ✺
❯
❯
❃
➒
❞P
→➣↔
↕
b
✤✥✦
✭✮✤✧✥
( a) b =
➜
❀❁✯
✰
✰
✻
✳✥❁✬
➝
➙ ❩❞
❃
❃
(a b )
❴
✳✮✤✧✥
❃
❃
❵
a ( b)
❴
➝
❀❁➛
✫✥✦
➍
a, b
➍
➎➏➐➑
c
✫P✷
458
① ✁✂✄
❀☎✆
✈☎✆✠✝
✆✝✞
✟☎✆✠✝
✙✓✚✝✘✍✛✜
✡✝☛☞
✌✍
☎✘✢✘☎✜☎✣✝✕
a1iˆ a2 ˆj a3 kˆ
✑
✎✞✏
✌✍
✎✞✏
✑
✍✤ ✝✥✕
❢✝✞ ✕✝
b1iˆ b2 ˆj b3 kˆ ,
✱✒✏
✑
✑
☎✆✱
❢✓✱
❢✔✏
✕✖
✗✘☞✝
❢✔
✧
a b = (a1iˆ a2 ˆj a3 kˆ) (b1iˆ b2 ˆj b3 kˆ)
✧
✑
✦
✦
✑
✑
✑
= a1iˆ (b1iˆ b2 ˆj b3 kˆ) a2 ˆj (b1iˆ b2 ˆj b3 kˆ) + a3 kˆ (b1iˆ b2 ˆj b3 kˆ)
= a1b1 (iˆ iˆ) a1b2 (iˆ ˆj ) a1b3 (iˆ kˆ) a2b1 ( ˆj iˆ) a2b2 ( ˆj ˆj ) a2b3 ( ˆj kˆ)
✦
✑
✦
✑
✦
✑
✦
✑
✑
✦
✑
✦
✑
+ a3b1 (kˆ iˆ) a3b2 (kˆ ˆj ) a3b3 (kˆ kˆ)
(
= a 1b 1 + a 2 b 2 + a 3b 3
✦
✦
✑
✯
❜✟
a b = a1b1
✍✤☞ ✝✫
10.6.2
✰
✮
✲✳
✴✵✶✷✸
✳✸
a2 b2
✰
✵✳✴✹
✑
✑
✑
✦
✑
✦
✦
✑
✗✍❀✓★
♠ ✕
✯
✦
✑
✙✓✚✝✩✎♠
✪
(
✈✝✔✫
✬
☞✝
✗✍❀✝✞✙
☞✫✘✞
✍✫
✍✤✐ ✝✞ ✚✝
✭
☞✝
✗✍❀✝✞ ✙
☞✫✘✞
✍✫
)
)
a3b3
✺✻✼✸✸
✽✺
✴✸✾✸
✽✿❁✸✻✽
(Projection of a vector on a line)
❃❃❃
❄
✎✝✘
✜❡☎❂✱
☎☞
✱☞
AB
✟☎✆✠✝
☎☞✟❡
☎✆❅☛
l
✫✞✣ ✝✝
❆✎✝✘
✜❡☎❂✱❇
✒✞ ✛
✟✝❈✝
✒✝✎✝✒✕♠
❍❍❍
■
☞✝✞ ✚✝
✖✘✝✕✝
❢✔ ❞
❆✈✝✒❊✛ ☎✕
✪❋●✬❋
✆✞ ☎✣✝✱❇
✕✖
AB
l
☞✝
☎✆✠✝✝
✎✞✏
❉
❏
✍✫
✍✤✐ ✝✞ ✍
✱☞
✟☎✆✠✝
p
❆✎✝✘
✜❡☎❂✱❇
▲▲▲
▼
❢✔ ☎❂✟☞✝ ✍☎✫✎✝✚✝
❜✟
✖✝✕
✍✫
☎✘◆✝♠✫
| AB | cos
❢✔
cos
☎☞
❑ ❢✔ ✈✝✔✫ ☎❂✟☞❡ ☎✆✠✝✝ ☞✝
❜✟☞✝
✍☎✫✎✝✚✝
| p |,
☞❡ ☎✆✠✝✝ ✒✞✛ ✟✎✝✘ ✈❈✝✒✝ ☎✒✍✫❡✕ ❢✝✞ ✘✝
◗
❉ ✩✘✝❖✎☞
❢✔ ✈❈✝✒✝
P✚✝✝❖✎☞❞
✟☎✆✠✝
p
☞✝✞
✍✤✐ ✝✞ ✍
✟☎✆✠✝ ☞❢✕✞
☎✘❘✆❅☛
✫✞✣ ✝✝
l
✍✫
✟☎✆✠✝
AB
☞✝
✍✤✐ ✝✞✍
☞❢✜✝✕✝
❢✔❞
[
✈ ✝✒ ❊✛ ☎ ✕
✎✞✏
✟✞
10.20 (i)
✍✤❖❀ ✞☞
✟✞
(iv)
✈ ✝✒ ❊✛ ☎ ✕
✕☞
✎✞✏
✟☎✆ ✠✝
AB
]
❱❲❳❨❩❬❭
❪❫❴❵❫
✗✆✝❢✫✚✝✕❙
❚❚❚
❯
❃❃❃
❄
☎✘✢ ✘ ☎✜ ☎✣ ✝ ✕
❢✔✏
❍❍❍
■
◗
✈✝✔✫
l
☞✝
✫ ✞✣ ✝✝
l
✍✫
✍ ✤ ✐ ✝✞ ✍
✟☎ ✆✠✝
AC
❢✔❞
459
❧ ✁✂✄ ☎✆✝✞ ✟✄✠
✐✡☛☞✌✍✌
1.
l
❥✎✏✑✑
✥✎✒
✓✔✕✖✗✘✑
p̂
✙✖✗
❡✑✚✑✛
✜✖✗✘✑
✢✣
✤✑✎
l
❥✎ ✏✑✑
a
✱✛ ✜✖✗✘✑
b,
✛✑ ✗✬✜ ❥✎ ✜✖✗✘✑
✦❥ ✦❞★✑✎✦
b
,
|b |
✯
a bˆ,
✮
✭
a. pˆ
✦❞★ ✑✎ ✦
✜✎
✦❞✑✪✤
a
✓✈✑✥✑
1
(a b )
|b |
✯
✯
✯
✯
✰✲❦✳❦
✜✎
✢✑✎✤ ✑ ✢✣❣
✷✷✷
✯
✷✷✷
✯
✙✖✗
✛✑
✯
✩
✫
✦❞✑✪✤
✴✵
a
✜✖✗✘✑
✢✑✎✤✑ ✢✣ ❣
2.
✩
✧
✦❥
= 0,
✶
AB
✤✑✎
✛✑
✷✷✷
✯
AB
✦❞★✑✎✦ ✜✖✗✘✑ ✸✥✙✹
✢✑✎✺✑ ✓✑✣ ❥
✙✖✗
=
✶
AB
✻ ✤✑✎
✛✑ ✦❞★✑✎✦
✷✷✷✯
BA
✜✖✗✘✑
✼✵
✾
✙✖✗
✽
=
✙✖✗ ❆
✖✔❏✔✖❑✖✏✑✤
3
AB
2
a a1iˆ a2 ˆj a3 kˆ
✷✷✷
✯
✽
=
✾
✓✈✑✥✑
2
,
❢❀❁❂❃❄❅
✢✑✎✺✑❣
✤✑✎
✛✑
▲✦
❡✎✹
✦❞✑✪✤
cos
✛■
❉
▼✑
❊
✫
a
✓✔✕✖✗✘✑
a
✜✢✑✙✤✑
❘
❖
a1
, cos
|a|
❘
✓✈✑✑❳✤♦
✢✣✹
❡✎✹
✓✖❙✑❵✙❬✤
❤♥
✗✑✎
✛✑✎●✑
❭■①
✛✑
❜③
✖✗✙✑
✖✛✙✑
✜✖✗✘✑✑✎✹
②✑✤
✥✎✒
❉
▼✑
✖✗✥♦✒ ❋✛✑✎✜ ✑❍✔
✫
a
✙✖✗
a3
|a|
◗ ❖
❘
Ø❡✘✑❲
a1, a2
❨✑❩✛
✓✑✣ ❥
a3
OX, OY
Ø❡✘✑❲
✤✈✑✑
x, y,
✱✥✹
OZ
z
✥✎✒
✓★✑
✱✛ ❡✑✚✑✛ ✜✖✗✘✑ ✢✣ ✤❭ ❍✜✛✑✎ ✖✗✥♦✒ ❋✛✑✎✜ ✑❍✔
✢✕✓ ✑
✢✣
a
cos iˆ cos ˆj cos kˆ
❪
❊
❫
♣qr
s
b
✯
✦✖❥❡✑●✑
Ø❡✘✑❲
t
✓✑✣ ❥
✉
✢✣
✤✈✑✑
a b 1,
✇
❉
❍✔
✜✖✗✘✑✑✎✹ ✥✎✒
✯
✯
✯
iˆ
a
|b | 2 .
✰❦❱❯
cos
④
1⑤
⑨
ˆj kˆ
✯
a.b
| a || b |
✯
✯
⑥
⑩ ❉
cos
④
1⑤
⑨
❶
❷
✯
❹✲❦❦
✓✤❲
ˆj kˆ
b iˆ
✥✎✒
1
2
⑦
⑥
⑩
❉
❷
❭■①
3
✛✑
✛✑✎●✑
✩
✩
a
✯
✯
✥✎✒
a b 1, | a | 1
✜✖✗✘✑
✜✖✗✘✑✑✎✹
❴
✢✣❣
✜✛✤✑
❶
❤❸
❊
✩
⑧ ❉
✗✑✎
❍✜✛■
✤✑✎
✯
| a |cos
❱
✰❦❯
✓✖✗✘✑
✯
❜③
✢✣✹
✛■✖▼✱❣
✯
♠❛✌❜❝✍✌
✢✑✎✺✑❣
✜✎
✩
♠❛✌❜❝✍✌
✫
a
✜✖✗✘✑
✯
▲✦
❘
✯
✦❞★✑✎✦ ✢✣❣ ❍✜✥✎✒ ✓✖✤✖❥❬✤
✥✎✒
a2
, and cos
|a|
P ❖
| a | cos , | a |cos
✖✛
a
✥✎✒
✜✖✗✘✑
✢✣ ❣
✜✛✤■
a.iˆ
| a || iˆ |
◆ ❖
✗■✖▼✱
✦❞★✑✎✦
✫
✥✎✒ ✓✔✕✖✗✘✑
✛■
❚✙✑✔
✥✎✒
✘✑✬✿✙
✖✗✥♦✒ ❋✛✑✎●✑
✥✎✒
❊
✯
❙✑■
✜✖✗✘✑
✯
❇ ✓✑✣ ❥ ❈ ✜✖✗✘✑
❘
✙✢
✦❞★✑✎✦
♣qr
s
b
✥✎✒
❭■①
✛✑
✛✑✎●✑
✶ ✖✔❏✔
cos =
✶
❺✑❥✑
✦❞✗❻✑
✢✣
❘
❘
a b
| a || b |
❼
❘
❘
✜✎
✦❞✑✪✤
✢✑✎✤✑
✢✣❣
②✑✤
✛■✖▼✱❣
460
① ✁✂✄
✝
a b = (iˆ ˆj kˆ) ( iˆ ˆj kˆ) 1 1 1
1
cos =
3
1
1
= cos
3
ˆj 3kˆ
b iˆ 3 ˆj 5kˆ ,
✝
✈☎
✞
✆
✟
✠
✟
✞
✡
✟
✟
✡ ✟
1
✚
❜☛☞✌✍✎
✈✔✛
✏✑
✈✜✓✢✣✤
♠✧★✩✪✫★
✒✓✔✕
✏✖ ✗
☞✘
✙
✘✓✕✥✓
✙
✮
✮
15
a 5iˆ
✬☞✭
✏✖ ✦
✯✰✱
❙
✔✓✕
✭r✓✓✲❜✍
☞✘
✝
✝
a b
☛☞✭r✓
✈✓✖ ✳
✴
✴
a b
✩❣
✌✗☎ ②✔✵ ✏✖ ✦
✏✑
✶✓✷✔✕
✏✖ ✗
☞✘
✭✓✕
r✓✸✹ ✬✕✔✳
✌✗☎②✔✵✵
☛☞✭r✓
✾
✈✓✖✳
❃
✮
✮
❜☛☞✌✍
✈☞✭r✓
✺✷✘✓
✻✼✥✓✷✒✽✌
✿
✿
❁
❁
✿
❂
❁
✿
✿
r✓✸✹ ✬
✏✖ ✦
✿
✿
❁
✿
❂
✿
❁
☛☞✭r✓
✮
✏✖✦
✗
✮
✴
✴
✮
✴
✴
a b
a b
a 2iˆ 3 ˆj 2kˆ
✈✔✛
✌✗☎②✔✵
❄❅❆
❇
16
❈
☛☞✭r✓
❂
❁
☛☞✭r✓
a
✘✓
b
☛☞✭r✓
b
☛☞✭r✓
iˆ 2 ˆj kˆ
✡
✞
✞
✒✳
✒✐❉✓✕✒
❊✓✔
✘✢☞✶✍✦
❍
❍
❍
✒✐ ❉✓✕✒
✒✳
1
1
(a b ) =
|b |
17
✘✓✎
❁
✴
❋
♠✧★✩✪✫★
✬☞✭
✝
✝
✩❣
✏✖ ✗
ˆj 3kˆ) ( iˆ 3 ˆj 5kˆ) 6iˆ 2 ˆj 8kˆ
ˆj 3kˆ) ( iˆ 3 ˆj 5kˆ) 4iˆ 4 ˆj 2kˆ
(a b ) (a b ) = (6iˆ 2 ˆj 8kˆ) (4iˆ 4 ˆj 2kˆ) 24 8 16 0
a b = (5iˆ
a b = (5iˆ
✬✏✓❀
♠✧★✩✪✫★
✏✓✕✔ ✕
✝
✝
(1)
2
(2)
■
✭✓✕
a
☛☞✭r✓
2
(1)
■
2
■
✮
✮
✬☞✭
(2 . 1 3 . 2 2 . 1)
■
●
❏
6
❜☛
✒✐✘✓✳
✏✖✗
5
6
3
✮
✮
b
✯✰✱
❙
10
❏
✡
✝
✝
| a | 2, | b | 3
☞✘
✏✖✦
✡
4
✈✓✖✳
a b
❊✓✔
✘✢☞✶✍✦
✆
❑
✔✓✕
✮
✮
|a b |
✟
✩❣
✏✑
❊✓✔
✒✓✔✕
✘✢☞✶✍✦
✏✖ ✗
☞✘
✮
✮
2
a b
❈
✮
❈
✮
= (a b ) (a b )
✿
✿
▲
✴ ✴
✴
✴
✴
✴
✴
✴
= a.a a b b a b b
= | a |2 2( a b ) | b |2
▼
◆
▼
✮
= (2) 2
♠✧★✩✪✫★
|a b| =
a
✍✘
✑✓◗✓✘
☛☞✭r✓
✏✖
❘✬✓✕✗☞✘
a
✞
✮
✮
✟
✮
✮
(x a) (x a) 8 ,
✈✓✖✳
✠
✞
✡
✔✓✕
|x|
❯
P
✩❣
2(4) (3) 2
✮
P
✬☞✭
✞
5
✟
18
✠
◆
✮
✮
❜☛☞✌✍
✟
❖
✮
✮
✮
✟
◆
✍✘
✑✓◗✓✘
☛☞✭r✓
✏✖✎
❜☛☞✌✍
✮
| a |❚ 1 . ✬✏
✮
✮
✜✓✢
☞✭✬✓
✏✼✈✓
✏✖
☞✘
✮
(x a) (x a) = 8
✟
✈❱✓②✓
❋
❋
❋
❋
❋
❋
❋
✠
✞
❋
x❲x ❳ x❲a ❨ a❲ x ❨a❲a = 8
✮
✈❱✓②✓
| x |2 1 = 8
✟
✮
✯❩✰✰❬❭ ⑦
| x |2 9
✮
❜☛☞✌✍
| x | = 3(
❘✬✓✕ ☞
✗ ✘
☛☞✭r✓
✘✓
✒☞✳✑✓✥✓
☛✭✖②
r✓✸✹ ✬✕✔✳
✏✓✕✔✓
✏✖
)
461
❧ ✁✂✄ ☎✆✝✞ ✟✄✠
✕
✕
19
♠✡☛☞✌✍☛
✥✎✏
a
✑✒✥✓✎✎✏✔
✕
✕
b,
✈✖❥
❙
♦✏✗
✒✘✙
✕
✕
| a b | | a | | b | (Cauchy-Schwartz
✛
✑✥✚♦
✜
)
✢✑✒✣✤✎ ❆
✱
✱
☞❣
✥♥
✦✧★✩
✢✑✒✣✤✎
✬✎✵✏
✦✚✔
✫✬
✕
✕
✦✣
✑✦✪
✭✬✮✯
✣✏ ✔
✕
✛
✷
✱
0
a
✰✒✥
✲✳❦✴❦
✱
0.
b
♦✎✭✵♦
✕
✣✏✔
★✑
★✑✒✘✙
✷
✦✣
✤❜✬✸✎
✤✹✵✏
✦✚ ✔
✒✭✶✎✒✵
|a | 0 |b |
✺
✒✤
✣✏✔
✕
✕
| a b | 0 | a ||b | .
✒✤
✦✚
✺
✵r
✦✣✏✔
✼
✼
|a b |
= | cos | 1
| a || b |
✻
✼
✼
✽
✕
✕
|a b |
✛
★✑✒✘✙
✜
✦✚❆
✒✣✘✵✎
✕
✕
| a ||b |
✾
❤✐♣qst✉ ✇①②③✇
♠✡☛☞✌✍☛
✕
✕
20
✥✎✏
a
✑✒✥✓✎✎✏✔
✕
b
✿❋✖✖
♦✏✗
✒✘✙
✕
✕
✣✎✸
✘♥✒✪✙
✒✤
✕
❀
0
b
❄✖
✣✏✔ ✑✦✪ ✫✬ ✑✏ ✭✬✮✯
✕
❀
❈
❈
= a a
❅
❉✰✎✏✔
?)
❆ ★✑✒✘✙
❀
❈
❅
❇
❈
❈
❈
b a b b
❅
✕
✕
❀
✛
❈
a b
✕
✕
✕
(a b ) (a b )
✷
❈
❇
✕
✕
❀
❇
❅
✕
= | a |2 2a b | b |2
❀
✕
(
❀
✛
✕
✕
✢✒✥✓✎
❀
✕
✕
(
❀
✛
✕
❊✧● ✎✸✬✗✘
❍✣
✒♦✒✸✣✰
✦✚
)
✕
| a |2 2 | a b | | b |2
❉✰✎✏✒
✔ ✤
x | x| x R)
■
❏
❑
✕
| a |2 2 | a || b | | b |2
❀
✾
(
✦✚
✵r
✕
| a b |2 = (a b ) 2
✾
)
✒❢✎❁✎✧✪❂✢✑✒✣✤✎
❀
✜
✕
| a | 0 |b |
✕
0
a
✥✎✏✸ ✎✏✔ ✒✭✶✎✒✵✰✎✏✔
✕
✕
|a b | |a | |b |(
✑✥✚♦
✕
✕
✥♥ ✦✧★✩ ✢✑✒✣✤✎❃
☞❣
✕
(
❀
▲✥✎✦✹●✎
▼◆
✑✏
)
✕
✕
= (| a | | b |) 2
✕
✕
✕
✕
|a b | |a | |b |
❀
✢✵❖
❀
✜
✰✒✥ ✒❢✎❁✎✧✪❂✢✑✒✣✤✎ ✣✏✔ ✑✒✣✤✎ ❲✎✹●✎ ✦✎✏✵ ♥ ✦✚ ❳▲✬✰✧❉
✩ ✵ ▲✥✎✦✹●✎ ❨❩ ✣✏✔❬ ✢✶✎✎✩✵❭
◗❘❚❯✍☛❱
P
✕
✕
❫
✕
| a b | = | a | |b |,
❀
❪
✕
❴❴❴
❵❵❵
❫
✵r
❵❵❵
❫
| AC | = | AB | | BC |
❪
➥r✥✧
A, B
♠✡☛☞✌✍☛
☞❣
✦✣
✢✎✚✹
21
C
✑✔✹❛
✏ ✎
✥✓✎✎✩★ ✙
✬❞✎❡✵
✤✹✵✏
✒✤
✥✓✎✎✩✵ ✎
➥r✥✧
✦✚❆
A ( 2iˆ 3 ˆj 5kˆ), B( iˆ 2 ˆj 3kˆ)
❝
❀
❀
❀
❀
✢✎✚✹
C(7iˆ kˆ)
❝
✦✚❖
✔
✕
❴❴❴
AB = (1 2)iˆ (2 3) ˆj (3 5) kˆ 3iˆ
❀
❀
❝
❀
❝
✷
❝
ˆj 2kˆ
❝
✑✔✹❛
✏ ✎
✦✚❆
462
① ✁✂✄
☎☎☎
✆
BC = (7 1)iˆ (0 2) ˆj ( 1 3)kˆ
✝
✞
✝
✞
✝
✝
✟
6iˆ 2 ˆj 4kˆ
✝
✝
✠✠✠
✡
ˆ
ˆ
AC = (7 2)iˆ (0 3) ˆj ( 1 5)k 9iˆ 3 ˆj 6k
✞
✞
✝
✕✕✕
✖
✗✗✗
✘
✟
✝
✝
✌✌✌
✍
2 14
| AC | 3 14
✈✎✏❥
✗✗✗
✘
✞
✜✢✣✤
A, B
C
✙✥✦✧
✑❧ ✧★✩✥
✪✦❧✫
☎☎☎
✆
✙✥✦✧
C
❂★✿
✒❁✥❀✥✤✻
❃✥✺❄✥✥★ ❅
✼✥
✒✹✶✥❆✳✥
✹✪✺❧
✼✧✚★
✽
✡
✡
✑✒✣❃✥✥★❧
❂★✿
✢✺❱
a
✼✥
b
r◆❖❖
❂★✿
❲✥✚
✼✥★✳✥
iˆ 2 ˆj 3kˆ
3.
✑✒✣❃✥
4.
✑✒✣❃✥
iˆ 3 ˆj 7kˆ
✼✥❞
5.
✣❃✥✥❆❜ ✔
✒✼
✒✹♥✹✒✓✒✩✥✚
ˆj
✐✧
❴
❳❨❩
❬
iˆ ˆj
✑✒✣❃✥
❴
✒✣✔
✪✤ ✔
kˆ
3iˆ 2 ˆj
✑✒✣❃✥✥★❧
✞
✐✧❧✚✤ ✒✐✿✧ ❀✥✺
✜✢✣✤
A,
❯
❯
2
P◗❘
✪✦❧
a b
✙✥✦ ✧
❙
❚
6
✡
✡
✪✦
✚✥★
a
r◆❖❖
b
✼✺✒✻✔✫
2.
iˆ
✆
0
✾
❏❑▲▼
3
♦✶❃✥✛
✐✒✧✶✥✳✥
☎☎☎✆
✽
✪✦❧ ✫
❇❈ ❉❊❋●❍■
✣✥★
☎☎☎
✆
AB BC CA
♠✣✥✪✧✳✥ ✴✵ ✶★❧ ✷✸✥✹ ✣✺✒✻✔ ✒✼
❢✭✮✯✰✱✲
✬
1.
✝
AC = | AB | | BC |
❜✑✒✓✔
B
✝
✌✌✌
✍
☛☛☛
☞
| AB | = 14, BC
✙✚✛
✞
✼✥
❭
✐❪❫ ✥★✐
✢✺❱
❲✥✚
ˆj 8kˆ
✐✧
✚✺✹
✑✒✣❃✥✥★❧
✶★❧
✞
✼✥
❲✥✚
✼✥★ ✳✥
✼✺✒✻✔✫
✼✺✒✻✔✫
✝
7iˆ
✑✒✣❃✥
❂★✿
✐❪❫✥★✐
✑★
❲✥✚
✼✺✒✻✔✫
✐❪❵✸★✼
✶✥❁✥✼
✑✒✣❃✥
✪✦❞
1 ˆ
1
1 ˆ
(2i 3 ˆj 6kˆ), (3iˆ 6 ˆj 2kˆ),
(6i 2 ˆj 3kˆ)
7
7
7
❛
✸✪
6.
❀✥✺
❛
❝
✣❃✥✥❆❜ ✔
✒✼
☞
☞
✸★
✑✒✣❃✥
☞
☞
☞
☞
❦
8.
✑✒✣❃✥✥★ ❧
✼✥
9.
10.
✞
✼✥★✳✥
a
| a | 8|b |
✼✥
✶✥✹
❲✥✚
✔✼
✘
✸✒✣
✐✧
a
✓❧✢
♣❖q
s
60°
✪✦
✶✥❁✥✼
b
❂★✿
✚t✥✥
✐✒✧✶✥✳✥ ❲✥✚
❜✹✼✥
✙✒✣❃✥
✑✒✣❃✥
a,
☞
☞
✪✥★
✚✥★
|a|
✼✺✒✻✔❞ ✸✒✣ ❜✹❂★✿
✔❂❧
|b |
2 ˆj 3kˆ, b
✚✥★
⑨ ✼✥
✶✥✹
❲✥✚
✒✓✔
iˆ
✐✒✧✶✥✳✥
1
2
( x a ) ( x a ) 12
✉✤ ✳✥✹✐✿✓
☞
❂★✿
✘
2iˆ
✪✦❞
✪✦❧✫
❲✥✚
✼✺✒✻✔✫
✼✺✒✻✔✫
✇
✸✒✣
✓❧ ✢ ❂✚❤
☞
☞
❳❨❬❩
❂★✿
✣❣ ✑✧★
✡
✡
✣✥★
✔✼
❝
☞
7. (3a 5b ) (2a 7b )
✝
❛
✐✧❡✐✧
☞
(a b ) (a b ) 8
✸✒✣
❛
2 ˆj
✼✺✒✻✔✫
kˆ
✢✺❱
✪✦ ✫
☞
②
✑✶✥✹ ✪✦ ✙✥✦ ✧ ❜✹ ❂★✿
☞
☞
③
④
⑤
✘
❳❨❩
❬
c
3iˆ
ˆj
⑥
✪✥★
✚✥★
| x|
❲✥✚
✼✺✒✻✔✫
✡
✡
❜✑ ✐❪✼✥✧ ✪✦ ✒✼
a
⑦ ⑧
✇
b,c
463
❧ ✁✂✄ ☎✆✝✞ ✟✄✠
✙
✙
11.
♥✡☛☛☞✌✍ ✎✏
✙
12.
✔✎♥
13.
✙
✰
✔✎♥
0,
a b
✈✚✛
❙
✈✚✛
❙
♦✑✜
✤
✤
✤
✤
✤
✤
|a |b |b |a , |a |b |b |a
✎✢✍
✣
✐✖ ✢✘✦ ✧★✩
✥
✪
b
✕☛✑ ✗✎♥✡☛
✰
♦✑✜ ✦☛✖✑ ✫✑✘ ✬✔☛
✰
✰
a, b , c
✤
✤
b
✙
✙
0
a a
✰
a
♥☛✑ ✡☛✒✓✔✑✕✖ ✗✎♥✡☛☛✑✘
✫☛❡☛✏ ✗✎♥✡☛ ✌✗ ✐✱✏☛✖ ✧★ ✎✏
a b
✲
✰
✲
✎✭✮✏✮☛☞ ✎✭✏☛✢☛
✰
c
✳
0
✰
✰
a b
✕☛✑
✴
✰
✲
✯☛ ✗✏✕☛ ✧★
✰
✰
b c
✴
✲
?
✰
c a
✴
✏☛ ✫☛✭
❑☛✕ ✏✵✎✯✍✩
✰
✰
14.
✔✎♥
15.
✰
0
a
✍✏
✽♥☛✧✖✾☛
✔✎♥
✎✏✗✵
✰
b
✿☛✖☛
✼✐✭✑
♦✑✜
0
a b
r❝
✽❀☛✖
ABC
✎❡☛❂☛✺✯
✰
✰
0,
✶✷✸✹✸
✏✵
✐✺✎✮❁
✡☛✵✮☛☞
✐✖✘✕✺
✎♦✢☛✑✫
ABC
❑☛✕
16.
♥✡☛☛☞✌ ✍
✎✏
❈✦♥✺
17.
♥✡☛☛☞✌ ✍
✎✏
✗✎♥✡☛
✡☛✵✮☛☛✑❋ ✏✵
✔✎♥
✖❇✭☛
✡☛✒ ✓✔✑✕✖
✗✎♥✡☛
(A)
■❏▲
✐✎✖❩❬✑♥
❭❪❫❴
✐❜✎✕
✼❛☛
✫✑✘
A, B, C
BC
✍♦✘
♦✑✜
✦✵❇
✏☛✑✾☛
✏☛
✗✘✖❉
✑ ☛
✧★ ✩
✧★ ✘
✕☛✑
✎❡☛❂☛✺✯
♦✑✜
]
✧★
✧★ ✩
✘
4kˆ
✶✸❊❥
a
✏☛
✐✎✖✫☛✾☛
(B)
◗❏
✫✑✘
▼◆■❖❏
❍
‘a’
✼❛☛
✍✏
=–1
✍✏
✗✫✏☛✑✾☛
●
✧★
✼☛★ ✖
❍ ✍✏
✡☛✒✓✔✕✑✖
✼✎♥✡☛
✧★
a
❍
✕☛✑
✍✏
✫☛❡☛✏
♥✎❛☛✾☛☛♦✕✵☞
✐❜✎✕
✗✫✏☛✑✎✾☛✏
❣☛ ✺ ✫ ☛✏ ✖
✯✦
❍
[Vector (or cross) product of two vectors]
❤✐✱☛✫☛✎✾☛✏❦
✫✑✘
(D) a = 1/| |
❍
♥✎❛☛✾☛☛♦✕✵☞
✏☛ ✑ ♦☛✫☛♦ ✕☞
✎✭♥✑☞✡☛☛✘✏
(C) a = | |
❘❚❯❏❱❲❳❨
✎❡☛❵✎♦✫✵✔
x-
y(ii)]
✼☛♦♣✜ ✎✕ ❭❪❫❴❴
✭✧✵✘
✏✖✕✑ ✧★ ✩
✘
✧✫✭✑
♥✎❛☛✾☛☛♦✕✵☞
✗✑ ♥✒ ✖ ❢✭☛✻✫✏
BA
✗✎♥✡☛☛✑ ✘
❆❆❆
✙
✼☛★✖
✗✎♥✡☛
▼◆■❖❏❏▲P
✎♥✡☛☛
✼☛♦✡✔✏
(1, 2, 3), (–1, 0, 0), (0, 1, 2)
Ø✫✡☛❃
A(1, 2, 7), B(2, 6, 3)
C(3, 10, –1)
2iˆ ˆj kˆ, iˆ 3 ˆj 5kˆ
3iˆ 4 ˆj
=1
✫✑✘
✧☛✑✭☛
✔✎♥
❢✭☛✻✫✏
✏✵
✍✏
✧★
❍
10.6.3
[
❄
●
18.
z-
[ ABC,
✏✵✎✯✍✩
✗✻✔
✏✵✎✯✍✩
❅❅❅✰
❄
✏☛
♥☛✍q
❢✭☛✻✫✏
✎✭♥✑✡
☞ ☛☛✘✏
y-
✐✑✘ ❇
✼♠✱♠ ✕
✧☛❞☛
✏✵
✼❛☛
✧☛✑
✐❜✎✕
✐✖
✧★ ✩
✉✇①② ③④⑤ ⑥⑦⑧⑨⑨
✢☛✔☛
✯☛✕✵
✧★
✽q♠ ✎✢✔☛✑ ✘ ✏☛✑
s
✼❛☛ ✏✵ ✕✖✐✜ ♦✺✜
✘ ✕✢ ✎✏✔☛ ✯☛✕☛ ✧★ ✕☛✑ ✼q♠t
✒ ☛ ❢✭☛✻✫✏
✏✵
❇❇☛☞
✯☛ ✕☛
[
✼☛♦♣✜ ✎✕
❢✭☛✻✫✏
z-
✧★
✏✵
✕☛ ✑
❞☛✵✩
❢✭☛✻✫✏
10.22(i)]
x-
✼❛☛
✌✗
✏✵
✩
✎♥✡☛☛
✼❛☛ ✏✵ ✼☛✑✖ ✗✘♦✜
✑ ✕ ✏✖✕☛
464
① ✁✂✄
✕
✕
a
✐☎✆✝✞✞✟✞✞ ✠ ♥✡☛ ☞✡✌ ✍✎☛ ✏✑ ✒✓♥☞✡✡☛ ✔
✤
✤
✓✚✓♥❢✥✦
✧✡✏✡
✓❞✎✡
★✩
✪✡✩✑
a
n̂
❜✒
✛✼ ❞✡✑
✺❞
✮✡✻✡❞
✬
✫
✣
♦☛ ✜
✈✳❥
❙
b
★✩
✧✡☛
nˆ
✽✾✳✳
♦☛ ✜
✴✵✶
❞✡
✺❞
✪❉✡✡❢✏❊ ❊
❃❄❅❆❇❈
✮☛✔
✓❞
✛✓✑✯✡✡✓✥✡✏
♥✓✿✡✙✡✡♦✏✵❢
♥✓✿✡✙✡✡ ♦✏✵❢
b,
✈✳❥
❙
✛❀✓✏
✛❀✓ ✏
✷ ✸ ✷ ✹
★✩❣
✢✔ ✴
★✩❣
❞✑✏☛
★✩ ✔
✬
a
✒✓♥☞✡
0
★✩ ✪✡✩✑
❞✡☛✙✡
✛✑
✓✚✓ ✮❢✏
⑧⑨⑩❶❷❸❹ ❺❻❼❽❾
✬
a
❞✡☛
♥✡☛✚✡☛✔
❞✡☛
✬
❁✪✡ ♦❂ ✜✓ ✏
✭✛
✬
✒✓♥☞✡
✬
✬
a, b
✒☛
✣
✬
✬
✎★✡ ✰
✤
✤
a b
❞✡ ✒✓♥☞✡ ✗✘✙✡✚✛✜✢
a b = | a || b | sin nˆ
✬
✓❞✎✡ ✧✡✏✡ ★✩ ✧★✡✰ ✱✲
b,
r❋✖✖
b
❧s
●
✏✑✛✜ ❍✡✘ ✮✡✚☛
❞✵
✛✑
✎★
n̂
❞✵ ✓♥☞✡✡
▼
0
✮☛ ✔
✶✢✏✵ ★✩❣
✬
✬
✬
✎✓♥
0
a
✬
0,
b
✈✾✳■✳
✕
✕
a b
▲
✏✴ ❏ ✛✓✑✯✡✡✓✥✡✏ ✚★✵✔ ★✩ ✪✡✩✑ ❜✒ ✓❑❉✡✓✏ ✮✔☛ ★✮
✕
✛✓✑✯✡✡✓✥✡✏
❞✑✏☛ ★✩❣
✔
✐◆ ❖ P ✞◗✞❘
✬
✬
1. a
2.
b
✺❞
★✩❣
✒✓♥☞✡
✤
✤
✮✡✚
❚❯❱
❲
♥✡☛
☞✡✌ ✍✎☛ ✏✑
✈✳❥
❙
b
✺❞
♦☛✜
♥✌ ✒✑☛
✒✮✡✔✏✑
★✩✔
✒✓♥☞✡
✤
a b
✣
✏✴
✒✔✑☛ ❨✡❈
❁✪❉✡♦✡
✕
✕
❬
❳
0
✪✡✩✑
✎✓♥
♦☛ ✜♦✢
✎✓♥
▲
✓♦✓☞✡✥✦✏❪
✓❑❉✡✓✏
✮☛ ✔
✕
✕
✬
a a 0
,
▲
▼
❦
✬
❴
★✵
♥✡☛ ✚✡☛ ✔
✬
✬
❫
✪❉✡✡❢✏❊❊
a b
❭
✬
✬
a ( a) 0 ,
✪✡✩✑
✓✧✒✒☛
❏ ❡ ❤
❩
★✩✔
❬
❬
a b = 0
✕
❵
✓❑❉✡✓✏✎✡☛ ✔
❛✎✡☛✓
✔ ❞
sin
✮☛ ✔
✛✼❉✡✮
4.
✛✼☛✿✡✙✡ ❆ ✪✡✩✑ ❇ ♦☛ ✜ ✒✔♥✯✡❢ ✮☛ ✔ ✛✑❑✛✑ ✢✔✴♦✏❊ ✮✡✻✡❞ ✒✓♥☞✡✡☛ ✔
✓✢✺
✏✡☛
2
(
✪✡♦❂ ✜✓✏
☞✡✌ ✍ ✎
★✡☛
✧✡✏✡
✈✳❙❥
kˆ
✏❉✡✡
✓❝✏✵✎
★✩❣
a b | a || b |
❫
❃❄❅❆q
),
❵
★✮
✛✡✏☛
★✩✔
✣
❫
ˆj
❫
ˆj
kˆ kˆ 0
❫
✛✼✡②✏
★✡☛ ✏✡
❵
kˆ, ˆj kˆ iˆ, kˆ iˆ
❫
✒✓♥☞✡ ✗✘✙✡✚✛✜✢ ❞✵ ✒★✡✎✏✡ ✒☛ ♥✡☛ ✒✓♥☞✡✡☛ ✔
✮☛ ✔
♦☛ ✜
✬
❵
❵
❫
❵
ˆj
✤
✤
✭✛
iˆ, ˆj
✓❞
iˆ iˆ =
iˆ ˆj =
❏ ✓✚✇✚✓✢✓❨✡✏
❏
✬
✎✓♥
♠ ♣
=0
✮☛ ✔
✓❑❉✡✓✏
✮✡✚
❏ ❞✡
3.
5.
✤
✤
b
✬
✬
a
a
✢✵✓✧✺
a
t✉❯❯
b
⑧⑨⑩❶ ❷❸❹ ❺❻❼❽❿
♦☛ ✜ ✴✵✶ ❞✡ ❞✡☛✙✡
★✩
④
④
|a b |
sin =
| a || b |
③
❏
6.
④
④
✬
✬
♦✡❑✏♦ ✮☛ ✔
✬
✗✘✙✡✚✛✜✢ ⑥✮ ✓♦✓✚✮✎ ✚★✵✔ ★✡☛ ✏✡ ★✩ ❛✎✡☛✔✓ ❞
✬
a b | a || b | sin nˆ ,
❫
❵
✫
✬
✧★✡✰
✬
a, b
✈✳❙❥
nˆ
✬
✕
✕
✎★ ✒♦❢ ♥✡ ✒⑤✎ ★✩ ✓❞ ✒✓♥☞✡
a b =
▲
❴
✬
b a
❫
✺❞ ♥✓✿✡✙✡✡♦✏✵❢ ✛❀✓✏ ❞✡☛ ✓✚⑦✮✏ ❞✑✏☛
465
❧ ✁✂✄ ☎✆✝✞ ✟✄✠
✒
✒
❣✡ ☛
☞✌✍✍✎ ✏ ⑦
✒
✒
, a
✑
✓✔
b
❞✕
✒
b a | a || b | sin nˆ1 ,
✰
✱
✲
✒
☞✌✍✍✎ ✏⑦
✑✻
✏✖✗✘
✙✚✕✛
✒
✒
✒
b, a
✭❣✍t
✚✜
nˆ1
✈✳✴
❙
❣✍✢✏✍
✵❞
❣✡ ✣
☞✍✤✥ ✘✦✏
✶✦✷✍✸✍✍✤✏✕✎
✒
b
a
✓✔
❞✕
☞✍✢ ✖
✙✚✕✛
✚✜
❣✍✢ ✏✍
❣✡
✗✹✦✏
i
✧★✩✪✫✬ ✮
❞✍✢
✦✺✦✜✎ ✏
✭✯✦❞
❞✖✏✢
❣✡☛
ii
☞✍✤✥✘✦✏
✧★✩✪✫✬
✮✣
✼✽✾✿❀❁❂ ❃❄❅❆❇
✒
✒
☞✏❈ ✛✦✶
✶✍✢ ✺✍✢ ☛ ❞✍♥❊✍
●✢
✺✕✙✢
✤✢ ✘
✏❉
✏✖✗■✘
❞✕
✦✶❏❑
❣✍✢ ♥✍
☞✌✍✍✎ ✏⑦
n̂
nˆ1
P
P
✗◆✢ ✷✍✸✍
✐
☞✍✡✖
▲ ▼
●✢
❣✕ ❞✍♥❊✍ ✤✢ ✘ ✏❉ ✜✢☛ ❣✡ ☛ ✏✍✢
■
❍✗✖ ❞✕ ✏✖✗ ✘ ✦✶❏❑
❣✍✢♥✍
nˆ
☞✍✡✖
✈✳✴
❙
n̂1
nˆ1
❞✍♥❊✍
nˆ
✒
✒
✲
❘
✤✢ ✘
●☛✶❚✍✎
✒
◗
✒
✒
| a || b | sin nˆ1
✲
✱ ◗
✒
b a
✰
✜✢ ☛
kˆ, kˆ
ˆj
iˆ
✈✳✴
❙
iˆ kˆ
ˆj
❯❱
❲
✒
✒
a
✛✦✶
❞✍♥❊✍
❖
ˆj iˆ
8.
✶✍✢ ✺✍✢ ☛ ✵❞
a b = | a || b | sin nˆ
=
7.
b
✈✳✴
❙
✗✖ ❉☛✯ ❣✍✢☛♥✢ ✗✖☛✏❋
✗◆❞✍✖
❜●
a
❣✜ ✛❣ ✜✍✺ ❉✢✏✢ ❣✡☛ ✦❞
b
✈✳✴
❙
1
|a b |
2
✦❢✍❚✍❋✭
❞✕
●☛❉❳✺
❚✍❋✭✍☞✍✢☛
❞✍✢
✦✺❨✦✗✏
❞✖✏✢
❣✡☛
✏✍✢
✦❢✍❚✍❋✭
❞✍
✷✍✢❢✍✗✘❉
❬
❬
❩
✤✢ ✘
❨✗
✜✢☛
✗◆✍♦✏
❣✍✢ ✏✍
❣✡ ✣
✦❢✍❚✍❋✭ ✤✢✘ ✷✍✢❢✍✗✘❉ ❞✕ ✗✦✖❚✍✍❏✍✍ ✤✢✘ ☞✺❋●✍✖ ❣✜ ☞✍✤✥✘✦✏ ✧★✩✪❘
●✢
✗✍✏✢
❣✡☛
✦❞
✦❢✍❚✍❋✭
ABC
❞✍
✒
✗✖☛✏❋
AB | b | (
☞✏❈
✦❢✍❚✍❋✭
✦✶✛✍
✱
❣❋☞✍
❣✡
)
=
✷✍✢❢✍✗✘❉
9.
CD = | a | sin
a
✷✍✢❢✍✗✘❉
=
1
| b || a | sin
2
✑
1
|a b |
2
❫
❫
❪
❫
✒
✒
✛✦✶
❞✍
✼✽✾✿ ❀❁❂ ❃❄❅❆❴
❭
☞✍✡ ✖
❫
ABC
1
AB CD .
2
✈✳✴
❙
b
✒
✒
✷✍✢ ❢✍✗✘❉
✎
❞✍
●✜✍☛ ✏✖ ✙✏❋❚✍✎ ❋✭ ❞✕ ●☛❉ ❳✺ ❚✍❋✭✍☞✍✢ ☛ ❞✍✢ ✦✺❨✦✗✏ ❞✖✏✢ ❣✡☛ ✏✍✢ ●✜✍☛✏✖ ✙✏❋❚ ✍❋✭
|a b |
✰
✤✢✘
❨✗
✜✢☛
✗◆✍♦✏
❣✍✢ ✏✍
❣✡ ✣
466
① ✁✂✄
✈☎✆✝✞✟✠
♣✠✘ ✙☎✕ ✚✛
✡☛☞✌✍
✎✏
ABCD
✑✒
✖☎
✓☎✠✏
✑✔✕
✟✖
✎✒☎✕ ✠✗
= AB . DE.
❞☎✏ ✜☎✓✞✢
✤
AB | b | (
✓ ✗ ✕ ✠✘
✣
✟❢✥☎
✑✘ ✈☎
✑✔
),
✈☎✔✗
★
DE | a | sin
✦
✎ ✒☎ ✕ ✠✗
✤
✵✶ ✷ ✸✹✺✻
✤
✑✒
ABCD
✤
★
★
✥✟❢
✤
★
✟❢❀
✢r✟✛❀ ❢☎✏
★
★
★
✑✘ ❀
❖❍✸■❍✸
✑✔✕
✎✟❢✬☎
❋✯✖☎
✠✫
✑✒
✓☎✠✏
a
✓✗
✭✘ ✮☎☎✏✕
✟✆✠✗✮☎
✖☎✏
✟✯✥✒
✈✟✙☎✲✥✳✠
✖✗✏✭
✕ ✏✴
(Distributivity of vector product
✠r✯
✎✟❢✬☎
✑✔✕
✈☎✔✗
✿ ❀✖
✑✔
✠☎✏
★
★
★
a ( b)
❃
b
❄❅❆
❙
✎✟❢✬☎
✈✟❢✬☎
a c
❃
✒☎✯
c
✼✽✾❥
❂
✤
✒✑✰✆✓✱ ✮☎✚
✥☎✏✭✓✞✢
(a b ) = ( a ) b
(ii)
❢☎✏
★
a, b
★
★
✖☎
✆✏✞
★
(i) a ( b c ) = a b
❁
=
❞☎ ✏ ✜☎ ✓ ✞✢
▲▼◆P◗❘❚ ❯❱❲❳❨
✭✘ ✮☎✯✓✞✢
✭✘ ✮☎✯✓✞✢
over addition)
✖☎
✤
|a b |
✪
✎✟❢✬☎
✎✟❢✬☎
★
✈✠✩
♣✠✘ ✙☎ ✘ ✚✛
| b || a | sin
✈✫
✧
❇☎❈✖
✭✘✮☎✯✓✞✢
❉✓
✒✏ ✕
a1iˆ a2 ˆj a3 kˆ
❂
❊✒✬☎✩
iˆ
ˆj
kˆ
a b = a1
b1
a2
b2
a3
b3
❃
❃
●
❂
⑥☎✗☎
✟❢✥☎
✈☎✔✗
✛☎
b1iˆ b2 ˆj b3 kˆ
❂
✎✖✠☎
❂
✑✔ ✴
✑✔ ✕
★
a b = (a1iˆ a2 ˆj a3 kˆ) (b1iˆ b2 ˆj b3 kˆ)
★
❁
❂
❂
❁
❂
❂
= a1b1 (iˆ iˆ) a1b2 (iˆ ˆj ) a1b3 (iˆ kˆ) a2b1 ( ˆj iˆ)
❁
+ a2b2 ( ˆj
❂
❁
❂
❁
❂
ˆj ) a2b3 ( ˆj kˆ)
❁
❂
❁
+ a3b1 ( kˆ iˆ) a3b2 (kˆ ˆj ) a3b3 (kˆ kˆ)
❁
❂
❁
❂
❁
= a1b2 (iˆ ˆj ) a1b3 (kˆ iˆ) a2b1 (iˆ ˆj )
❁
❁
❁
❑
❁
❑
+ a2b3 ( ˆj kˆ) a3b1 ( kˆ iˆ) a3b2 ( ˆj kˆ)
❁
❂
❁
❑
❁
(
✭✘ ✮☎❏✒✚
✡
✎✏
)
467
❧ ✁✂✄ ☎✆✝✞ ✟✄✠
(
❉❀ ✡❢
s❛ ❞
iˆ iˆ
ˆj
kˆ kˆ
ˆj
0
iˆ kˆ
✈✡❥
❙
kˆ iˆ, ˆj iˆ
iˆ
ˆj
✈✡❙ ❥
kˆ
ˆj
ˆj kˆ)
= a1b2 kˆ a1b3 ˆj a2b1kˆ a2b3iˆ a3b1 ˆj a3b2iˆ
☛
(
♠✌✍✎✏✑✍
22
☞
iˆ
❉❀ ✡❛❢
s ❞
a3b2 )iˆ (a1b3
☛
iˆ
ˆj
kˆ
= a1
b1
a2
b2
a3
b3
✕
☛
☛
✕
ˆj 3kˆ
2iˆ
☞
☛
kˆ, ˆj kˆ iˆ
ˆj
= (a2b3
a
✒✓✔
☛
✈✡❙ ❥
kˆ iˆ
✈✡❙ ❥
a3b1 ) ˆj (a1b2
☞
3iˆ 5 ˆj 2kˆ,
b
ˆj )
a2b1 )kˆ
☛
✕
r✡s
✕
|a b |
✥✖✗
✘✙✓✚✛✜
✎❣ ✒✢✖✣
iˆ ˆj kˆ
a b = 2 1 3
3 5 2
= iˆ( 2 15) ( 4 9) ˆj (10 – 3) kˆ
✦
✦
✤
✧
★
✭
✭
♠✌✍✎✏✑✍
23
a
✚✢✖✣
✪
iˆ
☞
☞
✢❡
✴✖✗✲
✢✰✳
✓✘
✕
✛✘
✮✓✔✯✖✽
✚✖✲
✕
a b
✪
a b
✕
☞
❡✲ ✳
iˆ 2 ˆj 3kˆ
☞
4kˆ
✮✲
✕
a
✈✡❙ ❥
✔✖✲♥✖✲
✴✱
✹✳✺
✢✰✽
✫❋✖✙●❍
❡✖✼✖✘
✹✳ ✺ ✷✗✻
✴✵✶✒✲ ✘
✕
ˆj
b
✓♥✾♥✓✹✓✿✖✗
✕
✕
❃
☛
✕
✫✗✬
✷✲ ✸
☞
❡✖✼✖ ✘
✮✓✔✯✖
✢✰✳✜
☞
|c | =
✫✺
☞
507
iˆ ˆj kˆ
(a b ) ( a b ) = 2 3 4
0 1 2
✕
✪ ☛
2kˆ
✕
✕
✈✡❥
❙
(7) 2
(a b )
2iˆ 3 ˆj
a b
✕
✪
☞
17iˆ 13 ˆj 7 kˆ
✩
✕
✕
ˆj kˆ, b
☞
★
(13) 2
✫✖✰ ✱
✕
✕
✎❣
☞
★
✕
✕
✕
✘✙✓✚✛
☛
(a b )
✮✓✔✯✖
★
( 17) 2
a b =
✫✗✬
★
4 16 4
✩
✩
❁✖✱✖
✴✵✔ ❂✖
2iˆ 4 ˆj 2kˆ ( c,
❄
❊
24
❊
2 6
✮✓✔✯✖
■
c
=
|c |
■
1ˆ
i
6
❏
❑
2 ˆ
j
6
✢✰
❏
1 ˆ
k
6
✢✰✜
❅❦❆ ②❤❇t❈
)
✥✖✗
468
① ✁✂✄
✧
✧
❡✔✩✔☞
1 ˆ
i
6
✌☛✓✚✔
24
C(2, 3, 1)
♠✯✠✰✱✟ ✠
✥✔✣✒
2 ˆ
j
6
☛✩✔✲✔✪ ✳
✛☞
1 ˆ
k
6
☞✔
✢❡
✑ ✔ ✎✕
1
| AB AC |
2
✽✽✽
✾
✶✔✎
✑✒✖✎✪ ✫✢
✬
✢✣ ✖
☛☞
✧
✻✻✻
2kˆ
ˆj
AB
☞✔ ✛☞
✮
✹✗✓✪
✑☛✒❞✔✔❡
✢✣✤
A(1, 1, 1), B(1, 2, 3)
AC iˆ
✈★❥
❙
2 ˆj .
☛ ✓✛
✢✪✛
☛ ✩✔✲ ✔✪ ✳
☞✔
✴✔✕ ✩ ✔✑ ✵✏
3iˆ
ˆj 4kˆ
✢✣✤
iˆ ˆj kˆ
AB AC = 0 1 2
1 2 0
✥✗
✧
✻✻✻
✭
✧
✻✻✻
25
♠✯✠✰✱✟✠
❈
16 4 1
✭
✥✲✔✍✷❄
✴✔✕ ✩✔✑✵✏
1
21
2
✿ ❀
✌❡✔✖✎✒
b
iˆ
ˆj kˆ
✰✺
☛☞✌✍
✌❡✔✖✎✒
❂
❂
❀
21
❃
✢✣✤
⑥✔✒✔
✸
r✎✪✲✔✪✳
☞✔
✴✔✕✩✔✑✵✏
✓✍
❣❜✸
✢✣ ✤
✖
r✎✪✲✔✪✸ ✳
☞✍
✌✖✏ ❆❇
✶✔✎
☞✍☛✳✛
✲ ✔✪ ✳✔✛ ✜
☛✳✌☞✍
✌✖✏❆❇
✲✔✪✳✔✛✜
a
b
❊❋●
❍
iˆ
❉
a b = 3
1
✥✗
❑
✎✔✕
5iˆ
▼
❅✌☞ ✔
✭
✥✔✘✚✫☞
✴✔✕ ✩✔✑✵✏
42
kˆ
1 4
1 1
▲
ˆj 4kˆ
◆
✧
✧
|a b | =
❜✌☛✏✛
ˆj
✢✣✤
25 1 16
❂
❂
✧
✧
✢✣✖
✢✣ ✤
❉
a
❉
❉
✑■ ☞✔✒
❁
✧
❅✌
✥✔✣ ✒
✢✔✕ ✎✔
4iˆ 2 ˆj kˆ
✧
✻✻✻
| AB AC | =
❜✌☛✏✛
❜✌
✑✒ ✓✐✌✒✔ ✏✖✗✘✎✙
✢✣✤
✖
✧
✻✻✻
✑■ ✔❏✎
✭
✚✔✍✷✔✸
✧
✧
✧
(a b ) ( a b )
☛✳✌✘✕ ✵
☞✍☛✳✛
✧
a b
✈★❥
❙
✽✽✽
✾
✼
✥✎✦
✧
✧
✢✔✕❣✔✤
✴✔✕ ✩✔✑✵✏
✧
✻✻✻
✰✺
a b
☛☞✌✍ ✎✏ ✑✒ ✓✔✕ ✏✖✗✘✎✙ ☛✓✚✔✔✛✜ ✢✔✕✎✍ ✢✣✤
✖ ✥✎✦
❢✆✝✞✟✠✡
☎
❃
42
◆
✴✔✕✩✔✑ ✵✏
|a b |
✭
⑥✔✒ ✔
469
❧ ✁✂✄ ☎✆✝✞ ✟✄✠
✐✡ ☛☞✌✍✎✏
1.
2.
iˆ 7 ˆj 7 kˆ
✕
a
❀ ✖♥
✕
✕
✕
✕
✕
|a b |
r❦s
❑✗✘
✙✚✛✜✢✣
✕
✕
a b
a b
b iˆ 2 ˆj 2kˆ
✤✛✥✦✗
3iˆ 2 ˆj 2kˆ
b
✈❦❙❥
✑✒✓✔
✈❦❥
❙
✕
3iˆ 2 ˆj 2kˆ
a
✙✚ ❞✧★ ✛✥✦✗✗ ✩✪✧ ✩✗✫✗✙ ✤✛✥✦✗ ❑✗✘ ✙✚✛✜✢ ✜✬✗✭
✕
✮✗✯✰
3.
✬✯✣
✱✛✥
✢✙
✩✗✫✗✙
✲
a , iˆ ♦✳✴ ✵✶❋✶
✤✛✥✦✗
, ˆj
3
♦✴✳ ✵✶❋✶
✷✶✸✹
4
kˆ
✺✪✻
✤✗✼✗
✺✪✻
❅✗❆✙
✢✙
✽✱✾ ✿
✙✗✪ ❁✗
❂
❇✗✚
❑✗✘
✙✚✛✜✢✣
✕
★✿✗✘✗
✬✯
✘✗✪
❃ ✙✗
✕
✕
✥✦✗✗❈ ❄✢
✙✚✛✜✢
✮✗✯✰
✕
✕
5.
❍ ✮✗✯✰ ➭ ❑✗✘
6.
✛✥✱✗
❉
✛✙
❊
●
✙✚✛✜✢■
✱✛✥
✤✙✘✪
(2iˆ 6 ˆj 27 kˆ) (iˆ
●
●
a b
❖
✱✛✥
a
10.
0
✙✚
a
✕
kˆ) 0
● ▲
◗
◗
a
✤✛✥✦✗
▼
b
❘❚❯
❱
✺✪✻
★✗✰✪ ✩✪ ✧
✕
✮✗❲
●
Ø✩✦✗❩
✬✯✧
✈❪❦❫❦
❲❢✛ ❨❆
✘★
✥✦✗✗❈ ❄✢
❭
❭
b
0
a (b
❊
✛✙
◗
◗
✘★
●
✕
●
✕
●
✕
✕
●
✕
●
●
⑥✗✰✗
▼
❊
●
✕
❊
◗
a b
❴
✛✿♣✗❈✛ ✰✘
❵
0
✬✗✪✘ ✗ ✬✯✣ ❳✱✗ ✛✺❞✗✪✩
✤❣✱ ✬✯❛ ❜✥✗✬✰❁✗ ✤✛✬✘ ✮❲✿✪
A(1, 1, 2), B(2, 3, 5)
❞✚✛✜✢
a
✤✛✥✦✗
✈❦❥
❙
b
✤✛✥✦✗
(A)
/6
✢✙
– iˆ
③
❲q✙✗✰
✺✪ ✻
✦✗✚❨✗✗✪ ④
✬✯✧
⑦
⑦
|a| 3
✛✙
a
b
(B) /4
A, B, C
✈❦❥
❙
✺✪✻
★✚❡
✙✗
✙✗✪ ❁✗
(C)
✮✗✯✰
D
✛✜✿✺✪✻
1ˆ
1ˆ
1ˆ
j 4kˆ, iˆ
j 4kˆ , iˆ
j 4kˆ
2
2
2
1
2
(C) 2
(A)
✱✛✥
▼
2
,
3
t
t
❄✤
③
✮✗✱✘
⑦
✬✯
a
iˆ
❉
ˆj 3kˆ
●
⑦
✬✯✣
✧
✮✗✯✰
✉✇①
②
|b |
✕
✕
✘★
a b
❊
✢✙
✕
✕
✩✗✫✗✙
C (1, 5, 5)
✮✗✯✰
✬✯✧✣
✕
✕
✩✗✿
✺✪✻
✙✚✛✜✢✣
2iˆ 7 ˆj kˆ
❉
●
c) a b a c
✕
▼
❳✱✗ ✛✿❨✙❨✗❈
a1iˆ a2 ˆj a3 kˆ, b1iˆ b2 ˆj b3 kˆ, c1iˆ c2 ˆj c3 kˆ
✕
a, b , c
✢✙ ✤✩✗✧✘✰ ❡✘❢❇✗❢❈✜ ✙✗ ❝✗✪✫✗❲✻❞ ❑✗✘ ✙✚✛✜✢ ✛✜✤✙✚ ✤✧❞❤✿ ❇✗❢✜✗✢✭ ✤✛✥✦✗
b
12.
● ❏
◆
0.
P
✢✙ ✛✫✗❇✗❢✜ ✙✗ ❝✗✪✫ ✗❲✻❞ ❑✗✘ ✙✚✛✜✢ ✛✜✤✺✪✻ ✦✗✚❨✗❈
✕
11.
✬❢✢
❭
❭
❜♠✗✰
9.
✛✥✢
ˆj
❊
◆
◆
✮✗✯✰
✕
8.
✤✪
✬✯✧
✕
✩✪✧
✤✬✗✱✘✗
❊
0
a b
?
✩✗✿ ❞✚✛✜✢ ✤✛✥✦✗
❬❲
❄✤✙✚
✕
✕
◆
◆
✬❢✮ ✗ ✬✯ ✛✙
✛✿✙✗❞
❑✗✘
(a b ) ( a b ) = 2( a b )
4.
7.
✩✗✿
⑧
⑨
③
✬✯❩
/3
(D)
✛⑤✼✗✛✘
✮✗✯✰
(B) 1
(D) 4
✤✛✥✦✗
– iˆ
⑩
③
/2
Ø✩✦✗❩
1ˆ
j 4kˆ ,
2
⑦
✬✯✧
✙✗
❝✗✪✫ ✗❲✻❞
✬✯❩
470
① ✁✂✄
❢☎❢☎✆
♠☞✌✍✎✏✌
26 XY-
r✑
✒✓✔
✕✖✗✘
✧
✪
✍❣
✒✗❡
✮✗✯✰✱✛r
✑✘✛✦✤
✯✓ ✱
r
✛✚
✮❡✈ ✕✗✹
✫✒
✒✗✙✗✚
✕✛✜✢✗
✝✞✟✠✡☛✟
✛✑✛✣✗✤✥
✧
x i y j , XY-
★
r✑
✩
✺✗r✓
✫✬✔
✛✚
x = cos
✒✓✔
✤✚
✒✗✙✗✚
✕✛✜✢✗
y = sin (
✻ ✮✗✬✹
✻
✫✬
✭✮✗✯✰ ✱✛r
✲✳✴✵✶✷✥
r✸
✽
❉✼✗✓✔ ✛✚
| r | = 1).
❜✕✛✑✤
✫✒
✕✛✜✢✗
✽
r
✚✗✓ ❞
❁❁❁
❂
❂
r
✯✓✱
♦✺
✒✓✔
✛✑✣✗
✕✚r✓
✾❀
OP = cos iˆ sin ˆj
✿
❃
❄
... (1)
❃
✫✬✔ ✥
❈
|r | =
▲✺❅❆r❇
cos 2
❊ ❋
sin 2
1
❊ ❀
●❍■❏ ❑▼◆ ❖P◗❘❙
✦✬ ✕✓t✦✬ ✕✓ ✻❚
2
2
x +y =1
r✑
✒✓✔
✒✗✙✗✚
27
3iˆ 2 ˆj 3kˆ
✛❡❦✒❡
✕✓
2 ,
❯
r✚
✺✛✹✯✛r❱r
✫✗✓r ✗
✫✬
❲✸✜✈
P(
✮✗✯✰✱✛r
✲✳✴✵✶
)
✯✗✒✗✯r❱
✚✘✛✦✤
✼✛✜
❵❛❝❤
✛✚
✕✛✜✢✗
✺❬ ✗❪r
✫✗✓r✗
A, B, C
❲✸✜✈ ✮✗✓✔
iˆ 6 ˆj kˆ
AB
CD
✫✬ ❞ r✗✓
✮✗✬ ✹
✖✗✘
❡✗✓ ❆
✚✗✓ ❳✗
✚✘✛✦✤
✛✚ ✼✛✜
❧
, AB
D,
✮✗✬ ✹
✯✓✱
✕✹✑ ✹✓✣ ✗✗✮✗✓✔
✕✔ ✹✣
✓ ✗
✮✗✬✹
✛▲❫✗✛r
AB
✕✛✜✢✗
r❫✗✗
iˆ
❴✒✢✗❇
✯✰r
XY-
CD
✯✓ ✱
✸✘✐
✚✗
ˆj kˆ, 2iˆ 5 ˆj ,
✚✗✓ ❳✗ ❥✗r
✚✘✛✦✤✥
✫✬✔ ✥
CD,
✯✓ ✱
✸✘✐
✚✗
✚✗✓ ❳✗
✫✬
r✗✓
❧♥
AB
♣♣♣
q
s✉✇
②
CD
✯✓ ✱
✸✘✐
✫✬ ✥
③③③
④
✮✸
✒✓✔
✫✬ ✥
♣♣♣
q
✍❣
✛✜✢✗✗
✚✗ ✮❡✈✹✓✣ ✗❳✗ ✚✹r✗ ✫✬ ✮✗✬✹ ❜✕✒✔✓ ✕✖✗✘ ✕✔ ✖✗✗✛✯r ✛✜✢✗✗✤❨ ✕✛❩✒✛✑r ✫✬✔ ✥ ✮r❇ ✭✲✷ ✕✓
✺❬❭✼✓ ✚
♠☞✌✍✎✏✌
0
AB = B
✚✗
✛▲❫✗✛r
= (2iˆ 5 ˆj ) (iˆ
❄
⑤
–A
ˆj kˆ) iˆ 4 ˆj kˆ
✕✛✜✢✗
❄
❄
✚✗
⑥
❄
✛▲❫✗✛r
⑤
✕✛✜✢✗
✚✗
471
❧ ✁✂✄ ☎✆✝✞ ✟✄✠
✍✍✍
✎
CD = 2iˆ 8 ˆj 2kˆ
✔✕✖✗✘
(4) 2
( 1) 2
✏
✑
✒
3 2
|CD | 6 2
✈✛❙✜
✦✦✦
✧
✦✦✦
✧
AB . CD
cos =
|AB||CD|
✢✣✤
✦✦✦
✧ ✦✦✦
✧
✥
1( 2) 4( 8) ( 1)(2)
★
=
❉✫✗✬✭ ☛✖
♦✬ ✸
✏
✍✍✍
✎
✙✙✙
✚
❜✡✓
(1) 2
| AB | =
❜✡☛☞✌
✡✭✘✬✹ ✗
0
✮ ✥ ✮ ✯
✔✕✗✰✣
❜✡✡✬
✱✗✬✣ ✗
✱✲
★
✩
1
CD
2
❂❂❂
❃
★
✪
(3 2)(6 2)
= .
✳
☛✖
✴
✫✱
❀✵✗✗✶✣ ✗
✱✲
AB
☛✖
❂❂❂
❃
AB
❜●❡✬✭
28
✎
✎
❡✗●
✡✖✣✬
AB
☛✖
CD
✢✗✲ ✘
✡✭✘✬ ✹✗
✎
a, b
☞✓☛❍✌
✖✱
❜✡✡✬
✡✬
✔✕■✫✬✖ ❪
✢❏✫
❀✗✬
c
✈✛❙✜
☛❀✫✗
✱◗✢ ✗
✱✲
✡☛❀✵✗✗✬✭
✎
✎
❈❣
1
✪ ★
✣r✗✗
CD
✌✖
❀✷✡✘✬
❄❄❄
❅
✍✍✍
✎
❪
✡☛❀✵✗
a (b
☛✖
❘
✡☛❀✵✗
✫✗✬❑✔✸☞
✎
✎
✏
✔✕✖ ✗✘
❜✡
✔✘
✎
✎
✎
✏
✱✲✭
✎
✒
❖
✎
c |2 = (a b
❚
❚
❚
❱
❚
❚
❯
❚
❚
P✗✣
✖✓☛❍✌✺
0
c)
❚
❚
❱
❖
◆
✎
c ) (a b
= a a a (b
❯
◆
✒
✎
✎
✎
✒
|a b c|
✣✗✬ ❪
✎
| a | 3, | b | 4, | c | 5
☛✖
0, c (a b )
❚
❚
✱✭ ✲
✎
a)
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✎
✏
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c ) = 0, b (c
|a b
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♦✬✸
✱✲✭ ✺
✎
✣✓●
❖
✢✗✲ ✘
36
36
★
✱✲✭ ✺
❢✻✼✽✾✿❁
♠❆❇❈❊❋❇
,
✩
❚
❚
❚
❚
c ) b b b (a c )
❱
❯
❱
❯
❱
❚ ❚
+ c .(a b ) c .c
❱
❱
✎
✎
✎
= | a |2 | b |2 | c |2
= 9 + 16 + 25 = 50
✏
✎
✎
♠❆❇ ❈❊❋❇
✏
29
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| a | 1, | b | 4
❉✫✗✬✭ ☛✖
a b
❴
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c
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0,
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✎
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c
5 2
❅
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a b
✔ ✕☛✣▲✭✐
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✎
✎
✎
❅
❅
❳
✎
a b b c
❬✛s ✜✛❭❫✛
❜✡☛☞✌
❤❛
✎
a a
❘
❦
❥
✎
✎
✏
✱❡
❦
❦
❥
✔✗✣✬
c
❨
0
✖ ✗✬
✡✭ ✣ ◗❞❩
✖ ✘✣✬
✱✲✭✺
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✎
c a
✖✗
❡✗●
P✗✣
✖✓☛❍✌✺
✱✲✭
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c =0
❝
✎
✎
a b a c =0
❘
✏
✎
✎
❘
✎
✎
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❘
❜✡☛☞✌
②
✔◗ ●✤
❲
✚
✚
❴
50
✎
a a b
✢r✗♦✗
c| =
✎
✈✛❙✜
✚
✚
❈❣
a, b
✡☛❀✵✗
✏
✎
✎
✎
✎
✎
|a b
❜✡☛☞✌
✏
✏
❘
②
②
②
q
♥
b a b c =0
✇t
①
①
✉
a
2
♣ ♥
1
... (1)
472
① ✁✂✄
✠
✠
✞
✭✖✗✘
✕
✓
✈✆✚✒
✭✙✗
✭✛✗
✑✆✜
✢✆✜ ✣✥
✤ ✜
✠
✠
✠
✞
✕
a c
✎✏✑✆✒
✔
2
b
✡
☛ ✡
16
... (2)
✕
b c =–4
... (3)
✓
✎✒
✦✧
✠
✠
☞
✠
✟
✕
❜✌✍
✠
a b b c =
✈☎✆✝✆
✎✆★✜
✦✚✩
✪✑
✠
2 ( a b b c a c ) = – 21
✞
✟
✞
✟
✞
2 = – 21, i.e.,
❀✆
➭
♠✬✮✯✰✱✮
✠
✲✳
3iˆ
❀✪✴
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✶✩ ✷✝★✸
✎✒✵✎✒
ˆj – 3kˆ ,
2iˆ
❊
❉
✝✜ ❂
✌✧✆✩★ ✒
✈✆✚✒
✦✚
2
✧✆✥
✶✍✪✢❇
✪✑
❍
1
✠
✈✷
2
2
,
❇✑
,
✠
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❃
1
✠
2.
3.
P (x1, y1, z1)
r✆★
✑✍✪✢❇●
❏
✶✩ ✷
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✌✆✎✜ ❞✆
♦✎
✈✪❄✆❅❀❆★
✧✜✩
✑✍✪✢❇
✢✦✆❈
1✘
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▼
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■
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2
=
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i
j
2
2
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✈✆✚ ✒
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✈✆✚ ✒
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❭❪
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1ˆ
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2
❫❴❫❴❵
✪✴✺✆✆ ✝✜ ❂
Q (x2, y2, z2)
✶✣✤✑ ✍ ✎✪✺s✧ ✪✴✺✆✆
km
✪✴✺✆✆ ✧✜✩ ✛
✝✜ ❂
3ˆ
j – 3kˆ
2
❚
❭❛❝❡❨❴❢❤
✝✆✧✆✝★✿ ✪✴✺✆✆
✌✆☎✆
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✑✆ ✑✆✜ ✾✆
✷✥✆✥✜ ✝✆✶✆
✪✶✪❧✆❇●
➥✷✴♥
❇✑
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✎✒
✈❞✆ ✑✍ ✐✥✆❦✧✑
✌✪✴✺✆
✎❁✪★
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▲
x-
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✴✪❞✆✾✆✆✝★✍✿
✠
✝✜ ❂
2
= (2 3 )iˆ (1
❱❲❳❨❳
1. XY-
✑✍
✠
1
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=
P
✠
kˆ,
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✠
1
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iˆ, ˆj
❃ ✑✆✜
3(2 3 ) (1
✈☎✆✆✿★✸
✠
✶✩ ✷✝★✸
■ ❃ ❑❃
❊
❆❀✆✜✩✪✑
✠
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2
✫
✌✪✴✺✆✆✜✩
✝✜ ❂
✠
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✧✆✹✆✑
=
❋
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❍
✯❣
➭
✧✜✩ t
km
✑✆✜
✪✧✶✆✥✜
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✌✪✴✺✆
✝✜❂
✈✪✴✺✆
s✶★✍ ✦✚● ✉✌✝✜❂ ✎✺s✆★✸ ✝✦ ✉✇✆✒
♣✆q✑
✌✜
✈✆✚✒
30°
✎✪✒✧✆✾✆
✎✪✺s✧ ✑✍
s✶★✍ ✦✚ ✈✆✚ ✒ ♦✑ ✢✆★✍ ✦✚● ✎✏✵ ☎✆✆✥ ✝✜ ❂ ✎✏✆ ✒✩✪❄✆✑ ➥✷✴♥ ✌✜ ✶✣✤✑ ✍ ✑✆ ✪✝✵☎✆✆✎✥
r✆★ ✑✍✪✢❇●
4.
5. x
6.
④
④
a
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✑✆
②
b
✝✦
✧✆✥
⑥
✌✪✴✺✆✆✜✩
r✆★
c,
a
✑✍✪✢❇
✠
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④
③
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✎✪✒✧✆✾✆
⑦
b
✪✶❇
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❜✑✆❜✿
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✠
| a | |b | | c |?
x(iˆ ˆj kˆ)
✟
kˆ
✟
✝✜ ❂
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473
❧ ✁✂✄ ☎✆✝✞ ✟✄✠
7.
8.
☞
✙✚
☛✒✎✎♥✥✙
✡✚
10.
11.
♦✎✮✏
✚✎
✦✧☛★
✙✚
✖✎✛✎✚
☛✒✎✎♥✥✙
✡✚
OX, OY
3
a
✑✡☛✒✎
1
,
✜✎r
iˆ
✑✡☛✒✎
✑✡☛✒✎
✖✎✛✎✚
❊
❊
♦✏ ✕
a
✑✏
16.
19
☞
●❍✌✌
17.
✯▲✒✭✎✏✗
c
✡✚
✖✏✗
a
<
✮✢✡✣✙
a
✖✎✛✎✚
(A)
✯✘
✑✡☛✒✎
❭
❪ ❫
✮✗✧
❉★❣✎✭✯✕✮
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4
⑥✎✘✎
♦✏ ✕
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■
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✚✢
✡☛♦✽ ✕✾✚✎✏ ✑✎✥✭
2iˆ
ˆj
4kˆ .
✙✚ ✙✏ ✑✎ ✑✡☛✒✎
✔
c d
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✜✎r
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a
☞
❑
✑✬✢
☞
0, b
▼◆✎✘
❏
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b
♦✏ ✕♦✮
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b
❀✡☛
✚✎
✸❀✭
✚✢✡✣✙✤
✧✢✸
✚✎
✚✎✏❣✎ ❘ ✬✪
✖✎✛✎✚
r✎✏
✮✗✧ ♦r✽
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❭
❪ ❫
✬✪✤
✗
✔
a b
❂
0
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✚✎
✚✎✏ ❣✎
❬ ✬✪
(D)
❪ ❫
❚
❯
(B) 0
❳ ❲ ❳
(D) 0
❩ ❘ ❩ ❨
✬✪ ✗
▼✭♦✏ ✕
2
✩✎✪✘
✧✢✸
❀✡☛✱
(B)
c
☞
a, b
❁
❁
☛✎✏
❁
❋
0
✔
♦✏ ✕
❋
☞
❑
✔
❖P❙◗
a b
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✖✏ ✗
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✪
✥✑♦✏ ✕ ✡♦✚❣✎♥ ♦✏ ✕
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✧✘✎✧✘
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2
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❞✎✢
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☛✎✏✭✎✏ ✗
❯
❱ ❲ ❱
(C) 0 <
❀✬
iˆ 2 ˆj 3kˆ
✈✌❥
✍
✧✘✎✧✘
7 kˆ
3iˆ 2 ˆj
❆
✩✡☛✒✎
❆
✬✪
✑✡☛✒✎✎✏ ✗
(A) 0
✑✎✴✎
(a b ) (a b ) | a |2
✡✚
✬★✩✎
♦✏ ✕
☛✎✏
✚✢✡✣✙✤
✻✎✏ ✛✎✯✕✮
✥✑✚✎
2iˆ 4 ˆj 5kˆ
✑✡☛✒✎✎✏✗
✔
❀✡☛
B
✩✎✪ ✘
❁
✚✢✡✣✙
r✚
✜✎r
2iˆ 4 ˆj 5kˆ
✩✻✎✎✏ ✗ ♦✏ ✕
b
✈✌✍❥
✑✎✴✎
☞
☞
✡☛❀✎
✑✡☛✒✎
☞
☞
❀✬
16
OZ
✙♦✗
☞
✡✑❢
✬✪
✑✖✎✭ ✯✡✘✖✎❣✎✎✏✗ ♦✎✮✏ ✯✘✳✯✘ ✮✗✧ ♦r✽ ✑✡☛✒✎ ✬✪ ✗ r✎✏ ☛✒✎✎♥✥✙ ✡✚ ✑✡☛✒✎
a, b
✑✡☛✒✎✎✏ ✗
✑✗✘✏ ✫✎
❊
a, b , c
❀✡☛
C (11, 3, 7)
✩✎✪✘
✬✪✤
✚✎❅
❄
♦✏✕
1
,
✣✎✏
ˆj kˆ
❄
3c
✚✎✏ ✡✖✮✎✭✏ ♦✎✮✢ ✘✏ ✫✎✎ ✚✎✏ ✰✱ ✷ ♦✏ ✕ ✩✭★✯✎r ✖✏ ✧✎✲
✚✢✡✣✙✤
☞
✚✢✡✣✙
✔
✓
✬✪✤
☞
✜✎r
2a – b
✚✢✡✣✙✤
✡✳✴✎✡r
✚✎
3 3
ˆi 4 ˆj 2kˆ, b
☞
✖✎✭ ✮✢✡✣✙
d
15.
R
✔
✔
r✎✏ ✑✡☛✒✎
☞
☞
✖✶❀
❁
14.
✜✎r
Q (a – 3b )
✈✌❥
✍
✦✧☛★
✑✖✎✗r✘
1
13.
✩✭★✯✎r
✙✚ ✑✖✎✗r✘ ✸r★❞✎★♥✣ ✚✢ ✑✗✮✹✭ ❞✎★✣✎✙✺
✚✎✏ ✿❀✎✙✺
12.
b)
kˆ ,
iˆ 2 ˆj
✚✢✡✣✙✤
☞
☞
RQ
✘✏ ✫✎✎✫✎✗✵
♦✎✮✎
✚✘✭✏
✚✘✭✏
✜✎r
☞
c
✈✌✍❥
A (1, – 2, – 8), B (5, 0, – 2)
✦✧☛★
P (2a
☛✎✏ ✦✧☛★✩✎✏ ✗
ˆj 3kˆ
2iˆ
✑✡☛✒✎
✖✎✛✎✚
✡♦❞✎✎✡✣r
✡♦❞✎✎✡✣r
☞
kˆ , b
ˆj
✑✖✎✗r✘
✚✎✏
9.
iˆ
a
❀✡☛
3
(C)
❭
❪ ❫
2
2
3
❭
r✎✏
a b
❋
✙✚
474
① ✁✂✄
18. iˆ.( ˆj kˆ)
(A) 0
☎
19.
ˆj.(iˆ kˆ) kˆ.(iˆ ˆj )
(B) –1
✆
☎
✆
☞✝✌
a
✍☛☞✎✝✝✌ ✏
❞✝
✞✝✟
✠✡
(C) 1
(D) 3
✑
✑
❀☛☞
☎
✈✒❙❥
(A) 0
✜
✜
b
♦✌ ✓
✔✕✖
❞✝✌✗ ✝
❞✝
✣
(B)
✘ ✠✡
4
✙
✣
(C)
✜
✜
|a b | |a b |
❣✝✌
✚
(D)
2
t✔
✛
✘ ✔❝✝✔❝
✠✡ ✢
✤
❧✥✦✧✥ ★✥
✩
✱❞
x2
✩
✮
z2
☛✬✭✝☛❣
❞✕
✱❞
✲❜ ✼✝✌ ✲
☛✟✽☛✲❣
❞✝✌
✍☛☞✎✝
☛✟❢✟☛✹☛✾✝❣
✜
OP( r )
✍☛☞✎✝
✚
xiˆ
✚
✮
yjˆ
✮
zkˆ
✠✡
✰✝✡ ❝
✲☛❝✞✝✗✝
✠✡ ✳
✱❞ ✍☛☞✎✝ ♦✌✓ ✰☛☞✎✝ ✴✝✵❞
✶✍♦✌ ✓
✩
y2
✮
✯✯✯
✜
P(x, y, z)
✪✔☞✫
❞✝
✲☛❝✞✝✗✝
✽✲
✞✌ ✏
✶✍♦✌ ✓ ☛☞♦✷ ✓✸✰✟✫✲✝❣ ❞✠✹✝❣✌
✰✼✝✝✌ ✏ ♦✌✓ ✍✝✭✝
✏
✠✡✳
❞❝❣✌
(r),
✍✏✔☛
✏ ✿❣
✠✡ ✏ ✰✝✡ ❝ ✺✞✝✻❣
a, b, c
☛☞♦✷ ✓✸✰✟✫✲✝❣
✰✝✡ ❝
(l, m, n)
☛☞♦✷✓✸❞✝✌ ✍ ✝✶✟
✠✡✏✢
l
❁
a
,
r
❁
b
, n
r
✲❝
❄✟❞✝
m
❁
c
r
✑
✩
✩
☛❂✝❃✝✫t
☞✝✌
❞✕
✍✠✸✰✝☛☞✞
☛t✍❞✕
✩
❣✕✟✝✌ ✏
❃✝✫t ✝✰✝✌ ✏
✍☛☞✎✝✝✌✏
✍✏ ✹❆✟
❃✝✫t ✝✱❇
❞✝✌
✺✞
❀✝✌✻
❞✝
☛☞✱
✠✫✱
✞✌ ✏
✹✌ ✟✌
✱❞
✱✌✍✌
✍☛☞✎✝
✠✡✏✳
✍✞✝✏❣ ❝
✍☛☞✎✝
✖❣✫❃ ✝✫t
♥
| |
✱❞ ✍☛☞✎✝ ❞✝ ✰☛☞✎✝ ❈ ✍✌ ✻✫✗ ✝✟ ✶✍♦✌ ✓ ✲☛❝✞✝✗✝ ❞✝✌
✠✡
✰✝✡ ❝
✰✭✝♦✝
❈ ❞✝
✞✝✟
☛♦✲❝✕❣
✿✟✝❊✞❞
❝✾✝❣✝
✰✭✝♦✝
❋✗✝✝❊✞❞
✠✝✌✟✌
❉
♦✌ ✓
0
❀✝✌ ✻
♦✌ ✓
✠✡✳
☛♦❞✗✝♥
♦✌ ✓ ✻✫✗ ✝t
✰✟✫✍ ✝❝
✍✌
✲❜ ✝❅❣
✠✝✌❣ ✝
✠✡
✞✌ ✏ ✲☛❝♦☛❣♥❣ ❞❝ ☞✌❣ ✝
✶✍❞✕
☛☞✎✝✝
❞✝✌
✍☛☞✎✝
✠✡✳
✍✞✝✟
✠✡✳
■
●
✩
✩
☛☞✱
✠✫✱
☛✔☞✫✰✝✌ ✏
✍☛☞✎✝
P
✰✝✡ ❝
a
Q
♦✌ ✓
☛✹✱
✍☛☞✎✝
aˆ
❍
a
|a|
■
❏
❪
a
☛☞✎✝✝
❞✕
☛✬✭✝☛❣
✍☛☞✎✝
a
✺✞✎✝✢
✞✝❂✝❞
✜
✜
☛t✟♦✌ ✓
✞✌✏
❑▲▼
◆
b
✏
✠✡❪
❞✝✌
☛✞✹✝✟✌
♦✝✹✕
♦✌ ✓
✰✟✫✲✝❣
✞✌ ✏ ☛♦❃✝✝☛t❣
❞❝✟✌
♦✝✹✌
✪✔☞✫
R
na mb
(i)
m n
❖
❞✝ ☛✬✭✝☛❣
✍☛☞✎✝
❖
P
P
mb na
(ii)
m n
◗
☛♦❃✝✝t✟
✲❝
◗
✔✝❘
☛♦❃✝✝t✟
✲❝❪
❞✝✌
P
P
m:n
❝✌✾✝✝
♦✌ ✓
✽✲
✞✌ ✏
✲❜ ✝❅❣
✠✝✌❣ ✝
✠✡ ✳
✰✏❣ ✢
475
❧ ✁✂✄ ☎✆✝✞ ✟✄✠
✑
✑
✡
♥☛☞
a
✌✍ ♥ ✎☛☛ ☞ ✏
✑
✑
♦☞ ✔
✕✖✗
✘☛
✘☛ ☞ ✙ ☛
✚
✑
✪
✬✧
✭☞ ✏
✧✮☛✯✜
❣☛☞✜☛
❣✛✰
♦☞ ✔
✕✖✗
✘☛
✱✍♥
♥☛☞
✌✍♥✎☛☛☞ ✏
✸
✸
a
✈✒✓
❙
b
✹
✴
✴
✶
✶
✌☞
✧✮☛ ✯✜
b
✈✒✓
❙
a
❣✛
✜☛☞
✌✍♥✎☛☛☞ ✏
❣✛✰
♦☞ ✔
✕✖✗
✘☛
✘☛☞✙☛
✚ ❣✛
✜☛☞
✢✣✘☛
✌✍♥✎☛
✥✦✙☛✣✧✔★
♦☞✔ ✬✧
✭☞✏ ✧✮☛ ✯✜ ❣☛☞ ✜☛
n̂
❣✛✰ ✺❣☛✻
✼✘
✼☞✌☛
✭☛✽☛✘ ✌✍♥✎☛
✑
✑
✘☛☞
✑
✱✍♥
❣✦ ✤☛
❣☛☞✜☛
✑
✌✭✘☛☞✍ ✙☛✘
✡
✍♥✱☛
✲
✑
✑
✷
a
a b
‘ ’, cos =
| a || b |
✘☛☞✙☛
a b = | a || b | sin nˆ
✑
✥ ✦ ✙☛ ✣✧ ✔★
✑
✑
✡
a b
✱✍♥
✤✍ ♥ ✎☛
✵
b
✈✒✓
❙
✢ ✣ ✘☛
✳
✶
✶
a
✜☛ ☞
✳
✫ ♦☞ ✔
✑
✑
❣✛
✑
a b | a || b | cos
✩
b
✈✒✓
❙
✪
✌✍❞✭✍★✜
✍✣♥☞ ❂✎☛☛✏✘
✘✾✣☞
✧❃✍✜
a1iˆ a2 ˆj a3 kˆ
❄
❄
♦☛★☞
✘☛☞
✍✣✍✭❂✜
✑
✜❀☛☛
b
✸
a b = (a1 b1 ) iˆ (a2
✸
❄
❅
✜★
❄
❄
✪
♦☞ ✔
★✏✕ ♦✜✿
✘✾✜☞
❣✛
a ,b
✜❀☛☛
✈✒❙✓
nˆ
❣✛ ✺☛☞
♥✍❁☛✙☛☛♦✜✖❂
❣✛✏ ✰
b1iˆ b2 ˆj b3 kˆ
❄
✤☛✛✾
❄
b2 ) ˆj ( a3
❄
❄
✼✘
✤✍♥✎☛
❣✛
✜☛☞
b3 ) kˆ
a = (❈a1 )iˆ ❄ (❈a2 ) ˆj ❄ (❈a3 )kˆ
❇
❆
✳
✳
a.b = a1b1
iˆ
a b = a1
a2
a2 b2
❊
❉
a3b3
kˆ
c1
c2
ˆj
b1
b2
✳
✳
✤☛✛✾
❉
❋●❍■❏❑❍▲▼
◆❖P◗❘❑❚❍❯
✌✍♥✎☛ ✎☛❱♥ ✘☛ ❲✱✦❳✧❨✣ ★✛✍❩✣ ❬☛☛❭☛☛ ♦☞ ✔ ✼✘ ✎☛❱♥ ♦☞ ❪❩✌ ❫
✤❀☛❂
❣✛
❣❵✜✥✜
✤☛✌✧☛✌
✭☛✣✖
✘✾✣☛✰
✺☛✜✖
✤☛❛✦ ✍✣✘
❣✛❤
♣❂q
✣☞
✍✌❃☛✏✜
♦☞ ✔
Caspar Wessel
✺✕
Argand (1768-1822 )
✌✍♥✎☛
♣✌
✕☛✜
✘☛
♦✙☛❂✣
❬☛✮ ❜ ✙☛✖✱
vectus
✍♦✗☛✾
❫❝✐❥❦♠❝❡❝❡
✍✘✱☛
✍✘
✼✘
❴ ✌☞ ❣✦✤ ☛ ❣✛ ✍✺✌✘☛
✘✖
✍✜✍❀☛
✌✣✿
✤☛✛✾
Jean Robert
♣❂q❴
✍✣♥☞ ❂✎☛☛✏✘
✜★
✭☞ ✏
✍✘✌✖
a + ib
William Rowen Hamilton (1805-1865 )
"Lectures on Quaternions" (1853 )
(quaternians) [
✾☞ r☛☛r☛✏s
✘✖
✌❣☛✱✜☛
✌☞
✼✘
✌✍❞✭t
✘☛
✌✏r✱☛
✉✱☛✍✭✜✖✱
✤❀☛❂
✍✣♦❂ ✗✣
♣❂q
✧✮ ✱☛☞✥
✌✕✌☞
✧☛★✣
✘✾✜☞
✧❣★☞
❣✦ ✼
✍✘✱☛
❀☛☛✰
✭☞ ✏ ✍♥❭❩
✗✜✦ ❭❩✱✖✱☛☞ ✏
a b iˆ c ˆj d kˆ, iˆ, ˆj , kˆ
✘✛✌☞
♣❂q
✺☛ ✌✘✜☛ ❣✛✰ ✼✘ ✤☛✱✍✾✎☛ ✥✍✙☛✜✇❤
✧✦ ❵ ✜✘❤
❝❡❢❢
✾☞r☛☛r☛✏s
♦✦ ✔①
♦☞ ✔
✬✧
♦☛★☞
♦☞ ✔
✍★✼
✍✣✍✎✗✜
✗☛✾
✌✍♥✎☛
✍♥❭❩
✍✘✱☛
✣☞ ✤✧✣✖
✎☛❱♥
✘☛
✍✣✱✭☛☞ ✏
✘☛
✌✏r✱☛✤☛☞ ✏
✘☛
✕✖✺✖✱
♦☛❵✜✍♦✘
♦☞ ✔
476
① ✁✂✄
❧☎✆✝✞✟
]
❞✠ ✡☛☞ ☎✌✍✎ ☞✏☞✑ ❧☞✒✓✔✔✕✖ ❞✔✕ ☞✗✔✘☞✏☎✠✟ ✙✖✚☞✛✜✔ ☎✕✖ ✢✆✣✔✔ ❞✛✎✕ ❞✠ ❧☎✤✟✔ ❞✔ ✥❞
✡❣ ✦✔✔✧ ✚✦✔✔☞★
✡☎ ✟✡✔✩ ✪❧ ✫✔✚ ❞✔ ☞✬✭ ✙✏✓✟
❞✛✕✢
✖ ✕ ☞❞ ❧☞✒✓✔
Plato (384-322
Aristotle (427-348
)
✟✔✕✢★✯❣ ❞✔ ☞✏✞✔✛ ✫✡✆✚✘ ☞✒✎✔✕✖ ★✡❣✕ ❧✕
✒✔✓✔❜✔☞✎❞
✪❜❧✔
✏☛ ♥✔☞✎❞
✙✔☛✛
★✰ ✏❜
❞✠ ❧✖❞✌★✎✔ ✙✔☛✛
✪❜❧✔ ★✰ ✏❜✱ ✏✕✯ ✥❞ ☞✓✔✲✟ ✥✏✖ ✟✰✎✔✎✠
✏✕✯
❞✔❣
❧✕
✬✔✎❞✔✛✠ ❞✠ ❞✌★✎✔ ✦✔✠ ☞❞ ✒✔✕ ✙✦✔✏✔ ✙☞✑❞ ✫❣✔✕✖ ❞✠ ❧✖✟t
✆ ✚ ☞✭✟✔
✏✕✯
☞✎✟☎✔✎✆❧✔✛
✟✔✕✢
★✛
❞✛✎✕
★♦✔✴✚
❞✠
✬✔
❧❞✚✠
✡☛ ✧
✫❣✔✕✖
✫❣✔✕✖ ❞✔ ✟✔✕✢ ❧☞✒✓✔ ❝★ ☎✕✖ ☞❞✟✔ ✬✔ ❧❞✚✔ ✡☛✵ ❞✠ ✶✔✔✕✬
✮✎✏✕✯
✏✕✯
✡✠
✦✔✔✧
✮❧
❧☎✟
✪❧
✮✎❞✔✕ ❧☎✔✖✚✛ ✞✚✆✳✔✆❜✬
❧✖✟✔✕✬✎
❞✔
❧✡✠
☞✎✟☎✵
☞❞
Sterin Simon(1548-1620 )
"DeBeghinselen
✪❜✷
⑥✔✛✔
❣✖✫ ✏✚② ✫❣✔✕✖ ❞✠ ☞✤✦✔☞✚ ☎✕ ✖ ❞✠ ✢✪❜ ✧ ❧✎② ✸✹✺✻ ☎✕✖ ✮✼✡✔✕✎
✖ ✕ ✙★✎✠ ✓✔✔✕✑★✆✤✚❞✵
der Weeghconst"
✽✏✬✎
❞✛✎✕
❞✠
✏✕✯
❞❣✔
☞❧✾✔✖ ✚✱
☎✕✖
✫❣✔✕✖
✏✕✯
✟✔✕✢★✯❣
✏✕✯
✿✟✔☞☎✚✠✟
☞❧✾✔✖✚ ❞✔ ☞✏✓❣✕✲✔✣✔ ☞❞✟✔ ✦✔✔ ☞✬❧✏✕✯ ❞✔✛✣✔ ✟✔✖☞✗✔❞✠ ✏✕✯ ☞✏❞✔❧ ☎✕✖ ✥❞ ☎✆✶✟ ★☞✛✏✚❜✎ ✡✆✙ ✔✧
★✛✖ ✚✆
✪❧✏✕✯
✫✔✒
✳✔✠
✸✺✺❁
❧✎②
☎✕ ✖
(1839-1903 )
✪❜✷
❧☞✒✓✔✔✕✖
✥❞
✙✔☛ ✛
❞✠
✐✟✔★❞
✙☎✕☞✛ ❞✠
✥❞
♦✕
✙✖✢✬
❧✖ ❞✌★✎✔
✳✔✔☛ ☞ ✚❞
✏✕✯
☞✎☎✔❜✣✔
☎✕✖
❀❁❁
✏✲✔❜
❣✢
✢✥✧
Josaih Willard Gibbs
Oliver Heaviside (1850-1925 )
✓✔✔ ✤✗✔✠
✥ ✏✖
✢☞✣✔ ✚♥
✪❜✷
✙☞✳✔✟✖✚✔
✎✕
✥❞
✞✚✆✲✍✟✠ ✏✕✯ ✏✔✤✚☞✏❞ ✽✙☞✒✓✔✱ ✳✔✔✢ ❞✔✕ ❞✔✌★☞✎❞ ✽❧☞✒✓✔✱ ✳✔✔✢ ❧✕ ★♣✦✔✏②✯ ❞✛✚✕ ✡✆✥ ❧☞✒✓✔
☞✏✓❣✕✲✔✣✔
❞✔
❧♣✬✎
Vector Analysis"
✥✏✖ ❧✖☞✜✔✴✚
☞✏✏✛✣✔
D. Heaviside
✟✔✕✢✒✔✎
☞✒✟✔
✙✔☛✛
☞❞✟✔
✦✔✔✧
❧✎②
✎✔☎❞ ✥❞ ✓✔✔✕✑
☞✒✟✔
✡✆✙✔
✦✔✔✧
✸✺✺✸
★✆☞✤✚❞✔
✚✦✔✔☞★
✸✺✺❢
✙✔☛ ✛
✉★✏✔✪❜ ✧ ✪❧
❧☞✒✓✔✔✕✖
P.G. Tait (1831-1901 )
✪❜✷
✏✕✯
Gibbs
✎✕
"Entitled Element of
★✆✤✚❞ ☎✕✖ ❧☞✒✓✔✔✕✖ ❞✔ ✥❞ ✭☎✫✾
♦ ✔✕✢
✙✎✆★✟
❞✔
☞✎❝★✣✔
❞✛✎✕
❞✠
❞✠☞✚❜
❞✔✕ ★♦✔ ✴✚ ✡☛ ☞✬✼✡✔✕✎
✖ ✕ ✪❧ ☞✏✲✔✟ ✏✕✯ ☞❣✥ ❧✔✦✔❜❞
✡☛ ✧
—
☎✕✖
❂
—
v Ł; k;
f=k&fo e h;
❏❏
T; kfe fr
✾✿❀❁❂❂ ❃❄❅❂❆❇❄❈❆❉❊ ❋❂❈❅❂●❁❍■
✤✤✥✤
d { kk
Hkfe d k
XI
e ] o ’ y f "kd
v kj f =k& f o e h;
fo f/
✁✂✄ ☎✆✝✞✟✠ ✡✆☛✄☞ ✆✌☎✍✎✂✄☎✍✎✞✏✍✑ ✞✟✝✄✟✎✞✆✟ ✞✒ ✟✆✎
☞✄✍✒✆✟✞✟✠ ✓✔✎ ✞☎✍✠✞✟✍✎✞✆✟✕ ✖ ✗✕✘✙✚✛✜✢✗✣
✦✧★✩✪✫✬✭✮✩✯✫★✰
rd
T; kf e f r d k v Ł; ; u d j r l e ;
f o "k; k o Q i f j p ;
l hf e r
j [ kk g A b l
e g e u Lo ;
i Lr d
f } & f o e h;
d k o Qo y
d kr h;
v Ł; k;
e g e us
o Q fi N y
l f n ’ kk d h e y l d Yi u kv k d k v Ł; ; u f d ; k g A v c g e l f n ’ kksa
o Q c ht x f . kr
T; kf e f r
v R; r
d k f =k& f o e h;
e bl
T; kf e f r
mi kx e d k mn n ’ ;
l j y , o l # f p i . k ( l x kg ; )
bl
v Ł; k;
f n d & d kT; k o
e
ge
e mi ; kx d j x A f =k& f o e h;
g fd
; g b l o Q v Ł; ; u d ks
c u k n r k g A*
n k flc n v k d k f e y ku
f n o Q& v u i kr
d k v Ł; ; u
o ky h j [ kk o Q
d jx
v kj
f o f HkUu
f LFkf r ; k e v r f j { k e j [ kkv k v kj r y k o Q l e hd j . kk] n k j [ kkv k]
nk r y k o
f o "ke r y h;
l
, d
j [ kk v kj
, d
ry
o Q c hp
j [ kkv k o Q c hp U; u r e n j h o , d
n j h o Q f o "k;
d k d k. k]
ry
d h, d
n ks
flc n q
e Hkh f o p kj f o e ’ k d j x A mi j kDr i f j . kke k e l s
v f / d k’ k i f j . kke k d k l f n ’ kk o Q : i e i kI r d j r g A r Fkkf i g e
❯❱❲❳❨❩❬❭❪❫❴❱❬
❵❛❜❝❜❞❛❜❡❢❣
b u d k d kr h;
: i e Hkh v u o kn d j x t k d ky kr j e f LFkf r d k Li "V T; kf e r h;
f p =k. k i Lr r
d j
l o Qx kA
✤✤✥✱
✷✶✩✯✫✵✫✸ ✶✹✯★✳✰
j [ kk o Q f n o Q&d ks
l kbu v kj f n o Q&v u i kr
v Ł; k;
1 0 e ] Le j . k d hf t , ] f d
e y flc n l
v kj f o ’ y "k. kkRe d
✦✲✯✪✳✮✩✯✫★✴✫✵✯★✳✵ ✶★✬✲✯✪✳✮✩✯✫★
L
✺, ✻ ✼✽
x t j u o ky h l f n ’ k j [ kk
} kj k
v kj
v { kksa
a] b v kj g c u k, x , d k. k f n o Q& d k. k d g y kr g r c b u d k. kk d h d kl kb u u ke r %
cosa, cosb v kj cosg j [ kk L o Q f n o Q& d kl kb u ( direction cosines or dc's) d g y kr h g SaA
o Q l kFk
*
e’ k
For various activities in three dimensional geometry, one may refer to the Book
❑✗▲✍✟▼ ◆✆✆❖✌✆☞ ▼✄✒✞✠✟✞✟✠✚✍✎✂✄☎✍✎✞✏✒P✍✓✆☞✍✎✆☞◗ ✞✟❘✏✂✆✆✑✒❙❚ NCERT, 2005
478
x f. kr
; fn g e
p-g
l
c ny
Ld
h f n ’ kk f o i j hr d j n r
t kr
g A bl
g r k f n o Q& d k. k] v i u l i j d k e v Fkkr
i d kj ] f n o Q& d kl kb u o Q f p
c ny
t kr
p-a, p-b v
kj
gA
v ko Qfr 1 1 -1
Ł; ku n hf t , ] v r f j { k e n h x b j [ kk d k n k f o i j hr f n ’ kkv k e c < k l d r g v kj b l f y ,
f n o Q& d kl kb u o Q n k l e g
l eg
,✁
o Q f y , ] g e K kr
g A bl fy ,
v r f j { k e K kr
j [ kk d k , d
v kj ✂ o Q } kj k f u f n "V f d ,
t kr
l
, d
l f n ’ k j [ kk y u k p kf g , A b u v f } r h;
, d
g kr
flc n l
f n o Q& d kl kb u d ks
u g h x t j r h g r k b l d h f n o Q& d kl kb u d k K kr
n h x b j [ kk o Q l e kr j , d
l f n ’ k j [ kk o Q f n o Q& v u i kr
l e g l e ku ( o g h)
K kr
d jr
g D; kf d
j [ kk [ khp r g A v c e y
fV Ii . kh
e ku y hf t ,
, d
flc n l
bue a
s
n k l e kr j j [ kkv k o Q f n o Q& v u i kr k o Q
gA
j [ kk o Q f n o Q& d kl kb u o Q l e ku i kr h l [ ; kv k d k j [ kk o Q f n o Q& v u i kr
ratios or ✄ ☎✆✝) d g r
f d l h ’ kU; r j l Î ✡
☛
flc n l
f n o Q& d kl kb u o Q v f } r h;
gA
fV I i . kh v r f j { k e n h x b j [ kk ; f n e y
d j u o Q fy , ] g e e y
j [ kk o Q f y ,
bl o Q
g A ; fn , d
o Q fy , ✞
o QN y [ kd
j [ kk o Q f n o Q& d kl kb u
= l , ✟ =l✁
v kj ✠
, ✁, ✂ o
= l✂
f n o Q& v u i kr k d k f n o Q& l [ ; k,
j [ kk o Q f n o Q& v u i kr
✞
=
✞
✁
✟
, ✟,
=
✠
✂
✠
f n o Q& v u i kr
Hkh d g r
gA
v kj j [ kk d h f n o Q& d kl kb u
✌☞
(direction
✞, ✟, ✠ g k r c
( e ku
y hf t ,
),
☞
, d
, ✁, ✂
v p j gA
gAr c
f=k&fo e h;
= ✁✂, ✄ =
bl fy ,
2
i jrq
✂
bl fy ,
2
(✁
2
2
,
=
☎✂ ✆
+✄ +✆ = 1
+ ☎ 2 + ✝ 2) = 1
✂
1
✞
=
✁
☛
☛
f d l h j [ kk o Q f y ,
f n o Q& v u i kr k d k , d
v r % fd l h , d
✔✔✕✖ ✕✔
2
✓☞
2
✟☎
2
☞
,✎ ✑ ✒
2
2
✟✝
2
(✠ .✝.✥✡ )
l ] j [ kk d h f n o Q& d kl kb u
✍ ✑✒
... (1)
✝✂
2
; k
v r % ( 1)
479
T; kfe fr
✓✌
☛
2
✓☞
, d
✌
,✏ ✑ ✒
✓✌
☛
2
✓☞
2
2
✓✌
¹0
Hkh
j [ kk o Q f n o Q& v u i kr k o Q n k l e g Hkh l e ku i kr h g kx A
j [ kk o Q f n o Q& v u i kr k o Q v l [ ;
j [ kk d h fn o Q& d kl kbu e
2
e ’ k% ✁ , ☎ , ✝ g ] r k ✂✁ , ✂☎ , ✂✝; ✂
; f n j [ kk o Q f n o Q& v u i kr
l e g g A bl fy ,
2
l e g g kr
gA
✗✘✙ ✚✛ ✜✢✣ ✤ ✦ ✙ ✜✧✙✙✤ ✜★✙ ✩✢✪✙✫ ✜✢✣ ✤ ✫✣ ✬✢ ✤✙✬
l c a/
✣ ✭ ✛ ✚✢✤✙✮
e ku y hf t ,
flc n l
, d
RS d
j [ kk
( v ko Qfr
j [ kk [ khf p ,
✯
l
-v
{k i j
gA ey
v kj b l
i j , d
PA
[ khf p ,
y c
1 1 -2 ) A
OP = ✲. r k cos a ✳
OA
OP
✲
= ✄✲ v
2
✯ + ✰ +
2
bl fy ,
i jrq
✯
v r%
✚
2
✴
+
✰
2
✵ ✶
✴
✯
✳
✰
b l h i d kj
✔✔✕✖ ✕✖
P
y hf t , A
, ✄, ✆
h f n o Q& d kl kb u
n h x b j [ kk o Q l e kr j , d
P(✯, ✰, ✱)
flc n
; fn
fd
+
. ft
l l
✯
=
✲
2
+
i kI r g kr k g A
kj ✱ = ✆ ✲.
2
✱
2
✱
=
=
2
✲
(2+
✄
✆
2
)
2
✲
v ko Qfr 1 1 -2
✴
✵ ✤ ✷ ✔
n k flc n v k d k fe y ku o ky h j [ kk d h fn o Q&d kl kbu ✗✸✢✪✙✫✜✢✣ ✤ ✫✣ ✬✢✤✙✬ ✣ ✭ ✛ ✚✢✤✙
✹✛ ✬✬✢✤✺ ✜★✪✣ ✻✺★ ✜✧✣ ✹✣✢✤✜✬✮
D; kf d
nk fn,
P(✯1, ✰ 1, ✱1)
flc n v k l
g kd j
t ku o ky h j [ kk v f } r h;
Q( ✯2, ✰ 2, ✱2) l x t j u o ky h j [ kk d h
d j l d r g ( v ko Qfr 1 1 -3 ( a) A
e ku y hf t , f d j [ kk PQ d h f n o Q& d kl kb u , ✄ , ✆ g
e ’ k% a, b ✾ g c u kr h g A
v kj
g kr h g A b l f y ,
nk fn,
x,
f n o Q& d kl kb u d k f u Eu i d kj l
flc n v ka
s
K kr
v kj ; g ✯✼ ✰ v kj ✱✽ v { k o Ql kFk d k. k
480
x f. kr
P
e ku y hf t ,
v U;
(v
y ac
[ khf p ,
Ql
ksQS d
v kS
j
t
11.3 ( b)) b l
ko Qfr
s y c [ khf p ,
ks
Ni
j
t ks
XY- r y
R r Fkk S i j f e y r g A P l , d
k. k f =kHkqt PNQ e s,a ÐPQN = g
d ks
fe y r k g A v c l e d
fy ,
v ko Qfr 1 1 -3
cosg =
cosa =
b l h i d kj
v r % flc n v k
P(✆ 1, ✝ 1,
1
)
r Fkk
NQ
PQ
- 1
PQ
2
✁
✂ 2 - ✂1
PQ
Q(✆ 2, ✝ 2, 2)
PQ =
✄2
fV I i . kh flc n v k
i d kj l
fy ,
P(✆ 1, ✝1, 1)
t k l d r
( ✆2 - ✆1 ) 2
r Fkk
d k d k. k c u kr h g
gy
✌
e ku y hf t ,
= cos 30° =
PQ
-
- 1
PQ
2
PQ
j [ kk[ kM
fd
f n o Q& d kl kb u
gA
2
✝1 ) ✠ ✞ 2
d k t kM u o ky
-
1✟
2
j [ kk[ kM o Q f n o Q& v u i kr
f u Eu
gA
j [ kk ✆ , ✝ r Fkk
2
✥
-v
1
,
; k ✆ ✥ ✆
1
{ kk d h / u kRe d
r k f n o Q& d kl kb u K kr
j [ kk d h f n o Q& d kl kb u ☛
3
2
✠ ( ✝2
Q(✆ 2, ✝ 2, 2)
✆ 2 ✥ ✆ 1, ✝ 2 ✥ ✝ 1,
mn kg j . k ✡ ; f n , d
✄ 2 - ✄1
d k t kM u o ky
- ✄1
,
PQ
- ✆1
,
PQ
✆2
t g k¡
cos b ☎
v kj
2
,
✝ 1 ✥ ✝ 2,
1
✥
f n ’ kk o Q l kFk
2
e ’ k%
90°, 60° r Fkk 30°
d hf t , A
, ☞, ✌
gAr c ☛
= cos 90° = 0, ☞ = cos 60° =
1
,
2
f=k&fo e h;
mn kg j . k
gy
; fn , d
481
T; kfe fr
j [ kk o Q f n o Q& v u i kr 2 ] & 1 ] & 2 g r k b l d h f n o Q& d kl kb u K kr d hf t , A
f n o Q& d kl kb u f u Eu o r
ga
S
2
2 ✁ (-1) ✁ ( -2)
2
2
2
-2
-1
,
2 ✂ ( -1) ✂ ( -2)
2
2
2
,
2
2 ✂ ✄ -1☎ ✂ ( -2) 2
2
2 -1 -2
, ,
3 3 3
v Fkkr ~
mn kg j . k ✆ n k flc n v k
(✥ 2, 4, ✥ 5)
(1, 2, 3)
v kj
d k f e y ku o ky h j [ kk d h f n o Q& d kl kb u K kr
d hft , A
gy
ge
t ku r
g
fd
n k flc n v k
P(✝1, ✞1, ✟1)
v kj
Q(✝ 2, ✞2, ✟2)
d k f e y ku
o ky h j [ kk d h
f n o Q& d kl kb u
✝ 2 - ✝1 ✞ 2 - ✞1 ✟ 2 - ✟1
,
,
PQ
PQ
PQ
; gk
P
( ✝2 - ✝1 ) 2 ✂ ( ✞ 2 - ✞1 ) 2 ✂ ✠ ✟2 - ✟1✡
PQ =
g ] t g k¡
Q
v kj
e ’ k%
( ✥ 2, 4, ✥ 5) v
kj
bl fy ,
(1, 2, 3) g A
(1 - ( -2)) 2 ✂ (2 - 4) 2 ✂ (3 - ( -5)) 2 =
PQ =
bl fy ,
,
77
mn kg j . k ☛
✝-v
{k
✝, ✞
✞-v
b l h i d kj
{k
mn kg j . k ☞ n ’ kkb ,
Av
kj
v kj
e ’ k%
d h f n o Q& d kl kb u
gy
B
kj
Cd
kj
8
,
77
{ kk d h f n o Q& d kl kb u K kr
fd
flc n
77
d hf t , A
A (2, 3, ✥ 4), B (1, ✥ 2, 3) v
kj
C (3, 8, ✥ 11) l
✝-v
{k
10, ✥ 14
gA
fy ,
j [ kg A
d k f e y ku o ky h j [ kk o Q f n o Q& v u i kr
Fkkr
✥ 1, ✥ 5, 7 g A
k f e y ku o ky h j [ kk o Q f n o Q& v u i kr
3 ✥ 1, 8 + 2, ✥ 11 ✥ 3, v
AB v kj BC o Q f n o Q& v u i kr l e ku i kr h g A v r % AB v
BC n ku k e B mHk; f u "B g A v r % A, B, v kj C l j [ k flc n g A
Li "V g f d
AB v
✟- v
-2
✝ , ✞ v kj ✟- v { k o Q l kFk 0°, 90° v kj 90° o Q d k. k c u kr k g A b l
cos 0°, cos 90°, cos 90° v Fkkr 1,0,0 g A
v kj ✟- v { k d h f n o Q& d kl kb u
e ’ k% 0, 1, 0 v kj 0, 0, 1 g A
1 ✥ 2, ✥ 2 ✥ 3, 3 + 4 v
Bv
77
n k flc n v k d k f e y ku o ky h j [ kk d h f n o Q& d kl kb u g %
3
gy
2
Fkkr ,~ 2,
kj
BC l
e kr j g A i j r q
482
x f. kr
i ’ u ko y h 1 1 -1
✁
j [ kk ✂ , ✄ v kj
; fn , d
f n o Q& d kl kb u K kr
✆✁
, d
✝✁
; fn , d
✞✁
n ’ kkb ,
✟✁
, d
☎- v
j [ kk d h f n o Q& d kl kb u K kr
j [ kk o Q f n o Q& v u i kr
fd
flc n
f =kHkt
d { kk
e f } & f o e h;
ry
; g fn,
(ii)
; g nk fn,
✥ 18, 12, ✥ 4, g
kj
x,
r k b l d h f n o Q& d kl kb u D; k g \
K kr
j [ kg A
d hf t ,
; fn
f =kHkt
o Q ’ kh"kZ flc n q
(✥ 5, ✥ 5, ✥ 2) g AS
☞✌✍✎✏✑✒✓✔ ✓✕ ✏ ✖✒✔✗ ✒✔ ✘ ✙✏✚✗✛
l e hd j . kk d k K kr
flc n l
k. k c u kr h g r k b l d h
t k f u n ’ kk{ kk o Q l kFk l e ku d k. k c u kr h g A
e j [ kkv k d k v Ł; ; u d j u o Q i ’ p kr
j [ kk v f } r h; r % f u / kf j r
(i)
d hf t ,
j [ kk d k l e hd j . k
o Q l f n ’ k r Fkk d kr h;
, d
90°, 135°, 45° o Q d
d h Hkt kv k d h f n o Q& d kl kb u
r fj { k e
XI
e ’ k%
(2, 3, 4), (✥ 1, ✥ 2, 1), (5, 8, 7) l
(3, 5, ✥ 4), (✥ 1, 1, 2) v
✠✠✡☛ v
{ k o Q l kFk
d hf t , A
v c g e v r fj { k e , d
j [ kk
d jxA
g kr h g ] ; f n
n h x b f n ’ kk l
flc n v k l
g kd j t kr h g ] ; k
g kd j t kr h g A
✢
✁✝✁ fn , x , flc n ✜ l t ku o ky h r Fkk fn , x , l fn ’ k ✣ o Q l e kr j j [ kk d k l e hd j .
✢
✤✦✧★✩✪✫✬✭ ✬✮ ✩ ✯✫✭✰ ✪✱✲✬★✳✱ ✩ ✳✫✴✰✭ ✵✬✫✭✪ ✜ ✩✭✶ ✵✩✲✩✯✯✰✯ ✪✬ ✩ ✳✫✴✰✭ ✴✰✷✪✬✲ ✣ ✸
l e d kf . kd
f u n ’ kk{ k f u d k;
e ku y hf t ,
fd
flc n
Ad
oQ ey
k l fn’
flc n
o ky h r Fkk f n ,
j[
A l t ku
kk ✻ g A e ku
y hf t ,
P
d k f LFkf r
rc
l,
✽✽✽✢
AP
d
l fn’ k
l fn’
✢
k ✺
o kLr f o d
✢
✼
x,
flc n
O
o Q l ki { k
g A e ku y hf t ,
l fn’ k
✢
✺
fd
o Q l e kr j
j f LFkr f d l h Lo PN flc n q
g ( v ko Qfr
1 1 -4 ) A
✽✽✽✢
✢
o Ql e kr j g v Fkkr ~AP = l ✺
,
t g k¡
l [ ; k gA
✽✽✽✢ ✽✽✽✢ ✽✽✽✢
AP = OP ✥ OA
✢
✢ ✢
l✺ = ✼ - ✹
i jrq
v Fkkr ~
f o y ke r % i kp y
✻i
fd
✢
k ✹
k
loQi
R; d
e ku o Q f y ,
v ko Qfr 1 1 -4
; g l e hd j . k j [ kk o Q f d l h flc n
P
d h f LFkf r
i n ku
d j r k g A v r % j [ kk d k l f n ’ k l e hd j . k g %
✢
✢
✢
✲ = ✩✾l ✣
... (1)
f=k&fo e h;
✠
fV I i . kh ; fn
o Q fn o Q& v uiqkr
l fn ’ k : i
✞ ✁✄✂ ✟ ✂☎ ✟ ✆✝✂ g r k j [ kk o Q f no Q& v u i kr ✁ , , ✆ g
✠
✁ , , ✆ g ksar ks ✞ ✁✄✂ ✟ ☎✂ ✟ ✆✝✂ j [ kk o Q l e kr j g kx kA ;
l d kr h;
: i
O; Ri Uu d j u k
483
T; kfe fr
v kj f o y ke r % ; f n , d
j [ kk
✠
gk
d k
| |u
l e > k t k, A
✡☛☞ ✌✍✎✏✑✍✒✓ ✒✔ ✕✏✌✑☞ ✖✍✏✓ ✗✒✌✘ ✔✌✒✘ ✙☞✚✑✒✌
✗✒✌✘✛
e ku
✁, , ✆
y hf t ,
fd
g e ku y hf t
v kj
b u e ku k d k ( 1 )
A o Q f u n ’ kkd (✜ 1, ✢1, ✣1) g v kj j [
, f d l h flc n P o Q f u n ’ kkd (✜ , ✢ , ✣) g A r c
✠
✠
✤ ✞ ✜✄✂ ✟ ✢✂☎ ✟ ✣✝✂ ; ✁ ✞ ✜1 ✄✂ ✟ ✢ 1 ✂☎ ✟ ✣1 ✝✂
✠
✞ ✁ ✄✂ ✟ ☎✂ ✟ ✆ ✝✂
fn,
flc n q
e i f r LFkkf i r d j o Q
✄✂ , ✂☎
v kj
✥✦ , o Q x . kkd
kk d h f n o Q& d kl kb u
k d h r y u k d j u i j g e i kr s
g fd
;
j [ kk o Q i kp y
l
✜ = ✜ 1 + l✁ ; ✢ = ✢ 1 + l ; ✣ = ✣1+ l✆
e hd j . k g A ( 2 ) l
i kp y l d k f o y ki u d j u
✧ - ✧1
★
; g j [ kk d k d kr h;
✩ - ✩1
=
✫
✭
✪ - ✪1
gy
(5, 2, ✲ 4)
l
l e hd j . kk d k K kr
g e K kr
g%
... (3)
✬
✮, ✯ , ✰
g ] r k j [ kk d k l e hd j . k
✜ - ✜1
✢ - ✢1
✣ - ✣1
✞
=
✮
✯
✰
r Fkk d kr h;
i j ] g e i kr
l e hd j . k g A
fV I i . kh ; f n j [ kk d h f n o Q& d kl kb u
mn kg j . k ✱ flc n
... (2)
t ku o ky h r Fkk l f n ’ k
gA
3✄✂ ✟ 2 ☎✂ - 8 ✝✂
o Q l e kr j j [ kk d k l f n ’ k
d hf t , A
g ] fd
✴
✳ = 5✄✂ ✟ 2 ☎✂ - 4 ✝✂ v
✠
kj
✞ 3✄✂ ✟ 2 ✂☎ - 8 ✝✂
b l f y , ] j [ kk d k l f n ’ k l e hd j . k g %
p fd
j [ kk i j
✠
✤ = 5 ✄✂ ✟ 2 ✂☎ - 4 ✝✂ ✟ l ( 3 ✄✂ ✟ 2 ✂☎ - 8 ✝✂ ) [(1) l ]s
✠
f LFkr f d l h flc n P(✜ , ✢ , ✣) d h f LFkf r l f n ’ k ✤ g ] b l f y ,
✜ ✄✂ ✟ ✢ ✂☎ ✟ ✣ ✝✂ = 5✄✂ ✟ 2 ✂☎ - 4 ✝✂ ✟ l ( 3 ✄✂ ✟ 2 ☎✂ - 8 ✝✂ )
= (5 ✟ 3l) ✷✄ ✟ (2 ✟ 2 l) ✵☎ ✟ (- 4 - 8l ) ✝✶
484
ld
x f. kr
k f o y ki u d j u i j g e
i kr
g
fd
-5
✁- 2 ✂✄ 4
☎
=
3
2
-8
t k j [ kk o Q l e hd j . k d k d kr h;
✆✆✝✞✝✟ n k fn , x , flc n v k l
✍✘✙✏☞✗✘ ✍✚✏ ✗✎✛✔✑ ✕✏✎✑✍✖✜
e ku y hf t ,
✤
✣
v kj
✤
✥
, d
P
t ku o ky h j [ kk d k l e hd j . k
A( ✢1, ✁ 1, ✂1)
✤
✦
P
r Fkk
j [ kk i j f LFkr
, d
Lo PN
j [ kk i j
g
✪✪✪✫ ✫ ✫
AB ✩ ✬ - ★
g
; f n v kj
flc n
; fn
P
v kj
B( ✢2, ✁2, ✂2), o Q f LFkf r
g e i kr
l
d kr h;
l j [s k l fn ’ k g A b l fy ,
o Qo y
; fn
... (1)
: i
v ko Qfr 1 1 -5
O; Ri Uu d j u k
g fd
✤
✤
✦ ☎ ✢✰✯ ✄ ✁ ✯✱ ✄ ✂ ✲✯ , ✣ ☎ ✢1✰✯ ✄ ✁ 1 ✯✱ ✄ ✂1 ✲✯ ,
b u e ku k d k ( 1 )
e
e ’ k%
; fn
t k j [ kk d k l f n ’ k l e hd j . k g A
l fn ’ k : i
l fn’ k
d k f LFkf r
o Qo y
✤ ✤
✤ ✤
✦ - ✣ ☎ l (✥ - ✣ )
✤ ✤
✤ ✤
✙ ☎ ✌ ✄ l (✭ ✌ ) , l Î ✮
; k
v kj
✠✡☛☞✌✍✎✏✑ ✏✒✌ ✓✎✑✔ ✕✌✖✖✎✑✗
11.5) A
ko Qfr
e ku y hf t ,
l fn’ k g A r c
✪✪✪✫ ✫ ✫
AP ✩ ✧ - ★
gA
j [ kk i j f LFkr n k flc n v k
(v
g
: i
i f r LFkkf i r
d ju i j ge
i kr
g
v kj
✤
✥ ☎ ✢2 ✰✯ ✄ ✁ 2 ✯✱ ✄ ✂ 2 ✲✯
fd
✵ ✶✽ ✻ ✷ ✳✸ ✻ ✹ ✺✴ ✼ ✵1 ✶✽ ✻ ✷1 ✳✸ ✻ ✹1 ✺✴ ✻ l [( ✵ 2 - ✵1 )✶✽ ✻ ( ✷ 2 - ✷1 ) ✳✸ ✻ ( ✹2 - ✹1 ) ✴✺ ]
❀❁ , ✿❁, ✾❁
ld
o Q x . kkd k d h r y u k d j u i j g e i kr
g fd
✢ = ✢1 + l ( ✢2 ❂ ✢1); ✁ = ✁ 1 + l (✁ 2 ❂ ✁ 1); ✂ = ✂1 + l ( ✂2 ❂ ✂1)
k f o y ki u d j u i j g e
✢ - ✢1
✢2 - ✢1
☎
i kr
✁ - ✁1
✁2 - ✁1
t k j [ kk o Q l e hd j . k d k d kr h;
mn kg j . k ❃ flc n v k
g
(❂ 1, 0, 2)
fd
☎
: i
v kj
✂ - ✂1
✂2 - ✂1
gA
(3, 4, 6)
l
g kd j t ku o ky h j [ kk d k l f n ’ k l e hd j . k K kr
d hft , A
gy
e ku y hf t ,
✤
✣
v kj
✤
✥
flc n v k
A( ❂ 1, 0, 2)
v kj
B(3, 4, 6)
o Q f LFkf r
l fn’ k g A
f=k&fo e h;
✝
☎ - ✁✂ ✆ 2 ✄✂
✝
✞ ☎ 3 ✂✁ ✆ 4 ✂✟ ✆ 6 ✄✂
✝ ✝
✞ - ☎ 4 ✁✂ ✆ 4 ✂✟ ✆ 4 ✄✂
rc
v kj
bl fy ,
e ku y hf t ,
fd
l f n ’ k l e hd j . k
mn kg j . k
✡
, d
j[
j [ kk i j f LFkr
f d l h Lo PN flc n
P
d k f LFkf r
✝
✠ ☎ - ✁✂✆ 2 ✄✂ ✆ l (4 ✁✂ ✆ 4 ✂✟ ✆ 4 ✄✂ )
☛ ✍3 ☞- 5 ✌ ✍ 6
✎
✎
kk d k d kr h; l e hd j . k
2
4
2
l fn’ k
g A bl
✝
✠
T; kfe fr
485
g A v r % j [ kk d k
j [ kk d k l f n ’ k l e hd j . k
K kr d hf t , A
g y fn,
x,
l e hd j . k d k e ku d
☛ - ☛1
✏
l
r y u k d j u i j g e i kr
✎
: i
☞ - ☞1
g fd
✑
✒
✓1 = ✥ 3, ✔1 = 5, ✕1 = ✥ 6;
i d kj v Hkh"V j [ kk flc n q( ✥
bl
l e kr j g A
e ku y hf t ,
l e hd j . k
fd
✌ - ✌1
✎
3, 5, ✥ 6)
j [ kk i j f LFkr
l
g kd j t kr h g
f d l h flc n d h f LFkf r
= 2, ✞ = 4, ✖ = 2
r Fkk l f n ’ k 2 ✁✂ ✆ 4 ✂✟ ✆ 2 ✄✂ o Q
✝
l f n ’ k ✠ g r k j [ kk d k l f n ’ k
✝
✠ ☎ ( - 3 ✁✂ ✆ 5 ✟✂ - 6 ✄✂) + l (2 ✁✂ ✆ 4 ✂✟ ✆ 2 ✄✂ )
} kj k i n k g A
✗✗✘✙
n k j [ kkv k o Q e Ł;
fd
, ✞ 1, ✖ 1 v
1
o Q c hp
✖1 v
kj
kj
1 2
cosq =
i u%
✚✛✜✢✣✤ ✦✤✧★✤✤✜ ✧★✩ ✣✪✜✤✫✬
L1 v kSj L2 e y flc n l x t j u o ky h n k j [ kk, g f t u o sQ f n o Q& v u i kr e ’ k%
,
✞ 2, ✖2, g A i u % e ku y hf t , f d L1 i j , d flc n P r Fkk L2 i j , d flc n Q g A
2
1 1 -6 e f n , x , l f n ’ k OP v kj OQ i j f o p kj d hf t , A e ku y hf t , f d OP v kj OQ
U; u d k. k q g A v c Le j . k d hf t , f d l f n ’ kk OP v kj OQ o Q ?kVd
e ’ k%
, ✞ 1,
1
, ✞ 2, ✖2 g A b l f y , mu o Q c hp d k d k. k q
2
e ku y hf t ,
v ko Qfr
d k. k
2
2
1 ✆ ✞1
sin q o Q :
i
✆ ✞1 ✞2 ✆ ✖1 ✖2
✆ ✖12
=
l
} kj k i n k g A
✆ ✞22 ✆ ✖22
e ] j [ kkv k o Q c hp
sin q = 1 - cos2 q
1-
2
2
d k d k. k
i n k gS
2
1 2 ✆ ✞1✞2 ✆ ✖1 ✖2 )
2
2
2
2
2
2
1 ✆ ✞1 ✆ ✖1 ✮ ✭ 2 ✆ ✞2 ✆ ✖2
(
✭
✮
v ko Qfr 1 1 -6
486
x f. kr
2
2
2
2
2
2
2
☎1 ✞ ✆1 ✞ ✝1 ✁ ☎2 ✞ ✆2 ✞ ✝ 2 ✁ - ✂ ☎1☎ 2 ✞ ✆1✆2 ✞ ✝1 ✝ 2 ✄
=
2
2
2
☎1 ✞ ✆1 ✞ ✝1 ✁
(☎1
=
✟
✆2
-
☎2 ✆1 )
(☎12
fV I i . kh
ml
f LFkf r
e
2
2
2
2
☎2 ✞ ✆2 ✞ ✝ 2 ✁
( ✆1 ✝2 -
✞
2
2
✞ ✆1 ✞ ✝1
t c
✞ (✝1 ☎2
-
✝ 2 ☎1
)2
L2
kj
ey
flc n
l
ug h x t j r h g
L1 v kS L 2 o Q l e kr j ] e y flc n l x t j u o ky h j [ kk, e ’ k% L¢1 o L¢ 2 y r
L1 v kj L2 o Q f n o Q& v u i kr k o Q c t k; f n o Q& d kl kb u n h x b g k t l L1 o Q f y ,
L2 o Q f y , ✠2, ✡ 2, ☛ 2 r k ( 1 ) v kj ( 2 ) f u Eu f y f [ kr i k: i y x A
cos q = |✠1 ✠2 + ✡ 1✡ 2 +
sin q =
v kj
✂ ✠1 ✡ 2
☛ ☛
1
2
| ( D;
... (2)
2
2
2
☎2 ✞ ✆2 ✞ ✝ 2
L1 v
j [ kk,
)2
✆2 ✝1
g A ; f n j [ kkv ka
s
✠
1
, ✡ 1, ☛ 1 v
2
2
2
2
2
2
✠1 ✞ ✡1 ✞ ☛1 ☞1 ☞ ✠2 ✞ ✡ 2 ✞ ☛ 2
kf d
2
- ✠2 ✡1 ✄ - (✡1 ☛ 2 - ✡ 2 ☛1 ) 2 ✞ (☛1 ✠2 - ☛ 2 ✠1 ) 2
r k ge
)
kj
... (3)
... (4)
f n o Q& v u i kr ☎ , ✆ , ✝ v kj ☎ , ✆ , ✝ o ky h j [ kk, ¡
1
1
1
2
2
2
(i)
y cor
(ii)
l e kr j g ] ; f n
g ] ; fn
q = 90°, v
q = 0, v
Fkkr
Fkkr
( 1)
( 2)
l s
✌ ✌ ✏ ✑ ✑ ✏ ✒ ✒ ✓ ✔
✍ ✎
✍ ✎
✍ ✎
l
✌✕
=
✌✖
✑✕
☞
✑✖
✒✕
✒✖
v c g e n k j [ kkv k o Q c hp d k d k. k K kr d j x f t u o Q l e hd j . k f n ,
✘
✘
✘
✘
✘
= ☎1 ✞ l ✆1 v kj ✗ = ☎2 ✞ m ✆2 o Q c hp U; u d k. k q g S
✘ ✘
✆1 ✙✆ 2
cosq = ✘ ✘
rc
✆1 ✆ 2
x,
g A ; f n mu j [ kkv ka
s
✘
✗
d kr h;
: i
e ; f n j [ kkv k%
✚
☎1
✚
v kj
2,
- ✚2
☎2
o Q c hp
☎
- ✚1
✆
2
, ✝2 g
d k d k. k q g S t g k j s[ kk,
=
=
( 1)
✛
- ✛1
☞
✜ - ✜1
✆1
✛
- ✛2
✆2
o
... (1)
✝1
☞
( 2)
✜
- ✜2
... (2)
✝2
o Q f n o Q& v u i kr
rc
cos q =
☎1 ☎2 ✞ ✆1 ✆2 ✞ ✝1 ✝2
2
2
2
☎1 ✞ ✆1 ✞ ✝1
2
2
2
☎2 ✞ ✆2 ✞ ✝ 2
e ’ k% ☎ 1 , ✆ ✝ r Fkk
1, 1
f=k&fo e h;
mn kg j . k
fn,
487
j [ kk& ; Xe
✂
✁ = 3 ☎✄ ✞ 2 ✄✆ - 4 ✝✄ ✞ l (☎✄ ✞ 2 ✄✆ ✞ 2✝✄)
✂
✁ = 5 ☎✄ - 2 ✆✄ ✞ m (3 ☎✄ ✞ 2 ✆✄ ✞ 6 ✝✄)
v kj
o Q e Ł;
gy
x,
T; kfe fr
d k. k K kr
d hf t ,
✂
✟1 = ☎✄ ✞ 2 ✆✄ ✞ 2 ✝✄
e ku y hf t ,
v kj
✂
✟2 = 3 ☎✄ ✞ 2 ✆✄ ✞ 6✝✄
q g ] bl fy ,
✑ ✑
(☛✠ ✎ 2 ✠☞ ✎ 2✌✠ ) ✍(3 ☛✠ ✎ 2 ✠☞ ✎ 6 ✌✠ )
✡1 ✍✡2
cos q = ✑ ✑ ✏
1 ✎ 4 ✎ 4 9 ✎ 4 ✎ 36
✡1 ✡2
n ku k j [ kkv k o Q e Ł;
=
d k. k
3 ✞ 4 ✞ 12 19
✒
3´ 7
21
æ 19 ö
q = cos✥1 ç ✓
è 21 ø
v r%
mn kg j . k ✔✕ j [ kk& ; Xe %
✖✞3
✗ -1 ✘ ✞ 3
✒
=
3
5
4
✖ ✞1
✗ -4 ✘-5
✒
=
1
1
2
v kj
o Q e Ł;
d k. k K kr
d hf t , A
gy
i g y h j [ kk o Q f n o Q& v u i kr
c hp
d k d k. k
q gk
rc
cos q =
v r % v Hkh"V d k. k
✚✚✛✜
3.1 ✞ 5.1✞ 4.2
3 ✞5 ✞ 4
2
2
2
æ8 3 ö
cos✥1 çç
✙
✙
è 15 ø
n k j [ kkv k o Q e Ł;
v r fj { k e
3 ] 5 ] 4 v kj n l j h j [ kk o Q f n o Q& v u i kr 1 ] 1 ] 2 g A ; f n mu o Q
; f n n k j [ kk,
1 ✞1 ✞ 2
2
i f r PN n d j r h g
l e kr j g r k mu o Q c hp
v Fkkr
flc n l
j [ kk o Q , d
✒
16
50 6
✒
16
5 2 6
✒
8 3
15
✢✣ ✤✦✧★✩✪★ ✫✬✪★✭✮✯✩ ✰✩★✱✩✩✮ ★✱✦ ✲✬✮✩✪✳
v r f j { k e ; f n n k j [ kk,
, d
2
gA
U; u r e n j h
i j Li j
2
r k mu o Q c hp
d h U; u r e n j h ’ kU;
d h U; u r e n j h] mu o Q c hp
n l j h j [ kk i j [ khp k x ; k y c A
y cor
g A v kj
n j h g kx h
488
x f. kr
b l o Q v f r f j Dr
g kr h g
g kr h
t ks u r k i f r PN sn h v kj
g A o kL r o
v l e r y h;
(
v r f j { k e ] , l h Hkh j s[ kk, ¡
u g h l e kr j
, l h j s[ kkv ka
s o sQ ; qXe
g kr s g v kS
j b Ug a
s f o "ke r y h;
sk ew lin es)
v ko Qfr
e
j [ kk, ¡
d gr s gS
A m n kg j . kr ; k g e
,✁
1 1 -7 e
✂- v
v kj
{ k o Q v ufn’ k
e ’ k% 1 ] 3 ] 2 b d kb o Q v kd kj o ky d e j i j
f o p kj d j r
gA
GE N r
DB, A o Q
j [ kk
j [ kk
o Q f o d . k o Q v u f n ’ k g v kj
B hd
¯ i j
Nr
o Q d ku
v ko Qfr 1 1 -7
l s
x t j r h g b n ho kj o Q f o d . k o Q v u f n ’ k g A ;
j [ kk,
f o "ke r y h;
g D; kf d
o l e kr j u g h g v kj
d Hkh f e y r h Hkh u g h g A
n k j [ kkv k o Q c hp
, d
U; u r e n j h l
g e kj k v f Hki k;
flc n d k n l j h j [ kk i j f LFkr v U;
U; u r e n j h j [ kk[ kM n ku k f o "ke r y h;
✄✄☎✆☎✄
n k fo "ke r y h;
, d
, l
flc n d k f e y ku l
j [ kkv k i j y c
j [ kkv k o Q c hp
d h nj h
j [ kk[ kM l
i kI r
g k r kf d
l s
gS
g A e ku y hf t ,
f t u o Q l e hd j . k ( v ko Qfr
v kj
✕2
n k f o "ke r y h;
11.8) f u Eu f y f [
j [ kk,
kr g %
✗
✗
✗
✖ = ✘1 ✚ l ✙1
✗
✗
✗
✖ = ✘2 ✚ m ✙2
j [ kk ✕ i j d kb flc n
1
flc n
kj
T
... (1)
... (2)
Sft
f t l d h f LFkf r
l f n ’ k d k i f j e k. k]
b l d h y c kb U; u r e g kA
✝✞✟✠✡☛☞✌✍ ✎✍✡✏✍✍☞ ✡✏✑ ✠✒✍✏ ✓✟☞✍✠✔
✕1 v
d jr
j [ kk i j f LFkr
g kx kA
v c g e j [ kkv k o Q c hp d h U; u r e n j h f u Eu f y f [ kr f o f /
K kr
g t k , d
l d h f LFkf r l f n ’ k
l fn’
ST
✜
k✛
2
✜
✛1
v kj
✕2 i
j d kb Z
v ko Qfr 1 1 -8
g ] y hf t , A r c U; u r e n j h
d k U; u r e
n j h d h f n ’ kk e
i { ki
d h e ki
o Q l e ku g kx k ( v u PN n
10.6.2)A
; fn
✕1 v
kj
✕2
o Q c hp d h U; u r e n j h l f n ’ k
d h f n ’ kk e b d kb l f n ’ k
✣✤
bl
✢✢✢✗
PQ
g r k ; g n ku k
✗
✙1
v kj
✗
✙2
i j y c g kx hA
✢✢✢✗
PQ
i d kj g kx h f d
✗
✗
✙1 ´ ✙2
✗
✣✤ = ✗
| ✙1 ´ ✙2 |
... (3)
f=k&fo e h;
489
T; kfe fr
✁
PQ = ✂ ✄☎
rc
✁
t g k¡ ✂ , U; u r e n j h l f n ’ k d k i f j e k. k g A e ku y hf t ,
ST
✁
v kj
PQ
o Q c hp d k d k. k
q g]
PQ = ST |cos q|
✁
i jrq
=
✁
✂ ✄☎ ✆(✝ 2
=
✁
- ✝1 )
✂ ST
☛
bl fy ,
✁
PQ ✆ST
✁
✁
cos q =
| PQ | | ST |
☛
☛
✁
kf d
☛
(✟1 ´ ✟2 ) ✡(✠ 2 - ✠1)
☛
☛
((3)
ST ✟1 ´ ✟2
✂
; k
: i
✁
✁
✁
( ☞1 ´ ☞2 ) ✆(✝ 2 - ✝1)
✁
✁
| ☞1 ´ ☞2 |
=
(Cartesian Form)
j [ kkv k%
✌1 :
✌2
v kj
o Q c hp
o Q } kj k)
= PQ = ST |cos q|
✁
:
✍
- ✍1
✝1
✍
- ✍2
✝2
=
=
✎
- ✎1
✞
☞1
✎
- ✎2
✞
☞2
✏
- ✏1
✑1
✏
- ✏2
✑2
d h U; u r e n j h g %
✒2
- ✒1
✓2
- ✓1 ✔ 2 - ✔1
✕1
✖1
✕2
✖2
✗1
✗2
( ✖1✗2 - ✖2 ✗1 ) ✘ (✗1✕ 2 - ✗2 ✕1 ) ✘ (✕1✖2 - ✕ 2 ✖1 )2
2
2
✁
✁
ST ✞ ✝2 - ✝1 )
v Hkh"V U; u r e n j h
✂
d kr h;
( D;
gA
rc
490
x f. kr
✁✂✁✄
l e kr j
j [ kkv k o Q c hp
; f n n k j [ kk, ¡ ✓ ; f n
✓2 l
1
e kr j g
d h nj h
r k o l e r y h;
v kj
g] t gk
✓1
i j flc n
d k f LFkf r l f n ’ k
D; ksafd
M ky
nj h
x,
✓ 1,
S
✕
✖2
d k f LFkf r
l fn’
y c d k i kn
✓2 l e r y
P g rc
g kr h g A e ku k n h x b j [ kk,
✕
✕
✕
✔ = ✖1 ✘ l ✗
✕
✕
✕
✔ = ✖2 ✘ m ✗
✚
k ✙ v kj ✓ i j flc n T
2
1
... (1)
✥ (2)
h;
g A ; f n flc n
j [ kkv k
✓1 v
kS
j
✓2
T
l
✓1
i j
o Q c hp d h
= |T P|
e ku y hf t , f d
t g k j [ kkv k
l f n ’ kk
✓1 v
kj
✛✛✕
ST v
✓2 o Q r y
✕
✗ o Q c hp d k d k. k q g A r c ]
✕ ✛✛✕
✕ ✛✛✕
✗ ´ ST = ( | ✗ || ST | sin q) ✢✜
kj
✢✜
✛✛✕
✕
✕
ST = ✖2 - ✖1
i j y c b d kb l f n ’ k
i jrq
bl fy ,
( 3)
l
g e i kr
bl fy ,
K kr
v ko Qfr 1 1 -9
... (3)
gA
g fd
✕
✕
✕ ✕
✗ ´ (✖2 - ✖1 ) = | ✗ | PT ✢✜
✕ ✕
✕
✕
| ✗ ´ (✖ 2 - ✖1 ) | = | ✗ | PT ✣1
v Fkkr ~
j [ kkv k o Q c hp
( D;
PT = ST sin q)
kf d
(as | ✦✤ | = 1)
U; u r e n j h
✕ ✕ ✕
✛✛✛✕
✗ ´ (✖ 2 - ✖1 )
✕
✧ = | PT | ★
|✗ |
mn kg j . k
j [ kkv k
✓1 v
kj
✓2 o Q c hp
v kj
gy
i kr
e ’ k%
11.9)
g ( v ko Qfr
v kj
☎✆✝✞✟✠✡☛☞ ✌☞✟✍☞☞✡ ✎✠✏✠✑✑☞✑ ✑✝✡☞✞✒
l e hd j . k ( 1 )
g
fd
o ( 2)
d h
d h U; u r e n j h K kr
d hf t ,
gA
f t u o Q l f n ’ k l e hd j . k g S%
✕
✔ = ✩✜ ✘ ✜✪ ✘ l (2 ✩✜ - ✜✪ ✘ ✫✜ )
✕
✔ = 2 ✩✜ ✘ ✜✪ - ✫✜ ✘ m (3 ✩✜ - 5 ✜✪ ✘ 2 ✫✜ )
✕
✕
✕
✕
✕
✕
✔ = ✖1 ✘ l ✗1 v kj ✔ ★ ✖2 ✘ m ✗ 2 , l r y u k d
... (1)
... (2)
ju i j ge
✕
✕
✖1 = ✩✜ ✘ ✜✪ , ✗1 ★ 2 ✩✜ - ✜✪ ✘ ✫✜
✕
✕
✖2 = 2 ✤✬ + ✜✪ ✭ ✮✤ v kj ✗2 = 3 ✤✬ ✭ 5 ✪✜ + 2 ✮✤
f=k&fo e h;
✁
T; kfe fr
491
✁
✂ ✂
1 = ✄ -☎
✁
✁
✆1 ´ ✆2 = ( 2 ✄✂ - ✂✝ ✞ ☎✂ ) ´ ( 3 ✄✂ - 5 ✂✝ ✞ 2 ☎✂ )
bl fy ,
2
v kj
-
✄✂ ✝✂ ☎✂
2 -1 1
=
✟ 3 ✄✂ - ✝✂ - 7 ☎✂
3 -5 2
bl
✁ ✁
| ✆1 ´ ✆2 | =
i d kj
bl fy ,
n h x b j [ kkv k o Q c hp
✠ =
9 ✞1 ✞ 49 ✟
59
d h U; u r e n j h
✁
✁
✁
✁
( ✆1 ´ ✆2 ) ✡( 2 - 1 )
✁
✁
| ✆1 ´ ✆2 |
| 3- 0 ✞ 7 |
✟
59
✟
✌1 v kj ✌2 :
✁
✍ = ✄✂ ✞ 2 ✝✂ - 4 ☎✂ ✞ l ( 2 ✄✂ ✞ 3 ✝✂ ✞ 6 ☎✂ )
✁
✍ = 3✄✂ ✞ 3 ✝✂ - 5 ☎✂ ✞ m ( 2 ✄✂ ✞ 3 ✝✂ ✞ 6 ☎✂ ) o Q c hp
10
59
mn kg j . k ☛☞ f u Eu f y f [ kr n h x b j [ kkv k
v kj
gy
n ku k j [ kk,
✁
1
bl fy ,
l e kr j g A ( D; k\ )
= ✄✂ ✞ 2 ✝✂ - 4 ☎✂ ,
j [ kkv k o Q c hp
✁
2
ge
i kI r
g fd
= 3✄✂ ✞ 3 ✝✂ - 5 ☎✂
v kj
U; u r e n j h K kr
d hf t , A
✁
✆ = 2✄✂ ✞ 3 ✝✂ ✞ 6 ☎✂
d h nj h
✄✂ ✂✝ ☎✂
2 3 6
2 1 -1
✁ ✁
✁
✆ ´ ( 2 - 1)
✁
=
✠=
|✆ |
4 ✞ 9 ✞ 36
=
| - 9✄✂ ✞ 14 ✝✂ - 4 ☎✂ |
49
✟
293
49
✟
293
7
gA
i ’ u ko y h 1 1 -2
☛✎
n ’ kkb ,
j [ kk,
☞✎
n ’ kkb ,
fd
f n o Q& d kl kb u
i j Li j
l
o ky h r hu
gA
flc n v ksa(1,
fd
(3, 5, 6)
y cor
12 -3 - 4
4 12 3
3 - 4 12
,
,
;
,
,
;
,
,
13 13 13
13 13 13 13 13 13
✥ 1, 2), (3, 4, ✥ 2) l
t ku o ky h j [ kk i j y c g A
g kd j t ku o ky h j [ kk flc n v k
(0, 3, 2) v
kj
492
x f. kr
✁
n ’ kkb ,
(1, 2, 5)
✂✁
flc n
flc n v ksa(4,
fd
l
g kd j t ku o ky h j s[ kk] flc n v ks ( ✥
7, 8), (2, 3, 4) l
1 , ✥ 2, 1),
t ku o ky h j [ kk o Q l e kr j g A
(1, 2, 3)
l
3 ✆✝ ✞ 2 ☎✝ - 2 ✄✝ o Q
xq
l e kr j g A
✟✁
flc n f t l d h f LFkf r
2 ✆✝ - ☎ ✞ 4 ✄✝ l
l fn’ k
o ky h j [ kk d k l f n ’ k v kj d kr h;
✠✁
ml
j [ kk d k d kr h;
✍✁
, d
j [ kk d k d kr h;
: i k e l e hd j . k K kr
l e hd j . k K kr
✡ ✞3 ☛- 4 ☞✞8
✌
✌
3
5
6
✆✝ ✞ 2 ☎✝ - ✄✝
xq
d hf t ,
t k flc n
d h f n ’ kk e t ku s
d hf t , A
( ✥ 2, 4, ✥ 5)
l
t kr h g
v kj
o Q l e kr j g A
l e hd j . k
✡ -5 ☛✞ 4 ☞ -6
✌
✌
3
7
2
g A b l d k l f n ’ k l e hd j . k K kr
d hft , A
✎✁
ey
(5, ✥ 2, 3)
flc n v kj
l
t ku o ky h j [ kk d k l f n ’ k r Fkk d kr h;
d hft , A
✏✁
flc n v k
(3, ✥ 2, ✥ 5), v
e l e hd j . k d k K kr
✑✒✁
f u Eu f y f [ kr
✑✑✁
f u Eu f y f [ kr
kj
(3, ✥ 2, 6)
l
xq
d hf t , A
j [ kk& ; Xe k o Q c hp
d k d k. k K kr
d hf t , %
✔
(i) ✓ ✌ 2 ✆✝ - 5 ✝☎ ✞ ✄✝ ✞ l (3 ✆✝ ✞ 2 ☎✝ ✞ 6 ✄✝ ) v kj
✔
✓ ✌ 7 ✆✝ - 6 ✄✝ ✞ m ( ✆✝ ✞ 2 ✝☎ ✞ 2 ✄✝ )
✔
(ii) ✓ ✌ 3 ✆✝ ✞ ☎✝ - 2 ✄✝ ✞ l ( ✆✝ - ☎✝ - 2 ✄✝ ) v kj
✔
✓ ✌ 2 ✆✝ - ☎✝ - 56 ✄✝ ✞ m (3 ✆✝ - 5 ✝☎ - 4 ✄✝ )
✑✚✁ ✛
j [ kk& ; Xe k o Q c hp
(i)
✕ - 2 ✖ -1 ✗ ✘ 3
v
✙
✙
-3
2
5
(ii)
✡ ☛ ☞
✌ ✌
2 2 1
d k e ku K kr
v kj
v kS
d hf t ,
d k d k. k K kr
kS
d hf t , %
✕✘2 ✖ -4 ✗-5
✙
✙
-1
8
4
✡ -5 ☛ -2 ☞ -3
✌
✌
4
1
8
r kf d
j [ kk,
7 - 7✡ ☛ - 5 6 - ☞
✌
✌
3✛
1
5
1 - ✡ 7 ☛ - 14 ☞ - 3
✌
✌
3
2✛
2
i j Li j y c
g kA
: i k e l e hd j . k K kr
f=k&fo e h;
493
✁✂
f n [ kkb ,
✟✂
j [ kkv
✄ ☎ ✆
✞ ✞ i j Li j y c g A
1 2 3
✍
✠ ) + l (☛✠ - ✠☞ ✝ ✌✠ ) v kj ✡✍ ✞ 2 ☛✠ - ✠☞ - ✌✠ ✝ m (2 ☛✠ ✝ ✠☞ ✝ 2 ✌✠ ) o Q
ksa ✡ ✞ ( ☛✠ ✝ 2 ☞✠ ✝ ✌
c hp
d h U; u r e
✎✂
j [ kkv k
fd
j [ kk, ¡
✄ -5 ☎✝ 2 ✆
✞
✞
7
-5
1
T; kfe fr
n j h K kr
v kj
d hf t , %
✏ ✓1 ✑ ✓ 1 ✒ ✓ 1
✔
✔
-6
7
1
✄ -3 ☎-5 ✆ -7
✞
✞
-2
1
1
v kj
o Q c hp d h U; u r e n j h K kr
d hft , A
✕✂
j [ kk, ] f t u o Q l f n ’ k l e hd j . k f u Eu f y f [ kr
✖✂
j [ kk, ] f t u d h l f n ’ k l e hd j . k f u Eu f y f [ kr
g ] o Q c hp
✍
✡ ✞ (1 - ✗ ) ✠☛ ✝ (✗ - 2) ✠☞ ✝ (3 - 2 ✗) ✌✠
✍
✡ ✞ (✘ ✝ 1) ☛✠ ✝ (2 ✘ - 1) ✠☞ - (2 ✘ ✝ 1) ✌✠
✍
✡ ✞ (☛✠ ✝ 2 ☞✠ ✝ 3 ✌✠ ) + l (☛✠ - 3 ✠☞ ✝ 2 ✌✠ )
✙✙✚✛
, d
l ery
l ery
(i)
g ] o Q c hp
v kj
v kj
d h U; u r e
n j h K kr
d hf t , %
✍
✡ ✞ 4 ☛✠ ✝ 5 ✠☞ ✝ 6 ✌✠ ✝ m (2 ☛✠ ✝ 3 ☞✠ ✝ ✌✠ )
d h U; u r e K kr
d hf t , %
✜✢✣✤✥✦✧
d k v f} r h;
: i l
K kr f d ; k t k l d r k g S; f n f u Eu f y f[ kr e l
l e r y d k v f Hky c v kj e y flc n l
d kb , d ’ kr K kr g k%
l e r y d h n j h K kr g ] v Fkkr v f Hky c : i e l e r y
d k l e hd j . k
(ii)
; g , d
flc n l
(iii)
; g fn,
x,
xq
r hu v l j [ k flc n v k l
xq
v c g e l e r y k o Q l f n ’ k v kj d kr h;
✂✕✂
v fHky c
, d
l er y
; fn
✹✹✹✍
ON
v ufn’ k
d k l e hd j . k
ft l d h e y
flc n l
ew
y flc n l
✼✠
e k=kd
✺✺✺✻
ry
i j y c
g r Fkk
y cor
nj h
✸ ( ✸ ¹ 0)
g ( v ko Qfr
oQ
✹✹✹✍
ON = ✸ ✼✠
v f Hky c l f n ’ k g r c
f d l e r y i j d kb flc n
ON
d jxA
★✩✪✫✬✭✮✯✰ ✯✱ ✬ ✲✳✬✰✴ ✮✰ ✰✯✵✶✬✳ ✱✯✵✶✷
✺✺✺✻
NP , ON
v r%
e l er y
i j f o p kj d hf t ,
g A e ku y hf t ,
✺✺✺✻
: i
l e hd j . kk d k i kI r
P
g A bl f y , ]
i j y c gA
✺✺✺✻ ✺✺✺✻
NP ✽ON = 0
e ku y hf t ,
P
✹✹✹✍ ✍
NP = ✡ - ✸ ✼✠ ( D;
d h f L Fkf r
kf d
l fn ’ k
✍
✡
✹✹✹✍ ✹✹✹✍ ✹✹✹✍
ON ✝ NP ✞ OP )
... (1)
g S r ks
v ko Qfr
1 1 -1 0
1 1 -1 0 ) A
494
bl
x f. kr
i d kj
( 1)
d k : i
☎
f u Eu f y f [ kr g %
Ù
Ù
( - ✁ ✂ ) ✄✁ ✂ = 0
☎
Ù
Ù
; k
( - ✁ ✂ ) ✄✂ = 0 (✁ ¹ 0)
; k
☎ Ù
✄✂
-✁
v Fkkr ~
☎ Ù
✄✂
=
Ù
Ù
✂ ✄✂
=0
Ù
✆ (D; ksad
Ù
✝ ✞✝ ✟ 1)
✥ (2)
; g l er y
d k l f n ’ k l e hd j . k g A
d kr h;Z : i
✠✡ ☛☞✌✍ ✎✏☛✑ ✒✓☞✔✕
d k l f n ’ k l e hd j . k g t g k ✖
✗ l er y
l er y
i j d kb flc n
P(✘ , ✙, ✚)
OP =
☎ Ù
✄✂
✗
✖
=
o Q e ku k d k ( 2 )
e
★, ✩ , ✂ g A r c
i f r LFkkf i r
d ju i j
g e i kr
g]
✙ ✜✣ ✧ ✚ ✤✜ ) ✄( ★ ✜✢ ✧ ✩ ✜✣ ✧ ✂ ✤✜ ) ✦ ✁
✪✫ ✬ ✭✮ ✬ ✯✰ ✱ ✲
; g l er y
d k d kr h;
fV I i . kh
l er y
✦ ✘ ✢✜ ✧ ✙ ✣✜ ✧ ✚ ✤✜
★ ✢✜ ✧ ✩ ✣✜ ✧ ✂ ✤✜
(✘ ✢✜ ✧
✳
☎
✜ d h f n o Q& d kl kb u
✂
v Fkkr ~
l er y
gAr c
✛✛✛☎
e ku y hf t ,
o Q v f Hky c b d kb l f n ’ k g A e ku y hf t ,
... (3)
l e hd j . k g A
l e hd j . k ( 3 )
i n f ’ kr
d k l f n ’ k l e hd j . k g r k ✴✘
d jrk g
fd
+ ✵✙ + ✶✚ = ✁
; fn
l er y
☎
✄( ✴ ✜✢ ✧ ✵ ✜✣ ✧ ✶ ✤✜ ) = ✁ , d
d k d kr h;
l e hd j . k g
t g k¡
✴ , ✵ v kj ✶ l e r y o Q v f Hky c o Q f n o Q& v u i kr g A
mn kg j . k ✷✸ ml
v kj e y
gy
flc n l
e ku y hf t ,
6
l e r y d k l f n ’ k l e hd j . k K kr
b l d k v f Hky c l f n ’ k
☎
✂ ✦
2 ✢✜ - 3
✜✣ ✧
✜ ✦
✂
|
2
4 ✤✜
☎
✂
☎
✂|
✻✼
d hf t ,
t k ey
flc n l
29
d h nj h i j g S
- 3 ✼✺ ✽ 4 ✹✼ g A
gA r c
=
2 ✢✜ - 3 ✜✣
4
✧
✧
4 ✤✜
9 ✧ 16
✦
2 ✢✜ - 3
✣✜ ✧ 4 ✤✜
29
f=k&fo e h;
bl fy ,
l er y
d k v Hkh"V l e hd j . k
æ 2
✠
✠
mn kg j . k ✡☛ l e r y
f n o Q& d kl kb u K kr
l er y
l e hd j . k d k b l
l er y
bl l
D; kf d
i d kj
2
ry
✝ ( - 3) ✝ 4
i j g e i kI r
3
æ
è
6 ✂
✁
7
✠
✄✂ ✝ 2 ☎✂ |
✘✏
d h ey
M ky x ,
y c b d kb l f n ’ k d h
-
2
3
7
✝
2
7
✂✄ ✝
✞ ✌ oQ: i
6 ✂
✁
7
✝
3
7
✂✄ ✝
-3
2
2
✝
fd ; k t k l d r k g %
... (1)
✝
Hkkx d j u i j g e i kr
( - 3)
2
✝4
2
ö
☎✂ ✟
ø
✞
4
7
g fd
1
7
=
d k gA
2 ✂
☎ l e r y o Qy c
7
-6 3 2 gA
, ,
7
7 7
d h ey
flc n l
2, ✥ 3, 4
b d kb l f n ’ k g t k e y
n j h K kr
g bl fy ,
2
2
✝ ( - 3) ✝ 4
v Fkkr
2
2
flc n q
d hf t , A
b l d h f n o Q& d kl kb u g %
4
,
2 ✏ ✥ 3✑ + 4 ✒ ✥ 6 = 0
,v
2
Fkkr ~
2✏ ✥ 3 ✑ + 4 ✒ = 6
,
-3
29
d ks
29
29
l
,
4
29
Hkkx d j u s
g%
✓ ✖
-3
29
29
+ ✙✑ + ✍✒ = ✌ , o Q : i e
6
l er y
i j e y flc n l
36 ✝ 9
=
✍✂ d h f n o Q& d kl kb u
,
l e hd j . k
d jr
gA
29
=1
. ✍✂
2
v kj ; g
✂✄ ✝ 2
3
o Q v f Hky c o Q f n o Q& v u i kr
2
bl fy ,
✝
☞
☎)
2✏ ✥ 3✑ + 4✒ ✥ 6 = 0
2
2
✝
✆ç -
✍✂ ✞
Li "V g f d
mn kg j . k ✡✎ l e r y
gy
- 6 ✁✂
d k l e hd j . k
x t j r k g A bl
ø
29
i d kj O; Dr
o Q n ku k i { kk d k 7 l
✠
l
✆(
| - 6 ✁✂
v c
t k fd
=0
6
d hf t , A
o Q K kr
( 1)
29
ö
☎✂ ✟ ✞
4
✂✄ ✝
✆(6 ✁✂ - 3 ✄✂ - 2 ☎✂ ) ✝ 1
✠
bl fy ,
-3
✁✂ ✝
✆ç
è 29
gy
495
T; kfe fr
flc n l
nj h
29
gA
✔ ✖
4
29
g t gk ey
✕ ✗
flc n l
6
29
l er y
d h nj h ✌ g A bl fy ,
496
x f. kr
✁
mn kg j . k
i j M ky
gy
x,
e y flc n l
y c o Q i kn o Q f u n ’ kkd
e ku y hf t ,
i kn
P
2✂ ✥ 3 ✄ + 4 ☎ ✥ 6 = 0
l er y
e y flc n l
l er y
y c oQ
11.11)A
ko Qfr
✂1, ✄1, ☎1
r c j [ kk OP o Q f n o Q& v u i kr
d hf t , A
i n M ky x ,
(✂1, ✄1, ☎1) g ( v
o Q f u n ’ kkd
K kr
gA
l e r y d h l e hd j . k d k v f Hky c o Q : i e f y [ ku i j g e
v ko Qfr
i kr
g
2
29
t gk
1 1 -1 1
fd
OP
D; kf d
3
✂-
29
2
o Q f n o Q& v u i kr
, d
-3
,
29
4
✄✆
29
29
f n o Q& v u i kr
-3✟
, ☎1 ✝
✂1 =
b u e ku k d k l e r y
o Q l e hd j . k e i f r LFkkf i r
. kh
f n o Q& d kl kb u
✏✁✏✑
, d
; fn
ey
flc n
✌, ✍ , ✎ g k r c
fn ,
29
æ 12 , -18 , 24 ö
ç
✡
è 29 29 29 ø
v r % y c o Q i kn o Q f u n ’ kkd
☛fV I i
, ✄1 =
l
gA v r %
y c d k i kn
4✟
29
d ju i j
g e i kr
g fd
✟=
6
29
gA
☞
g k v kj
(✌☞ , ✍☞ , ✎☞ )
g kr k g A
l er y
l fn ’ k o Q v u y c
l e ku i kr h g kr
☎1
=✟
4
29
v Fkkr ~
29
29
gA
29
✞1
✄1
✝
=
-3
2
29
29
2✠
6
4
,
j [ kk o Q f n o Q& d kl kb u v kj
☎✝
d h nj h
r Fkk fn ,
l er y
o Q v f Hky c
flc n l s
g kd j t ku o ky l e r y d k l e hd j . k ✒✓✔✕✖✗✘✙✚ ✙✛ ✖
✜ ✢✖ ✚✣ ✜ ✣✤✜✣✚✦✘✧✕✢✖ ✤ ✗ ✙ ✖ ★✘✩✣✚ ✩✣✧✗✙ ✤ ✖✚✦
✜✖✪✪✘✚★ ✗✫✤✙✕★✫ ✖ ★✘✩✣✚ ✜✙✘✚✗✬
v r fj { k e ] , d
gk l d r
i d kj
g
i jr
d k o Qo y
v ko ‘Qfr
fn,
, d
, d
11- 12) A
x ,
l fn’ k o Q v uy c v uo Q l e r y
fn,
l er y
x,
flc n
P(✂ 1, ✄1, ☎1)
l s bl
d k v f L r Ro g kr k g ( n sf [ k,
v ko Qfr
1 1 -1 2
d h
f=k&fo e h;
e ku y hf t ,
l fn’ k
y hf t ,
✁
g] l
fd
( v ko Qfr
fd
l er y
, d
flc n
t kr k g v kj l f n ’ k
l e r y i j f d l h flc n
✂✁
N
P
A,
f t l d h f LFkf r
o Q v u y c g A e ku
✁
✄ gS
d k f LFkf r l f n ’ k
1 1 -1 3 ) A
r c flc n qP l e r y
✂✂✂✁ ✂✁
; f n AP , N
✂✂✂✁ ✁ ✁
AP ✝ ✄ - . b l
i j
e f LFkr
y ac
g]
g kr k g ] ; f n v kj
v Fkkr
o Qo y
✂✂✂✁ ☎✆
AP . N = 0 .
i j ar q
fy ,
v ko Qfr
1 1 -1 3
✁ ✁ ✁
( ✄ - ) ✞N ✝ 0
; g l er y
d k l f n ’ k l e hd j . k g A
d kr h;Z : i
✟✠✡☛☞✌ ✍✎✡✏ ✑✒☛✓✔
e ku y hf t ,
fd
f n o Q& v u i kr
v c
f n ; k flc n
mn kg j . k ✰✱ ml
d k v f Hky c
f n o Q& v u i kr
l fn’ k
l er y
d k l
; k
( 1)
v kj
l er y
d kb flc n qP
i j
v kj
( ✕ , ✖ , ✗)
☎✆
N
g r Fkk
oQ
✁
N ✝ A ✙✘ ✜ B ✚✘ ✜ C ✛✘
l e r y d k l f n ’ k v kj d kr h; l e hd j . k K kr d hf t , ] t k flc n
2, 3, ✢ 1
g e t ku r g f d
bl fy ,
A (✕1, ✖1, ✗1)
é✣ ✕ - ✕1 ✤ ✙✘ ✜ ✣ ✖ - ✖1 ✤ ✚✘ ✜ ✣ ✗ - ✗1 ✤ ✛✘ ù ✞(A ✙✘ ✜ B ✚✘ ✜ C ✛✘) ✝ 0
ë
û
✦✟✧ ★ ✧ ✩✔ ✪ ✫✟✬ ★ ✬ ✩✔ ✪ ✠ ✟✭ ★ ✭✩✔ ✮ ✯
v Fkkr ~
gy
✥ (1)
A, B r Fkk C g ] r c
✁
✁
✝ ✕1 ✙✘ ✜ ✖1 ✘✚ ✜ ✗1 ✛✘, ✄ ✝ ✕ ✙✘✜ ✖ ✚✘ ✜ ✗ ✛✘
✁ ✁ ✁
( ✄ ✢ ) ✞N= 0
bl fy ,
g v kj
497
T; kfe fr
d k d kr h;
: i
flc n
(5, 2, ✢ 4) d
k f LFkf r
l fn’ k
✁
✝ 5 ✙✘ ✜ 2 ✘✚ - 4✛✘
✁
N =2 ✘✙ + 3 ✘✚ - ✛✘ g A
✁ ✁ ✁
f n ’ k l e hd j . k ( ✄ - ) ✞N ✝ 0 l i n k g A
✁
[ ✄ - (5 ✙✘ ✜ 2 ✘✚ - 4 ✛✘ )] ✞(2 ✙✘ ✜ 3 ✚✘ - ✛✘ ) ✝ 0
e
: i kr j . k d j u i j g e i kr
g ] fd
[( ✕ ✢ 5) ✙✘ ✜ ( ✖ - 2) ✘✚ ✜ (✗ ✜ 4) ✛✘] ✞(2 ✙✘ ✜ 3 ✘✚ - ✛✘) ✝ 0
2( ✕ - 5) ✜ 3( ✖ - 2) - 1( ✗ ✜ 4) ✝ 0
2 ✕ + 3 ✖ ✢ ✗ = 20
; k
v Fkkr ~
t k l er y
d k d kr h;
(5, 2, ✢ 4) l
t kr k
o ky h j [ kk i j y c g A
l e hd j . k g A
g v kj l e r y o Q y c
... (1)
498
x f. kr
✁✂✁✄
r hu v l j [ kh;
flc n v k l
g kd j
t ku o ky s
l er y
d k l e hd j . k ☎✆✝✞✟✠✡☛☞ ☛✌ ✟ ✍ ✎✟☞✏
✍✟✑✑✡☞✒ ✠✓✔☛✞✒✓ ✠✓✔✏✏ ☞☛☞✕✖☛✎✎✡☞✏✟✔ ✍☛✡☞✠✑✗
e ku y hf t ,
l ery
i j
f LFkr
r hu
v l j [ k flc n qv ksa
✙ ✙
✙
R, S v kSj T o Q f LFkf r l f n ’ k e ’ k% ✘ , ✚ v kSj ✛ g aS
( v ko Qfr 11.14) A
✜✜✜✢
✣✣✣✙
l f n ’ k RS v kj RT f n , l e r y e g A b l f y ,
✣✣✣✙ ✣✣✣✙
l f n ’ k RS ´ RT flc n v k R, S v kj T d k v Ur f o "V
d j u o ky l e r y
d kb flc n
Pd
i j y c g kx kA e ku y hf t ,
v ko Qfr
ea
s
✙
✤ g A b l f y , R l t ku o ky
✙ ✙ ✣✣✣✙ ✣✣✣✙
(✤ - ✘ ) ✥(RS´ RT) = 0 g A
✙ ✙
✙ ✙
✙ ✙
✦ ✔ - ✟ ✧ ✁ ★✦✫ ✩ ✟ ✧ ´ ✦ ✖ ✩ ✟ ✧✪ ✬ ✭
k f LFkf r l f n ’ k
d k l e hd j . k
; k
; g r hu v l j [ k flc n v k l
i k: i
l er y
x t j u o ky
l er y
1 1 -1 4
✣✣✣✙ ✣✣✣✙
RS ´ RT
r Fkk l f n ’ k
i j y c] l er y
✮ (1)
o Q l e hd j . k d k l f n ’ k
gA
✯fV I i
. kh mi j kDr i f ; k e r hu v l j [ k flc n d g u k D; k v ko ’ ; d
g \ ; f n flc n , d
g kx ( v ko Qfr
;
g h j [ kk i j f LFkr g r c ml l
1 1 -1 5 ) A
l er y
, d
i Lr d
flc n v k
R, S
v kj
T
i Lr d
o Q i "B k o Q c /a u o ky
d kr h;Z : i
e ku y hf t ,
e ku y hf t ,
o Q i "Bk d h Hkkf r
d k v r f o "V d j u
LFkku d k l n L;
gA
v ko Qfr
1 1 -1 5
✦✰✱✲✳✴ ✵✶✱✷ ✸✹✲✺✧
R, S v kj T o Q f u n ’ kkd e ’ k% (✻1, ✼1, ✽1), (✻2, ✼2, ✽2) v kj (✻3, ✼3, ✽3) g A
✙
f d l e r y i j f d l h flc n P o Q f u n ’ kkd ( ✻, ✼, ✽) o b l d k f LFkf r l f n ’ k ✤ g A r c
✣✣✣✙
RP = ( ✻ ✾ ✻1) ✿❀ + (✼ ✾ ✼1) ❁❂ + ( ✽ ✾ ✽1) ❃✿
✣✣✣✙
RS = ( ✻2 ✾ ✻1) ✿❀ + (✼2 ✾ ✼1) ❁❂ + (✽2 ✾ ✽1) ❃✿
✣✣✣✙
RT = ( ✻3 ✾ ✻1) ❀✿ + (✼3 ✾ ✼1) ❂❁ + ( ✽3 ✾ ✽1) ❃✿
o Q l e hd j . k ( 1 )
❈ - ❈❄
t k r hu flc n v k
i k: i
g kx t g k¡
o ky h j [ kk
flc n v k
b u e ku k d k l f n ’ k i k: i
d kr h;
x t j u o ky d b l e r y
❈❅ - ❈❄
❈❆ - ❈❄
( ✻1, ✼1, ✽1), ( ✻2,
gA
❉ - ❉1
e i f r LFkki u d j u i j
❊ - ❊1
❉❅ - ❉❄ ❇ ❅ ❋ ❊❄ ✬ ✭
❉❆ - ❄❉ ❇ ❆ ❋ ❊❄
✼2, ✽2) v kj ( ✻3, ✼3, ✽3)
l
xq
g e i kr
g fd
f=k&fo e h;
mn g kj . k
✁
flc n v k
l e hd j . k K kr
gy
rc
T (5, 3,✥ 3) l
t ku o ky l e r y d k l f n ’ k
✠
☛
v kj
✠
✠
✠
✄ ✞ 2 ☎✂ ✟ 5 ✆✂ - 3 ✝✂ , ✡ ✞ - 2 ☎✂ - 3 ✆✂ ✟ 5 ✝✂ , ☛ ✞ 5 ☎✂ ✟ 3 ✂✆ - 3 ✝✂
l
t ku o ky
v Fkkr ~
l er y
l er y
d k l f n ’ k l e hd j . k f u Eu f y f [ kr
g%
✠ ✠ ✍✍✍✠ ✍✍✍✠
= 0 ( D; k?sa )
(☞ - ✄ ) ✌(RS
✠ ´✠RT) ✠ ✠
✠ ✠
(☞ - ✄ ) ✌[(✡ - ✄ ) ´( ☛ - ✄ )] = 0
✠
[☞ - (2 ☎✂ ✟ 5 ✆✂ - 3 ✝✂)]✌[( -4 ✂☎ - 8 ✂✆ ✟ 8 ✝✂ )´ (3 ✂☎ - 2 ✆✂)]✞ 0
; k
✎✏✎✑
✙★✦✔✖✩
kj
d hf t , A
e ku y hf t ,
✠ ✠
✄, ✡
R(2, 5, ✥ 3), S(✥ 2, ✥ 3, 5) v
499
T; kfe fr
o Q l e hd j . k d k v r % [ kM&: i
✒✓✔✕✖✗✘✖✙✕ ✚✛✗✜ ✛✚ ✕✢✖ ✖✣✤✦✕✧✛✔ ✛✚ ✦
bl
v u PN n e ] g e l e r y o Q l e hd j . k d k] ml o Q } kj k f u n ’ kk{ kk i j d V v r % [ kM o Q : i
K kr
d j x A e ku y hf t ,
l er y
A✪ + B✫ + C✬ + D = 0 (D ¹ 0)
e ku y hf t , l e r y
Li "Vr % l e r y
} kj k ✪ ,
✪, ✫ v
✫, v
kjS ✬- v { kksl
e ’ k% flc n v k
... (1)
gA
kjS ✬✭ v { kk i j d V v r % [ kM
e ’ k%
✄, ✡
v kj
( ✄ , 0, 0), (0, ✡ , 0), v
☛ (v
; kA
b u e ku k d k l e r y o Q l e hd j . k ( 1 )
d ju i j ge
11.16) g A
☛) i
j fe y r k g A
kjS (0, 0,
i kr
=
e i f r LFkkf i r d j u v kj
g fd
v ko Qfr
✪ ✫ ✬
✟ ✟ =1
✄ ✡ ☛
t k v r % [ kM : i
mn kg j . k
ko Qfr
-D
✮
-D
B✡ + D = 0 ; k B =
✡
-D
C☛ + D = 0 ; k C =
☛
A✄ + D = 0
bl fy ,
l jy
ea
s
d k l e hd j . k
✯
ml
e l er y
l er y
1 1 -1 6
... (2)
d k v Hkh"V l e hd j . k g A
d k l e hd j . k K kr
d hf t ,
t ks✪ ,
✫
v kj
✬- v
{ kk i j
e ’ k% 2 ] 3 v kj
4 v r % [ kM d kVr k g A
gy
e ku y hf t , ] l e r y
; g k¡
✪ ✫ ✬
✟ ✟ =1
✄ ✡ ☛
✄ = 2, ✡ = 3, ☛ = 4 K kr g A
d k l e hd j . k g A
... (1)
500
x f. kr
,✁
✂ o Q bu
v kj
✄ ☎ ✆
✝ ✝ ✞1
2 3 4
; k
e ku k d k ( 1 )
e i f r LFkkf i r
6✄ + 4 ☎ + 3✆ = 12
d ju i j ge
i kI r d j r
p1 v kSj p2 n k l e r y ] f t u o Q l
★
★
✦ . ✧✥1 = ✩ 1 v kSj ✦ . ✧✥2 = ✩ 2 g b u o Q i
y hf t ,
e ’ k%
j [ kk i j
f L Fkr
f d l h flc n d k f LFkf r
l e hd j . kk d k l r q"V d j x k ( v ko Qf r
★ ★
✪ ✫✧1 = ✩ 1 v
loQl
b l hf y ,
D; kf d
✭
✬
Hkh o kLr f o d
kj
★
n ka
su ka
s
g ] r ks
★ ★
✪ ✫✧2 = ✩ 2
e ku k o Q f y ,
☞✌✍✎✏✑ ✒✎✓✓✔✏✕
f r PN n u
j [ kk i j f LFkr f d l h flc n d h f LFkf r l f n ’ k ✪
; fn b l
t ku o ky k l e r y
e hd j . k
l fn’ k bu
11.17)A
d k v Hkh"V l e hd j . k
gA
✟✟✠✡✠☛ n k fn , l e r y k o Q i fr PN n u l g kd j
✖✗✘✙✚✕✗ ✖✗✑ ✔✏✖✑✘✓✑✛✖✔✙✏ ✙✜ ✖✢✙ ✕✔✣✑✏ ✒✍✎✏✑✓✤
e ku
l er y
g e i kr
v ko Qfr
1 1 -1 7
g fd
★ ★
★
✪ ✫(✧1 ✝ l✧ 2 ) = ✩1 ✝ l✩ 2
Lo PN g b l f y ,
; g j [ kk o Q f d l h flc n d k l r "V d j r k g A
✴ ✴
✴
✮ ✱( ✯1 ✲ l✯2 ) ✳ ✰1 ✲ l ✰ 2
l e r y p d k fu: fi r d j r k g t k ,
3
★
✦
, p1 v kj p2, o Q l e hd j . kk d k l r "V d j r k g r k o g p3 d k v o ’ ;
d kb l f n ’ k
★ ★
★ ★
v r % l e r y k ✦ ✫✧1 = ✩1 v kj ✦ ✫✧2 ✞ ✩ 2 o Q i f r PN n u j [ kk l
t ku o ky f d l
★ ★
★
l e hd j . k ✘ ✫✷ ✏✵ ✝ l✏✶ ✸ ✹ ✺ ✼ l✺ g A
✻
✽
bl
i d kj l e hd j . k
d kr h;Z : i
✷✾✿❀❁❂ ❃❄✿❅ ❆❇❀❈✸
d kr h;
o Q fy ,
: i
e ku k
d k i fj o fr r
: i
; fn
l r "V d j x kA
h l er y
d k
... (1)
✴
✯1 = A1 ❉✥✝ B 2 ✥❊ ✝ C1 ❋✥
✴
✯2 = A2 ❉✥ ✝ B 2 ✥❊ ✝ C 2 ❋✥
★
✦ = ✄❉✥ ✝ ☎ ✥❊ ✝ ✆ ❋✥
v kj
r k ( 1)
l k g fd
g%
✄ (A1 + lA2) + ☎ (B1 + lB2) + ✆ (C1 + lC2) = ✩ 1 + l✩ 2
✷●✻❍ ✼ ■ ✻❏ ✼ ✾ ✻❑ ▲ ✺ ✻✸ ✼ l✷●✽❍ ✼ ■ ✽❏ ✼ ✾ ✽❑ ▲ ✺ ✽✸ ✹ ▼
; k
t k i R; d
l
l e hd j . k g A
o Q fy ,
fn,
l e r y k o Q i f r PN n u j [ kk l
g kd j t ku o ky f d l h l e r y
... (2)
d k d kr h;
f=k&fo e h;
✁
mn kg j . k
(1,1,1)
gy
l
; gk
bl fy ,
l
l er y k
t ku o ky
✡
✄ ✞(☎✂ ✟ ✂✆ ✟ ✝✂ ) ✠ 6 v
kj
✡
✄ ✞(2☎✂ ✟ 3 ✂✆ ✟ 4✝✂ ) ✠ - 5, o Q i
l e r y d k l f n ’ k l e hd j . k K kr
✡
✡
☛1 ✠ ☎✂ ✟ ✆✂ ✟ ✝✂ v kj ☛2 = 2☎✂ ✟ 3 ✂✆ ✟ 4 ✝✂
✡ ✡
✡
=k ✄ ✞(☛1 ✟ l☛2 ) ✠ ☞1 ✟ l ☞ 2 d k i ; kx d
T; kfe fr
501
f r PN n u r Fkk flc n q
d hf t , A
v kj
☞1 = 6
v kj
☞2 = ✥5
gA
ju i j]
✡
✄ ✞[☎✂ ✟ ✆✂ ✟ ✝✂ ✟ l (2☎✂ ✟ 3 ✆✂ ✟ 4 ✝✂ )] = 6 - 5 l
✡
✄ ✞[(1 ✟ 2 l) ☎✂ ✟ (1 ✟ 3 l) ✆✂ ✟ (1 ✟ 4 l)] ✝✂ = 6 - 5 l
; k
t gk
l,
d
o kLr f o d
✌ (1)
l [ ; k gA
✡
✄ ✠ ✍☎✂ ✟ ✎ ✆✂ ✟ ✏ ✝✂ , j [
ku i j g e i kr
g fd
(✍☎✂ ✟ ✎ ✆✂ ✟ ✏ ✝✂)✞[(1 ✟ 2l )☎✂ ✟ (1 ✟ 3l ) ✂✆ ✟ (1✟ 4l ) ✝✂ ]✠ 6 - 5l
; k
(1 + 2l ) ✍ + (1 + 3l) ✎ + (1 + 4l) ✏ = 6 ✥ 5l
; k
(✍ + ✎ + ✏ ✥ 6 ) + l (2✍ + 3 ✎ + 4 ✏ + 5) = 0
v c i ’ u ku l kj
v Hkh"V l e r y
flc n ( 1 ] 1 ] 1 )
l
t kr k g ] v r % ; g flc n ] ( 2 )
... (2)
d k l r "V d j x k
v Fkkr ~
(1 + 1 + 1 ✥ 6) + l (2 + 3 + 4 + 5) = 0
l=
; k
l o Q bl
e ku d k ( 1 )
3
14
e i f r LFkkf i r
d ju i j ge
i kr
g ] fd
æ 6ö ù
✙ éæ 3 ö æ 9 ö
✒ ✖ ê ç1 ✗ ✘✓✑ ✗ ç1 ✗ ✘ ✔✑ ✗ ç 1 ✗ ✘✕✑ ú = 6 - 15
è
ø
è
ø
è 7ø û
7
14
ë
14
69
★ æ 10 23 ✚ 13 ✚ö
✛ ✤ ç ✚✜ ✦
✢ ✦ ✣✧ =
è 7
14
14
7 ø
✡
✄ ✞(20☎✂ ✟ 23 ✆✂ ✟ 26 ✝✂) = 69
; k
; k
t k l er y
✩✩✪✫
d k v Hkh"V l f n ’ k l e hd j . k g A
n k j [ kkv k d k l g &r y h;
e ku y hf t ,
fd
n k K kr
g ku k
✬✭✮✯✰✱✲✱✳✴✵✶ ✮✷ ✵✸✮ ✰✴✲✹✺✻
j [ kk, %
✡
✡
✡
✄ = ✼1 ✟ l ✽1
... (1)
502
x f. kr
✁
r Fkk
j [ kk ( 1 )
flc n
A,
flc n
B ft
✁
✁
✂2 ☎ m✄2 g aS
=
✁
l f n ’ k ✂2 g ] l
l d h f LFkf r
✆✆✆✁
AB =
K kr
j [ kk,
l g & r y h;
✁
✁
✂2 - ✂1
d kr h;Z : i
✞✟ ✠✡☛☞ ✌✍✠✎ ✏✑✡✒✓
e ku y hf t ,
fd
fd
✆✆✆✁ ✁
g ] ; f n v kj o Qo y
✆✆✆✁ ✁
✁
AB.(✄1 ´ ✄2 )
Av
flc n v k
✁
✁
✄1 v kj ✄2
... (2)
✁
g kd j t kr h g r Fkk ✄ o Q l e kr j g A j [ kk ( 2 )
1
✁
g kd j t kr h g r Fkk ✄ o Q l e kr j g A r c
2
✁
f t l d h f LFkf r l f n ’ k ✂1 g ] l
kj
=0
B o Q fun’
✁
-
; k ( ✂2
1
✁
kj ✄2 l g & r y h;
✁ ✁
✁
✂1 ) ✝(✄1 ´ ✄2 )
e ’ k% ( ✔
kkd
e ’ k% ✂
o Q f n o Q& v u i kr
AB, ✄1 v
; fn
1
g A v Fkkr ~
=0
, ✕ 1, ✖1) v
kj
( ✔2, ✕ 2, ✖2)
g A e ku y hf t ,
✄ 1, ✗1 r Fkk ✂ 2, ✄ 2, ✗2 g A r c
,
✆✆✆✁
AB ✜ ( ✔2 - ✔1 ) ✙✘ ☎ ( ✕ 2 - ✕1 ) ✘✚ ☎ (✖ 2 - ✖1 )✛✘
K kr
j [ kk,
e O; Dr
✁
✁
✄1 ✜ ✂1 ✙✘ ☎ ✄1 ✘✚ ☎ ✗1 ✛✘ ; v kj ✄2 ✜ ✂2 ✘✙ ☎ ✄2 ✘✚ ☎ ✗2 ✛✘
✆✆✆✁ ✁ ✁
l g & r y h; g ] ; f n v kj o Qo y ; f n AB✝( ✄ ´✄ ) ✜ 0 f t l f u Eu f y f [ kr d kr h;
1
2
d j
l d r
gA
✔2 - ✔1
mn kg j . k ✢✣ n ’ kkb ,
✔
+3
✜
✥3
gy
fd
✄1
✂2
✄2
✔1
j [ kk,
l kj f . kd
l e & r y h;
✖ 2 - ✖1
✗1
... (4)
✜
0
+1
✩
✤1
✧
-2 ★ - 5
✩
2
5
= ✥ 3, ✕1 = 1, ✖1 = 5,
✂1
= ✥ 3, ✄ 1 = 1,
✗2
✖
-5
5
r Fkk
✦
l g & r y h;
✗1 =
= ✥ 1, ✕ 2 = 2, ✖2 = 5, ✂ 2 = ✥ 1, ✄ 2 = 2, ✗2 = 5
y u i j
✪2
bl fy ,
- ✕1
✂1
-1
✜
1
✕
✔2
✕2
j [ kk, ¡
; g k g e K kr g f d
v c f u Eu f y f [ kr
: i
- ✪1
g e i kr
✫2
- ✫1
g fd
✬2
- ✬1
✭1
✮1
✯1
✭2
✮2
✯2
gA
✰
2 1 0
-3 1 5
-1 2 5
✰0
5
gA
f=k&fo e h;
✁✂
n k l e r y k o Q c hp
d k d k. k ✄☎✆✝✞✟
i fj Hkk"kk ✑ n k l e r y k o Q c hp
( v ko Qfr
(v
ko Qfr
flc n l
fd
l e r y k]
l e r y k i j [ khp
x,
✠✟✡☛✟✟✆ ✡☛☞ ✌✞✍✆✟✎✏
d k d k. k mu o Q v f Hky c k o Q e Ł; LFk d k. k } kj k i f j Hkkf "kr
11.18 (a))A Ł; ku n hf t , f d
11.18 (b)) Hkh mu o Q c hp d k d
e ku y hf t ,
503
T; kfe fr
; f n n k l e r y k o Q c hp
d k d k. k
q
g rk
gS
180 ✥ q
k.s k g A g e U; u d k. k d k g h l e r y kso Q c hp d k d k. k y x A
✕ ✖
✒ ✔✓1 = ✗ 1 v
✕ ✕
✒ ✔✓ 2 ✘ ✗ 2
kj
v f Hky c k o Q c hp
d k d k. k
o Q c hp d k d k. k q g A r c f d l h l ko Z
q
gA
✛ ✛
✙1 ✚✙2
cos q = ✛
✛
| ✙1 | | ✙2 |
rc
v ko Qfr 1 1 -1 8
✜fV I i . kh
✢✣2 l e kr j g A
d kr h;Z : i
e ku y hf t ,
o Q c hp
d
n ku k l e r y
i j Li j
y cor
; fn
✢✣1 . ✢✣2 = 0
v kj
l e kr j
g
; fn
✢✣1
v kj
l e r y k%
A1 ✲ + B1 ✳ + C1✴ + D1 = 0
k d k. k q g A
cos q =
1.
; fn
✤✦ ✧★✩✪✫✬✧✭ ✮✯★✰✱
r k l e r y k o Q v f Hky c o Q f n o Q& v u i kr
✜fV Ii
g
v kj
e ’ k%
A2✲ + B2 ✳ + C2 ✴ + D2 = 0
A1, B1, C1 v
A2, B2, C2 g A
kj
bl fy ,
A1 A 2 ✵ B1 B 2 ✵ C1 C 2
A21
✵ B21 ✵ C12
A22 ✵ B 22 ✵ C 22
. kh
n ku ks l e r y
i j Li j
y c
g
rc
cos q = A1A2 + B1B2 + C1C2 = 0
q = 90°
v kj
bl
rj g
cos q = 0.
v r %
504
x f. kr
2.
; f n n ku k l e r y
l e kr j
mn kg j . k ✁✁ n k l e r y k 2 ✂
} kj k K kr
g r ks
A1
A2
B1
B2
+ ✄ ✥ 2☎ = 5 v
kj
3 ✂ ✥ 6 ✄ ✥ 2☎ = 7 o Q c hp
n k l e r y k o Q c hp
fn,
x,
l e hd j . kk l
d k d k. k o g h g t k mu o Q v f Hky c k o Q c hp
N1 = 2 ✟✞ ☛ ✞✠ - 2 ✡✞
cos q =
bl fy ,
d k d k. k g A l e r y k o Q
l e r y k o Q l f n ’ k v f Hky c
✆✝
✓✔ ✓✔
✓✔N1 ✏N✓✔2
| N1 | |N 2 |
✆✝
N2
v kj
3 ✟✞ - 6 ✞✠ - 2 ✡✞
gA
✒
æ4ö
(2 ✌☞ ✑ ☞✍ - 2 ✎☞ ) ✏(3 ☞✌ - 6 ☞✍ - 2 ✎☞ )
= ç ✕
4 ✑ 1 ✑ 4 9 ✑ 36 ✑ 4
è 21 ø
v kj
2✂ + 2 ✄ ✥ 2☎ = 5 o Q c hp
æ4ö
q = co s✖1 ç ✕
è 21ø
v r%
mn kg j . k ✁✗ n ksl e r y k 3 ✂ ✥
l e r y k d h K kr
6 ✄ + 2☎ = 7
d ju i j ge
i kr
d k d k. k K kr d hf t , A
l e hd j . kk d h r y u k l e hd j . kka
s
A1 ✂ + B1 ✄ + C1 ☎ + D1 = 0
l
d k d k. k l f n ’ k f o f /
d hf t , A
gy
gy
C1
C2
v kj
A2 ✂ + B2 ✄ + C2 ☎ + D2 = 0
A1 = 3, B1 = ✥ 6, C1 = 2
g fd %
A2 = 2, B2 = 2, C2 = ✥ 2
i u%
cos q =
=
bl fy ,
3 ´ 2 ☛ ( -6) (2) ☛ (2) (-2)
✘ 32 ☛ (- 6)2 ☛ (-2)2 ✙
-10
5
7´2 3
7 3
æ5 3ö
q = cos-1 çç
✚✚
è 21 ø
5 3
21
✘ 2 2 ☛ 22 ☛ (-2)2 ✙
f=k&fo e h;
✁✂
l ery
fn ,
x,
flc n d h n j h
✒✓✔✕✖✗✘ ✙✗✘✚✛
l fn ’ k : i
flc n qP f t l d k f LFkf r
, d
(v
l
11.19)
ko Qf r
l f n’ k
✢
✜
✄☎✆✝✞✟✠✡☛ ☞✌ ✟ ✍☞✆✠✞ ✌✎☞✏ ✟ ✍✑✟✠☛ )
v kj
, d
b d kb l f n ’ k
P
l
★✩
gAv
l er y
v Fkkr ~
v r %] e y
( v ko Qfr
l ery
p1
f t l d k l e hd j . k
✢
✤ ✦✥✣ = ✧
i j f o p kj d hf t , A
v ko Qfr
i u % flc n
505
T; kfe fr
flc n l
p1 o Q l
e kr j
l er y
1 1 -1 9
p2 i
j f o p kj
d hf t , A l e r y
✢ ✢
r % b l d k l e hd j . k ( ✤ - ✜ ) ✦✥
✣ ✪ 0 gA
✭
✭
✫ ✬★✩ = ✮ ✬★✩
✳
bl
l e r y d h n j h ON¢ = | ✰ ✲✱✯ | g A b l
P
fy ,
l
p2 o Q v
l er y
p1 l
f Hky c
nj h
11.21 (a))
✳
PQ = ON ✴ ON¢ ✵ |✧ ✴ ✰ ✲✱✯ |
g] t k , d
flc n l
K kr
d k i f j . kke LFkkf i r
✶fV Ii
1.
l e r y i j y c d h y c kb g A v ko Qfr
d j l d r
11.19 (b)
o Q fy ,
. kh
; fn l e r y
r k y kf c d
p2 d
nj h
k l e hd j . k
✢ ✢
| ✜.N -✧|
✢
✢ ✷✢
✤ . N ✪ ✧ , o Q:
i
d kg] t gk
✢
N
l e r y i j v f Hky c g S
gA
|N|
2.
ey
g e b l h i d kj
gA
flc n
Ol
l er y
✢ ✢
✤ .N✪ ✧
d h nj h
|✸|
✹✺
|N|
g
( D;
kf d
✢
✜ = 0)A
506
x f. kr
✁ ✂✄☎✆ ✝ ✞ ✂✟ ✠✡✄☛ ☞
d kr h;Z : i
e ku y hf t ,
d kr h;
P(✌ 1, ✍1, ✎1)
fd
, d
✑
l f n ’ k ✏ g r Fkk f n ,
f n ; k flc n g f t l d k f LFkf r
l er y
d k
l e hd j . k
A✌ + B✍ + C✎ = D g S
✑
✏
rc
=
✒
✌1 ✓✒ ✖ ✍ 1 ✒✔ ✖ ✎1 ✕
✑
N = A ✓✒ ✖ B ✒✔ ✖ C ✕✒
v r % ( 1)
o Q } kj k
P
l
l er y
( ✌1 ✓✒
i j
y c d h y c kb Z
✖ ✍ 1 ✒✔ ✖ ✎1 ✕✒
) ✗( A ✒✓
A2
B2
✖
A
=
B
✖
✖
gy
(2, 5, ✥ 3)
B
✌1 ✖
2
✍1 ✖ C ✎1
2
2
✖B ✖C
6 ✓✒ - 3 ✒✔
✑
✑
; g k¡ ✏ ✛
bl fy ,
✑
✚ ✗(
d h l er y
✖
2 ✓✒ ✖ 5 ✒✔ - 3 ✕✒ , N ✛ 6 ✓✒ - 3 ✒✔ ✖ 2 ✕✒ v
flc n q(2, 5, ✥ 3) d h f n , l e r y l n j h g %
| (2 ✓✒ ✖ 5
✒✔
- 3 ✕✒ ) ✗(6 ✒✓ - 3
| 6 ✒✓ - 3
=
✢✢ ✣✢✤ , d
j [ kk v kj
, d
l er y
C ✕✒ ) - D
C2
A
mn kg j . k ✘✙ flc n
✒✔ ✖
✒✔ ✖
✒✔ ✖ 2 ✕✒
2 ✕✒ ) = 4
kj ✜
o Q c hp
✖
l
n j h K kr
= 4.
|
9✖4
✛
13
7
d k d k. k ✦✧★ ✩✪ ✫ ✬✫✭✮✫ ✫★ ✯ ✪✰ ★✫
✯★✱ ✯ ✲✪✯★ ✫✳
i fj Hkk"kk ✘ , d
j [ kk v kj l e r y
j [ kk v kj
, d
o Q v f Hky c o Q c hp
(complementary angle)
l fn ’ k : i
e ku y hf t ,
l er y
l er y
i j d g kr k g
o Q c hp
d k d k. k]
o Q d k. k d k d k. k
(v
ko Qfr
11.20)A
✴✆ ✵☎✡✄ ✠✡✄☛ ☞
fd
✑
✑
j [ kk d k l e hd j . k ✚ ✛ ✏ ✖
l
✑
✶ g
d hf t , A
2 ✕✒ ) - 4|
| 12 - 15 - 6 - 4 |
36
-D
r Fkk
✑ ✑
d k l e hd j . k ✚ ✗✷ ✛ ✜ g A r c j [ kk v kj l e r y o Q
v ko Qfr
1 1 -2 0
f=k&fo e h;
v f Hky c o Q c hp
q,
d k d k. k
f u Eu f y f [ kr
l =k } kj k O; Dr
T; kfe fr
507
fd ; k t k l d r k g A
✄ ✄
✂✁
cos q = ✄ ✄
| | ✂| ✁ |
v kj b l
o Q c hp d k d k. k f, 90° ✥ q, } kj k i n k g v
sin (90° ✥ q) = cos q
✄ ✄
✂✁
✂✁
☎1
sin f = ✄ ✄ ; k f = sin
| | |✁ |
✁
i d kj
j [ kk v kj l e r y
v Fkkr ]
mn kg j . k ✆✝ j [ kk
✞✟1
2
=
✠
☛
3
✡ -3
6
v kj l e r y
Fkkr ~
10 ☞ + 2 ✌ ✥ 11 ✍ = 3 o Qc hp
d k d k. k K kr
d hft , A
gy
j [ kk v kj l e r y o Q v f Hky c o Q c hp d k d k. kq g A f n , x ,
e ku y hf t , f d
o Q l e hd j . kk d k l f n ’ k : i
e O; Dr
j [ kk r Fkk l e r y
d ju i j ge
✄
✎ = ( ✥ ✑✏ ✔ 3 ✒✏ ) ✔ l ( 2 ✑✏ ✔ 3 ✏✓ ✔ 6 ✒✏ )
✄
✎ ✂( 10 ✏✑ ✔ 2 ✏✓ - 11 ✒✏ ) = 3 i kI r d j r g A
✄
✄
= 2 ✑✏ ✔ 3 ✏✓ ✔ 6 ✒✏ v kj ✁ ✕ 10 ✑✏ ✔ 2 ✏✓ - 11 ✒✏
v kj
; g k¡
sin f =
v r%
=
(2 ✑✏ ✔ 3 ✏✓ ✔ 6 ✒✏ ) ✂(10 ✏✑ ✔ 2 ✏✓ - 11 ✒✏ )
2 2 ✔ 32 ✔ 6 2
10 2 ✔ 22 ✔ 112
- 40
-8
8
=
=
7 ´ 15
21
21
; k
æ 8 ö
f = sin -1 ç ✖
è 21 ø
i ’ u ko y h 1 1 -3
✗✘
f u Eu f y f [ kr
n j h K kr
i ’ uk e l
i R; d
e l e r y o Q v f Hky c d h f n o Q& d kl kb u v kj e y
(a) ✍ = 2
(c) 2 ☞ + 3 ✌ ✥ ✍ = 5
✆✘
ml
l er y
l fn’ k
flc n l s
d hf t , %
(b) ☞ + ✌ + ✍ = 1
(d) 5 ✌ + 8 = 0
d k l f n ’ k l e hd j . k K kr
3 ✑✏ ✔ 5 ✏✓ - 6 ✒✏ i
d hf t , ] t k e y
j v f Hky c g A
flc n l
7 e k=kd
nj h i j
g ] v kj
508
x f. kr
✁
f u Eu f y f [ kr
(a)
(c)
✌✁
✑✁
✒✁
l e r y k d k d kr h;
l e hd j . k K kr
d hf t , %
✡
✡
(b) ✄ ✞(2 ✂☎ ✟ 3 ✂✆ - 4 ✝✂ ) ✠ 1
✄ ✞(☎✂ ✟ ✂✆ - ✝✂ ) ✠ 2
✡
✄ ✞[(☛ - 2☞) ☎✂ ✟ (3 - ☞ ) ✂✆ ✟ (2 ☛ ✟ ☞ ) ✝✂ ] ✠ 15
f u Eu f y f [ kr
f LFkf r ; k e ] e y
flc n l
[ khp
(a) 2✍ + 3✎ + 4 ✏ ✥ 12 = 0
(c) ✍ + ✎ + ✏ = 1
f u Eu f y f [ kr
i fr c / k o Q v r x r
0, ✥ 2)
(a)
flc n q(1,
(b)
flc n q (1,4,
6)
l
l
y c o Q i kn o Q f u n ’ kkd
l e r y k d k l f n ’ k , o d kr h;
t kr k g k v kj
t kr k g k v kj
mu l e r y k d k l e hd j . k K kr
x,
K kr
d hf t , A
(b) 3✎ + 4 ✏ ✥ 6 = 0
(d) 5✎ + 8 = 0
☎✂ ✟ ✂✆ - ✝✂
☎✂ - 2 ✂✆ ✟ ✝✂
d hf t ,
l er y
l er y
l e hd j . k K kr
d hf t ,
t k%
i j v f Hky c g A
i j v f Hky c l f n ’ k g A
t k f u Eu f y f [ kr
r hu flc n v k l
(a) (1, 1, ✥ 1), (6, 4, ✥ 5), (✥ 4, ✥ 2, 3)
(b) (1, 1, 0), (1, 2, 1), (✥ 2, 2, ✥ 1)
✓✁ l e r y 2✍ + ✎ ✥ ✏ = 5 } kj k d kV x , v r % [ kM k d k K kr
✔✁ ml l e r y d k l e hd j . k K kr d hf t , f t l d k ✎ & v { k i j v
xt jr k gA
d hf t , A
r %[ kM 3 v kj t k r y
Z OX
o Q l e kr j g A
✕✁
ml
l er y
d k l e hd j . k K kr
✍+✎+✏✥ 2 =0
✖✗✁
✖✖✁
ml
l er y
d k
d hf t ,
o Q i f r PN n u r Fkk flc n
l f n ’ k l e hd j . k K kr
✡
✄ .( 2 ☎✂ ✟ 5 ✂✆ ✟ 3 ✝✂ ) ✠ 9
t ks l e r y ksa3 ✍
(2, 2, 1)
d hf t ,
l
✡
✄ .( 2 ☎✂ ✟ 2 ✂✆ - 3 ✝✂ ) ✠ 7 ,
(2, 1, 3)
l
g kd j t kr k g A
✍ + ✎ + ✏ = 1 v kj 2 ✍ + 3✎ + 4 ✏ = 5 o Q i f r PN n u j [ kk l g kd j t
✍ ✥ ✎ + ✏ = 0 i j y c o r r y d k l e hd j . k K kr d hf t , A
✡
✖✘✁ l e r y k] f t u o Q l f n ’ k l e hd j . k ✄ ✞(2 ☎✂ ✟ 2 ✂✆ - 3 ✝✂ ) ✠ 5 v kj
✡
✄ ✞(3 ✂☎ - 3 ✂✆ ✟ 5 ✝✂ ) ✠ 3 g ] o Q c hp d k d k. k K kr d hf t , A
✖ ✁ f u Eu f y f [ kr i ’ u k e K kr d hf t , f d D; k f n , x , l e r y k o Q ; Xe l e kr j
ry k
g ] v kj ml
K kr
(a)
(b)
(c)
(d)
(e)
f LFkf r
e] t c ;
u r k l e kr j
v kj
g kd j t kr k g A
t k l er y k
o Q i f r PN n u j [ kk v kj
✥ ✎ + 2✏ ✥ 4 = 0
g v kj u g h y c o r
d hf t , A
7✍ + 5✎ + 6 ✏ + 30 = 0 v kj 3✍ ✥ ✎ ✥ 10 ✏ + 4 = 0
2✍ + ✎ + 3 ✏ ✥ 2 = 0 v kj ✍ ✥ 2✎ + 5 = 0
2✍ ✥ 2✎ + 4 ✏ + 5 = 0 v kj 3✍ ✥ 3✎ + 6 ✏ ✥ 1 = 0
2✍ ✥ ✎ + 3 ✏ ✥ 1 = 0 v kj 2✍ ✥ ✎ + 3 ✏ + 3 = 0
4✍ + 8✎ + ✏ ✥ 8 = 0 v kj ✎ + ✏ ✥ 4 = 0
ku o ky
r Fkk r y
g v Fko k y c o r ~
r k mu o Q c hp
d k d k. k
f=k&fo e h;
✁✂
f u Eu f y f [ kr
i ’ u k e i R; d
fn,
x,
flc n l
flc n q
(a)
(b)
(c)
(d)
gy
?ku ] , d
, d
l xr
l e r y k d h n j h K kr
d hf t , A
3 ✄ ✥ 4 ☎ + 12 ✆ = 3
2 ✄ ✥ ☎ + 2✆ + 3 = 0
✄ + 2 ☎ ✥ 2✆ = 9
2 ✄ ✥ 3☎ + 6✆ ✥ 2 = 0
fo fo /
✝✞
x,
509
l ery
(0, 0, 0)
(3, ✥ 2, 1)
(2, 3, ✥ 5)
( ✥ 6, 0, 0)
mn kg j . k
fn,
T; kfe fr
mn kg j . k
?ku o Q fo d . kk o Q l kFk a,
b, g, d,
4
cos2 a + cos2 b + cos2 g + cos2 d =
3
, d j [ kk] , d
l e d kf . kd
"kVi Q
y d h;
d k. k c u kr h g Sr k f l … d hf t , f d
g kr k g ft l d h
y c kb ] p kM
S kb v kj ¯ p kb l e ku g kr g A
e ku y hf t ,
Hkt k
✟
OADBEFCG , d ?ku
( v ko Qfr 11.21)A
fd
OE, AF, BG v
CD p kj
kj
O r Fkk E d k
OE o Q f n o Q& d kl
fo d . k g A
n k flc n v k
f e y ku
fo d . k
kb u
✟ -0
✟ ✠✟ ✠ ✟
2
1
v Fkkr ~
3
g A b l h i d kj
e ku y hf t ,
2
1
,
3
1
3
3
2
v ko Qfr
1 1 -2 1
✟-0
,
2
✟ ✠ ✟2 ✠ ✟2
2
,
1
3
;
h f n o Q& d kl kb u
1
3
,✥
OE, AF, BG, v
✡, ☛ , ☞ g A
1
cos a =
cos g =
CD d
kj
n h x b j [ kk t k
g ] d h f n o Q& d kl kb u
✟ ✠ ✟ ✠✟
2
v Fkkr ~
3
1
,
✟ -0
,
2
OE
o ky h j [ kk
1
,
AF, BG v
✥
rc
f t l d h i R; d
y c kb d h g
3
1
3
1
3
kj
e ’ k%
1
,
CD,
( ✡ ✥ ☛ + ☞ ); cos d =
3
,
3
o Q l kFk
e ’ k%
3
1
1
3
1
( ✡ + ☛ + ☞ ); cos b =
v kj
1
3
1
3
a, b, g, v
(✥ ✡ + ☛ + ☞ )
(✡ + ☛ ✥ ☞ )
,✥
,gA
kj
dd
k. k c u kr h
510
x f. kr
o x d j o Q t kM u i j
g e i kr
g fd
cos a + cos b + cos2 g + cos2 d
2
2
1
[ ( + ✁ + ✂ )2 + (✥ + ✁ + ✂ )2 ] + ( ✥ ✁ + ✂ )2 + ( + ✁ ✥✂ )2]
3
1
4
= [ 4 ( 2 + ✁ 2 + ✂2 ) ] =
( D; kf d 2 + ✁ 2 + ✂ 2 = 1)
3
3
=
mn kg j . k ✄ ☎ ml
l er y k
gy
ry
d k l e hd j . k K kr
2✆ + 3✝ ✥ 2✞ = 5
fn,
x,
d hf t ,
ft l e
+ 2✝ ✥ 3✞ = 8
v kj ✆
flc n d k v r f o "V d j u o ky
l er y
e l
flc n
i R; d
i fr c /
+ 3✝ ✥ 2 ✞ = 5
d k i ; kx
v kj ✆ +
2 ✝ ✥ 3 ✞ = 8, o Q l
d j u i j g e i kr
d ju i j
v kj t ks
kFk ( 1 )
... (1)
gA
} kj k i n k l e r y
i j y c g ku o Q
g fd
2A + 3B ✥ 2C = 0
b u l e hd j . kk d k g y
v r f o "V g
i j y c gA
d k l e hd j . k
A (✆ ✥ 1) + B(✝ + 1) + C ( ✞ ✥ 2) = 0
l e r y ksa2 ✆
(1, ✥ 1, 2)
g e i kr
A + 2B ✥ 3C = 0
v kj
A = ✥ 5C v
g fd
kj
B = 4C
v r % v Hkh"V l e hd j . k g %
✥
5✆ ✥ 4✝ ✥ ✞ = 7
v Fkkr ~
mn kg j . k ✄ ✟ flc n
f u / kf j r
gy
5C ( ✆ ✥ 1) + 4 C (✝ + 1) + C(✞ ✥ 2) = 0
l er y
e ku y hf t ,
P(6, 5, 9)
d h n j h K kr
fd
l
flc n v k
A (3, ✥ 1, 2), B (5, 2, 4)
l er y
e r hu flc n
A, B, r Fkk C
g A flc n
✠✠✠
✡
D gA ge
v Hkh"V n j h
PD K kr
d j uh g t g k
✠✠✠✡
AP = 3
✠✠✠✡
✠✠✠✡
✍
✌ ✎
6
✠✠✠✡
AB ´ AC =
v kj
✠✠✠✡
AB ´ AC
✥
1, 6)
} kj k
d k
o Q v u f n ’ k b d kb l f n ’ k
l er y
i j y c
d k i kn
✠✠✠✡
AB ´ AC
o Q v uf n ’ k bd kb Zl fn ’ k r Fkk
✍
☞ ✎
7
AP
✍
☛
✏
✓
✏
✑
✏
✒
2
3 2
-4
0 4
=
l
i j i { ki
gA
✠✠✠
✡
✠✠✠
✡
i u%
PD , AP
P
✠✠✠✡
✠✠✠✡
PD = AB ´ AC
v r%
C(✥ 1,
v kj
d hf t , A
3 ✑✏ - 4
✔
✏✒ ✕
34
12 ✑✏ - 16 ✒✏ ✕ 12 ✓✏
3 ✓✏
d k v f n’ k x . kui Qy g A
f=k&fo e h;
=
fo d Yi r % flc n q A,
d h l er y
l
Bv
n j h K kr
mn kg j . k ✞✟ n ’ kkb ,
kj
Cl
fd
P
xq
j [ kk, ¡
✠
v kj
; g k K kr
3 34
17
d hf t , A
✠
- ✡ ✝☛
☞-✡
✌ -✡-☛
✍
=
a-d
a
a✝ d
-✎✝✏
☞-✎
✌ -✎ -✏
✍
=
b- g
b
b✝g
l g & r y h;
gA
g fd
v kj
✠1
= ✡✥☛
✠2
= ✎✥✏
☞1
✌1
= ✡
= ✡ +☛
☞2
✌2
= ✎
= ✎ +✏
✡1
= a✥ d
✡2
= b✥ g
✎1
= a
= a+d
✎2
= b
= b+g
✏1
v kj
✏2
v c l kj f . kd
✠2
- ✠1
☞2
- ☞1
- ✌1
✌2
✡1
✎1
✏1
✡2
✎2
✏2
Lr Hk e
t kM u i j
✎
=
-✏ - ✡ ✝ ☛
a -d
b-g
✎
- ✡ ✎ ✝✏ - ✡ - ☛
a
a✝ d
b
b✝g
i j f o p kj d hf t , A
r hl j Lr Hk d k i g y
✎
2
-✡
a
b
D; kf d
511
✁
3 ✆✂ - 4 ✂☎ ✝ 3 ✄✂
PD = ( 3 ✆✂ ✝ 6 ✂☎ ✝ 7 ✄✂ ) .
34
v r%
gy
T; kfe fr
i Fke
v kj f } r h;
g e i kr
✎
gA
- ✡ ✎ ✝✏ - ✡ - ☛
a
a ✝d
=0
b
b✝g
Lr Hk l e ku g A v r % n ku k j [ kk,
l g & r y h;
gA
512
x f. kr
mn kg j . k
o ky h j [ kk
gy
✁ ml flc n o Q f u n ’ kkd K kr
XY- r y d k d kVr h g A
flc n v k
v Fkkr ~
e ku y hf t ,
; g flc n v o ’ ;
oQ: i
A(3, 4, 1) v
kj
B(5, 1, 6) d
k fe y ku s
... (1)
d k f LFkf r
e gA
g h l e hd j . k ( 1 )
( D; k?as )
d k l r "V d j r k g A
✡ ✆☎ ✟ ✠ ✝☎ = (3 ✟ 2 l ) ✆☎ ✟ ( 4 - 3 l) ☎✝ ✟ ( 1 ✟ 5 l ) ✞☎
v Fkkr ~
☎✆ , ☎✝
t g k flc nvq k
A v kj B l t ku o ky h j [ kk d k l f n ’ k l e hd j . k%
✄
✂ = 3 ✆☎ ✟ 4 ✝☎ ✟ ✞☎ ✟ l [ (5 - 3)✆☎ ✟ (1 - 4) ✝☎ ✟ ( 6 - 1) ✞☎ ]
✄
✂ = 3 ✆☎ ✟ 4 ✝☎ ✟ ✞☎ ✟ l ( 2 ☎✆ - 3 ☎✝ ✟ 5 ✞☎ ) g S
P o g flc n g t g k j [ kk AB, XY- r y d k i f r PN n d j r h g A r c flc n P
✡ ✆☎ ✟ ✠ ✝☎
l fn’ k
d hft ,
v kj
✞☎ , o Q x . kkd
k d h r y uk d j u i j g e
i kr
ga
S
✡= 3+2l
✠=4✥3l
0= 1 +5 l
mi j kDr
l e hd j . kk d k g y
d j u i j g e i kr
g fd
13
5
✡=
æ 13 23 ö
, 0☞
ç ,
5
è5
ø
v r % v Hkh"V flc n o Q f u n ’ kkd
v Ł; k;
✌✍
f n [ kkb ,
fd
( 4] 3]&1)
✎✍
ey
l
flc n
f u / kf j r
l
11
( 2]
j [ kk i j
i j
1]
v kj
fo fo /
1)
y c
f e y ku
i ’ u ko y h
o ky h j [ kk]
flc n v k ( 3 ]
5
&1)
v kj
y c gA
✏ 1 , ✑ 1, ✒ 1
v kj
✏ 2, ✑ 2, ✒ 2 g k r k f n [
kkb ,
fd
j [ kk d h f n o Q& d kl kb u
✑ 1 ✒ 2 ✥ ✑ 2 ✒ 1, ✒ 1 ✏2 ✥ ✒ 2 ✏1, ✏1 ✑ 2 ✥ ✏2 ✥ ✑ 1
✍
23
5
gA
; f n n k i j Li j y c j [ kkv k d h f n o Q& d kl kb u
b u n ku k i j
✠ ☛
mu j s[ kkv k o Q e Ł;
d k. k K kr
gA
d hf t , ] f t u o Q f n o Q& v u i kr
✓, ✔, ✕ v
kS
j
✔ ✥ ✕, ✕ ✥ ✓,
✓ ✥ ✔gA
✖✍ ✡ - v { k o Q l e kr j r Fkk e y & flc n l t ku o ky h j [ kk d k l e hd j . k K kr d hf t , A
✗✍ ; f n flc n v k A, B, C, v kj D o Q f u n ’ kkd e ’ k% (1, 2, 3), (4, 5, 7), (✥ 4, 3, ✥ 6) v
(2, 9, 2) g r k AB v kj CD j [ kkv k o Q c hp d k d k. k K kr d hf t , A
kj
f=k&fo e h;
✁
rk
✞✁
✂ -1
✄ - 2 ☎ -3
✝
✝
-3
2✆
2
; f n j [ kk,
✆
flc n
d k e ku K kr
d hf t , A
(1, 2, 3)
t ku o ky h r Fkk r y
l
l f n ’ k l e hd j . k K kr
✍✁
(✎ , ✏ , ✑)
flc n
l
v kS
✂ -1 ✄ - 1 ☎ - 6
✝
✝
3✆
1
-5
✌
☞ . ( ✟☛ ✠ 2 ✟✡ - 5 ✆✟ ) ✠ 9 ✝ 0 i
513
T; kfe fr
i j Li j y c g ka
s
j y cor
j [ kk d k
d hf t , A
t ku o ky
✌
☞ ✒(☛✟ ✠ ✟✡ ✠ ✆✟ ) ✝ 2
r Fkk r y
o Q l e kr j r y
d k l e hd j . k K kr
d hft , A
✓✁
✌
☞ ✝ 6 ☛✟ ✠ 2 ✟✡ ✠ 2 ✆✟ ✠ l (✟☛ - 2 ✟✡ ✠ 2 ✆✟ )
j [ kkv k
v kj
✌
☞ ✝ - 4 ☛✟ - ✆✟ ✠ m (3 ☛✟ - 2 ✟✡ - 2 ✆✟ ) o Q c hp d h U; u r e n j h K kr d
✔✕✁ ml flc n o Q f u n ’ kkd K kr d hf t , t g k flc n v k (5, 1, 6) v kj (3, 4,1)
j [ kk YZ- r y d k d kVr h g A
✔✔✁
✔✖✁
ml
flc n o Q f u n ’ kkd
j [ kk
ZX- r y
ml
flc n o Q f u n ’ kkd Kkr d hf t ,
2✂ + ✄ + ☎ = 7
( ✥ 1, 3, 2)
flc n
i R; d
✔✘✁
✔✚✁
✔ ✁
i j
✔✞✁
( ✥ 3, 0, 1) l
i j f LFkr
g k] r k
l er y k
✌
☞ ✒(☛✟ ✠ ✡✟ ✠ ✆✟ ) ✝ 1 v
o ky
r Fkk
✂-v
; fn
O ey
l er y k
K kr
flc n
(5, 1, 6)
(3, 4, 1)
v kj
d k f e y ku o ky h
kj
d k e ku K kr
{ k o Q l e kr j
P
flc n r Fkk flc n
ry
kj
(2, ✥ 3, 1) l
xq
✂ + 2✄ + 3 ☎ = 5
v kj
3✂ + 3 ✄ + ☎ = 0
e l s
d hf t , A
er y
✌
☞ ✒(3 ☛✟ ✠ 4 ✡✟ - 12 ✆✟ ) ✠ 13 ✝ 0 l
l e ku n j h
d hf t , A
✌
☞ ✒(2 ✟☛ ✠ 3 ✟✡ - ✆✟ ) ✠ 4 ✝ 0
kj
ry
(3, ✥ 4, ✥ 5) v
t kr h g A
d k l e hd j . k K kr
(1, 1, ✙ ) v
✙
t g k flc n v k
o Q i kj
t ku o ky r Fkk l e r y k
; f n flc n
v r f o "V
✔✍✁
l
y c l er y
oQy cor
t g k flc n v k
d k f e y ku o ky h
d k d kVr h g A
j [ kk] l e r y
✔✗✁
K kr d hf t ,
hf t , A
d k l e hd j . k K kr
o Q f u n ’ kkd
d k l e hd j . k K kr
d hf t
o Q i f r PN n u j [ kk l
t ku s
d hf t , A
(1, 2, ✥ 3), g
, A.
r k flc n
P
l
t ku o ky
r Fkk
✌
✌
☞ ✒(☛✟ ✠ 2 ✡✟ ✠ 3 ✆✟ ) - 4 ✝ 0 v kj ☞ ✒(2 ☛✟ ✠ ✡✟ - ✆✟) ✠ 5 ✝ 0 o Q i f r PN n u
✌
d j u o ky r Fkk r y ☞ ✒(5 ☛✟ ✠ 3 ✟✡ - 6 ✆✟ ) ✠ 8 ✝ 0 o Q y c o r r y d k l
OP
j [ kk d ks
e hd j . k
d hf t , A
( ✥ 1, ✥ 5, ✥ 10)
✌
☞ ✒(☛✟ - ✟✡ ✠ ✆✟ ) ✝ 5 o Q i
l
j [ kk
✌
☞ ✝ 2 ☛✟ - ✡✟ ✠ 2 ✆✟ ✠ l (3 ☛✟ ✠ 4 ✡✟ ✠ 2 ✆✟ ) v
f r PN n u flc n o Q e Ł;
d h n j h K kr
d hf t , A
kj
l er y
514
x f. kr
✁✂
☛ ✄ ✄
✟(✆ - ✝ ✠ 2 ✞✄ ) ✡ 5
t ku o ky h r Fkk l e r y ks ☎
(1, 2, 3) l
flc n
l e kr j j [ kk d k l f n ’ k l e hd j . k K kr
☞✌✂
(1, 2, ✥ 4)
flc n
✒ - 15
3
☞ ✂
; fn , d
d hf t ,
i ’ uk
22
☞☞✂
v kj
✑ - 29
=
8
l er y
l
t ku
✍
i j y c
-5
✒ -8
n ku k j [ kkv k
3
✍
✑ ✎ 19
v kj b l d h e y flc n l
✍
- 16
j [ kk d k l f n ’ k l e hd j . k K kr
✓, ✔, ✕ g
o Q v r %[ kM
☛
☎ ✟(3 ✆✄ ✠ ✄✝ ✠ ✞✄) ✡ 6 o Q
d hf t , A
o ky h v kj
✏- 5
v kjS
nj h
✏ -10
7
v kj
d hf t , A
✖
b d kb g r k f l …
1
1
1
1
✠
✠
✡ 2
✓2 ✔2 ✕ 2
✖
fd
23 e
l g h m kj d k p u ko d hf t , A
nk l e r y k
2✗ + 3✘ + 4 ✙ = 4
(A) 2 b d
kb Z
4✗ + 6✘ + 8 ✙ = 12
v kj
(B) 4 b d
(C) 8 b d
kb Z
o Q c hp d h n j h g %
(D)
kb Z
2
b d kb Z
29
☞✚✂
l er y
(A)
2✗ ✥ ✘ + 4 ✙ = 5
i j Li j
(C) ✘ - v
5✗ ✥ 2.5 ✘ + 10 ✙ = 6 g %
(B) l e kr j
v kj
y c
{ k i j i f r PN n u d j r
(D)
gA
æ
è
flc n q ç 0, 0,
5ö
✛l
4ø
xt jr
gA
l kj k’ k
✜
, d
j [ kk d h fn o Q&d kl kbu j [ kk } kj k f u n ’ kk{ kk d h / u f n ’ kk o Q l kFk c u k,
d k. kka
s
d h d kl kb u g kr h g A
✜
; fn , d
✜
n k flc n v k
j [ kk d h f n o Q& d kl kb u
P (✗ 1, ✘ 1, ✙1)
t gk P
aQ
✜
, d
=
✢2 + ✣ 2 + ✤ 2 = 1
d k f e y ku o ky h j [ kk d h f n o Q& d kl kb u
ga
S
( ✭ 2 - ✭1 ) 2 ✮ ( ✬ 2 - ✬1 ) 2 ✮ ✩✫ 2 - ✫1 ✪2
j [ kk d k fn o Q&v u i kr
g kr h g A
g rk
Q (✗ 2, ✘2, ✙2)
v kj
✦ 2 - ✦1 ✧ 2 - ✧1 ★2 - ★1
,
,
PQ
PQ
PQ
✢, ✣ , ✤
o l [ ; k,
g t k j [ kk d h f n o Q& d kl kb u o Q l e ku i kr h
f=k&fo e h;
; fn , d
✁, ✂ , ✄
j [ kk d h f n o Q& d kl kb u
☎, ✆, ✝
✆
✝
;✄=
☎ ✞✆ ✞✝
☎ ✞✆ ✞✝
✁ = ☎ ✞ ☎✆ ✞ ✝ ; ✂ =
2
fo "ke r y h;
; g j [ kk,
2
j [ kk,
2
v kj f n o Q& v u i kr
2
v r f j { k d h o j [ kk,
2
2
f o "ke r y h;
l
e gA
✁ ,✂ , ✄
1
1
1
j [ kkv k e l
✁ , ✂ ,✄
v kj
2
2
i R; d
☎,✆,✝
1
1
o Q l e kr j [ khp h x b n k i f r PN n h j [ kkv ka
s
1
2
2
2
1
2
1
2
f n o Q& v u i kr k o ky h n k j [ kkv k o Q c hp
✟ ✟ ☛✠ ✠ ☛✡ ✡
✟ ☛ ✠ ☛✡ ✟ ☛ ✠ ☛ ✡
✍
✌
☞
✆
✎✍ ✏ ☎✍ ✞ l ✆✍
(✑ , ✒ , ✓ )
✁, ✂ , ✄
✑-✑ ✏ ✒ - ✒ ✏✓ - ✓
✁
✂
✄ ✍ ✍
☎ ✆
✍
✎✍ ✏ ☎✍ ✞ l (✆ - ☎✍)
(✑ , ✒ , ✓ )
(✑ , ✒ , ✓ )
✑-✑ ✏ ✒-✒ ✏ ✓-✓
✑ - ✑ ✒ - ✒ ✓ ✍- ✓ ✍ ✍ ✍
✎✍ ✏ ☎✍ ✞ l ✆ ✎ ✏ ☎ ✞ l ✆ ,
✍ ✔✆✍
✆
cos q ✏ ✍
✍
| ✆ | |✆ |
1
2
1
K kr flc n f t l d h f LFkf r l f n ’ k
flc n
1
1
l
1
1
2
1
2
1
2
1
g l
xq
2
1
2
2
2
2
2
2
2
o Q l e kr j j [ kk
gA
d k l f n ’ k l e hd j . k
t ku o ky h j [ kk f t l d h f n o Q& d kl kb u
1
1
g ] d k l e hd j . k
gA
n k flc n v k f t u o Q f LFkf r l f n ’ k
l e hd j . k
v kj
g l
t ku o ky h j [ kk o Q l e hd j . k d k l f n ’ k
gA
n k flc n v k
1
1
1
2
d k U; u d k. k
rc
cos q =
, d
U; u d k. k q g S
✁✁ ✂ ✂ + ✄✄|
cos q = | 1 2 +
☎,✆,✝
v kj
f d l h flc n ( o j h; r k e y
f n o Q& d kl kb u o ky h n k j [ kkv k o Q c hp
2
rc
qg
2
f o f HkUu r y k e g kr h g A
flc n d h)
; fn
2
t k u r k l e kr j g v kj u g h i f r PN n h g A
j [ kkv k o Q c hp d k d k. k o g d k. k g t k , d
; fn
g r ks
2
fo "ke r y h;
o Q c hp
515
T; kfe fr
1
v kj
1
1
2
1
; f n n k j [ kkv ksa
1
1
2
1
2
2
2
2
1
gA
2
l
t ku
o ky h j [ kk d k d kr h;
l e hd j . k
1
1
v kj
2
2
o Q c hp
d k U; u d k. k
q
g
r ks
516
x f. kr
✝ -✝ ✞ ☎- ☎ ✞ ✂ -✂
✆
✄
✁
✌-✌ ✍ ☛-☛ ✍ ✠ -✠
☞
✡
✟
cos q = |✎ ✎ + ✏ ✏ + ✑ ✑ |.
1
; f n n k j [ kkv ka
s
1
1
2
1 2
n k j [ kkv k
o Q c hp
1
2
1
1
✔✕ ✗ ✓✕ ✖ m ✒✕
✕ ✕ ✕ ✕
(✒ ´ ✒ ) ✘(✓ ✥ ✓ )
✕ ✕
|✒ ´✒ |
v kj
1
2 o Q c hp
2
2
2
1
2
✙ - ✙ ✗ ✚ - ✚ ✗ ✛ -✛
✓
✒
✜
1
1
U; u r e
rc
d h U; u r e n w
j h o g j [ kk[ kM g t k n ku k j [ kkv ks i j
1
c hp
g
2
j [ kkv k o Q c hp
1
n k j [ kkv ksa
q
d k d k. k
2
✔✕ ✗ ✓✕ ✖ l ✒✕
gA
v kj
1
2
2
n k f o "ke r y h;
y c
1
2
2
1
1
1
1
U; u r e n j h
gA
✙-✙ ✗✚-✚
✓
✒
2
v kj
2
1
1
2
1
2
n k l e kr j j [ kkv k
1
1
2
2
2
2 1
2
1 2
2 1
l e r y ] ft l d h e y
flc n l
g ] d k l fn’ k : i
l e r y ] ft l d h e y
l e hd j . k
1
nj h
2
2 1
d h nj h
gA
r Fkk l e r y
e l e hd j . k
flc n l
gA
o Q c hp
2
nj h
g ] d k l e hd j . k
flc n f t l d k f LFkf r
1 2
v kj
1
2
, d
oQ
2
1
1
2
1 2
, d
2
nj h
2
l fn’ k
✛ -✛
✜
=
2
✢ -✢ ✣ -✣ ✤ -✤
✦
✧
★
✦
✧
★
(✧★ - ✧ ★ ) ✩ ( ★✦ - ★ ✦ ) ✩ (✦ ✧ - ✦ ✧ )
✯
✔✕ ✗ ✓✕ ✖ l ✒✕
✪✯ ✭ ✫✯ ✮ m ✬
✒✕ ´ (✓✕✕ - ✓✕ )
|✒ |
✰
✑✱
✳✸ ✶✴✲ ✷ ✵
✰
✎, ✏ , ✑
✎✙ + ✏✚ + ✑✛ = ✰
✕
✺
N
✹
✕ ✕ ✻✕
( ✔ - ✓ ).N ✗0
, d
2
i j ey
flc n l
v f Hky c b d kb Z
gA
r Fkk l e r y
o Q v f Hky c
d h f n o Q& d kl kb u
gA
l fn’ k
gA
l
t ku o ky k v kj
l fn’ k
i j
y c l er y
d k
f=k&fo e h;
, d
fn,
B, C
x,
✁✂ ✄
( 1, 1, 1)
flc n
t ku o ky v kj , d
n h x b j [ kk f t l o Q f n o Q& v u i kr
✁ ✥ ✁ ) + B (✂ ✥ ✂ ) + C (✄ ✥ ✄ ) = 0
(✁ , ✂ , ✄ ), (✁ , ✂ , ✄ )
(✁ , ✂ , ✄ )
g ] i j y c l e r y d k l e hd j . k A (
r hu v l j [ k flc n v k
1
1
1
2
2
1
1
v kj
2
3
3
l
3
1
t ku o ky
517
T; kfe fr
A,
gA
l er y
d k
l er y
d k
l e hd j . k g %
✁-✁ ✂- ✂ ✄- ✄
✁ -✁ ✂ - ✂ ✄ - ✄ =0
✁ -✁ ✂ - ✂ ✄ -✄
✆✝, ☎✝ ✞✝
✝ ✝ ✝ ✝ ✝ ✝
( ✟ - ✆ ) . [ (☎ - ✆ ) ´ ( ✞ - ✆ ) ] ✠ 0
(✆ , 0, 0), (0, ☎ , 0)
(0, 0, ✞)
✑ ☛ ✎ ☛✌ ✡1
✏ ✍ ☞
✝✟ ✔✒✝ ✠ ✓ ✟✝ ✔✒✝ ✠ ✓
✟✝ ✔(✒✝ ✕ l ✒✝ ) ✠ ✓ ✕ l ✓ , l
1
1
1
2
1
2
1
2
1
3
1
3
1
3
1
r hu flc n v k f t u o Q f LFkf r
l fn’ k
v kj
d k v r f o "V d j u
o ky
l f n ’ k l e hd j . k
, d
l er y
t k f u n ’ kk{ kk d ks
v kj
i j
d kVr k g ] d k
gA
l e hd j . k
l e r y ksa
1 v kS
1
l e hd j . k
1
2
2
2
o Q i f r PN n u
2 g St g k
1
l
, d
x t j u o ky
i kp y
l er y
d k l fn’ k
gA
l e r y ksa
✁+ B ✂+ C ✄+ D = 0
A ✁+B ✂+ C ✄+ D = 0
(A ✁ + B ✂ + C ✄ + D ) + l(A ✁ + B ✂ + C ✄ + D ) = 0
✟✝ ✠ ✆✝ ✕ l☎✝ ✟✝ ✠ ✆✝ ✕ m ☎✝
✝ ✝ ✝ ✝
(✆ - ✆ )✔(☎ ´ ☎ ) = 0
A( ✁ , ✂ , ✄ )
B(✁ , ✂ , ✄ )
✛ -✛ ✚ - ✚ ✙ - ✙
✘
✗
✖ =0
✘
✗
✖
✜✧ ✤✢✧ ✦ ✣ ✟✝ ✔✒✝ ✠ ✓
✫ ✫
| ✩ ✪✩ |
q
q = cos★ ✫ ✫
| ✩ || ✩ |
A1
1
v kj
1
2
2
o Q i f r PN n u l
1
1
2
2
x t j u o ky
1
1
n k j [ kk,
l er y
d k l e hd j . k
1
1
1
2
v kj
2
2
; f n mi j kDr j [ kk,
2
g ; fn
f c n v ksa
1
2
1
1
1
1
2
1
1
2
2
2
1
1
1
2
1
2
1
2
2
1
2
l g & r y h;
gA
g ; fn
2
r Fkk
2
2
2
l
xq
1
1
nk r y ft l o Q l fn’ k : i
g rc
2
1 v kj
2
2 g r Fkk b u o Q c hp d k U; u d k. k
518
x f. kr
j [ kk
✁✝ ☎ ✂✝ ✆ l ✄✝
✁✝ ✡✟✞ ☎ ✠
v kj r y
o Q c hp
sin f
✝
☎ |✄✄✝|✡|✟✟✞✞ |
A☛ + B☞ + C✌+ D = 0
☛+B ☞ +C ✌+D = 0
A A ✆ B B ✆C C
q = cos
A ✆B ✆C
A ✆B ✆C
ry k
1
1
A2
1
2
2
1
e]
2
2
1
✝
| ✠ - ✂ ✡✟✞ |
(☛ , ☞ , ✌ )
l fn’ k : i
, d
flc n
gA
, d
flc n
1
1
o Q c hp
2
1
-1
1
d h ry
rc
r Fkk
1
2
fg
d k U; u d k. k
1
2
2
1
q
d k U; u d k. k
1
2
2
2
2
l fn’ k
✂✝
☛ ☞ ✌
A☛ ✆ B☞ ✆ C✌ ✆ D
A ✆B ✆C
g]
A + B +C +D =0
1
2
rc
2
2
2
f t l d k f LFkf r
1
g
l
1
2
✍✎✍
2
gA
l
ry
nj h
✁✝ ✡✟✞ ☎ ✠
l
nj h
❆❇❈❉❈ 12
❊❋●■❏❑ ▲▼❏◆❖▼❏P◗ Linear Programming
The mathematical experience of the student is incomplete if he never had
the opportunity to solve a problem invented by himself. — G POLYA
❍✁✂✄☎✆✁ (Introduction)
❢✝✞✟✠ ✡☛☞☞✌☞✍✎ ✏✍✎ ✑✏ ✒✓❢✔☞✡ ✕✏✠✡✒✖☞☞✍✎ ✌☞✒✓ ❢✗✘ ✝✙❢✚ ❢✗✘ ✡✠
✕✏❧✛☞✌☞✍✎ ✏✍✎ ✜✘✢✍✣ ✌✘✤✝✛✙ ☞✥✍ ✝✒ ❢✢✦☞✒✧❢✢✏★☞✩ ✡✒ ✦✤✢✣✍ ✑✓✪✎ ✡☛☞☞
XI ✏✍ ✎ ✑✏✘✍ ✗☞✍ ✦✒ ✒☞❢★☞✛☞✍✎ ✢☞✟✍ ✒✓❢✔☞✡ ✌✕❢✏✡☞✌☞✍✎ ✌☞✒✓ ✒✓❢✔☞✡
✌✕❢✏✡☞✌☞✍✎ ✢✍✣ ❢✘✡☞✛☞✍✎ ✢✍✣ ✌☞✟✔✍ ☞✠✛ ❢✘✈✝✖☞ ✕✍ ✑✟ ❢✘✡☞✟✘✍ ✢✍✣
❢✢✫☞✛ ✏✍✎ ✌✬✛✛✘ ✡✒ ✦✤✢✍✣ ✑✓✪✎ ✥❢✖☞✚ ✏✍✎ ✡✭✩ ✌✘✤✝✛✙ ☞✥✍ ☞✍✎ ✏✍✎
✌✕❢✏✡☞✌☞✍✎✮✕✏✠✡✒✖☞☞✍✎ ✢✍✣ ❢✘✡☞✛ ✕❢✯✏❢✟✚ ✑✓✪✎ ✭✕ ✌✬✛☞✛ ✏✍✎ ✑✏
✒✓❢✔☞✡ ✌✕❢✏✡☞✌☞✍✎✮✕✏✠✡✒✖☞☞✍✎ ✢✍✣ ❢✘✡☞✛☞✍✎ ✡☞ ✘✠✦✍ ✗✠ ✥✭✩ ✢✤✣✞
✢☞❧✚❢✢✡ ♦✠✢✘ ✡✠ ✕✏❧✛☞✌☞✍✎ ✡☞✍ ✑✟ ✡✒✘✍ ✏✍✎ ✜✝✛☞✥✍ ✡✒✍✥✎ ✍✪
✱✡ ✝✣✘✠✩✦✒ ✰✛☞✝☞✒✠ ✗☞✍ ✢❧✚✤✌☞✍✎ ♦✓✕✍ ✏✍✲☞ ✌☞✒✓ ✢✤✣✳✕✛☞✍✎ ✡☞
✰✛✢✕☞✛ ✡✒✚☞ ✑✓✪ ❢✘✢✍★☞ ✢✍✣ ❢✟✱ ✜✕✢✍✣ ✝☞✕ Rs ✺✴✵✴✴✴ ✌☞✒✓
✒✔☞✘✍ ✢✍✣ ❢✟✱ ✢✍✣✢✟ ❥✴ ✢❧✚✤✌☞✎✍✎ ✢✍✣ ❢✟✱ ❧❛☞☞✘ ✑✓✪ ✱✡ ✏✍✲☞ ✝✒
L. Kantorovich
Rs ✷✺✴✴ ✌☞✓✒ ✱✡ ✢✤✣✕✠✩ ✝✒ Rs ✺✴✴ ✡✠ ✟☞✥✚ ✌☞✚✠ ✑✓✪ ✢✑
✌✘✤✏☞✘ ✟✥☞✚☞ ✑✓ ❢✡ ✱✡ ✏✍✲☞ ✡☞✍ ✶✍✦✡✒ ✢✑ Rs ✷✺✴ ✌☞✒✓ ✱✡ ✢✤✣✕✠✩ ✡☞✍ ✶✍✦✘✍ ✕✍ Rs ✼✺ ✡☞
✟☞②☞ ✡✏☞ ✕✡✚☞ ✑✓✪ ✏☞✘ ✟✠❢♦✱ ❢✡ ✢✑ ✕②☞✠ ✢❧✚✤✌☞✎✍✎ ✡☞✍ ✶✍✦ ✕✡✚☞ ✑✓ ❢♦✘✡☞✍ ❢✡ ✢✑ ✔☞✒✠✗✚☞
✑✓ ✚✶ ✢✑ ♦☞✘✘☞ ✦☞✑✚☞ ✑✓ ❢✡ ❢✡✚✘✠ ✏✍✲☞ ✍✎ ✱✢✎ ✢✤✣❢✕✩✛☞✍✎ ✡☞✍ ✔☞✒✠✗✘☞ ✦☞❢✑✱ ✚☞❢✡ ✜✝✟❡✸ ❢✘✢✍★☞
✒☞❢★☞ ✝✒ ✜✕✡☞ ✕✡✟ ✟☞②☞ ✌❢✸✡✚✏ ✑☞✪✍
✭✕ ✝✙✡☞✒ ✡✠ ✕✏❧✛☞✌☞✍✎ ❢♦✘✏✍✎ ✕☞✏☞❜✛ ✝✙✡☞✒ ✡✠ ✕✏❧✛☞✌☞✍✎ ✏✍✎ ✟☞②☞ ✡☞ ✌❢✸✡✚✏✠✡✒✖☞ ✌☞✓✒
✟☞✥✚ ✡☞ ❜✛✹✘✚✏✠✡✒✖☞ ✔☞☞✍♦✘✍ ✡☞ ✝✙✛☞✕ ❢✡✛☞ ♦☞✚☞ ✑✓✵ ✭✫✻✚✏✡☞✒✠ ✕✏❧✛☞✱✽ ✡✑✟☞✚✠ ✑✓✪✎ ✌✚✾
✭✫✻✚✏✡☞✒✠ ✕✏❧✛☞ ✏✍✎ ✌❢✸✡✚✏ ✟☞②☞✵ ❜✛✹✘✚✏ ✟☞✥✚ ✛☞ ✕✎✕☞✸✘☞✍✎ ✡☞ ❜✛✹✘✚✏ ✜✝✛☞✥✍ ✕❢✯✏❢✟✚ ✑✓✪
✒✓❢✔☞✡ ✝✙☞✍✥✙☞✏✘ ✕✏❧✛☞✱✽ ✱✡ ❢✢★☞✍✫☞ ✟✍❢✡✘ ✱✡ ✏✑✿✢✝✹✖☞✩ ✝✙✡☞✒ ✡✠ ✭✫✻✚✏✡☞✒✠ ✕✏❧✛☞ ✑✓ ✌☞✒✓
✜✝✒☞✍♠✚ ✜❢❀✟❢✔☞✚ ✭✫✻✚✏✡☞✒✠ ✕✏❧✛☞ ②☞✠ ✱✡ ✒✓❢✔☞✡ ✝✙☞✍✥✙☞✏✘ ✕✏❧✛☞ ✑✓✪ ✜❁☞✍✥✵ ✢☞❢✖☞❂✛✵
✝✙✶✸✎ ✘ ❢✢✐☞✘ ✌☞❢✗ ✏✍✎ ❢✢❧✚❃✚ ✕✤✕✥✎ ✚✚☞ ✢✍✣ ✡☞✒✖☞ ✒✓❢✔☞✡ ✝✙☞✍✥✙☞✏✘ ✕✏❧✛☞✱✽ ✌❄✛❢✸✡ ✏✑✿✢ ✡✠ ✑✓✪✎
✭✕ ✌✬✛☞✛ ✏✍✎✵ ✑✏ ✢✤✣✞ ✒✓❢✔☞✡ ✝✙☞✥✍ ✙☞✏✘ ✕✏❧✛☞✱✽ ✌☞✒✓ ✜✘✡☞ ✌☞✟✔✍ ☞✠ ❢✢❢✸ ❅☞✒☞ ✑✟ ❢✘✡☞✟✘✍
✡☞ ✌✬✛✛✘ ✡✒✍✥✎ ✍✪ ✛❁❢✝ ✭✕ ✝✙✡☞✒ ✕✏❧✛☞✌☞✍✎ ✡☞ ✑✟ ❢✘✡☞✟✘✍ ✢✍✣ ❢✟✱ ✌❜✛ ❢✢❢✸✛☞✽ ②☞✠ ✑✓✪✎
12.1
520
① ✁✂✄
12.2
Problem and its Mathematical Formulation)
(Linear Programming
❥☎✆✝✞✟ ✠✡✞☛☞✞
✡ ✌✍ ✎✌✏✑✞ ✒✞☎ ❥ ✓✎✟✞ ☞✆✔✞✕✖✑ ✎✗✘✞✖✟❥✔✞
❣✙ ✚✛✜✢ ✣✤✥✢✦ ✣✤✙✧✢★ ✩✛✦✢✪✫✬ ✩✭✢❣✦✮✢ ✤✪✯ ✰✢✱✢ ✛✲✢✦✳✴✢ ✵✦✬✪ ❣✶✳ ✷✢✪ ✣✵ ✭✢✪ ✥✦ ✦✢✣✧✢✸✢✪✳ ✤✢✹✺ ✰✙✻✸✢
✤✪ ✯ ♦✣✮✢✬✺✸ ✰✼✽✢✺✵✦✮✢ ✚✱✢✤✢ ♦✣✮✢✬✺✸ ✛✲✣ ✬✾✛ ✵✢ ✙✢♦★✭ ✧✢★ ✜ ✵✦✪ ♦✢✿ ❀✰ ✩✭✢❣✦✮✢ ✙✪✳ ❣✙✜✪ ❁✸✢✜✛✼ ✤★✵
✭✪ ♥✢✢
✣✵
(i)
❖✸✢✛✢✦✺
❀✰✤✪✯
(ii)
✚✛✜✺
❂✜ ✦✢✣✧✢
✚✣✬✣✦✫✬
✤❣
✵✢✪
✣✜✤✪ ✧✢
✙✢✜
✬✵ ✰✺✣✙✬ ❣✶
❉❊❋❊❊❊
✩✛✹♠❂
✹✺✣✷❛
✣✵
✤❣
✵✢✪❀★
✣✤✣✴✢❜✜
✹✺✣✷❛
✚✛✜✺
❣✢✪ ♦✢✿
✸✢
✭✢✪ ✜✢✪✳
✸✢✪✷✜✢❅✙✵
✤✪✯
✰✳✸✢✪ ✷✜✢✪✳
✣✤✣❂✸✢✪✳
✰✪
✙✪✳
✣✜✤✪✧✢
✣✤✣✴✢❜✜
✵✦
✹✢✴✢
✰✵✬✢
✵✙✢
❣✶✿
✰✤✪✯ ♦✢✿
✩✰✪
✚✢✶ ✦ ✴✢✺
✥✸✜
✤❣
Rs
✵✢✪ ❀★
✜❣✺✳
✰✵✹
✚✬❏ ❣✙
✵✦
✛✢✰
♥✢✦✺✭✬✢
✙✪❃✢✢✪ ✳
✚✣❂✵✬✙
✤✻✬❄ ✚✢✳✪ ✳
●❊
✵✢✪
✦♥✢✜✪
✤✪✯
✣✹❛
✻✱✢✢✜
✵✢✪
✤✪✯ ✤✹
♥✢✦✺✭
✙✪ ❃✢✢✪✳
✰✵✬✢
❣✶✿
✤✪✯
♥✢✦✺✭✜✪
❀✰
✣✻✱✢✣✬
✵✢
✙✪ ✳
✣✜✧✥✸
✩✰✵✢
✵✦✬✢
✰✵✹
✙✪ ❃✢
✜
✵✺
❉❊❋ ❊❊❊
✰✵✹
✹✢✴✢
❍❣❄✬ ✰✢✦✺
✹✢✴✢
♥✢✦✺✭✵✦
✦✢✣✧✢
✤✪✯ ✤✹
✤❄✯ ❡✰✸✢❈
50,000 ÷ 500,
✙✪ ✳
Rs 60 × 75
❣✶
✚✱✢✢★ ✬✈
✰✳✴✢✢✤✜✢❛❈ ❣✶✳ ✿ ✩✭✢❣✦✮✢
✛✢✰
Rs
✤✪✯
❣✺
♥✢✦✺✭✜✪
✚✱✢✢★ ✬✈
✚✢✶✦
✣✤✣✴✢❜✜
❖✸✢✛✢✦✺
✣✜✤✪ ✧✢
❑▲▼▼
✣✹❛
●❊ ✤✻✬❄ ✚✢✪✪✳
Rs (10 × 250 + 50 × 75),
◆✢✬ ✵✦✬✪ ❣✶✳ ✣✵ ✛✯✜✺★ ✥✦
✰✵✬✢
❣✶ ❋
✹✢✴✢
❣✢✪♦✢✿
✵✦ ✰✵✬✢ ❣✶ ❋ ✫✸✢✪ ✣
✳ ✵ ✩✰✤✪✯
✩✰✵✢
✣✜✤✪ ✧✢
✩✰✤✪✯
■❊❊
✵✢
✥✸✜
✤❄✯ ❡✰✸✢❈
❣✺
✵✦✬✢
❣✶✿
♥✢✦✺✭
❣✺
❣✢✪ ♦✢✿
✵✢
✵✢✪
✻✱✢✢✜
Rs 6250
✣✤✣✴✢❜✜ ✥✸✜ ✣✤✣❂✸✢✪✳
✸✢✪✷✜✢✚✢✪ ✳
❍✢❁✸
✤❣ ■❊ ✙✪❃✢✢✪ ✳ ✚✢✶ ✦ ❉❊ ✤❄✯ ❡✰✸✢❈ ♥✢✦✺✭✜✪
✵✢✪ ✦♥✢✜✪
✚✱✢✢★ ✬✈
✬❍
✰✵✬✢
✤✪ ✯✤✹ ●❊ ✜♦✢✪✳ ✵✢✪ ❣✺ ✦♥✢ ✰✵✬✢ ❣✶✿ ✚✬❏ ✤❣ ●❊ ✤❄✯ ❡✰✸✢❈ ✙✢✽✢ ♥✢✦✺✭✜✪ ✤✪✯ ✣✹❛
✣✷✰✰✪
❛✪✰✺
✣✵
✩✛✹♠❂
❣✶✿ ✛✦✳✬❄ ✤❣
20
✸✢
✸✢
✙✢✜
✬✱✢✢
✤❄ ✯✰✺★
50,000 ÷ 2500,
Rs (250 × 20)
Rs 5000
✙✪✳
✤❄✯✣✰★ ✸✢✪✳
❣✶✿
✤❣
❀✰✣✹❛
✵✢
✤✪✯
✸✢
✤❄✯❆ ✚✣❂✵ ✙❣❇✤✛✼✮✢★ ✣✻✱✢✣✬✸✢❈ ✸✢ ❖✸✤✦✢✪ ❂✢✪ ✳ ✵✢ ✴✢✺ ✰✙✢✤✪✧✢ ❣✶ ✷✶✰✪ ✩✰✵✢ ✣✜✤✪ ✧✢ ✚✣❂✵✬✙
Rs
✤❣
✙✪ ❃✢✢✪✳
✚✛✜✢✵✦
✩✛✹♠❂ ❣✶✿
❀✰
✣✻✱✢✣✬
❀❅✸✢✣✭✿
✤✪✯
✣✤✣✴✢❜✜
P✢✦✢
✹✢✴✢
✚✛✜✺
✵✙✢
❂✜ ✦✢✣✧✢
✵✢
✰✤✪✯ ♦✢✿
✚❍ ✰✙✻✸✢ ✸❣ ❣✶ ✣✵ ✩✰✪ ✚✛✜✺ ❂✜ ✦✢✣✧✢ ✵✢✪ ✚✣❂✵✬✙ ✹✢✴✢ ✛✲✢◗✬ ✵✦✜✪ ✤✪✯ ✣✹❛ ✣✵✰ ✛✲✵✢✦
✣✜✤✪ ✧✢
✵✦✜✢
✛✲ ✸✢✰
✵✦✜✢
12.2.1
✙✢✜
x
✣✵
✚✢✶✦
❱❯
✙✪ ❃✢✢✪✳
y
x 0
(
y 0
✫✸✢✳ ✪✣ ✵
✙✪ ❃✢✢✪✳
❀✰
✛✲ ✧✜
✵✢
✩❇✢✦
✭✪ ✜✪
✤✪ ✯
✣✹❛
❣✙✪✳
✰✙✻✸✢
✵✢
♦✣✮✢✬✺✸ ✰✼✽✢✺✵✦✮✢
✵✦✜✪
✵✢
✥✢✣❣❛✿
❧❘❙❚❯
✹✺✣✷❛
✻✛❴❵✬❏
✥✢✣❣❛❢
✚✢✶✦
❲❳❨❯❩❬❚
❧❭❪❯❬❱❫❨❯
✵✺
x
❝✮✢✪✬✦
➼❞❦sr❤
✤❄ ✯✣✰★✸✢✪✳
✰✳♥ ✸✢
❣✶✳ ❋
✚✢✶ ✦
(Mathematical Formulation of the Problem)
y
✤❄ ✯✣✰★✸✢✪✳
✵✺
✰✳♥ ✸✢
❣✶
✣✷❜❣✪ ✳
✛✯✜✺★✥✦
❖✸✢✛✢✦✺
♥✢✦✺✭✬✢
❣✶✿
✚✱✢✢★ ✬✈
✐♣ q❤❦st
✵✺
... (1)
... (2)
)
✰✳♥ ✸✢
❝✮✢✢❅✙✵
✜❣✺✳
❣✢✪
✰✵✬✺
❣✶✿
❥ ✁✂
✄
❖
☛
☞ ✌☞
✍✎
✏
❖
☛
✑ ✒☞ ☛
✎
✓
✌
✍
✔
✕✖✗✘
✙
✧
❖
☛
✑
✍
☞
✦
✖
✜
✥
✔
☞
✥
✍
❖
☛
✑ ✒☞
☛
✎
✧
✑
✦
✌☞ ✒
✑
✦
✖✚
✍
☞
✕✛
☞
✏
☛
✜
★
☞ ✢
✩
✑
✔
✕✖✗
✘
✙
✑
Rs
☛
✜
✪
✺ ✣ ✤
✣ ✣ ✣
✫
✘
✔
✜
✥
✓
✗
☞
☎
✆
✄
✝
✕✚
✏
☛
✜
☞ ✢
☛
✜
521
✠
✡
✑
✦
✛
☞
✬
☞
✦
✟
✞
✄
✝
✗ ✍✚ ✦
✭
✣
✜
✥
✓
✗
☞
✦
✍
✗
✧
☞
✚
✦
☞
★
✮
✑
✦
✕
✩
▲
①
☞ ☞
✚
✗
☞
✱☞
✕
✯
☞
✎
❖
☛
✑
✍
✲
✘
✎☛
✫
✌
☞
✦
✖
✜
✥
✰
✳
✙✦
❖
☛
✘
✗ ✍✚ ✦
✌
✍
2500x + 500y
5x + y
☛
☞
✕✚
✑
✦
✛
☞
❖
x
✔
☛
✑
✒☞ ☛
✎
y
☞
✥
✍
✵ ✒
✧
✌❧
✗
★
✑
✦
✌✧
☞
✍
✧
✚
✗
☞
✔
Z = 250x + 75y
☞ ✢
❖
☛
✑
✍
☞
✦
✖
✕
✚
✌✧
★
✚ ✕
60
✴
✶
(
✫
①❧
✒
☞
✜
✗✜
☛
✱☞
✜
★
✚ ✕
✕
✖✗
✘
✙
✌❧
✗
☞ ✘
☞
✜
✙
❖
✌
✕✍
✑
✕
Z
✫
☞ ✘
✫
✔
✒
✕ ✙✗
✫
☞
✔
✫
✜
✥
✕
✧
✍✥
✕
✲
✫
✌
✑
✗✎
✕
✕
✗✜
✹
✘
✦
✜
✗
☞
✕
✕
☞
✑✛
☞
❁
✦
☞
✕
❂
✌
❖
☛
✘
✙
✜
☞
✦
✔
☞
✍
✥
✕
✸
✒
✦
✹
✘
✕✗
☛
☞
✸
☞
✒
✗
✘ ☞
✜
✥
✜
☞
✦
✸
☞
✘
✎
✜
✥
100
60
✾
0
✿
☞
✔
✕✖✗
✫
▲
❀
✘
✙✎ ✗ ✍
✗ ✍✚
✧
☞
✕✘ ☛
☞
✦
☞
✜
✥
✳
✑
✦
❖
☛
✑
✍
☞
✦
✖
❖
☛
✮
✘
✘
✙✎ ✗ ✍
✕✗☛
☞
✸
☞ ✘
☞
✧
☞
✕✘ ☛
☞
✦
✜
✥
✳
✑
✦
❖
☛
✑
✍
☞
✖
✦
❖
☛
✫
✜
✥
✦
✸
☞ ✘
✦
✜
✥
✰
✦
✶
✍
✑
✶
✍
✑☞
✑☞
✒
✙
✔
❂
☛
✒
✙
✍✥
✕
✍✥
✕
☞
✗
✮
☛
☞
✢
✯
☞
✎
✫
☞
✗
✔
✫
☛
✭
✎
✭
✎
✩
✒
✎
★
☞
✦
✩
✈
★
✘
✍
✮
✕✗
✫
✘
✍
✧
✪
✰
✱☞
✸
✼
✕✗
✮
✘
✦
①✮
✕✗
❀
☞
✱☞
✸
✼
✕
✗
☞
✔
✒
✕ ✙✗ ☞
✔
✭
☞
✦
✗
☞
✦
✍✥
✕
☞
✦
①
☞
✗
✌
❧
☞
✦
☞
❧
✙✚
✰
✍✥
✕
Z
✚
①
☞
✗
✌❧
☞
✦
✩
☞
❧
✙✚
✒✙
✕✗
✷
✦
✑
✜
✒
✙
★
✛
☛
✘
✙
☛
☞
❂
✌✧
✙
✫
☞
✗
✩
✔
✒
✕ ✙✗
✫
☛
☞
✙✦
☞
✔
①✕
✒✯
☞
✚ ✓
✧
☞
✦
✒
✙
✫
✘
✎☛
☞ ✘
✗ ✍✚
✦
❇
✪
✑
✦
✱☞
☞
✎
❅
☞
✜
✗
✜
✫
✍✥
✕
✒
✦
✒
✜
✕✗
✥
✓
✗
☞
❖
✜
✥
✵
❁
✫
✔
y
☞
✥
✍
✧
✪
✸
✥
✒
✦
✪
✘
✙
✑
①
✫
✒✒
✘
✜
✥
✰
✌❧
✕
☛
✑
✍
☞
✖
✦
❄
✘
✗
☞
✦
✪
☛
✜
✌❧
☞
✦
✜
✥
✕✗
❃
✒✘
❁
✪
✔
❣
✘
✦
✜
✥
✪
✘
✙
✒
✦
✘ ☞
✒
✱☞
✦
❣
✘
✍
✍✥
✕
✧
☞
✦
✮
✑
✦
✗
①
✫
✒
✘
✱☞
✌
✭
✰
✙
☞
✚
✫
☞
✗
❆
✜
✥
✔
☞
✍
✥
❈
☞
✗
✌
✐
✒
✦
❈
☞
✙✚
✶
✍
✧
★
✚ ✑
✶
✍
✫
✗✍
①❧
✸
✼
✕✗
✔
✚
✦
✗
❀
✘
✼
✖
✫
☞
✦
✫
✈
✫
✼
✕✖
✫
☞
✗
✸
☞
✦
✫
☞
✗
✭
✍✥
✕
✥
❃
☞ ✘ ☞
✭
✶
☛
✒✼
✖
☛
★
✚
✫
☛
✚ ✘
✙
x
✩
☛
☞
♠
✏
✸
☞
✦
✭
✍✥
✕
✒
✙
✔
✕✖
✗
)
✘
✽
✘
✾
❣
✏
✔
✕✖✗
☛
✦
✒
✦
✱
☛
✚
✫
▲
✭
✗
✧★
☞
✗
✥
✮
✘
✭
✕
✥
☞
✍
✱☞
✗
❣
✚
✑
✛
☞
✦
❁
☞ ✓
✸
0, y
✿
✩
✈
✩
✈
☞
✚
✫
☛
☞
✔
✍✥
✚
★
✌✧
✩
✒
✙
✑
✪
✧
✙✦
✌✧
✪
✧
✌
☞
✗
Z
★
☞
✗
✲
✑
✦
✭
✸
✚
✙✦
✑
✦
✧
☞
✦
✏
✙
... (4)
✜
✥
✭
✍✥
✕
☞ ✯
☞
)
✭
✕
x
✜
✙✦
☞
☞
✖
✦
✮
✎
✗✍
x+y
★
✮
✷ ✒
✗
☛
✑
✍
✹
✦
5x + y
✵ ✒
✕
✜
✥
✫
✌
)
☞
✦
✖
✳
☞ ✘
✱☞
✔
✘
☞
✭
✕
✲
✘
✎
☞
★
✚
✱☞
✼
✗☞
♦
✸
✜
☛
①✕
☛
☞
✕✚
★
✛
☞
✎
✩
✒
✙
✗ ✍✚
♦
✙✦
♠
✦
✑
✦
✛
☞
✫
✌
✷
☞
✕✚
✲
✑
✦
Z = 250x + 75y (
✌
❧
✐ ✻
✒
✦
✑
✍
... (3)
★
❖
☛
100
✴
x+y
✔
☞
✍
✥
50,000 (
✴
✘
☞
✌☛
✽
✜
✥
✕✗
①
✌☛
✽
✜
✥
✕
✗
✕✑✛
☞
❁
✦
☞
✌❧
☞
✦
☞
❧
✙
☛
☞
✕✑✛
☞
❁
✦
☞
❢
✕
☛
☞
☛
☞
✦
✸
✚
①✦
✔
☞
☞
❅
☞ ✘
✗ ✍✚
❉
❊
✼
☞ ✰
❣
✚
✦
✒
✦
✧
✪
✌✑ ✽
✜
✙
✔
✼
✫
①
✌
✕✍
✯
☞
☞
✕❁
● ❍■
❏
❑
☞ ✘
✗ ✍✦
▼◆
P
✦
◗
✸
✚ ✗
☞
✕✗
✭
★
☞
✗
❣
☛
✚
✌✧
✚
✮
✘
✙✎ ✗ ✍✱☞
✜
✫
✈
✌
✐
☞
✦
①
✕
✍
✥
✕
❂
✑
☞
✦
✚
☞
✜
✥
✌❧
☛
☞
✦
✭
✜
✙
✍✥
✕
✍✥
✕
♠
☞
✗
✷
✦
☞
✌❧
☛
✌❧
☞
✦
✌✧
✸
✼
✕✗
✚
✪
✌
✍
✩
☞
❧
✙
★
✛
☛
❋
☞
✦
①
☞
✗
Z = ax + by,
✭
✗
①
✏
✕✸
✚
✗
✒
✙
a, b
☞ ✘
☞
☞
✦
✫
☛
☞
✔
☞
✦
✲
✶
✗
☞
✥
✓
✗
☞
✦
✔
☞
✥
✌✶
☞
✕ ✍✗
✌
✒
✦
✹
①
✫
✫
✙✦
✜
✗
✍✦
✦
✱☞
✔
★
✗✜
✜
✜
✥
✰
✶
✍
✜
✥
✕
✸
✚ ✗
☞
✔
✕✖
✗
✘
✙✎ ✗
✍
☛
☞
522
① ✁✂✄
♠☎✆✝✞✟✠
✚✆
♠✡✝☛✆☞✝
Z = 250x + 75y
✌✞✍
✱✎
✆✏ ✑✒✝✎
♠✓✞✔✕
☎✖✗✘
☛✏✙
✚✆
x
y
✈✝✏✆
✑✘☞✝✝✥✕ ✎
✎☛✗✝✠✞ ☛✏✍ ✙
❖✛✜✢✣✤✦
✱✎
✆✏ ✑✒✝✎
✯✕✫✆✝✞✮
✎☛✗✝✠✞
✩✞
✠✎
✲❧✴
☛✏ ✍✙
☎✧ ✝★
✞ ✧ ✝✌✘
x
☎✧ ✑✠✭✍ ✮
✈✩✑✌✎✝✈✝✞✍
✩✌✪✕✝
✫✞✖
0, y
✰
✰
✩✌✵✶✚✕
✎✝
✚✆✝✞✍
0
☎✆
➼☞✝✞✠✆
✯✕✫✆✝✞✮
✆✏ ✑✒✝✎
✈✩✑✌✎✝✈✝✞✍
✯✕✫✆✝✞✮
✎☛✗✝✠✞
✎☛✗✝✠✞
☛✏✙
✍
✕✝
✩✌✬✎✆☞✝
♠☎✆✝✞ ✟✠
✕✝
☎✧✑✠✭✍ ✮
♠✡✝☛✆☞✝
✌✞✍
✲✳✴
☛✏ ✙
✍
❜✷✸✹✺ ✻✼ ✻✽✾✹ ✻✺✿✛✣❀❁ ✑✘✑✔✚✠ ✯✕✫✆✝✞✮ ✝✞✍ ✫✞✖ ✈✮✬✘ ✈✩✑✌✎✝✈✝✞✍ ✫✞✖ ✩✌✵✶✚✕ ❢✝✆✝ ✑✘✮✝✥ ✑✆✠ ✩✌✪✕✝
t✝✞
✚✆✝✞✍
✲✕❂✝✝
✩✌✪✕✝
✡✝✞
x
✚✆
✎☛✗✝✠✬
☛✏✙
y
✈✝✏✆
✆✏ ✑✒✝✎
✴
✌✞✍
✆✏ ✑✒✝✎
☎✧ ✝★
✞ ✧ ✝✌✘
☎✖✗✘
✩✌✪✕✝✱❉
✎✝✞
✱✎
✈✑✮✎✠✌
✑✫✑✔✝❇❈
✕✝
☎✧ ✎✝✆
❃✕❄✘✠✌
✎✬
✎✆✞❅
❆❇❈✠✌
❆❇❈✠✌
✩✵✩✍★✠
✩✵ ✩★
✍ ✠
✩✌✪✕✝
☛✏✙
✩✵✩✍★✠ ✩✌✪✕✝ ✯✕✝☎✝✆✬ ❢✝✆✝ ✌✞❊ ✝✝✞✍ ✠❂✝✝ ✫✵✖✑✩✥✕✝✞✍ ✎✬ ✒✝✆✬✡ ✌✞✍ ☎✧✕ ✵✟✠ ✱✎ ❆❇❈✠✌ ✩✵✩✍★✠ ✩✌✪✕✝ ✠❂✝✝
✆✏✑✒✝✎
☎✧ ✝★
✞ ✧ ✝✌✘
✈✭
❆✩
☛✌
✈❋✕✝✕
✎✬
✩✌✪✕✝
✑✫✫✞✚✘✝
✌✞✍
☛✌
✎✝
✎✆✞★
✍ ✞
✫✞✖✫✗
✱✎
✑✎
♠✡✝☛✆☞✝
✱✎
✈✝✗✞ ✒✝✬✕
☛✏✙
✆✏ ✑✒✝✎
✑✫✑✮
☎✧ ✝★
✞ ✧ ✝✌✘
✩✞
☛✬
✩✍ ✭✍✑✮✠
12.2.2
of Solving Linear Programming Problems)
❥●❍■❏❑
✎❞✝✝
XI,
✑✘✎✝✕✝✞✍
✌✞✍
✎✝
▲▼❏P
◆ ❏
▼ ◗❘
☛✌
✩✬✒✝
✈✝✆✞ ✒✝
✌✞✍ ✑✫✫✞✚✘
✎✬ ☛✵❆✥
✎✝✞
❢✝✆✝
✈✝✆✞✒✝
5x + y
x+y
x
y
❆✩
❆✩
❤
✰
✰
✚✵✫✖
✞
✒✝✬✍ ✚✠✞
✎✆✞✍★✙
✞
✌✞❊✝✝✞✍ ✈✝✏✆
✌✞✍
✑✎
✠❂✝✝
❑❏◆ ❳❨
✑✎✩
✈✭
☎✧ ✎✝✆
✈✝✆✞✒✝✬✕
✫✵✖✑✩✥✕✝✞✍
☛✌✞✍
❑❥❘◆
✌✞✍ ✑✘✫✞✔ ✝
✆✏ ✑✒✝✎
✎✬
60
0
0
... (2)
... (3)
... (4)
✲✳✴
✈✝✆✞✒✝
✩✞
✲❧✴
☎✧♣✕✞✎
♦✭✡✵
✲❫✝✕✝✍✑✎✠
✠✎
✫✞✖
♦✭✡✵✈ ✝✞✍
✯✕✝☎✝✆✬
❞✝✞✐✝✴
❢✝✆✝
✩✞
✑✎✩
x
☛✗
✩✌✪✕✝
☎✧ ✎✝✆
☛✗
✲✯✕✫✩✝✕✬✴
✈✝✏ ✆
y
❪✝✠
✎✆✠✞
✩✞
✎✝ ♠❡✗✞✒✝
✫✞✖
❛☎
☎✧✡❝✝
✩✍ ✭✍ ✑✮✠
☛✏✍ ✙
✆✏ ✑✒✝✎
✈✭
✎✆✞★
✍ ✙
✞
☛✌✞✍
☎✧♣✕✞✎
✎☛✗✝✠✝ ☛✏✙
♦✭✡✵
✩✌✪✕✝
✎✝
✯✕✫✆✝✞✮ ✝✞✍
✌✞✍
☛✏✙
✍
✎✝✞
✩✵✩★
✍ ✠
✩✵✩★
✍ ✠
☛✏ ✙
☛✗
rs✉✇②③④
✈✩✌✬✎✆☞✝
✈✘✵✶❫✞✡
✈✭ ☛✌ ❆✩
✎✝ ✩✵✩★
✍ ✠ ❞✝✞✐✝ ✎☛✗✝✠✝ ☛✏ ✲✈✝✫q✖✑✠ ✳❴❵✳✴✙ ❆✩
✎✝
t✝✠✝
(Graphical Method
❍❬❍❭
✑✫✎❡☎ ☎✧✪✠✵✠ ✎✆✠✝ ☛✏✙ ❆✩✑✗✱ ✕☛ ❞✝✞✐✝ ✩✌✪✕✝
❞✝✞✐✝
✑✎✕✝
✆☛✞★
✍ ✙
✞
❱❏❨◆■❏❩❯
✚✆✝✞✍
❢✝✆✝
✎✝✞
✑✘✕✠
✑✘✑✌✥✠
✫✵✖✑✩✥✕✝✞✍ ✌✞✍ ✑✘✫✞✔ ✝ ✎✆✘✞ ✫✞✖ ✑✗✱
❑❩
✈✩✌✬✎✆☞✝✝✞✍
... (1)
✎✝
✡✝✞
✑✫✑✮
100
✈✮✥✠✗✝✞✍ ✫✞✖ ♠❦✝✕✑✘❇♥
❞✝✞✐✝
☛✏
☛✏ ✍
✌✞❊ ✝✝✞✍ ✈✝✏✆
☛✗
✑✘✎✝✕
✈✩✌✬✎✆☞✝✝✞✍
✩❦✝✬
❤
❙◗❚❯❏❱❏◆❲
✩✌✪✕✝
12.1
✎✝
✈✝✆✞ ✒✝
✳❴❵❴
✩✌✪✕✝
✒✝✬✍✚✞✍❣
❥ ✁✂
✄
✈
☛ ☞
✌
✍
❧✛
❧✜
✢✣
✎
✏ ✑✏
✤
✥ ✦
✧
✥
✒✓ ✔
✕
✐
★ ✩
✓
✸
✵
✬
✭
✖✓ ✶
✔
✓ ✚
OABC
✽✓
✓
✔
✾
✈
✪
✍
✎
✘
✓ ☛
✫
✬
✓
✼
✪✪
✚
✲
☛
✔
✎
✏
✬
☛
✿
✛
❧
✜
✢✣
✌
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✓
✬
✎
✏
✪
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✔
✮
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☛ ✔
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❆
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✓ ✙
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✪✪
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✒✓ ✖
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✭✮
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☛ ✔
✼
✌
❆
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❖
▲
❏
✓
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✓
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❲ ❳
❋
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✓ ✓
✲
●
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✒✓
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❆
✙
✪
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✳
✴
✓ ☛
✔
✖
✵
x, y 0
✬
✭ ✖✓ ✶
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✷
✽✓ ❀
✓
❁
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✒✌
✼
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✌
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▼
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✽✓
✓
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■❏
★
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✭ ✖✓ ✶
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✓ ✚
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✓ ✓ ✲
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■
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★
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✪
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■❏
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P
▲
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✰
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✭✮
✔
✾
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❨
❊
✣
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✓ ☛ ✓
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★ ✓
✌
✖
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✭✮
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■❏
★
★
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★
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✓ ☛ ✔
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☛
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✙
❬
❧❧✜
✛
✢✣
✰
✒
✼
✪✪
✚
✲
☛
✽✓
✓
❣
✔
✍✚
☛ ✍
✼
✪✪
✚
✲
☛
✒
OABC
✽✓
✓
✌
✯
✔
✌
✒
✓
✯
✓ ☛ ✓
✔
✾
❑
■❏
★
✔
✕
✐
✾
✬
✒
✔
✌
✙
✈
✔
❣
♦
✔
✿
✬
✓
✙
✖
✰✔
✪✔
✈
✏ ☛
✚
❂
❀
✕
✮
✿
■❏
★
✪✔
✾
✒ ✖☛ ✓
✺ ❭
❈
✓
✯
✏
✒✓
❪
✘
✈
✎
✶
✒☛
✍
✍✓ ✏
❪
■❏
★
✌
❆
✚
✙
✬
✌
❀
✸
◗
☛ ✒
✍✚
✸
✫
✕✘
✏
✌
✔
✕
✐
★ ✩
✓
✍✓
✏
✭✓
✯
✔
✪
✗
✓
✺
❭
✚
✌
✙
✎
✒
✌
✍
♦
✔
✾
✭✔
✮
☛
✍
✌
✙
❆
P
✍✚
❪
✘
Z = 250x + 75y
✔
❪
✘
✾
✌
✍
✬
✭ ✖✓ ✶
✔
✓ ✔
✚
✯
✾
✔
☛ ❍
✓ ✓
OABC
✽✓
✓ ✔
★ ✓
❀
✎
✶✒☛
✍
❏
✌
✽✓
✓
✭✮
✔
✿
✬
✓
✪
✚
✲
☛
✫
✬
✓
✾
◗
▲
✈
✵
✽✓
✓ ✔
✼
✪
☛ ☞
✪✪
✚
✲
☛
✿
■
❏
★
✿
✈
✎✯
✰
❣
❙
✣
✼
✪✪
✚
✲
☛
✪
✍
✸
✔
❂ ❃
❄❂
✎
☛
❀
◆
▼
❜
❘
✾
✲
✔
✓
✐
✍✏
✽✓
✓
✾
❃
523
✠
✡
✌
✙
✚
❆
❋
●
❉
✈
✓ ✭✮
✪
✪✪
✚
✿
✽✓
✕
✓
✐
✺
✪
✚
✲
☛
❣
✌
✙
✎✱
✓ ✒
✓ ✔
✭✮
✔
✾
✭
✼
✘
✖✙
✾
✎
✒
☛
✟
✞
✄
✝
☞
✰
✒
✻
✗
✆
✄
✝
✽✓
✓
❉❊
✯
✓ ☛
✌
✚
✙
❀
❅ ✓ ✬
✓ ✚
❁
✒
✔
✎✯
✰
✺
✓
✖✓
✾
❧
✒ ✖☛
✭✮
✔
✹
✪
✗
✓
✼
✕✎
✖✗
✓ ✓
☎
✬
✕
✮
✯
✏
❣
■❏
★
✒✓ ✔
✎
✒✪
✕
✒
✐
✓ ✖
❵
✓ ☛
✒ ✖✏ ✔
✒
✓
✕
✐
✬
✓ ✪
✒
✖❆
✚
✔
✪
✎✫
❍
✓ ✎
☛
✺
✒✓
✔
✌
✯
✒
✖✏
✔
✭✮
✔
✎✯
✰
✌
✍
✎
✏ ✑✏
✿
✒ ✖✏
✔
✍✚
❝❡
❋
✦
❢
✍
❤
❴
✯
✎ ✪❞
✓ ✚
☛
✍✓
✏ ✓
✎
✒
✬
✭ ✖✓
✶
✔
✓ ✚
✔
✪✔
✪
✚
❏
✚
✿
✭✔
✮
✒✓ ✏
✔
❝
❡
❋
✦
❢
✔
♥
✕
✮✯
✏
✸
♦
✓
♦
✔
♣
✈
✓ ✙
✖
q
rst
✬
R
✈
✭
✰
✒
✖✙
✉
✇
✬
✎
★
✌
✙
❆
✲
✐
✓
✍✏
✯
✏
✌
❆
✙
y
✈
✓
✙
✖
✪✔
R
✎✫
❍
✓ ☛
✎✱
✓ ✒
✼
✕✎
✖
❏
❞
✕
✐
✓
R
P
✔
✚
✕
✮
x
♠
✖
✕
✖
✎
✒
✬
✎
★
✏
✍
✕
✓
✐
✔
✕✬
✓
✔
✪
✒
☛ ✓
✾
✒ ✖✚
✸
✏
✕
✐
✍
✪
✍
✫
✬
✓
Z
❈
❏
✬
✔
✓ ✚
✔
✭✮
✔
✎✱
✓ ✒
✈
✪
✍
❈
✓ ✔
✎
✒
✰✒
✖✙
❣
R
✎
✯
✰
✎✱
✓ ✒
✕
✓
✐
✲
✔
✓
✐
✍✏
✔
✫
✬
✓ ✈
✓ ✚
✾
✼
✪✪
✚
✲
☛
✓
✽✓
✔
*
✭
✮
✔
✎
✭
✘
✓ ✬
❛
✭
✿
✺
✾
✩
✓
✯
❏
✫
☛
✾
✌
✗
✓
✪✔
✍✓ ✏
✵
✬
❇
☛
✔
✌
✯
❏
✓
✌
✖
✌
❆
✙
✌
✙
✈
✓ ✖
✙
✍✓ ✏ ✓
❀
✈
✎
✶✒
☛ ✍
✬
✓
❣
✓ ✖
✓
✒✓
❀
❈
✿
☛ ✍
✔
✾
✕✫
☛ ✒
✹
✒ ✖✴
✓ ✓ ✚
✪
✍
✾
✪
❪
✘
✸
✖✙
✲✔
✕✕✎
☛
❣
✒✓
✔
✺
✒
✬
✒
✔
✌
✓ ✏
✔
✕
✓
✐
✔
✌
✓ ✔
☛
❏
✬
✌
❫
✌
✓
❪
✘
❦
✬
✏
❴
☛
✍
✔
❈
✌
✓
✾
☛ ✍
✍✓ ✏
✼
✪✪
✚
✲
☛
✽✓
✓ ✔
♠
✓ ✎✌
✰
❆
✲
✔
✐
✓
✍✏
✽✓
✪
✍
✫
✬
✓
✭✮
✔
✌
✙
❆
✎
✯
✰
Z, R
✺
❭
✔
✌
✓
✔
☛
❏
♦
✔
✸
✭✮
✔
✒✓ ✏
✔
✿
✬
✬
✕
✮
✯
✏
corner
❀
✾
R
✍✚
✾
✼
✪
✔
✿
■❏
★
✪
✚
✲
☛
★
✓ ✏
✔
✓ ✚
✸
♦
✓
✔
✈
Z = ax + by
✽✓
✓
✔
✌
✙
☛
❍
✎
✶✒
☛ ✍
✓
✓
✈
✓
✙
✖
❫
✬
✏
❴
☛
✍
✕✎
✖
❏
❞
✌
✙
☛
❏
✎✕
✮✖
✗
✓
✬
✎
★
✬
✌
✺ ❭
♦
✔
✍✓
✏
✖
✕
✖
✎✫
❍
✓ ☛
✌
✓ ☛
✔
✓
✬
✕
✮
✯
✏
✒✓
✎
✭①
✍✓ ✏
✌
✙
☛
✓
✔
✈
R
✎
✶✒☛
✍
✬
✓
❫
✬
✏
❴
☛ ✍
✸
✭✮
✔
✒✓ ✏
✔
✱
✓ ☛ ✓
✌
✙
❆
P
♦
✔
✬
❀
✘
✓ ❑
✺
❭
✈
✸
✌
✓
✲
✔
❣
✓ ✶
✓ ✖✗
✓ ☛
❴
✎✱
✓
✒
✒
✓
❀
❣
✌
✙
✖✙
✔
✺
❭
✎
✶☛
✘
✓ ❑
✍✓ ✏ ✓
✌
✙
❆
✬
✔
✓ ✚
❀
✈
✰✒
Z = ax + by
✎
✒
✵
✔
✕
✍
✐
✍✓ ✏
✒✓
✈
✎
✫
☛
✾
✬
■❏
★
✸
✭
✏ ✌
✿
✕
✖
✌
✓ ✏
✔
✓
♠
✓
✎
✌
✰
▼
✸
✚
✗
✓
❂
✕
✍
✐
✬
✔
✭✮
✔
❀
✈
✏ ✪
✓
✖
✺
✺
✕
✖
✓ ❇
✔
✎
✏ ★
❑
♦
✔
✓ ✓
✚
✒
❣
✌
✍✚
✔
✏
☛
❵
✓
☛
✿
★
✓
✌
✖
✒ ✖✏
✓
✾
■❏
★ ✈
✓ ✔
✚
✕
✖
✴
✓
✍✚
✪
✖
,Z
✔
✯
✒✓
✕✎
✖
✌
✙
❏
❞
✬
❍
✓ ✓
✍✓ ✏
✾
✪✪
✚
❀
✲
☛
✼
✓
✽✓
✔
✸
✭✮
✔
✒
✓ ✏
✔
✾
✬
(0, 0), (20, 0), (10, 50)
❵
✓ ☛
✒
✖✏ ✓
✌
✙
❆
■❏
★
O, A, B
(0, 60)
✈
✓
✖
✙
C
✾
✌
✚
✙
Ø
✈
✓
✖
✙
✍♦
✓ ☞
✈
✓ ✖
✙
■
❏
★ ✈
✓ ✚
✸
✒✓
✏
✔
✔
✭✮
✔
✾
✬
■❏
★
✌
❆
✚
✙
✈
❏
524
♦☎
① ✁✂✄
✆✝
✞✟✠✡☛
☎☞✌
⑤⑥⑤ ⑧
⑦ ⑨
☎❣
♦P◗✾✝▼ ❇✡❄❅
✆✝
♦❄◗
✾♦✾❚
1.
✠☛❃❄
✾✿☛❀❁✡❂✡
❘✡☛❀❙✿❄
❣❄ ❅
✞✟✡❲
❄ ✟ ✡❣✿
2.
❫✝
C (20,0)
5000
✠✡❄
❆❇♦✝✡❇❀
❯✡❱✡
✝❣✡✾♦❨❩
❫♠❄❈❇
✞◗❯✿
m,
3.
Ø❣❈✡✌
(i)
❉❵
(ii)
✾✿♦❄❈ ✡
✝P✝❲
❅ ❃
(b)
✠✡❄
Z
✝P✝❲
❅ ❃
✆✝❀
✞✟✠✡☛
❣✡✿
♦P◗ ❝
qrs✉✇②s
✾✿❜✿
❁✡❄❬✡
❆❇♦☛✡❄ ❚✡❄❅
✱✖
✭✖✙✓
✧✓
✪✥✘✭✲✑
✠❀✾❉❪
❭✡❃
❑✡☞☛
■●
❴❵❙P❑ ✡❄ ❅ ❊❈✡❀❨✡▼❏
✠✡❄
❑▲✡✡▼❃◆
❑✡☞☛
❫✝♦❄◗
✠✡❄✿❀❇
❴❵❙P ✠✡❄ ❙✡❄ ☛❄❘✡✡❑✡❄❅
✞✟✾❃❛❝❄❙
✞✟❞❇❄✠
❣✡✿
❑✾❚✠❃❣
✞✾☛❵t
♦❄◗
✒✓✔✕✓
✖✓
❣❄❖ ✡✡❄❅
❋●
✠❀ ✝❣❀✠☛❂✡✡❄❅ ✠✡❄ ☎❯
❡❇❤✿❃❣
❃▲✡✡
M
☎☞ ❍
❑✡☞☛
✠✡❄✿ ❀❇
m, Z
✞☛
❴❵❙P
✞✟❙✾❈✡▼❃
❣✡✿
♦❄◗
✠❀✾❉❪❳
❭✡❃
✠☛❃❄
❑✡☞☛
❑✾❚✠❃❣
❣✡✿✡
M
✾✠
❑✡☞☛
☎☞❳
❅
❡❇❤✿❃❣
❣✡✿
☎❅❳
☞
✠✡❄
❯❄❃❄
☎☞ ❅
✞➥♥ ❄
❑❡❇▲✡✡
Z
Z
✠✡
❡❇❤ ✿ ❃❣
❣✡✿
❁✡❄ ❬✡
❣❄❅
✠✡❄✆▼
ax + by > M
❇✾❙
✿
✝P✝❲
❅ ❃
♦❄◗
✐✡☛✡
✠✡❄✆▼
❴❵❙P
❑✾❚✠❃❣
❯❄❃ ❄
☎❅ ☞
❣✡✿
❇✾❙
❫❱✡❇✾✿❨♣
✐✡☛✡
✿☎❀❅
✞✟ ✡❦❃
✿☎❀❅
❑❚▼ ❧❃❯
✠✡❄ ✆▼
☎☞❳
ax + by < m
Z
☎☞❳
✠✡
❑❡❇▲✡✡
✐ ✡☛ ✡
✠✡
✞ ✟ ✡ ❦❃
✠✡❄✆▼
❘✡P ❯ ❄
❡❇❤✿❃❣
✐✡☛✡
✠✡❄✿ ❀❇
✾♦✾❚
♦❄◗
✾✿❜✿
☛☞✾ ❘✡✠
✞✟✡❲
✟ ❣✿
❄ ✡
✞❙✡❄❅
✠✡❄
✝❣❥❇✡
✠☛❄❲
❄
❅ ✌
❥✞❨❩
✠✡❄
☎❯
✠❀✾❉❪✌
❑❅❃❲▼❃
④
✖✓✔✗✘✙
✥✲✘✖★✳✓
✥✖✑✓
✢✦
✈✫
❣❄ ❅
❣✡✿
☎☞❳
★✦✭✩✓✖
✥✔
❁✡❄❬✡
, m,
❫❙✡☎☛❂✡✡❄ ❅
❑✡❯❄❘✡
③
✥✍✥ ✎ ✏✑
■●❏
☎☞✌
❅
✠✡
❑✾❚✠❃❣
❑✡☞☛
✿☎❀❅
x+y
*
**
✠✡
❴❵❙P
❑❚▼❃ ❯
❑❵
✞☛
❁✡❄❬✡
❊❋●❍
❑✾❚✠❃❣
✠❀✾❉❪❳
❭✡❃
❴❵❙P❑ ✡❄ ❅
➀
☎✡❄❲ ✡❳
✠✡ ✝P✝ ❅❲❃
Z = ax + by
✆✿
❇✡❄❉✿✡
❾s❿
❪❄✝❀ ✾❥▲✡✾❃ ❣❄❅ ❉❵ ✝P✝❅❲❃ ❁✡❄❬✡ ❑✞✾☛❵t ☎✡❄ ❃✡❄ ☎❣ ✾✿❜✿✾❯✾❘✡❃ ✾♦✾❚ ✠✡ ❫✞❇✡❄❲ ✠☛❃❄ ☎☞❳
❅
4. (a) M
☎❣
✠✡❄
⑤⑦⑧⑨
6250
✝❣❥❇✡
❴❵❙P
❸❶❹
B (10,50)
❇✡ ❃✡❄ ✾✿☛❀❁✡❂✡ ✝❄ ❑▲✡♦✡ ❙✡❄ ☛❄ ❘✡✡❑✡❄❅ ♦❄◗
✠☛♦❄◗
Z
❺s❻❼s❽
0
4500
✾✠
✠✡
❸❶❹
O (0,0)
A (0,60)
❑✾❚✠❃❣
✞❙✡❄❅
✾✿❜✿
☛☞ ✾❘✡✠
❣❄❅
☎☞ ❅
⑩s❶❷ s
✪❢✙✸✓✓
✛✼✽ ✓✙✓
✧✓
50
✚✛✜✍
... (1)
✒✓✔✕✓
✭✗✖✓✙
✣✥✔
✖✓
✖✓
✪✫✭★✛✴
✥✖✑✓
✢✦✰
✢✘
✥✍✥✎✏✑
✖✓✔✣✤
✚✛✜✍
✒✓✔ ✕✓
✖✢✑✔
✢✦✰
✎
✢✓✔ ✑✓
✫✭★✛✴
✢✦
✧✓✔
✖✢✓
✪✫✭★✛✴
✥✔
✜✓✔
✧✓✑✓
★✔✩✓✓✪✓✔✎
✖✓
✫✬✭✑✮✯✔✜✗
✢✦
✙✢
✱✖
✑✓✹✫✙✤
✙✭✜
✢✦
✭✖
✥✍✥✎✏✑
✵✶✑
✒✓✔ ✕✓
✚✛✜✍
✵✔✷
✢✦✰
✪✎ ✑✏✤✑
✭✖✥✘
✺✓✘
✫✭★✛✴
✭✜✻✓✓
✲✔✎
❥ ✁✂
✄
3x + y
x
☛
0, y
☞
✍
✌
✥
✚
✛
✎
✩
✌ ✤
✪✌
✒
④
✵
✶
✔
✌
✑
✑
✎
✜ ✢ ✣
✢
✓
✰
✕
✩
✌
④
❞
✒✤
✦
✌
✥
✤
✛
✎
... (3)
❞✌
★
✌ ✎
✦
❞
❞ ✰✑
✤
✍
✎✏
❞
✑
✩
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✎❞
✪
✌
✭✳
✑
✒
✒✌ ✓
✫
✜ ✬
✭✦
✴✑
✭✤
❞✌ ✤
✓
✕
★
✎✦
✻
❑
✦
✳
✎
✥
✎
✏
✑
✼
✸
✥
✤
✛
✌ ✤
✴
❞
★
P
❑ ✳
O, A, B
✍
✌ ✤
✦
✸
✍
★
✤
❞
✸
✍
❞✌
✤
✑
✘
❋
✎
P
❑
✳
❄
❣
✫
✮
P
✍
✌
❄
Z
✌ ✕
★
✎
✥
✤
✛
❞✌
✎✓
❑ ✲
✤
✒✌
✓
✥✰
✌
✤
✏
✌ ✤
✦
✥
✤
✛
◗
✌ ✌ ❞
✦
✔
✌
✑
❞✌
✎
✓
❞✌ ★
✭✎
✸
✹
④
✶
✵
✺
✗
✻
✥
✤
✛
✒
④
✱ ✌ ✰
✌
Z
✎✓ ✏
✌ ✎
✲
❞✌
✍
✎
✰✑
✏
❞
✭✭
✳
✦
✑
✒
✴✑
✒✌
✓
✰
✦
✤
✴
✤
Z
❉
❊❋ ●
♦
❅
❍
■❏
▲▼
(0, 0)
0
(30, 0)
120
(20, 30)
(0, 50)
110
50
✾
❀
✿
✰✑
✤
◆
❄
❆
❖
❖
✍
✎
✏
❞
✑
✒
12. 2
❁
❂
(0, 0), (30, 0), (20, 30)
◗
✌
✘
✍
✌
✰
✵
(0, 50)
✦
④
✵
✶
✦
✍
✎
✏
❞
✑
✒
✒✌
✓
✜ ✢
❯
✥
✎
✥
✤
✛
✰
✐
❄
❅
❆
❇ ❈
❘
✒
❞
✏
✌ ✦
✤
④
✶
✵
✏
✱ ✌
✰
✌
✎✓
✻ ❩
❨
✯★
✐✰
✐✰
✤
Z
❑
✳
❯
C
✰
✵
✬
❯ ❱
❲
❳
✓
★
✸
✸
♠
✓
✕
✐
❙
❚
✍
✌
✎
✥
✰
✌ ✤
✶
✈✽
✓
✕
✖
✗
✘
✯★
✐★
❃
❞
✌ ✤
❞✕
✎
OABC
✩
✌ ✪
✤
✌
❞✌
✔
✌
✫
✮ ✬
④
✵
✗
525
✠
✡
0
✷
✰✓
✤
✟
✞
✄
✝
... (2)
☞
✧
✌
✆
✄
✝
90
Z = 4x + y
❣✙
☎
✰
✎
❬✓
✵
✌ ❞
❩
✌ ✤
④
✶
✵
✴
✌ ✒
✐
❙
✓
✭✒
❙
★
✌
❞✌
✤
❭
❞✕✎
✖
✗
④
✻
✶
✍
✦
✑ ✴
✲
✑
❬✓
x + 2y
3x + 4y
x 0, y
☞
Z = 200 x + 500 y
❣
✙
✍
✌
✥
✚
✛
✎
✑
✜ ✢
✣
✮
❞✌
✒
✤
✦
❪
★
❫
✓
✑
✒
✧ ✌ ★
✌ ✎
✦
❞
✑
✒✌
✓
✔
✌
✩
✌ ✪
✤
✌
✫
✑
☞
☛
☞
10
... (1)
24
0
... (2)
... (3)
❞✕✎
✜ ✬
✭✤
✖
✗
✫
✮
✬
✥
✤
✛
✯★
✥✰
✌ ✤
✏
✌ ✦
✤
✥
✤
✛
✎
✓
❞
✌
★
✱
✌
✰
✌
✎
✓ ✏
✌ ✎
✲
✰✑
✭✭
✳
✦
✴✑
❱
ABC
(0, 6)
✖
✌ ✤
✩
✌ ✤
✪✌
✦
④
✵
✶
④
✵
✒
④
✎
✐
✓
✺
P
✰
A, B
Z = 200x + 500y
❞✌ ✤
✸
❑ ✳
✸
✹
④
✶
✵
✓
✕
★
P
❑ ✳
✍
✌
✤
✦
✍
✌
✸
❞✌
✍
✌ ✦
✤
✐✰
✰
✵
C
✒✌
✓
✥
✤
✛
✔
✌
✎✓
❑ ✲
✤
✑
◗
✌
✌ ❞
✦
❞
✰✑
✤
❘
✒
✦
④
✵
◗
✌ ✘
(0, 5), (4, 3)
✍
✌
✵
✰
526
① ✁✂✄
❉✤❊❋●❍
❂❃❄❅ ❆❇❈
✈☎✆
✝✞✟✠
♠✣✤✦✧★✤
✬✕✰✕
✡☛☞
3
✌✍
✎✏
✈✑✩✛ ✪✑✫✓
❢✓✭✏✑✛✮✑✛ ✴
✭✛ ✵
Z
❞✑
✬✭✬✮
✒✓✔ ✕☎✖
✯✛
✬✕✰✕
Rs
✖✑✕
✯✖✱✓✑
✦❣
✯✞✯✛
❞✑
✎✚✩✛
♦❊▲
▼◆ ❖P
(0, 5)
2500
(4, 3)
2300
(0, 6)
3000
◗✤❋
❘
✒✓✔ ✕☎✖
12.3
✥✌✗✗
❞✑✛
✎✘ ✑✙☎
✚✩
✚✑✛☎ ✑
✚✜✢
❞✫✬✲✳✆
✈✴☎✶✷☎
x + 3y
x+y
x
x 0, y
Z = 3x + 9y
Z
■❏✣❑
✒✓✔ ✕☎✖
✚✖
✈✑✜✏
✡✻✍
✯✛
✈✑✩✛ ✪✑ ✪✑✫✴✾☎✛ ✚✜✢
✴
✴ ☎ ✼✑✛ ✽ ✑
✯✠ ✯✶
✸
✹
✸
60
10
y
0
✹
✹
✈✬✮❞☎✖
✖✑✕
✡☛✍
❞✫
☎❞
ABCD
✺✑☎
...
...
...
...
(1)
(2)
(3)
(4)
✼✑✛ ✽ ✑
❞✑
❞✫✬✲✳✢
✏✜✬ ✪✑❞
✈✯✬✖❞✑✈✑✛ ✴
✭✛ ✵
✬✕❞✑✓
✴ ☎
✯✠✯✶
✭✛✵
❞✑✛ ✈✑✭✿ ✵✬☎ ✻✥❀☛ ✖✛ ✴ ✬✟✪✑✑✓✑ ✶✓✑ ✚✜✢ ✼✑✛✽ ✑ ✎✬✏✞❁ ✚✜✢ ❞✑✛ ✕✫✓
❉✤❊❋●❍
■❏✣❑
A (0, 10)
B (5, 5)
C (15, 15)
D (0, 20)
❂❃❄❅❆❇❈
12.4
Z
Z = 3x + 9y
♦❊▲
▼◆ ❖P
90
60
180
180
◗✤❋
❘
}
❘
❯❙❚ ❱❲❳
❨❩❬❭❲❳
(
❝❪❫ ❴❵❛❲❳
❪❜
)
❥ ✁✂
✄
A, B, C
➥☛
☞ ✌
✍
✎ ✏
✑
✢
✣
✤
♦
✏
✓
✮
✎
✒
✪
✦
✚
✸ ✹
✵
✍
✎ ✈
✒
❧
✎
❣
✚
✍
✷ ✥
✔
❣
✎ ✏
✥
✪
✘
❆
✫
❧
B
✌
❧✭✥
✌
★ ✩
✘✒
✱
✲ ✳
✕
✏
♦✓
✏
✲ ✴
✯
✰✎
❆
4
♦✓
✏
✏
➥☛
☞ ✌
❅
✏
✒
✏
✥
☞ ✎ ✏
✍
✑
❧
✣
❧
✦
✔
♦
✧
❣
✈
❏
✔
✍
✎ ♦✓
✏
✥
✱
✸ ✴
✌
✔
♦
✥
✕
(0, 20)
✫ ✬
✣
✔
✘✎
❡
✵
✚
✚✎
☛
☞ ✍
✌
✎ ✑
✏
✍
☛
Z
❣
✚
✘ ✒✥ ✏
❣
✑
✈
✜
✷ ✥
✘ ✒✥ ✏
C (15, 15)
✘
✏
✜
✈
✭
✎
✏
✎
✣
❞
✬
✶
✕
❣
✑
527
✠
✡
✍
❣
✑
✜
✈
D (0, 20)
✎
✒
✈
✬
✒
☞
✎
✏
✸ ✹ ❯
✲
✚✏
✑
✌
✭✥
➥
☛
☞
✌
✎ ✎
✏
✣
☛
☞
✬
✗
✌
✬
✢
✥
✕
C
✘
✒
✍
✎
✒
✈
✥
✕
✏
D
✫
✚
✚
✎ ✕
✕
D,
✍
✎
✒
✈
✩
❧
✪
❣
✑
✜
✈
✪
❆
✏
✔❂
✩
❍
✎
✬
❧
✒
✎
✑
✏
✦
❣
✥
✚✎ ✕
✐
❃
❄
✥
✣
✎ ✔
✪
✘
✍
✔
✚
✪
✣
✚✑
✏
☞
✎ ✏
✘
✎
✏
✕
✥
✘
✥
✚
✚
✎
✕
☞
✏
✕
✘
✒✥
✏
✢
✣
✘✎
✤
❣
✑
✜
✈
✥
✕
▲✕
✚
✚✎
✕
✔
✕
★
✔
❅
✔
✥
✎
◆
◆
✴
✯
✰✎
0, y
◆
✥
❘
✑
◆
P
✱
... (1)
3
... (2)
12
... (3)
0
... (4)
❧
✘
❱
✎ ✏
–5
♦✓
✏
✍
✣
✎
✪
✚
✥
✴
✎ ✑
✮
✘✒
❅
✔
✘
✔
☞
✣
✎
✣
✎
✎
❏
✔✕
✘
✎
❧❧
✎
✒✎
✭✣
✎
✎
✌
✯
❣
✜
✈
✔
✕
✒✪
✭✥
✯
✑
✎
✰✎
★ ❅
✎
✏
✮
✎
✪
✘
✘✎
✍
✎
✩
✔
❀
✏
❅
✎
✎
❧❧✭✥
✔
✘
✌
✪
✥
✑
❙
✯
✑
✒
☛
❲
❣
✈
✜
✪✣
✍
☛
❣
✚
✘
✎ ✏
✕
✬
➥
☛
☞ ✍
✌
✎
✏
✑
✒
Z
✪
✘✎
✚✎
✕
❍
✎
✧
✥
✎
✭✏
✘ ✒✏
✑
✛
❳
❊ ❩
❨
❬ ❭
❪ ❫❉
❴
(0, 5)
(0, 3)
(1, 0)
(6, 0)
❛❜
❝
❦
❤
♥
♣
12. 5
Z = – 50x + 20y
100
60
–50
– 300
❧
❵
❧
☛
✏
✏
✰✎
✎
✏
✬✔
✍
✏
✬
✢
❞
Z = –50x + 20y
★
✓
✢
✣
✤
♦❣
➥☛
☞
✍
✌
✎
✑
✏
✛
✱
✏
❧❧
✔
✥ ❆
✘
✏
➥
✫
❞
✪
✔
☞
✘
✎
✏
✕
✚
✎ ✕
✣
❞
C
✪
✣
✎
✸ ❈
✵
✚
✬
✶
✒
✣
✚❂
✩
✔
❀
❧
❣
✚
❧
✚
✑
✏
✥
✘
✫ ❑
✪
✘
✔
✬
✎
✒✎
✥
✎
✬❣
★
✏
❚
✈
✜
➥
✣
❞
✪✣
✎
CD
✎
✑
▼
❍
✎
✍
x
❣
☛
☞
✬
✶
✕
✳
☞
✎ ❣
✒✮
✎
2x – 3y
☛
✘✎ ✏
✫
✦
♦❣
❅
✎
✎
✪
✚
✑
✏
★ ❅
♦✏
✓
✥
✒
✎
❁
✏
✪
♦✎
✎ ✔
✍
✎
✫ ✬
✔
✘
☞
✎ ✏
✕
✎
✑
✏
✥
✭✖
✥
❧
➥
✢
✣
✤
✘✎
✍
✎ ✒
✈
✦
✕
✪✣
✎ ✏
3x + y
◗
✘✎ ✏
Z
✬
2x – y
❋
✪✣
✔
✒
✑
✩
✔
❀
★
✔
❂
✎ ✏
✑
☛
☞
(0, 10), (5, 5), (15,15)
✗
✎ ✛
✥
✎
✥
❇
✎
✎
✖
✎ ✕
✏
✦
♦ ✒✎
✏
➥
✚✎ ✕
✪
✎
✍
✔
✚
♦❣
✣
❖
✕
✰✎
❧
✚
✮
✳
✘
✎ ✏
♠ ❉
❊ ❋●
■
❊
✚✎
✙
✚
✟
✞
✄
✝
❣
✈
✜
✎
❣
✈
✪
✯
✎ ✏
★
❣
✑
✈
✜
✗
✎ ✎ ✘
✑
✧
✚
✥
✎
✔
✕
✒
✒❅
✎
➥☛
☞
✍
✌
✎
✏
✑
✔✕
☞ ✖
✏
✥
✘
✑
✘
✪
✯
★
♦
✏
✓
✆
✄
✝
✥
✎
❢
✺
✻
✼
✽
✾
✿
✔
❧✭✥
✌
✦
✘✎
✬
✶
❣
✍
❧
✏
Z
D
✎ ✈
✒
✥
✕
❧
✍
☎
✘
✚
528
① ✁✂✄
❜☎
☎✆✝✞✆✟
✡☛
✍✡
❉✓✆
Z
✌✆✠ ✓✡
✡✎ ✤
✸✠✹
✡☛
☎✍✌✠
☞✆✌
✡✎✏
✍✝✌✠
Z
✑✍
❞✣✛✛✚
✑②✪
✡☛
Z
✡✎ ✤
✑✍
✍✆✠✒✟✓
✥✓✦ ✒✌☛
✍✆
✥✓✦✒✌☛
✍✆
✑✒✺✒✑②✑✱✆✌
☛✆✒
✔✕✖✗
✘✙✚
✡✎ ✧
❞✣✛✛
✛✜
Z
✢✝
★✓✆✒
✍✆
✖✟✑✩✪
☎✕☎✠ ✍☛
✑✍
✓✑✖
☛✆✒
✫✆✠✬✆
❞✣✛✛
✢✑✝✕✭
✡✎ ✤
✡✆✠✌✆
☛✆✒
✴✆✟
✡✆✠
✳☎☛✟✍✝✞✆
☎✍✌✆
✍✆
✡✎
✳✆✎✝
✳✆②✠✱✆
✴✆✟✤
✒✡✟✏
✱✆✟✏ ✻✌✠
❜☎
☎☛✵✓✆
✍✆
✑✒✶✍✶✆✷
☞✆✌
✡✎✼
✏
– 50x + 20y < – 300
– 5x + 2y < – 30
✳✈✆✆✷✌✽
✩✆✰ ✻ ✍✟✑✩✪
✳✆✎✝
✡✎✏
✍✆ ☎✕☎✠ ✍☛ ☛✆✒ ✘✢✮☛✓
✠
✯ ☎✠✜ ✡✆✠✌✆✤ ②✠✑✍✒ ✡☛ ✓✡✆✰ ✖✠✱✆✌✠ ✡✎✏ ✑✍ ☎✗☎✲
✏ ✌ ✫✆✠✬✆ ✳✢✑✝✕✭
❜☎✑②✪
✍✝✒✠
☎✠
✓✑✖
✑✍
❁✴✆✓✑✒✶❂
❜☎☛✠✏
✾✆✝✆
✳✆②✠✱✆
✔✕✖✗
✡✎✚
✏
✢✮✆ ✿✌
Z
✌✕
✱✆✗② ✠ ✳❀✷✌②
✥✓✦✒✌☛
✍✆
✸
☛✆✒
✫✆✠✬✆
☎✗☎✲
✏ ✌
❞✣✛✛
✒✡✟✏
☛✠✏
❁✴✆✓✑✒✶❂
✡✆✠✲✆✤
✔✕✖✗
✳✥✓✈✆✆✚
Z
✡✎✏
✍✆
✓✆
✒✡✟✏
✥✓✦ ✒✌☛
☛✆✒ ❞ ✣✛✛ ✡✆✠✲✆✤
✩✎☎✆
✢✑✝✢✮ ✠ ✫✓
✑✍
✳✆✸t ✹✑✌
☛✏✠ ✥✓✦✒✌☛
❁✢✝✆✠❉✌
❃✯❄❅
☛✆✒
❁✖✆✡✝✞✆
✒✡✟✏
☛✠
✫✆✠✬✆
✸✠ ✹
5
♠❈❊❋●❍❊
☎✆✈✆
✲✓✆
✩✆✰✻
✳✆✢
✍✝
✡✎ ✧ ❜☎✸✠✹ ✑②✪✚
❁✴✆✓✑✒✶❂
✔✕✖✗
❇✓✸✝✆✠❀✆✠✏
✑✒✺✒✑②✑✱✆✌
✝✱✆✌✆
✓✡
✪✠☎✆
❉✓✆✠✏
✳✆✸t ✹✑✌
✘❃✜
☎✠
✍✆
✢✮✖❆✆
✸✠ ✹
☎✍✌✠
✡✎✏
✑✍
Z = – 50 x + 20 y,
– 50 x + 20 y > 100
✘✛✚ ❅✜
✳✏✌✲✷✌✚
■
❏
■
✘✣✜
✍✆
✳✆②✠✱✆
☞✆✌
✍✝
☎✍✌✠
Z = 3x + 2y
✍✆
✥✓✦✒✌☛✟✍✝✞✆
☎✠
✳✆✢
☎✆✈✆
☎✗☎✏✲✌
☎✏✌✶
✗ ❑
✡②
✍✝
✒✡✟✏
☎✸✠✹✤
✳✌✼✚
✱✆✟✏✑ ✻✪
... (1)
... (2)
... (3)
✘✳✆✸t ✹✑✌
❃✯❄✙✜✤
❉✓✆
✍✆✠❜ ✷
☎✗☎✲
✏ ✌
✡✎
❁✖✆✡✝✞✆✆✠✏ ☎✠ ✑✩✒✍✆
✌✍
✻✗✸✠✹
✍✝
☎☛✵✓✆
✍✆
✡✎ ✤
❢▲▼◆❖P◗
✡✎✏
✑✩☎✸✠ ✹
✑✸✸✠✻✒
✳✆❀✆✝
✢✝
✡☛
✡☛
✳✕
✸✗✹r
✝✎✑✱✆✍ ✢✮✆✲
✠ ✮✆☛✒ ☎☛✵✓✆✳✆✠✏ ✍✟ ☎✆☛✆✥✓ ✑✸❥✆✠✶✆✌✆✳✆✠✏
✍✆
❁❘②✠✱✆
✍✝✌✠
✡✎✏ ✤
(1)
☎✗☎✏✲✌ ✫✆✠✬✆ ☎✖✎ ✸ ❁❆✆② ✕✡✗ ✴✆✗✩ ✡✆✠✌✆
(2)
❁ ❙ ✠❥ ✓
✥✓✦ ✒✌☛✜
✢ ✹② ✒
✡②
✍✆
☎✗☎✲
✏ ✌
✳✑❀ ✍✌☛
✫✆✠✬✆
✸✠✹
✡✎ ✤
✘✓✆
❥✆✟✶✆✷
✢✝
✍✟✑✩✪✼
✑✍ ✪✠☎✆ ✍✆✠❜ ✷ ✔✕✖✗ ✒✡✟✏ ✡✎ ✩✆✠ ☎✴✆✟ ❇✓✸✝✆✠❀✆✠✏ ✍✆✠
✪✍
✸✠ ✹
✍✆ ✳✆✝✠✱✆
8
15
0
✡✎✧
❃✯❄✙
❇✓✸✝✆✠❀✆✠✏
✡✎ ✤
■
✳☎✑☛✍✆✳✆✠✏
Z = –50 x + 20 y,
✩✆✰✻ ✍✟✑✩✪ ✑✍ ❉✓✆
x+y
3x + 5y
x 0, y
❋❣
✡✎ ✤ ❜☎✑②✪✚
✡✎ ✤
❉✓✆
✳✑❀✍✌☛ ☛✆✒ ❃✛✛ ✝✱✆✌✆
☎✗☎✏✲✌
☛✠✏ ✑✖✱✆✆✓✆
✢✝
❚❯❱❲❳❨❩
12.6
✫✆✠✬✆
✡✎✧
❥ ✁✂
✄
✭
☛☞
✈
✌
✍
✌
✎✏
✴
✒
✑
✒✓
✔
☞
❞
✕
✮
☛✕
✭
✙
☞
✖
✯
☞
✌
✕
☞
❞
✙
✍ ✕
☛
✧★
✚ ✩
✶
☞
✦
✷
✗
✘
✙
✒✚
✛ ✜
✢
✌
✙
✎✣
❞
✑
❞
✎✰
✐
✙
✌
✖
✱
☞ ✍
✈
☞
✍
✎✰
✚
☞ ✍
✴
✒
✤ ✍
☛
✏
❞
✕ ✌
✮
☛✕
✭
✙
☞
✹✺
✻
✼
✖
✯
✽
✾✿
✕
☞
✕
✌
✚ ☞
✌
☛☞
✈
✍
✧
★
✚
✩
✍
✌
✦
✙
✱
✧
★
✚
✩
✭
✆
✄
✝
✢
☞
✟
✞
✄
✝
✦ ✪☞
✫
✑
❞
☞ ✌
✥✣
✌
✒
✬
✳
✤
☞ ✍ ✌
✥☞
✤ ✦
✏
529
✠
✡
✌
☛
✳
☞ ☞
✖
✦
✵
✱
☞
☛
☞
❞
✑
❀
✣
✌
✲
✗
❞
✙
✍
✥
☎
❁
☞
✍
✚
✸
✌
☞
✘
❂
❃
❁
❄
✰
✴
✸
☞ ✎
✣
✦✙
✒
✥
1.
✳
✒
✷✌
✒
✍ ❤
✍
✏✗
✒
❞
☞
☛
✎✰
☞ ✌
✸✰
☞
2.
❤
✍
✴
✥ ✏☞ ✌
✱
☞
✥✌
✣
✴
✥ ✏☞ ✌
✌
☛
☞
✌
✖
✈
✤
☛
✦✒❅
✬
❆
✱
☞
❤
✍
✥✏
☞
✴
✱
✌
✌
☞
✈
✒
✍
❤
✍
✥
✏☞
✴
✱
✌
☞
✥✣
✌
✮
✕ ✸
✫
✕
✈
✥✣
✌
☛
☞
❉
☛
☞
❤
✍
✥
✏☞
✴
✱
✌
✌
☞
✮
❤
✍
✴
✥ ✏☞ ✌
✥✌
✣
7.
✒☛
☛
☞
✈
✒✍
❤
✍
✯
✴
✥
❞
✏☞
✌
✱
☞
✥
☞ ✍
❤
✍
✥
✏☞
✴
✱
✌
✌
☞
❤
✍
✥ ✏☞
✴
✱
✌
✌
☞
❉
10.
❤
✍
✴
✥ ✏☞ ✌
✹✺
❇☞
✎✯
✏✗
✈
✷✌
✽ ▼
◆
✺
✒
☞
✎✰
☞
❪
❫ ❴ ❵
❛❜
❝❡
❜
❢
❛❣
✒☛✕
✍ ✌
❞
✉
❞
✌
❋
✥✣
✌
✱
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●
✱
☞
s
✌
✳
✏
☞
★
✍ ☞
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✶
☞
✴
☛✕
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●
☞
✒
✕
❇☞
✖
☞
✌
✕
☞
✈
☛
✏
✕ ✔
☞ ☞
❞
✖
✗
✘
✴
✒
✈
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✏❇☞
✴
✒
❞
☛✕
✦
☛
✏❇
☞
☛✦
✒❅
✬
❆
✕ ✔
☞ ☞
✴
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❞
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✦☛
✏❇☞
☛✦
✒❅
✬
❆
❞
☛✕
✴
✒
✽
❙
✽
❚
▼
✾❯
▼
❏
✦☛
✏❇☞
☛✦✒❅
✬
❆
❞
☛✕
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✍
✦
❩
✌
✦☛
✏❇☞
☛✦
✒❅
✬
❆
❞
✷
✎✰
☛
☞
✏
☛✦
✒
✍ ✒✢
❩
❇☞
☞ ✏
❞
✕
❞
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☞ ✚
✌
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☞
✒
☛
❆
✗
✙
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s
☞
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✌
✌
❞
❞
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✢
☞
✱
✖
q
☞
❋
✦
✍
✥✣
✌
☞
✕
☛
✏
✕
✌
✖
✗
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✒
✥
✙
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✒
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●
✱
s
✌
✍
❞
✎✰
✐
✙
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✌
✍
✸
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✌
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✖
✈
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☞ ✫
✱
✛ ✐
✎
☞ ✚ ✍
☞
❞✱
t
✌
☛
✌
✛ ✐
✎
☞ ✚ ✍
❅
☞
✷
✥✣
✌
✘
☛
☞
✌
✏✳
☞ ✍
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✒✤ ✬
✓
✔
☞ ☞ ✍
☞
s
✙
✈
✎✰
✐
✙
❞
✤
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☞
(Different Types of Linear
✽
❍
✱
✦★
❬
r
✬☛
✹▲
✺
✯
✷❩
✱
✵
✱
☞
✱
☞
❉
☛
☞
❦
❧
❫ ♥
♣
❞
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❞
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☛
☞
❖
◗
❘
✍
✌
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❅
❉
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❞
✥✣
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★
✚
✯
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P
✦☛
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❈
✕ ✸
✫
✕
✰
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✧
✙
✍
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♠
❭
☛
☛
☞
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✼
✕
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✸
✫
✕
✥✣
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❏ ❑ ✽
▲
❳
✥
✱
☞
✮
✥✌
✣
✱
☞
✯
☛☞
12.3
Programming Problems)
✖
✈
✩
✕ ✸
✫
✕
❈
❞
★
✚
✕
❉
✈
✒
✍
✧
❈
✈
✒
✍
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✥✣
✌
❉
9.
✚ ☞
❉
✈
✒
✍
✕
✮
✕ ✸✫
❈
8.
☛✦✒❅
✬
❉
❞
✌
✣
❊
✦☛
✏❇☞
❞
✙
✍
✈
✱
✌
✕
✯
✮
✙
✍ ✕
☛✦✒❅
✬
❆
❞
✙
✍
☛
☞
❉
✮
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✏
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✫
✕
❉
✲
✯
✮
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☞
✈
✱
✌
☛✦✒❅
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❉
✕ ✸
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✕
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✕
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✱
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❉
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☞
❈
✈
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6.
☛
✦✒❅
✬
❆
❞
✕
✴
✒
✈
✱
✥✣
✌
✯
✙
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✕
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✈
✒
✍
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✏❇☞
❉
✱
✕
✸
✫
❉
5.
❞
☛✕
✈
✱
✌
❈
✌
✴
✒
❉
❈
✈
✒
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4.
1.
✱
☞
☛
☞
❈
✥✩
✣♦
✙
☞
✕ ✸
✫
✕
❉
✈
✒
✍ ❤
✍
3.
☞ ☞
✈
✓
✈
✱
✌
❈
✳
✷
Z = 3x + 4y
x + y 4, x
0, y 0
Z = – 3x + 4 y
x + 2y 8, 3x + 2y 12, x
0, y 0
Z = 5x + 3y
3 x + 5y 15, 5x + 2y 10, x 0, y 0
Z = 3x + 5y
x + 3y 3, x + y 2, x, y 0
Z = 3x + 2y
x + 2y 10, 3x + y 15, x, y 0
Z = x + 2y
2x + y 3, x + 2y 6, x, y 0
Z
Z = 5x + 10 y
x + 2y 120, x + y 60, x – 2y 0, x, y 0
Z = x + 2y
x + 2y 100, 2x – y 0, 2x + y 200; x, y 0
Z = – x + 2y
x 3, x + y 5, x + 2y 6, y 0
Z=x+y
x – y –1, –x + y 0, x, y 0
✈
✒
✍
❈
✒
✚
❞
✍
✙
●
✒
✚
☛
☞
✌
✚ ✇
✒
✪
✌
530
① ✁✂✄
2.
✈☎✆☎✝
✞✟ ✠✟✡☛
✞☞✌✍☎✎✏
❜✑
✒✓ ✔✕✖
✔✗
✑✘✙✚✕✛✕✜✢
✘✜✢
✣✘
✤✕✥
✔✖✥✜
✣✦✢
✧✔
✧★✧✩✕✪✫
✒✓ ✔✕✖
★✜♦ ✬✕✭✔✮✒✕✜✯✕✔ ✥✰★ ✛✕✣✕✖ ✘✜✢ ✧✔✥✫✗ ✘✕✱✕✕ ✘✜✢ ✒✓ ✚✕✜✲ ✧✔✳ ✴✕✳✵ ✧✴✑✑✜ ✶✑✘✜✢ ✑✩✕✗ ✒✕✜✯✕✔ ✥✰★✕✜✢
✔✗
3.
✪✚❞✫✥✘
✐✻✝✼✆✽
✛✕★✷✚✔
✞✟✠✟ ✡☛
✘✕✱✕✕
✞☞✌✍☎✎✏
✔✘
❜✑
✑✜
✔✘
✒✓✔✕✖
✸✕✲✥
✔✗
✒✖
✒✓✕✹✥
✑✘✙✚✕✛✕✜✢
✘✜✢
✣✕✜✺
✣✘
✒✧✖★✣✫
✒✓ ✾✕✕✸✗
✔✕✜
✥✚
✔✖✥✜
✣✦✢ ✧✴✑✑✜ ✑✢ ✚✢✱✕✕✜✢ ✮ ✔✕✖❣✕✕✫✜ ✑✜ ✧★✧✩✕✪✫ ✙✿✕✕✫✕✜✢ ✒✖ ✧✙✿✕✥ ✧★✧✩✕✪✫ ❀✕✴✕✖✕✜✢ ✘✜✢ ✶✰✒✕❁✫✕✜✢ ✔✕✜ ✩✕✜✴✫✜
✘✜✢
✛❀
✣✘✜✢
✒✧✖★✣✫
❜✑
♠❅☎✆✝❆☎
❡✚✚
✒✓ ✔✕✖
✪✚❞✫✥✘
★❂♦ ❃
✔✗
✣✦✺
6 (
❧❉
✣✦
✣✦✢✺
✴❀✧✔
✧✔
✒✓ ✧✥
❜✑
✆■
✈☎✆☎✝ ✞✟ ✠✟✡☛ ✞☞✌✍☎
✘✕✱✕✔ ✣✕✜✺
✩✕✕✜ ❇✚
kg
II
✩✕✕✜❇✚
★✜♦
✒✓ ✔✕✖
✘✜✢
I
✩✕✕✜❇✚
❧
I
✩✕✕✜❇✚
✑✜
✧✫❖✫
A
✘✕✱✕✔
✔✕✜
✔✕
I
✑✕✖✾✕✗
❀✫✕✥✜
✔✕
A
Rs
✘✜✢
✪✚❞✫✥✘
x kg
(
✸✕✲✥②
Rs kg
❦♥q♥rs
✛✕✦✖
❦♥q♥rs
✛✥❏
✧✫❖✫✧✸✧❣✕✥
I
★✜♦
x kg
✛✕✦✖
✮
A
✤✕✥
❄✕✧✣✳
C
✩✕✕✜❇✚
✔✕ ✬✕✭✔ ✔✘
C
kg
Rs
❧ ✘✕✱✕✔ ✧★✭✕✧✘✫
✧★✭✕✧✘✫
II
✔✕✜
C
✒✓✧✥
❣✕✖✗❁✫✜
✘✜✢
✒✓ ✧✥
✣✦✺
●❉
kg
✧❁✚✕
✸✲✥✜
✔✗✧✴✳✺
II
✩✕✕✜❇✚
kg
✘✕✱✕✔
✔✕
y kg
✣✦✺ ✙✒✯✭✥❏
x 0 y 0.
❑
❑
✣✘ ✒✓❁▲✕
II
8
1
2
10
50
70
✑✜
✣✕✜✥✜
y kg
★✜♦
❭❘❪❫❲❴❚❘
1
✔✘
✒✓✕✹✥
❳❨❘❩❘❬
2
t
✔✗
❡✚★✖✕✜◗
✩✕✕✜❇✚
✒✓✧✥
❊
✛✕✦✖
II
(y)
2x + y
x + 2y
✩✕✕✜ ❇✚
✔✖✫✕
I
(x)
✧★✭✕✧✘✫
❄✕✧✣✳P
✘❞❍ ✚
kg
✒✓✧✥
✛✕✦✖
✛✕✦✖
❯❘❘❙❱❲
A
kg)
C
kg)
✧★✭✕✧✘✫
✣✸
✣✦✺
✢
(
✘✜✢
✒✓✧✥
❋❉
✧★✭✕✧✘✫
✧✘♣✾✕
A
kg
✘✕✱✕✔ ✧★✭✕✧✘✫
Ï❘❙❚
❄❞✵ ✧✔
✔✕✜
✳✔ ✛✕✣✕✖ ✧★✤✕✫✗ ❁✕✜ ✒✓ ✔✕✖ ★✜♦ ✩✕✕✜❇✚✕✜✢ ✔✕✜ ❜✑ ✒✓ ✔✕✖ ✧✘✸✕✫✕
✧★✭✕✧✘✫
❣✕✖✗❁✫✜
✧✘♣✾✕
):
✑✘✙✚✕✛✕✜✢
✔✕ ✬✕✭✔ ✔✘ ✑✜ ✔✘ ❈ ✘✕✱✕✔ ✛✕✦✖ ✧★✭✕✧✘✫
✘✜✢ ❊
✘✕✫✕ ✧✔ ✧✘♣✾✕ ✘✜✢ ✩✕✕✜❇✚
✛✕✵ ✔▼◆✕✜✢
✒✓✕✲
✜ ✕
✓ ✘✫
✖✦✧❣✕✔
❄✕✣✥✕ ✣✦ ✧✔ ✧✘♣✾✕ ✘✜✢ ✧★✭✕✧✘✫
✑✜ ✔✘
✣✕✜✺
❑
❑
✔✘
❈
✘✕✱✕✔
✔✕
★❂♦✸
❵❛❘❝❘❴❘❙❤
✛✕✦✖
❛❙ ❥
❙❤
✧★✭✕✧✘✫
C
★✜♦
❧❉
✘✕✱✕✔
✣✕✜✫✜
✣✦✢
8
10
❣✕✖✗❁✫✜
✘❞❍ ✚
Z
✣✦
✴✣✕✵
Z = 50x + 70y
✛✥❏
✑✘✙✚✕
✔✕
✧✫❖✫
❡✚★✖✕✜◗✕✜✢
★✜♦
✲✧✾✕✥✗✚
✑❞ ✱✕✗✔✖✾✕
✧✫❖✫✧✸✧❣✕✥
✣✦❏
✛✢✥✲❢✥
2x + y
❑
8
... (1)
❥ ✁✂
✄
x + 2y
x, y
Z = 50x + 70y
✈
❞☞
✙
(1)
✚
✛
✒✓ ❞
✌
✍
✔✕☞ ☞
✎
✏ ✑
✒✓
❞
(3)
✙✚
✔✕☞
☎
✆
✄
✝
✟
✞
✄
✝
531
✠
✡
10
0
☛
☛
... (2)
... (3)
❞✓✖✗
✘
r✜
✚
✈
✑
❞
✢ ✣
✚
☞
✛
✚
✤
✥
☞ ☞
☞
✔☞
✖
✏
✦
✙
✧
✙
✛
★✑
✩
☞
✖ ✔✑
✪
✚
☞
✚
☞
❞
✈
☞
r
✫
✜
☞
✬ ✭ ✮
✯
✖
✛
✑
✚
✰ ✣
✒
✖
☞
☞
✍
☞
★
✍
☞
①
✱
✲
❡
❘ ❬
❤
✐
❴
❦
❳
Z = 50x + 70y
◗
♥
(0,8)
(2,4)
(10,0)
✳
✴
✰ ✚
✣
✍
①
☞ ❀
✛
①
✒
✚
✚
①
✒
❞
✚
❣
☞ ✏
✓✍
✚
❧
❂
✈
✒
❣
✰
✧
✾
✭
❧
❃
❄
✾
❅
①
✱
✈
✔
✕☞
❞
☞
✪☞
✛
✍
✢ ✚
✣
☞
Z
✚
❁
✣
✛
✍
✹
✌
✍
✏
✎
✑
✒
12.7
C(10,0)
✈
☞
✱
✔
✙
❞
①
✲
✱
☞
✺
Z
✔
✙✚
❞✓
✖✗
✘
✽
❞
☞
✒
❂ ❃ ❄
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① ✁✂✄
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Rs 9,000
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Z = 10500x + 9000y
O (0, 0)
0
A ( 40, 0)
420000
B (30, 20)
495000
❲
✈✔✴❞✖✠
450000
12.8
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= Rs 10500
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B
A
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Rs 4,95,000
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✎❀
✤
✕
✒
✓
Rs
✐✢
B
✧ ✑
❑
✑
✍
✧
✑ ✧
✒
✕
✒
✓
✉
✐
✍
❂
❅ ✎
❉
✔
✏
✑ ✢
✣
✥
☛
✌
✍
✔ ✖
■
✘ ✘ ✘
✍
✏☛
✧
✒
r
☛ ✑ ✙
✈
✍
✏☛
✎
❑
✑
✚
✜
✑
✙
✣
✼
✬
✎
❣
✑ ✧
✔
A
❊
✍
✖
✍
✏
✧
✎❀
✧ ✑
x
✼
✕
✒
✓
✧
★✑
✦
✒
✏
✌✦
▲
✜
✑
✙
✣
☛ ❋
✑ ✑
✧
✎❀
B
✧ ✑
✼
✕
✒
✓
✧
★
✑
✦
✒
✏
✌
✦
▲
✜
✑
y
❞
✙
✣
= (Rs 8000 x + 12000 y)
Z = 8000 x + 12000 y
✩
❜
✌✍
✔ ✖
✕✓
▼
✔
✔ ✑
✑
✈
☛
☞
◆
✈
✥
✙
✎
◗✧
✑ ✢
✒
❢
✍✧
✜
✺
✐✑
✌
✐
❂
✿
❖
✑
✌✎
r
✕ ✢✑
✒
✻✑
✜
✑
✏✑
★✍
✼
☛
P
✜
✌❀
✕
✒
✓
✈
✦
☛
★
❙
❯
Z = 8000 x + 12000 y
✼
▲
✑
✏
✈
✌
✎
r
✏✑
✻
✏
✢
✈
✍
❁
✑ ✕
❨
✓
✍☛
✤ ❩❄
✦
✍✿ ▲ ✑ ✑
✜
❘
❘
❯
✼
✏☛
✎
✼
✏ ✢✻✑
✑
★✜
✑
✏
(
(
(
✍
✔
✍
▲ ✑ ☛
✌
✒
❁
✗
❃
✙
✣
✙
☞
✣
✑ ✧
✿
✍
✢
❢
✧
✒
❈
✔
❊
✍
✒
✑
❴
❵
❛
❝
✏
❡
✕ ✢✑
✒
✕ ✢✑
✒
r
✕✢
✑
✒
)
r
✜
❢
✜
... (1)
... (2)
... (3)
)
❞
✖
✍
✍
✏✑
☛ ✢
✧
✺
✑
❲
✍
✢
☛
✌
❲
✖
)
r
✜
❢
✑
➼
✻✑
❊
❪
❫
✏✑
❉
✐✑
✍
r
✑
✼
❬
✜
①
❚
★
❱
❃
❞
✎✒
180
60
30
0
❘
❇
✑
❇
✈
✍✧
☛
9x + 12y
3x + 4y
x + 3y
x 0, y
✣
✍
◗✧
✏ ✢✻✑
✺
✑
✒
✦
✈
❋
✑
✑
☛
✢
✼
✑
✌
✌
▼
★
✦
12.9
☛
▼
✌
✦
★
☛
✑
P
✑
✒
OABC
P
✑ ✒
❞
✑
✐
✍ ✢✥
❭
✙
✣
❳
❁
✑ ✜
✑ ✦
✍
✏
☛
❃
534
① ✁✂✄
✐☎ ✆✝✞ ✟
❞✞❡
✟✠✞ ✡☛✝
✣✍✥✠✠✝✠
☞✌✍✎
✐✏
✑✒✞✓ ✝
✧★✩✪✫✬
✵✞✔
12
✡✖
✟✠
❞✠✡
0 (0, 0)
0
A (20, 0)
160000
B (12, 6)
168000
C (0, 10)
120000
✐✏
✳✉✠✠
✡❞✶ ✡✠
✵✞✔
♦
✡✖✠✞ ❡
✕✠✷✠
✍✠✞
✵✞✔
✐☎ ✟✠✏
Q
B
✈✠✛✏ ✷✠✠✞❥ ✝
✣✵❆✠✣❞✡
2.
✸✟
P
✷✠✠✞❥ ✝
✵✞✔
✟☛ ✕✠✖✳
✚✛
✜✌✣✟
❢✝✶ ✡✳❞
✟☛
✐☎ ✟✠✏
Rs
✈✠✛ ✏
✑✆✐✠✍✡
✵✞✔
✵✞✔ ✟
✐✏
3.
✸✟
❞✞ ❡
✖✝✠
✚✛
✟✠✏✥✠✠✡✞
✣✕✸
✟✠✞
♥❖❍
✳✛✝✠✏
P✠❡ ❆✞
(i)
✜✛ ✢ ✠
✣✟
✣✡✤✡
✢✠✏✗✠☛
❞✞❡
✝✠❡✣ ❈✠✟
❞✞❡
✟✏✡✞
❞✞❡
✢✞
❆✞✣ ✡✢
◗
✵✞✔
P✠❡ ❆✞
g
✈✣✲✟✳❞
✐☎ ✟✠✏
✈✣✲✟✳❞
A
✐✠✳✞ ✚✛❡ ✴ ✈✳✦ ✵❡✔✐✡☛ ✟✠✞ ✡❞✶ ✡✠
✕✠✷✠
✟❞✠✡✞
✣❞✕✠✡✠ ✹✠✚✳☛
B
❞✠❈✠✟ ✣✵❆✠✣❞✡
❞✠❈✠✟❊
✈✠❆✠
✌✡✠✡✞
✏✛✵✔
✞ ❆
g
✳✉✠✠
P
kg
3
❞✞❡
✵✞✔
✚✛
✣✕✸
✹✠✣✚✸
✟✏✡✠
❞✠❈✠✟❊
✣❞❅✗✠ ❞✞ ❡
✣✟
P
✚✠✞❡ ✴ ✷✠✠✞❥ ✝
kg
A
✣✵❆✠✣❞✡
✐✏
◗
g
●❍
✟☛ ✕✠✖✳
A
✣✵❆✠✣❞✡
2
✈✠✛✏
✈✠❆✞
✣✕✸
✣▼✵✞✔ ❆
P✠❡ ❆✞
✢❞✝
g
✣✵❆✠✣❞✡
Rs
kg
kg
♦❉❊
✈✠✛ ✏
5
kg
✣✵❆✠✣❞✡
❞✠❈✠✟❊
✵✞✔
✝✠❡✣ ❈✠✟
❏✟☛
✵✢✠ ✟☛
✐✍✠✉✠✠✞ ▲
✌◆✕✞
♥ P✠❡ ❆✠
fat
✈✠✵✓✝✟✳✠
❞✠❈✠✟❊
✟☛
✌✡✳✞
✟✠
✟❞☛
✚✛❡ ✴
✢❞✝
✣✓✠◆✐✟✠✏
✢❞✝
✵✞✔
❙●
✚✠✞ ✳☛
✈✠✵✓✝✟✳✠ ✚✠✞ ✳☛
♥ ✣✟✕✠✞ ✵✢✠ ✢✞
✣✓✠◆✐✟✠✏
✳✉✠✠
✑✐✕❘✲
✳✉✠✠
✈❢✝
■
✵✢✠
✈✠❆✠ ✳✉✠✠ ❍❉
✵✞✔
✳✉✠✠
✳✉✠✠
✝✠❡ ✣❈✠✟
✝❡ ❈✠✠✞❡
✘✢
✜✠✞ ❍ ✣✟✕✠✞
✢❞✝
✣✵✣✷✠❢✡
✈✣✲✟
✟✠✞
✱
✚✛ ✴
✟☛✣✜✸✴
✣✕✸ ♥❉❉
✵✞ ✔✟✠✞❡
✣✟
P✠❡❆✠
✟✠✏✥✠✠✡✞
●❙
❞✞ ❡
✈✣✲✟✳❞ ✢❡✥✝✠ ✌✳✠✈✠✞
✣✕✝✠
✚✛
❂❃❄❃
✚✛ ✴ ✷✠✠✞❥ ✝
❇❉❊
●❉❉
✟✠✞
✟✠✞
✳✉✠✠
❋✠✳
✕✠✖✳
Q
A
11
kg
Q 4
✷✠✠✞ ❥✝
✍✶✢ ✏☛ ✐☎✟✠✏ ✵✞ ✔ ✵✞✔ ✟ ✵✞✔
✟☛
✖✘✙
✚✠✞ ✖✠✴
✈✵✝✵✠✞❡ ❞✞❡ ❇ ❞✠❈✠✟ ✣✵❆✠✣❞✡
✣❞❅✗✠
✟☛
Rs 1,68,000
✟✠ ✈✣✲✟✳❞ ❞✠✡
✺✻ ✼✽✾✿❀❁
✏✞✓ ✠❞✠
✖✗✠✡✠
Z = 8000 x + 12000 y
✭✮✯✰
B (12, 6)
Z
B
Rs 1,68,000
✈✠✛ ✏ ✈✣✲✟✳❞
1.
✟☛
✚✛ ✦
✖✝✠
✚❞ ✓✠☛❣✠✙
Z
✐✔✕✡
✌✡ ✢✟✳✞
✡✚☛❡
✸✟
✕✖✳✠
✟✠
P✠❡ ❆✞
✳✉✠✠
✚✛ ✴ ✵✞✔ ✟✠✞❡
✝✚ ❞✠✡
✏✚✞ ✖☛✴
❆✞✣ ✡✢
✚✛ ✴
✏✛✵✔
✞ ❆
✸✟
✢❞✝ ✕✖✳✠
✈✠✛ ✏
❡
✚✛ ❑
✚✛
✌✡✠✡✞
✣▼✵✞✔ ❆
✚✛ ✴
✣✓✠◆✐✟✠✏
✵✞✔
✌◆✕✞
✸✟
✣✍✡
✢❞✝
✵✞✔
✚✛❡ ✴
✡✚☛❡
✏✛✵✞✔ ❆✠✞❡ ✈✠✛ ✏ ✌◆✕✠✞❡ ✟✠✞ ✣✟✳✡☛ ✢❡ ✥✝✠ ❞✞❡ ✌✡✠✝✠ ✜✠✸ ✳✠✣✟ ✟✠✏✥✠✠✡✠ ✐✶ ✏☛ ❚✠❞✳✠ ✢✞ ✟✠✝✙
✟✏✞ ❯
(ii)
✝✣✍
✏✛✵✞✔ ❆
✈✣✲✟✳❞
4.
✸✟ ✣✡❞✠✙✗ ✠✟✳✠✙
✈✠✛ ✏
✕✠✷✠
✡❆
✌◆✕✞
❋✠✳
✈✠✛ ✏
✐✏ ✸✟ P✠❡ ❆✠ ✈✠✛ ✏ ❞✓✠☛✡
✕✠✷✠
✟☛✣✜✸
✌✠✞ ◆❆
B
✐✏
▼❞✓✠✦
✝✣✍
✟✠ ✣✡❞✠✙✗ ✠
Rs
✟✠✏✥✠✠✡✠
●❉
✐✶ ✏☛
✳✉✠✠
Rs
❚✠❞✳✠
✢✞
✟✏✳✠ ✚✛ ✴ ✸✟ ✐✛✵✔
✞ ❆
♥❉
✟✠✝✙
✡❆✠✞❡ ✵✞✔
✚✠✞❡
✳✠✞
✟✠✏✥✠✠✡✞
✟✠
✟✏✞ ✴
✣✡❞✠✙ ✗ ✠ ❞✞ ❡ ❞✓✠☛✡
A
✐✏ ◗ P✠❡ ❆✞ ✟✠❞ ✟✏✡✠ ✐❱❲✳✠ ✚✛❑ ✜✌✣✟ ✸✟ ✐✛✵✔
✞ ❆ ✌✠✞ ◆❆ ✵✞✔ ✣✡❞✠✙✗ ✠
❥ ✁✂
✄
❡☛
☞
✌
Rs
✍✎ ☞
✏
☛
✗
✶ ✦
✧ ★
✥✩
✪
✛
✥
❞
✘
✎
✕
✪
✲
✥✯
✎
✵
✴
✪✘
✕
✎ ✔
✖
✕
✎
✔
✖
✱
✥✎ ✪ ✰
✛
✵
✥✔
✖
✣
✫
☛
✏
✘
✛ ❡
✱
✷
5.
A
❡✑
✎ ✒✓
✬
✎ ✭
☛
✏✎
☞
☛
✘
✎
✔
✍
✎ ✏
☞
✵
♠
☛
✪✘
✯
❡
☞
☛
✛ ✎ ✪
✘
✕
✪
✕
✎
✣✑
✯
✰ ✎ ☛
✘
✛ ✎
❝
✽
✣
✫
☛
✪
✓
❡✎
✥✩
✘
✎ ✔
✣
✫
☛
✪
✪
✓
❡
✎
✒
✜
✾
✖
✪
❡✓
✏
A
✥☛
☞
✒✓
✣
✫
☛
✥✩
✪
✪
❡✓
✏
✮
✎
✒
✪✰
☛
✘
✥
✖
✣
✫
☛
✏
✓
✛ ☛
✣
✫
☛
Rs
✥✔
✵
❡
✎
✼
✛
✪✘
✘
✎ ✔
✘
✛ ✓ ☛
✥
✖
✣
✫
☛
✏
✘❡
✎ ✯
B
✣
✓ ☛
✪✣
✪
✮
✎
✥☞
✓
✪
✥☞
☛
✌
✘
☛
✜
✢
✖
✯
✪✰
✣
✫
☛
✟
✞
✄
✝
535
✠
✡
✣✜
✓
✏✎ ☞
☛
✥
✪
✩
✛
✪✰ ✓
✤☛
❡✑
✎
✒
✓
✎
☞
☛
✹
✓
✏
✕
✎ ✔
✖
✎
✎
✬
✎
✭
☛
✏
✣
✫
☛
✪
✘
✛
✓
☛
✥
✖
✣
✫
☛
✏
✤✣
✫
☛
✢
✓
✛
☛
✜
✢
☞
✖
✥✩
✽
❝
✯
✘
☛
✣
✫
☛
✼
✛
✕
✎ ✔
✖
✰ ❀
✤
✔
✒
✜
✪
✓
❡
✎
✿
✛
❡✓
✏
✜
✿
✛
✲
✎
❡
☞
☛
✰
✎
✼
✛
❡✑
✎
☛
❡✑
✎
✪ ❡✓
✏
✜
✿
✛
✼
✛
✒
✓ ✾
B
☛
✍
✎ ✏
☞
☛
✼
✛
❡
✘
✎ ❡
✣✫
☛
✪
✑
✎ ✒✓
✜
✢
✖
✛
✥✩
✯
☛
✘
✥
✤✮
✎
✒
✥☞
✴
✬
✓
✎
☛
✵♠
☛
✣
✫
☛
✥
✣
✖
✫
☛
✏
✣
✖
✫
☛
✏
✥✔
✪✬
✴
✎
✪
✼
✎ ✮
✎
✥
✣
✖
✫
☛
✏
✥✖
✥☞
✣
✫
☛
✏
✥
✘
✕
✪
✜
✢
✖
✗ ★
✜
✢
✖
✘
✎
✼
✎
B
☞
☛
✥✩
❝
✼ ❃
A
☛
✷
✜
✎ ✛
☛
✎
❞
✥
✎ ☛
✻
✘
✘
✎
✯
✱
✪
✮
✎
✯
✘
☛
A
✻
✓ ❡
✎ ✛ ✎
✥
✘❡
✎
✛ ✎
☞
☛
✜
✢
✖
✯
✜
✴
✵
✎ ✛
☛
✜
☞
✖
✾
❅
✎ ✛
❞
✤✤☛
✥
✯
✩
✵
✴
✎ ☞
✣
✫
☛
✻
✘
✎ ✎
✘
✎
✼
Rs
✷
✣
✫
☛
☛
❝
✎ ✪
✵
❡
✒✓
✎ ☞
✵
✎
✪
❂
✎ ✪
♦
✘
✛
✵
✎
☞
☛
✜
✢
✖
✸
✯
✻
❡
✛
✻
✓
✘
❡
✎ ✛
✎
✻
❞
✕
✪
✕
✎ ✔
✖
✪
✥✙ ✛
✚
✎
✻
✪
✕
✎ ✔
✖
✻
❡
☞
☛
✼
✎ ✮
✎
✓
✎
✷
✬
✎
✪
✕
✎ ✔
✖
✼
✛
✼
✶
✘
✔
✻
✿
❝
✎ ✎
❢
✪
✪
❡
✻
✎ ✪
✺
✜
❄
✥✩
❁
✎ ✪
✿
✣
✓
✷
✯
☛
✻
❡
☞
☛
✘ ✤✒
✣
✖
✫
☛
✏
✷
✛ ✎
✮
✎
✕
✎ ✔
✖
✘
✵
❡✑
✎
✼
✎
✵
✤❡☞
☛
✿
✣
❁
✎
✥
✵
✎
✻
❡
☞
☛
❝
✽
✣
✫
☛
✜
✎ ✛
☛
♦
✎
✥✩
✪
✛
✘
✎
✆
✄
✝
✴
✘
✛ ❡
✴
✘✒
✎
✥✔
❞
✎
✓ ☛
✶ ✦★
★
✴
✗
B
❡✑
✎ ✒✓
Rs
✥
✔
✻
✎
✎
✍✎ ✏
☞
✎
✳
☛
✺
✘
✗
☎
✘✒
✪
❂
✘
✛ ❡
✜
✎
☛
✛
✪
✘
✥✩
✪✛
✪✰
✓
❞
✎
✎
✕
✪
✘
✛ ❡
✼
✎
✮
✎
❅
✎ ✛
✴
✵
✘✒
✪
6.
✢
✵
✱
✘
✣
❄
✫✏
✵
✒
✔
✲
✘
✔
✲
✔
❝
✪
✷
✓
❡
✎ ✛ ✎
❇
✙
✓
☛
✲
✘
✎
✏
✓
☛
✼
✥
☞
✖
✕
✎ ✔
✖
✼ ✘
✙
✓ ☛
✕
✎ ✔
✖
✕
✎ ✔
✖
✌
✍
✎ ✏
☞
✓ ☛
✕
✎ ✖
✔
☛
✿
✥✩
✯
☛
✿
✥✩
✯
☛
✔
✍✎ ✏
☞
❡
✔
✍
✎ ☞
✏
☛
✕
✎ ✔
✖
✔
✼
☞
✥
✖
✘✒
✪
✬
✒
✘✒
✕
✎
✣
✑
✯
✤✒
✪
✘
✯
✎
☛
✕
✎ ✣✑
✯
✤✮
✎ ✒
✪✓
✪
✘
✛ ✎
✘✒
✕
✎
✓
✚
☛
✓ ✒
✥✙ ✛
✚
✵
✪
Rs
✪
✘
✼
✎
✓ ☛
✘✒
✕
✎ ✔
✖
✕
✎ ✔
✖
✮
✎
✕
✪
❡✑
✎ ✒✓
❡☞
✣
✫
☛
✕
✓ ❀
✑
✎ ✙
☛
✻
✿
❡
✪✛
☛
✧
✪❡
❡✓
✏
✌
✣
☛
✫
✍✎ ☞
✏
✓
✏
✘
✎
✏
✓ ☛
✘
✎
✏
✓ ☛
✕
✎ ✔
✖
✕
✎ ✔
✖
☛
✪
✕
✎ ✔
✖
✥✩
❡✓
✏
✯
✕
✪
☛
✘
✣
✫
☛
✪✓ ❡
✎
✽
❝
✼
✥
☞
✖
✣
✫
☛
✪
✓ ❡
✎
✎
❡
☞
☛
✳
✎
❡
☞
☛
✍✎ ✏
☞
☛
✽
❝
✘
✑
✎ ✙
☛
✣
✫
☛
✪
✓
❡
✎
✎
❡
☞
☛
✗
✍✎ ✏
☞
✎
✜
✎
☛
✓
✎
✛
B
✘
☛
✘✒
✓
✪
✕
✪
❣
❡✑
✎ ▲
☛
✥✩
✬
✒
✪
❡✓
✏
✓
❡
✎
☛
✜
✾
✖
✓
✚
☛
❡☞
☛
✓
✚
☛
✣
✫
☛
✪
✎
✓
☛
✘
✔✛
❡
☞
☛
✼
✣
✫
☛
✼
❡✑
✎
☛
✒✓
✵
✍✎ ✏
☞
✌
✥✩
✪✛
✪✰
✘
✎
☛
✣
✫
☛
✪
✼
✎ ✮
✎
✱
✼
✜
✎ ✛
☛
✎
✓
❞
✥
✼
❃
✜
✢
✖
✯
✜
✽
❝
✬
✛ ✎ ❈
✣✜
✪
✓
❡✎
✎
✘✒
✥✩
✪✛
✪✰
✒
A
✜
✢
✖
✛ ☛
✜
✢
☞
✖
✜
☞
✢
✖
B
✪✰ ✯
✎
✱
✍✎ ✏
☞
✻
✿
❡
✪✛
☛
✥
✘
✎
✔
✘
✎ ✔
✓
✥✩
✯
✥✩
✪✛
✿
❡
✪✛
✣
✫
☛
✥✩
✪✛
✿
❡
✪✛
✥✔
✎
✜
✖
✪✘
A
✷
✜
☞
✢
✖
✥✩
✯
✘
✎ ✏
✘
☛
✘
✥✩
✘
✎ ✔
❍
✻
❍
✪
✣
✫
☛
✪
✓ ☛
✣
✫
☛
✪
✼
●
✥✩
✘
✎
✔
✣✫
☛
✿
❡
✪✛
❁
✴
✵
✘
✎
✼
✎
✮
✎
✜
✎
✓
☛
✎
✜
✢
✖
●
☛
✻
✪
✵
✯
✷
✼
✣
✫
☛
●
✥✩
❞
✼
❃
Rs
❍
✪
●
✥✩
✲
✛ ☛
♦
✼
●
✘
✎ ✔
✪
✛
✎
✲
✎ ✙
☛
✵
✎ ✙
☛
✣
✫
☛
✗
Rs
✥✔
✲
✎ ✙
☛
✚
✴
✥✩
✘✒
✘
✛ ✓ ❏
☛
✪✘
✛
✓ ☛
✻
✿
❡
✪✛
❅
✎
✛
✘✒
✪
❍
✪
✎ ☞
☛
✘
✎
✘
✥
☞
✓ ✒
✵
❉
✎ ✪
✜
✘
✎ ✔
Rs
✯
☛
✔
✳
✘
✛ ❡
✽
❝
✪
✴
✎ ✎
✴
✰
✎
✿
✥✩
✵
✎ ✛
❍
✵
❡✒ ✘
✔
✲
✔
✜
✢
✖
✜
✎ ☛
❡✓
✏
✽✎
✘
✛
✻
✎
✤
✎
✰
✖
✎
✘✒❡
✛ ☛
☞
✽
❝
✯
✵
✘
✜
✎ ✛
☛
✒
✥✩
✪
✛ ✪✰
✘
✘
✎
✗ ★
✪
❂
★
✷
✧
❝
✽
✓
❡✎
✵
✘
✛
✎
✑
✎ ✙
☛
✪✬
✪
■
Rs
✼
✎ ✮
✎
⑥
✘
✥✩
❉
●
☛
■
✤❡
✯
✪
✜
✢
✖
✘
❣
✘
✘
✛ ❡
❋
❈ ✣
✙
❄
❞
✎ ✔✎
✜
✢
✖
✬
✪
✴
✎
✪
✥✔
✘
✜
✾
✖
✵
✧
✼
✥
☞
✖
❝
✼
✎
✼
❍
✪
✒
✴
❡
✛
✳
✼
✜
✎ ✛
☛
✒
✣
✑
✯
❝
✽
✪✓ ❡
✎
✣
✫
☛
✬
✓
✎ ✛
✎
❞
✘
✎ ✏
❞
✬
✓
✎
❊
✣
☞
✫✥
✔
✵
✓
✎
✵
✣
✫
☛
8.
✘
✛
✎
❝
✜
❄
✴
✪
✑
✎ ✙
☛
❇
✙
✥✔
✵
✓
✛ ☛
✘
✒
✿
✥✩
✯
☛
❣
✘
✘
☛
✲
★
✵
✻
✣
✫
☛
✵
✘
✳
✘
✛
✜
✢
✖
✣
❄
✫
✒
✚
✳
✘
✎ ✏
❞
7.
✼
✴
✘
✎ ✏
✚
✓ ☛
✕
✪
✣
✫
✖
✿
☛
✏
✵
✚
✓ ☛
❇
✙
❡
✎
✥✙
✖
❇
✙ ✚
✔
✲
❆
✎ ☛
✣
✫
☛
✪✓
✒
❊
✘
☞
✳
✧
✾
★
★ ★
✕
✎ ✔
✖
✵
✯
✏
❀
✔
Rs
❏
✘
❑
✙
♦
✲✒
★ ✾
★ ★ ★
❝
✿
☛
✘
✏✎
✥
✜
✎
☛
✓
❡✓
❀
✎
✕
✎ ✔
✖
✰ ✤
❀
✔✎
✻
✾
✬
☛
✴
✥
✎ ✏
☛
✬
☛
✼
✓ ❡
✓
❀
✎ ✾
✪
✓ ✘
✒
✴
✓
☛
✘
✒
✯
✎ ☛
✓
✎
✬
✓
✎ ✛ ✎
✜
✢
✖
✣✜
✕
✓
❄
❡
✎
✓
536
① ✁✂✄
②☎✆✝✆
✞✟
✠✡
✡☛☞✌✍ ✎✏✆✑☛
✡✒
✓✔✕ ②
✖✆✠✗✡
✖✆✘☎
✙✚✛
✜☎✆✑☛
✗✑
✢✠✣✡
✜✞✒☛
✞✆✑☎✒✤
✥✦✧ ✌✑✡
✥✦✡✆✏
✓✑✕ ✡☛☞✌✍✎✏✆✑☛ ✓✑✕ ✜☎✆✑☛ ✡✒ ✗☛ ♦✌✆ ★✆✝ ✡✒✠✩✪ ✠✩✗✑ ✗✆✟✫✆☎✆✏ ✢✠✣✡✝✖ ②✆✬✆ ✥✦✆☞✝ ✡✏✜✑ ✓✑✕ ✠②✪
Rs
✗☛ ☎✞
✦
✡✏✑☛ ✌✠✫ ❧✗✓✑✕ ✥✆✗ ✠✜✓✑ ✭✆ ✓✑✕ ✠②✪
✜✖✍ ✜✑
9.
❧✗✡✆
✥✏
Rs
②✆✬✆
✹✚✛✛
✼✛ ②✆♦✆ ✗✑ ✢✠✣✡ ✜✞✒☛ ✞✟ ✢✆✟ ✏ ✌✠✫ ✮✑ ✯ ✡✎✆✰ ✥
✱ ②
✥✆✑ ✎✑✲
✢✆✟✏
✜✖✍ ✜✑
A
✪✡ ✬✆✆✑ ✳✌ ✥✫✆✴✆✱ ✖✑☛ ✡✖ ✗✑ ✡✖ ✵✛ ✖✆✶✆✡ ✠✓✎✆✠✖✜
✫✆✑
✥✦✡✆✏
✡✒
A
Rs
②✆☎✝
✹
✢✆✟✏
F1
✬✆✆✑ ✳✌
✓✑✕
✚
✥✦✠✝
✖✆✶✆✡
♦✆✠✜✩
✖✆✶✆✡
F2
✢✆✟ ✏
❧✥②♠✣
F2
✞✟✤
F1
✬✆✆✑ ✳✌
✞✟ ✤
✡✒
✞✟☛ ✤
✥✦✠✝
F1
✬✆✆✑ ✳✌
✪✡
✡✒
❞✡✆❞✱
✖✑☛
Rs
✥✏
②✆✬✆
✞✆✑✤
✢✆✟ ✏ ✈✛✛ ✖✆✶✆✡ ♦✆✠✜✩ ✞✆✑✜✆ ✷✆✠✞✪✤
✡✒
Rs
②✆☎✝
❞✡✆❞✱ ✖✑☛
✡✖
✚✛✛✛
✗✑
✡✖
✡✖
✗✑
✺
✹
✥✦✠✝
✡✖
✖✆✶✆✡
✸
✖✆✶✆✡
✖✆✶✆✡
✠✓✎✆✠✖✜
F2
✢✆✟ ✏
✠✓✎✆✠✖✜
A
✢✆✟ ✏
✸
✖✆✶✆✡ ♦✆✠✜✩ ✞✟☛✤ ❞✗✡✆✑ ✪✡ ✏✟✠ ♦✆✡ ✥✦✆☎
✑ ✆✦ ✖✜ ✗✖✯✌✆ ✓✑ ✕ ❡✥ ✖✑ ☛ ✗✍ ✶✆✲✻ ✡✒✠✩✪✤ ❧✗ ✢✆✞✆✏
✡✆
✖✍ ✾ ✌ ★✆✝
✽✌✍ ✜✝✖
✡✒✠✩✪✿ ✠✩✗✖✑☛ ❞✜ ✫✆✑
✬✆✆✑ ✳✌✆✑☛
✡✆
❧✗✖✑☛
✠✖❀❁✆ ✞✟ ✢✆✟ ✏
✽✌✍✜✝✖
✥✆✑ ❂✆✡
✝✧✓ ✞✟☛✤
10.
✫✆✑
✥✦ ✡✆✏ ✓✑✕
F2
✖✑☛
✚
%
✜✆❞✎✉✆✑ ✩✜
F2
%
✢✆✟ ✏
✞✟✤
✈✛
✝✴✆✆
F1
10%
✖✑☛
✥✕✆✯✥✕✆✑✠ ✏✡
✥✕✆✯ ✥✕✆✑ ✠ ✏✡
Rs /kg
✓✆☛✠❄✝
✢✐ ②
✥✆✑ ❂✆✡
✠✜✐✜✠②✠♦✆✝
❤✆✑✶✆
✓✑✕
❈✸✿
✡✒
✢✆✓✭✌✡✝✆
✝✧✓
✡✆✑ ✜✒✌
✹❉
✠✖②
✠✖✎ ✒ð
✞✟✤
q
✝✴✆✆
✢✆✟ ✏
❈✛✿
✗✓✑ ✕✤
✽✌✍ ✜✝✖
✓✑✕
✠✜✡✆✌❢
✠②✪
✫✆✑ ✜✆✑☛
✚❉
✡✒
✡✒✖✝
✠✯✴✆✠✝✢✆✑ ☛
kg
Rs /kg
✡✆
✥✏✒❤✆❁✆
✡✏✜✑
kg
✜✆❞✎✉✆✑ ✩✜ ✢✆✟✏ ✈✹
✈✹
✺
✢✆✟ ✏
F2
✡✒
✡✒✖✝
)
P
❯◗❘◗❙ ❱❲❳❨◗
✢✆✞✆✏ ✝✟✌✆✏ ✡✏✝✆ ✞✟ ✤ ✬✆✆✑ ✳✌
✝✧✓
✖✆✶✆
✓✑✕
✓✑✕
✹
✖✆✶✆✡✿
✬✆✆✑ ✳✌
✞✟✤
❆
✠✜✐✜✠②✠♦✆✝
✥✏
❊✆✠✎✝
Q
✖✑☛
✞✆✑✝✆
(B) p = 2q
9(
❧✗✒
✡✒
❅✌✆
②✆☎✝
❆
✡✆✟✜
✖✆✜✆✠✡
❧✠✷✝
✥✦✠✝✲☛✣
✠✜✣✆✱✠ ✏✝
✞✟
✝✆✠✡
✡✆
✗✔✗☎
☛ ✝
✩✞✆✘
✢✠✣✡✝✖
(D) q = 3p
■❏❑▲▼◆❑
P
✢✆✟ ✏
Q
✡✆ ❧✥✌✆✑☎
✡✏✝✑ ✞✔✪ ✪✡ ✠✓✭✆✑ ❂✆
✥✦✧ ✌✑ ✡ ✥✟✓✕
✑ ✎ ❈✠✩✗✖✑ ☛ ✸✛ ☎✦✆✖ ✢☛✝❩✓❂✎ ✞✟❉ ✖✑ ☛ ✡✟✠✾✭✆✌✖ ✓✑✕
✡✆✑ ②✑✯ ✎✉✆✑ ②
✓✑✕
✞✟✤
✗✑
✞✟
✪✡ ✢✆✞✆✏✠✓✫❃ ✫✆✑ ✬✆✆✑ ✳✌✆✑☛
✡✆
❇
(C) p = 3q
❋●❋●❍
②✆✟✞
F1
✌✠✫
✢✆✟ ✏
④✲✫✔❢
(A) p = q
✩✲✠✡
✞✟✤
✥✕✆✯✥✕✆✑✠ ✏✡ ✢✐② ✞✟✤ ✝✴✆✆
2x + y 10, x + 3y 15, x, y 0
(0, 0), (5,0), (3, 4)
(0, 5)
Z = px + qy,
Z
✢✗✖✒✡✏❁✆
p, q > 0, p
✖✆✶✆✡
✢✐②
6%
✞✟ ✿ ✥✦✧ ✌✑ ✡ ✥✦✡✆✏ ✡✆ ✠✡✝✜✆ ❧✓✱ ✏✡ ❧✥✌✆✑☎ ✓✑✕ ✠②✪ ✷✆✠✞✪ ✝✆✠✡ ✽✌✍ ✜✝✖ ✖✍✾ ✌ ✥✏
✚
❖P◗❘❙❚◗
✜✆❞✎✉✆✑ ✩✜ ✢✆✟✏
✥✭✷✆✝❃ ✪✡ ✠✡✗✆✜ ✥✆✝✆ ✞✟ ✠✡ ❧✗✑ ✢✥✜✒ ✥✕✗② ✓✑✕ ✠②✪
✓✑✕
11.
F1
❧✓✱✏✡
✥✟✓✕
✑ ✎
✖✑☛
A
✓✑✕
✡✆✑ ②✯
✑ ✎✉✆✑ ② ✓✑✕ ✹ ✖✆✶✆✡ ✢✆✟ ✏ ✠✓✎✆✠✖✜
✓✑✕
✺
✖✆✶✆✡
✓✟✕✠✾✭✆✌✖
3
✢✆✟ ✏
✝✧✓
✠✓✎✆✠✖✜
✓✑ ✕
✸
A
✖✆✶✆✡✿
✓✑ ✕
②✆✟✞
✺
✖✆✶✆✡
✝✧✓
✓✑✕
✈✙
✢☛✝❩✓❂✎
✙✛
✞✟☛
✖✆✶✆✡✿
✖✆✶✆✡ ✢☛✝✠✓✱❂✎ ✞✟✤ ✢✆✞✆✏ ✖✑☛ ✡✖ ✗✑ ✡✖ ✙✹✛ ✖✆✶✆✡
✓✟✕✠✾✭✆✌✖✿ ②✆✟✞ ✝✧✓ ✓✑✕ ✡✖ ✗✑ ✡✖ ✹✺✛ ✖✆✶✆✡✿ ✢✆✟ ✏ ✡✆✑ ②✑✯ ✎✉✆✑ ② ✓✑✕ ✢✠✣✡ ✗✑ ✢✠✣✡ ✸✛✛ ✖✆✶✆✡
✢✥✑✠ ❤✆✝
✖✆✶✆✆
✡✆
✞✟ ☛✤
✥✦✧ ✌✑✡
✽✌✍ ✜✝✖
✬✆✆✑ ✳✌
✠✡✌✆
✓✑✕
✩✆
✠✡✝✜✑
✗✓✑ ✕✤
✥✟✓✕
✑ ✎✆✑☛
✡✆
❧✥✌✆✑☎
✠✡✌✆
✩✆✪
✝✆✠✡
✢✆✞✆✏
✖✑☛
✠✓✎✆✠✖✜
A
✡✒
❥ ✁✂
✄
❣☛
❡
✘
✐
✫ ✬
☞
☞
✌
✛
❡
☞
✧
✓
❢
✍
✌ ✱✌
✍✎
☞
✏
☞ ☞
✑
✒ ✓
✎☞
P
☞
✑
✔
✭✍ ✮☞
✩ ✚
✓
✲
✓
♦ ✖☞
✑
✈
Q
☞
✕
✖
✛✰
✯
♦
✑
✗
✈
✩
✔
☞
✚ ✎ ✖✮☞
✴
♦
✕
✗✍
4x + 20y 460 (
6x + 4y
✸
0, y
☞
✓
✪
300 (
❃
✎
☞
✑
♦
✖☞
✑
❢
♦
❞
✲
✓
✎☞
✲
✓
②
♦ ✖☞
✑
❢
✧
✙☞
✑
✎☞
✔
✫ ✣
✎
☞
✑
✌ ✚
✓
Z
)
✵
✈
✵
✈
)
♦ ✖☞
✑
✳
☞
✤
x
✈
y
☞
✕
✖
4x + y
✶
☞
☞
✩
✳
✵
✈
A)
✹
✎
☞
✥
✕
✦
✧
✘
★
❡
0, y
✪
0.
✪
✳
3x + 2y
✶
☞ ☞
✩
✪
☞
✭
✓
☞
✥
✕
✦
✌
✯
✩ ❡
✚✎✖
✮☞
✎
✚✍
✔
▼
❉
✫
❧
✈
☞
✑
✔
L, M
✈
☞
✕
✖
✎
✛
❧
✍
✌
N
✪
... (1)
115
... (2)
150
... (3)
0
... (4)
✎☞
✈
☞
✑
✦
❢
②
✩
❄ ✛❡✑
✸
0, y
✲
✓
✜
☞
✌
♦ ✖☞
✑
✼
✽ ✾
☞
✑
✔
✍
✣
❅
✩
✛
❧
✔
✛✭
✩
❍
❏
■
❑
▲
❆
☞
✰
✑
✍
✌ ✫
✑
✣
☞ ☞
✎
✔
✳
✩ ✎
♦
✗
✑
✾
❇ ☞
✓
☞
✔
✍✎
✩
✺
✻
✘✖
❈
✓
☞
✌
☞
✌
✍
✌ ✱
✌
✍
✍✜
☞
✩
✛☞ ✖✮☞
✚
❖
P ❘
◗
❙ ❚
❡✑
✔
✍
✫
✓
❯ ❱❲
❳
☞
✢
❡✣
☞
✤
✭
✓
❨ ❩
P
❙
☞
❬
P ❪
❭
✈
✩
✔
❀
✭
✩
✈
☞
♦
✗
✍
✩
✺
✫ ✚✍
☞
✑
✘✍ ✖
❉
❊
✥
✕
✦
12. 10
✳
♦
✗
✑
☞
✼
✿ ✾
✛
✑
✼
☞ ☞
✓
✪
80
(2, 72), (15, 20)
✈
(40, 15)
☞
✕
✖
②
✎☞
x
✙
✩
✤
✺
✻
✓
✼
✿ ✾
✛
✑
✪
x + 5y
✶
☞
☞
✩
✲
✓
❋●
✘✖
)
✳
❡✑
✢
❡✣
x
✚ ✎ ✖✮☞ ☞
✑
✔
✽
☞
0
✼
✽ ✾
❁ ❂
✩
②
✎☞
✑
✍ ♦ ✙☞ ✍
❡✌
✽
❡
✷
☞
✕
✥
Z = 6x + 3y (
✛
❡
✛✜
✔
✓
537
✠
✡
✥
✕
❢
✣
②
✪
✈
✍
✌ ✱✌
✎
✚
✟
✞
✄
✝
✭
✩
✪
✪
✘
✕
♦
✑
✗ ✙☞
✑
✔
✆
✄
✝
✳
☞
✑
✔
12x + 3y 240 (
x
♦
✑
✗
☎
✥
✕
✦
Z = 6x + 3y
(2, 72)
228
(15, 20)
150
(40, 15)
285
❫
✹
✓
✌
✯
✩
❡
✥
✕
✦
✔
◆
✌
▼
❉
✫
✈
❧
☞
✑
✔
538
① ✁✂✄
❧☎✆✝☎✞
❧✟✠
A
✘☎✤✞☞ ✈✣✦☎✈☛☞
✮✐✒☎✟✧
❞☎
♠✲✳✴✵✶✳
☛✟✖
✔✞☞
✆❄☎✔✞
✈✣✯☎✟✰☎
10
M
M
✡☎✟ ✔✞ ✡✕✗
☛✟✖ ❡
✈☞❊☞
✘☎✡☎✆
(15, 20)
✌✍✎✏
III
III
☛✯☎✞☞
N
N
✣✟✥
✐✛✍✤
✖
✺✻✼✽
✻
✾
✘☎✕✆
✘☎✕✆
✘☎✕✆
☛☎☞
✈❞✒☎
☛✟✖
②✧✞
✡✕✖✗
✡✕✗
M
☛☎☞✔✟
✈❞
✮❃✐☎✎
②☎✩☎
✘✈✤❞✔☛
❧☎✆✝☎✞
✴●
☛☎☞☎
✮❃✐☎✎☞
✐✛ ✎✜☎
✈❞
✐✆
✈✎❂
☛✟ ✖
❞☎
x
✡☛
❯
✢✒✣✆☎✟✤ ☎✟✖
ABCDE
❧☛✚✒☎
❅☎✖✦✟
☛✟✖
✐✛✎✜☎
✣✟✥ ♦✫ ✐✕✣✥
✟ ✦ ✘☎✕✆ ✩☎☎✟ ✪✒
❞☎
✑✒✓☞ ✔☛
❆②☞☎
♦✬
❅☎✖✦✟
♦✫✭
☛☎☞
❂❞
✔❞
Q
❆☎✈✡❂✗ ✈☞☛☎❈✝ ☎❞✔☎❈
✣✟✥
✣✟✥ ✬✭ ✐✕✣✥
✟ ✦
☛☎✱☎
✮❃✐☎✎☞
❆②☎❂
✢✒✣✆☎✟✤☎✟✖
✡☎✟✧☎✗
❞☎
✣✟✥
★☎☞✟
❞☎✆❄☎☎☞✟
❞✞ ❇☎☛✔☎
✣✟✥ ✣② ✎☎✟
✐✛ ❞☎✆
✮❃✐☎✎☞
☛✟✖
✔✞☞☎✟✖
☛✯☎✞☞☎✟✖
✣✟✥
❧✖✧ ✔ ②✧✟ ❧☛✒ ❉❅☎✖✦ ☎✖✟
✡✕✖ ✗
✸✵
●❍✳
☞✧
☞✧☎✟✖
✮❃✐☎✎
✈✍❞
❞☎
■❏✳✻❑✳✻ ▲
II
2
1
✘☎✕✆
❧✩☎✞ ✮❃✐☎✎
✺✿❁
★☎✔✟
N
✿▲✻▼
III
1
1.25
✐✆
Rs
❖✭✭
✐✛ ✈✔
✡✕✖✠ ✈★☞❞☎ ✮❃✐☎✎☞
✮❃✐☎✎☞
✈❞✒☎
★☎❂✠
❧✖ ❄✒☎
◗☛✯☎✙
☞✧
❞✞
✈❞✒☎ ✧✒☎
✈★❧❧✟
②☎✩☎
x
y
❞☎
✎✆
❧✟
②☎✩☎
❞☛☎✔✞
✡✕✠
✔✍
❣☎✔
❞✞✈★❂
✘✈✤❞✔☛✞❞✆✝☎
✡☎✟ ❢
✡☎✟ ✧ ☎❢
M
N
= Rs (600 x + 400 y)
✘☎✕✆
✧✈✝☎✔✞✒
❞☎
x + 2y
2x + y
x+
✫
❞☛
✣✟✥ ❂❞ ☞✧
✿❋✳✾✹
✐✛ ✈✔
◆✭✭
✣✟✥
❧✓ ✱☎✍❘
☞✧☎✟✖
5
y
4
0, y
❞✞
✘☎✕✆
✡✕✗
✈☞❊☞✈②✈❄☎✔
❚
12 (
12 (
❯
5(
❯
0
❚
✡✕✙
❞✞✈★❂
✡✕✖ ✗
☛✯☎✞☞
☛✯☎✞☞
❉♦❡ ❧✟
❉❳☎✒☎✖ ✈❞✔❡
❙✐
✘✈✤❞✔☛✞❞✆✝☎
✈☞❊☞✈②✈❄☎✔
✢✒✣✆☎✟✤
✘✔✙
②☎✩☎
Z = 600 x + 400 y
★✡☎t
P
A
✘✈✤❞✔☛
I
1
2
✈❞✔☞✟
✣✟✥
✮❃✐☎✎
✣✏✥ ②
❧☛✚✒☎
Rs
P✒☎
✡✕✗
✖
❞☎ ✮❃✐☎✎☞ ❞✆✔☎ ✡✕✠ ✈★☞☛✟✖ ✐✛ ❃✒✟ ❞ ✣✟✥ ✮❃✐☎✎☞ ☛✟ ✖ ✔✞☞☎✟✖ ☛✯☎✞☞☎✟✖ ❞✞ ✘☎✣✯✒❞✔☎
✈❞ ✮❧✣✟✥
✡✏❂
✐✛ ❃✒✟ ❞
✐✆
✐☎✔✟
✈✣✦☎✈☛☞
✘☎✕✆
❞☛ ❧✟
M
N
✮❃✐☎✎
★☎✒✗
☛✯☎✞☞✟ ✖
✐✛ ✈✔✈✎☞
♠✷✸✳✲
✣✡
✑✒✓☞✔☛
(Manufacturing problem)
I
II
✺✿❀❁✳
✣✟✥ ✐✛ ❃✒✟ ❞ ✮❃✐☎✎
✈②✈❄☎✔
✐✆
❞☎ ☛☎☞ ✑✒✓☞✔☛ ✔✍ ✡☎✟✧☎ ★✍✈❞ ✩☎☎✟ ✪✒
I, II
★✍✈❞
✣✟✥ ❧☎☛☎☞
❞☎
♠✷✸✳✲✹
☛✯☎✞☞✟ ✖
✡✕✗
Z
✡☛
✡✕
☛✯☎✞☞
I
II
✐✆
III
✐✆
✐✆
✢✒✣✆☎✟✤
)
)
... (1)
... (2)
)
... (3)
✢✒✣✆☎✟✤
✢✒✣✆☎✟✤
... (4)
❉❖❡
❞☎
✘☎②✟❄☎☞
✈★❧❞☎✟
✢✒✣✆☎✟ ✤☎✟✖
❞✆✔✟
❉♦❡
✡✕✖✗ ✘☎✣❱✥ ✈✔
❧✟
❉❖❡
✘✣②☎✟ ❞☞ ❞✞✈★❂ ✈❞ ❧✏❧✖✧ ✔ ❇☎✟ ✱☎ ✐✈✆✍❘ ✡✕✠ ❞☎✟☞ ✞✒ ✌✍✎✏✘ ☎✟✖
(5, 0) (6, 0), (4, 4), (0, 6)
✘☎✕✆
(0, 4)
✡✕✖ ✗
♦✬❲♦♦
✔❞
☛✟ ✖
❨☎✆☎
A, B, C, D
✈✎❄☎☎✒☎ ✧✒☎
✈☞✤☎❈ ✈✆✔
✘☎✕✆
E
❧✏❧✖✧ ✔ ❇☎✟✱☎
✈❞✒☎
✧✒☎
✡✕✗
✣✟✥ ✈☞✎✟❈ ✯☎☎✖ ❞ ◗☛✯☎✙
❥ ✁✂
✄
✈☛
❜ ✑
✒✓
✔
✑
✕✖
✗✘
✙ ✚
✛
✓
✔
✜
✢
✣
✓
✕✤
✓
✓
✔
✦
✥
Rs
✓
✔
✱
✲
✜
✩✒
✫
✓
❆
✓
(6, 0)
3600
(4, 4)
4000
(0, 6)
2400
(0, 4)
1600
✗
✘
✙ ✚
11
✒ ✓ ★✬
✓ ✓ ✑
✔
♥
✒ ❞✓ ✑
✔
❏❑
❍
❙
✓ ✓
✩
▲
●
✧
✭
❇✔
✷
✱
✲
✜
❈
▼◆
✳
❜
Z
❣
✢
✦
✩✫ ❉
✻
❖
◆
✑
♥
✧❊
✸
▼✾
❙
✓ ✓
✑ ✓
✜
✔
✒
✒
✕
✮
✓
❢
✭
✓
✩
✱
✒
✛
✓
❇
✣
❃
♥
❙
✓ ✓
✩
✧
✭
✒ ✓ ★✬
✓ ✓ ✑ ✓
✔
✜
✖
✒
✭
❄
✒
✕
✧✓
✙
✓
❯
◗
✮
❞
✣
❲
✑
✑
✪✑
✩✫ ✩✬ ✓
✭
✾
✵
❇✔
✩
❀
✒
✭
❞
❞✓ ✑
✱
✲
✳
❃
❈
✑
✰
✛
❁
❱
✓
✜
✔
✧★
✛
✓
✲
★
✧✓
✙ ✑
✩♥
❙
✓
✭
✓ ❞✓
✑
✒
✕
✑
✰
✓
✲
★
✑
✰
✱
✜
✲
✳
❂
✧
❊
✩
✭
✑
✰
✧
✩
★
❇
✱
✑
❩
✖
✖
✩
✑
✪✑
✮
✓ ★✯✓
✕
▼✶
✘
❬
✒ ★✑ ✓
✭ ✕✑
❨
✛
✧✓
✙
✱
✲
/
❡
✴
✵ ✶
✾
❭
✹
(Rs
✾
◆
✶
✒
✩✙ ✖
✒✓
✔
)
A
B
C
P
160
100
150
Q
100
120
100
P
❋
✓
✩
✛
✱
❉
✓
✲
★
✓
✰✖
✓
✱
✲
✳
✛
✩❀
✒
✭ ❞
✱
✜
✲
✳
Q
✐
✩
❁
✮
❞
✔
✜
❄
✒✓
✓
✲
★
✓ ★
✯✓
✕
❃ ❄
✛
✭
❣
❱
539
✠
✡
✛
✩❀✒
✭ ❞
❘
✓ ❞✓ ✑
✓
✟
✞
✄
✝
✷
Transportation Problem
A, B
C
❳
✓ ❞
✭ ✓
✴
✵
✮
✿
❣
❂
✧✓
✔
✛
✧✓
✙
P ✹
✵
✮✔
✩
❂
✒
✓
❃ ❄
✖
✔
✐
✩
❞✓ ✑
3000
❄
❅ ❅ ❅
✒✓
(5, 0)
❣
♠ ✼
✵ ●❍
■
✵
✙
✬ ✓
✭ ✔
✆
✄
✝
12.11
Z = 600 x + 400 y
✺ ✻✼
✽
❣ ❁
✙ ✔
✎
✏
Z = 600 x + 400 y
✧★
✴
✵ ✷
✶
✸ ✹
✱
❞
☞
✍
✌
☎
✙
❚
✧✓
✔
❞
✜
✔
❆
✓
✔
❚
✘
✩
✒
✔
P
✓
✔
♥
❙
✓ ✓ ✑ ✓
✔
✜
✧★
❚
✓
✭
✔
✛
✓
✲
★
✱
✲
✜
✳
Q
❜
✑
✒
✕
540
① ✁✂✄
✐☎✆✝✞✟
✟✠✡☛✠✠☞✞ ✌✞ ✍✟✎☞✞ ☞✏
✙✝✚☞ ✎✑
❣✢
✘✝✝
❯✝✠
✗✠✞✏✠✜
✥✦✧✥✦
★✠✡✠
✩✌
✐✍✡✖✗☞
✈✠✖✣ ✤✍✎
✌✠✑✠☞
✐☎✆✝✞✟
✌✑✪✝✠
✟✠✞
✍✒✐✠✞ ✟✠✞ ✓✠✞✔✠
✑✠☞✠
✍✟
✑✠✬
(8 – x – y)
x
✖✞✤
☞✏✠✞✮
☞✏✠✞✮
✟✠✞
C
y
✈✠✯✡
✍✒✐✠✞
✓✠✞✔✠
✑✞✮
✍✔✌✌✞ ✐✍✡✖✗☞
✘✝❯✎
✍✟✝✠
P
✌✞
A
❧✑✰✠✱
B
✈✠✯✡
✴
0, y
✴
0
✈✠✯✡
8–x–y
✴
0
✈✵✠✠✶✎✷
x
✴
0, y
✴
0
✈✠✯✡
x+y
✸
8
✍✒✐✠✞
A
5–x
✩✌✹
C
✴
✟✠✞
0,
✐☎✟✠✡
✟✠✞
A
✐✡
✓✠✞✔ ✞
✈✵✠✠✶✎✷
✌✠✑✠☞
✔✠
x
(5 – y)
✓✠✞✔ ✞ ✔✠✕❞✏ ✞✜
✟✹
✼✽ ✖✤
✞
✸
5
✈✠✯✡
✗✯✮
✌✠✺✎✠✍✗✟
✩✌✍✬✕
✐✍✡✖✗☞
✗✯✜
✓✠✞✔✠
✏✝✠✜
✎❢
Q
✌✞
✻
✗✯✜
☞✏
(5 – x)
☞✏✉
❯✝✠✞✍
✮ ✟
✍✒✐✠✞
A
✟✠✞
P
x
☞✏
✟✠✡☛✠✠☞✞
✓✠✞✔ ✞
✌✞
✔✠✕❞✏ ✜
✞
✪✐✾✿✎✱
✗✯✜
6 – (5 – x + 5 – y) = x + y – 4
☞✏
✟✠✡☛✠✠☞✞
Q
✌✞ ❧✑✰✠✱
✍✒✐✠✞
B
✈✠✯ ✡
✈✎✱
5–y
✴
0,
x+y–4
✴
0
y
✸
5,
x+y
✴
4
✈✵✠✠✶✎✷
✌✮✐❀
✚ ✠✶
✈✠✖✰✝✟✎✠
✟✠✡☛✠✠☞✞
✗✠✞✛
✮ ✳
✲❯✝✠✞✛
✔✠✕✏✠
x
✍✒✐✠✞
✙✝✚☞✎✑
✌✟✎✠
✍✒✐✠✞ ✟✠✞
✈✎✱
✈❢
✔✠
✘✝✝
12.12
☞✏✠✞ ✮ ✟✠✞ ✟✠✡☛✠✠☞✠
✎✟
✭✐
✍☞✫☞✍✬✍☛✠✎
❁❂❃❄❅❆❇
✔✠✕
✘✝✝✉
✔✠✞
Z
★✠✡✠
✍⑥✝✠
✏✝✠
✗✯
✍☞✫☞
✗✯✱
Z = 160 x + 100 y + 100 ( 5 – x) + 120 (5 – y) + 100 (x + y – 4) + 150 (8 – x – y)
= 10 (x – 7 y + 190)
❥ ✁✂
✄
❜ ☛☞ ✌
✍
☞
✛ ✜✛
☛
✎✏
✑
✒
✣✑
❢ ✪✒ ✙
✫
✒
✙
✚
✓
❢
✙
✬
☞
✔
✒
✕
✖
✭
✑
✚
✕
✓✮
✗
✘
✎
✚
✙
☞
✛ ✜✛ ☞ ✌ ☞
✢
✒
✯
0, y
x+y
x
y
x+y
Z = 10 (x – 7y + 190)
❖
✲
✳
☛
✙
❖
✴
✳
✗
✘
✥
✒
⑥
✒ ✪✒
❞
☞
✛
✑
✛
✱
✕
✎✖
✫
✒
✮
☞
✪
✕
✥
✪
✔
✒
✯
✌ ✒
✙
✥
✛
✥
✖
☞
✦
✍
✰
☞✥
☛❧
☛✓
✚
✕
❀
✒ ✕
✧
✒
✙
✕
✒
✧
★
☛✥ ✕
✖
✧
★
✩
✰
✰
✯
... (2)
5
... (3)
5
... (4)
4
... (5)
✥
✖☞
✦
✍
✵
✒
☞✥
✙
✶
✒
✭
✒ ✪
★
✘
ABCDEF
✵
✒
✶
✙
✒
▼
▲
P
◗
☛
✓
✚
✕
✼
✧
★
✚
✵
✒
✽✈
✧
✎
❜
✛
✙
✶✒
❢
✙
✬
✥
✒
✾
✭
❧
✒ ✙
✚
✘✪
1550
(3, 5)
1580
(5, 3)
1740
(5, 0)
1950
(4, 0)
1940
✾
❧
●
❍
❖
✴
✳
✘✪
Z
❖
✭
✒
❢
✬
✷
☞
✕
✽✈
☛
❧
(0, 5)
❇ ❈❉
❊
✧
★
✼
✧
★
(4, 0)
✵
✒
✶
✙
✒
✲
✸
✹
✲
✺
✳
12.13
✻
☞
✪
☛
☛
❧
✓
✚
✕
(0, 4)
✽✈
☛✙
✦
✒
Z = 10 (x – 7 y + 190)
1620
❁
❂ ❄
❃
❅ ❆
✔
✒ ✖
✥
✖
8
◆
✒ ✑
✒
☞
✚
✥
✕
(0, 4), (0, 5), (3, 5), (5, 3), (5, 0)
☛
✒
✪
✕
541
✠
✡
... (1)
✈
❢
✣✑
✤
✟
✞
✄
✝
0
■❏
❑
✭
☛✙
✆
✄
✝
✕
x
✣
✑
❢
✪
✒
✫
✙
✒
✚
✙
✕
☎
✛
✙
✖
Z
❋
✥
✒
❞
✑
✛
✱
●
✥
✒
❞
✑
✛
✱
✕
✎
✎✒ ✛
✲
✴
✴
✾
✑
✼
✧
★
✾
✭
❧
✎✒ ✛
✕
✎
✒
✙
✚
❀
✒
❢
✕
✙
✬
☞✛
✥
✪
✕ ✙
✿
✮
✙
✒ ✒
✧
★
✚
✥
✩
✚
542
① ✁✂✄
✈☎✆
✝✞✟☎✠
A, B
Ø✠✜✒✆ ☛✢✡✒✓
❖✭✭
Rs
✡☛☞✌✍✎
✥rr✤
☛✏✑✒☛☎
C
✈✒✙☞
✌✓✔
✈✎✕✖ ✒☞
Rs
C
✦✒✒✓❇✭
✠✒❃✒✒
✌✓✔
A
☎❆✌✒✓✫
✡✒✓✞✒✗
✠✒❃✒✗
✠✮ ❡✭
★✗
✫
✍✙✩
✥❏r
✠✒❃✒✗✣
A, B,
✡❅❆✭✓✗
✬✭✮ ✎☎✠
P
✍✒✓❉
☛✌✟✒☛✠✎
A,
C
✗✪
✗✪
✈✒✙☞
C
✎✛
✈✒✙☞
Q
✗✒☞✘✒✒✎✓
✖✓
✗✒
✵✺✻✼✲✸✽✾
A
♠✡✭✒✓✛
✗✪
✠✒❃✒✒ ✗✒ ✈☛❄✗☎✠✪✗☞❀✒ ✗☞✎✓ ✌✓✔
❈✒☛✍★❉
✍✒✓✎✒
✡❅✗ ✒☞
✖✓
✠✒❃✒✗
✈✒✙☞
Q
✈✒✍✒☞
Q
B
☎❆✌
A
✌✓✔ ✚ ✠✒❃✒✗✣ ☎❆✌
✡❅✗✒☞ ✗✒ ❈✒☞✒ ☛✧✖✗✒ ✠✮❡✭
✥✥❏■r
✗✒
✑✒✙ ❢✒✓✫
✠✒❃✒✗
✌✓✔
✡❅❆✭✓✗
✗✠
✍✒✓✫
✥
✑✒✙ ❢✓
X
✦✒✒✓ ❇✭✒✓✫
✌✓✔
✖✫✘✭✒
✗✪
P
B
Rs
C
✗✒✓ ☛✠❢✒☎✒ ❋☛✠●❀✒❍ ✍✙ ✩
✈✒✙☞
☎❆✌
✗✪ ✬✭✮✎☎✠ ✈✒✌✜✭✗☎✒★▲ Ø✠✜✒✆
✌✓✔
☛✠●❀✒
✗✠
▼
✷✸✷✸✹
✠✓✫ ☛✌✟✒☛✠✎
✡✒✓ ✞✒✗
☎❆✌
✡❅✗ ✒☞
✈✒✍✒☞☛✌✿◗ ✿✒✓
☛✌✟✒☛✠✎
✚
✠✓✫
☛✌✟✒☛✠✎
A
✗✪
✍✙ ❉
❊✭✒
■r✤ ✡❅☛☎ ✑✒✙ ❢✒ ✧✒✓☛✗
✗✒
✵✶
✡✙✌✔
✓ ✟✒✓✫
☛✗☎✎✓
✌✓✔ ■ ✠✒❃✒✗ ☞✘✒☎✒ ✍✙ ✧❑☛✗
☎❆✌
✳✴
❂✭✒✎ ✗✪☛✧★✩ ✈✒✍✒☞
★✗ ☛✗✖✒✎ ✿✒✓ ✡❅✗ ✒☞ ✌✓✔ ❈✒☞✓
✠✮❡✭
3.
✡☞
✡❅❆✭✓✗
✈☛❄✗☎✠
2.
✈✒✙☞
✍✒✓✛ ✒✩
♠✿✒✍☞❀✒ ❁
☛❢★
5, 0
✖✓
☎✗ r✣ ✤ ✈✒✙ ☞ ✥ ✎✛ ✦✒✓✧ ✒ ✧✒★✛✒✩ ✝✖✪ ☛✏✑✒☛☎ ✌✓✔ ✖✫✛ ☎ ✬✭✮ ✎☎✠ ✡☛☞✌✍✎
✯✰✱✲✱
1.
P
✗✒☞✘✒✒✎✒
P✒☎
✗✒
Y
✈✒✙☞
✥✤
✠✒❃✒✗✣
kg
✦✒✒✓ ❇✭✒✓✫
✗✪☛✧★
✬✭✮✎☎✠
✗✒✓ ✝✖
✡❅✗ ✒☞
B
☛✌✟✒☛✠✎
✠✓✫
☛✌✟✒☛✠✎✒✓✫
☎❆✌
✌✓✔
☎✪✎
✠✒❃✒✗
✍✙✩
☞✘✒☎✒
✥▼ ✠✒❃✒✗✣ ◆r ✠✒❃✒✗ ✈✒✙☞ ■◆
☛✠●❀✒
❊✭✒
✗✠
✠✒❃✒✒
✡❅❆✭✓✗
✌✓✔
✑✒✙ ❢✓
✗✒
✍✙ ❉
☛✠❢✒✎✒
✗✪
✗✪
✌✓✔ ■❏r ✠✒❃✒✗ ✈✒✙☞
■✤✤ ✡❅☛☎ ✑✒✙ ❢✒ ✍✙✣ ✡✒✓✞✒✗
☎✒☛✗
✠✮ ❡✭
✡❅✗ ✒☞ ✌✓✔ ❈✒☞✓✣ ☛✧✖✗✒
❈✒✍☎✒ ✍✙
✖✓
✗✠
☛✗
✥■
☛✎❘✎☛❢☛✘✒☎
☛✠●❀✒ ✠✓✫
✠✒❃✒✗
✖✒☞❀✒✪
✈✒✙☞
✠✓ ✫
✿✪
✛✝❙ ✍✙✩
❚❯❯❱❲❳
✦✒✒✓❇✭
✌✓✔
4.
★✗
X
☛❢★
✌✓✔ ✥
☛✎✠✒❙ ☎✒
✿✒✓
A
B
❨❩❬❯❨❭❪
❨❩❬❯❨❭❪
X
1
2
3
Y
2
2
1
kg
☛✠●❀✒
❨❩❬❯❨❭❪
✗✒ ✠✮❡✭
✗✒
Rs
✬✭✮✎☎✠
✡❅✗ ✒☞
✌✓✔
✥❫ ✈✒✙☞ ✦✒✒✓❇✭
✠✮❡✭
☛✘✒❢✒✙ ✎✓
P✒☎
A
y
✌✓✔ ✥
kg
C
✗✒ ✠✮❡✭
Rs
♠❵✓✜✭
✌✓✔
■✤ ✍✙✩ ✌✒✫☛❴☎ ✈✒✍✒☞
✗✪☛✧★✩
✈✒✙☞
B
❑✎✒☎✒
✍✙✩
✝✖
☛❢★
☛✎✠✒❙❀ ✒
✠✓✫
☎✪✎
✠✜✒✪✎✒✓✫ ✗✪ ✈✒✌✜✭✗☎✒ ✡✢❛☎✪ ✍✙ ✈✒✙☞ ✡❅❆✭✓✗ ✡❅✗ ✒☞ ✌✓✔ ☛✘✒❢✒✙ ✎✓ ✌✓✔ ☛✎✠✒❙❀ ✒ ✌✓✔ ☛❢★ ❢✛✒ ✖✠✭
❋☛✠✎✟✒✓✫
✠✓ ✫❍
☛✎❘✎☛❢☛✘✒☎
✍✙✩
❨❜❯❝❯❞❪ ❱
❩❱ ❣
❤✐❥❯❦
❭❧❯♥❪
I
II
III
A
12
18
6
B
6
0
9
❥ ✁✂
✄
✐☛
☞
✌
✍
✎
✏✑
✒ ✓✔
Rs
❢
✮
✖
✓
✕
✐
✬
✖✗✎
✘ ✏
✯ ✰✱ ✲
✙
✚
✒ ✛
✜✍
✐☛
B
✳
✥
✒
✒
✖
✘
✕
✒ ✬
✪
✖
✢
✐☛
✔
✣✤
✍
✎
✒ ✬
✣
✤
✍
A
✳
✕
✖
✗✎
✘
✏
✥
✒
✷
✖✔
5.
✒
✎✏
✒
✔ ✍
✣
✤
✍
✖✥
✦
✎
✩
✒ ✍
✩
✔
✍
✣
✒
✖ ✜✎ ✜
✩
✒ ✹
Rs
✕
✖✗✎
✘ ✏
✲ ✲ ✲
✕
❑
✒ ✘
✻
✌
✒
✕
✒ ✬
✪
✽
✘
✍
✿ ❀
✍
✒
✓
✎✏
✽
✍
✎✏
✐✍
❂
✒ ✒
✎✏
✎✓✖
✹
✦
✽
✍
✻
✽
✖
✗✎
✘
✖✎
✏✓
✎
✬
❀
✒
✐☛
✕
♥
✓✔
✔
✒
✛
❇
✒ ✬✒ ✛
✬✒ ✑
✒ ✔
✮
✎✓
✲ ❋
✏
✑
✒
❈
✍
✢
✓
✜✛
✍
✎
✒ ✍
✣
✤
✍
✍
✕
✎
✍
✐☛
✐☛
✾
✒
✌
✒ ✼✒
✓
✏
✿ ❀
✍
✒
✽
✎
✒ ✬
✔
✒ ✛
✍
✱ ✲ ❋
✖✗
✎
✘
B
D, E
✏
✓
✣
✤
✍
✵
✒ ✼✒
✒
✐☛
✎
✒ ✬
✱
✣
✤
✍
✖
✭
✒
✥
✒ ✪
✳
✎
✒
✖
✭ ✒
✎
✬✒
Rs
✐
✬
✣
✤
✍
✘ ✓
✽
✥
✒
✥
✒
✪
✔
✍
✎✓
✴
✺
✒
✩
✒
B
✕
✒ ✬
✪
✔ ✍
✍
✘ ✒ ✍
✢ ✑
✒
✒
✦
✖✎
✲
✐☛
✎
✒ ✬
✶
✖
✭ ✒
✥
✒ ✪
✔ ✍
✽
✎
✘
✒
✩
✪
✫
✲ ✲
✐☛
☞
✌
✎
✍
✐☛
✾
✒
✏
✿ ❀
✍
✒
✓
✣✤
✍
✳
✙
✎
✒
✥
✒
✒
✎✏
✒ ✌
✒
✹
✒
✽
✎
✘ ✒
✩
✪
✫
✢
✣
✤
✍
✖✎
✘
✒
✪
✬
✥
✒
✔
✒ ✍
✛
✎
✒ ✍
❍
❴
✣
✤
✍
✎
✬✘
✓
✩
✪
✫
✘
✾
✒ ✒
✖✐
✐☛
✾
✒
✏
✿ ❀
✍
✒
✓
✔ ❅
✍
✖✎
✘
✔ ✍
✖✜
F
✒
✪
✬
✽
✍
✌
✒ ✼
✒ ✒
✎
✬
❢
✸
✍
✍
✎✜
✔
✍
✎
✒
✍
✣
✬
✓
✌
✘ ✒
✢ ✘
✍
✍
✩
✛
✪
✫
✳
✹
✒ ✦❆
✘ ✒
✖✎
✥
✒
✒
✎
✒
✈
✒
✖✎
✘
✔
✒
❀
✒
✩
✪
❂
✒ ✏
✘ ✒
✲ ✲
✏✑
✒
❈
✱ ✲
✵
✖
❉
✣
✛
✜✥
✕
✒ ✬
✪
❊
✐
✬
✕
♥
✔
✣
✛
✜✥
✩
✛
✪
✧
✐✥
★
✗
✎
✬✒ ✔
✒
✖
❉
✣
✛
✜✥
✩
✪
✫
✧ ♥
✩
✛
✍
❋
✐
❇ ✘
✒
✩
✪
✖
✹
✔ ✎✓
✕
✒
✣✑
✌
✎
✘ ✒
✦❆
✫
✐☛
✖
✘
▼
✖
❉
▲
✣
✛
✜✥
◆
❖ ◗
P
❘
■▲
✣
✩
✔
♥
✌
●
✌
❙
❖ ❚❯
✌
✖✔
❱
❲
❲
❄
✔
✽✒ ✬
❀
✒
❳
❨■❲
❩ P
❬
B
D
6
4
E
3
2
F
2.50
3
❧❬
❴
✔
✘ ✏✓
✎
✬❀
✒
✣
✤
✍
✖✥
✦
✓
❭
P
A
❬
/
✐✖
✬
❴
✌
✌
✖
❂
✒
✘
✖ ✜✎
✜
✮
✒ ❇
✛
✒ ✬
✕
✖
❉
❫
❩
✣
✩
✔
✣
✤
✍
❍
✎
✒
✕
✒ ✬
❞ ✲
✕
✍
✖✥
✦
✿
✍
❀
✒ ✓
✳
✎✓
■▲
❏
✐✖
✬
✌
✖
✜✎ ✜
❄
✒
✛
❇
✒ ✬✒ ✛
✣✤
✍
✕
✔
✽✒
✬
✩
✪
❈
❪
❬
✺
✈
✕
✒ ✐✖
✘
✎
✒
✐
✖
✬✣ ✩
✔
✣
✤
✪
✽✍
✖
✎
✌
✒
✹
✒ ✦
❴
♥
✌
✔
✘ ✏
✐✖
✬
✣✩
✔
✈
✏❡✌
✎
❉
✌
✘ ✍
✖✥ ✜
✬
✒
✥
✩
✪
✎
✒ ✬✭
✒ ✒
✎✓
✩
✪
✛
✫
✔ ✍
✏
✛
✍
✎
✒
✬✭ ✒ ✒
✢ ✒ ✍
✔
✍
✮
✕
Rs
✐
✬
A
✖
✢
❁
✔ ✍
☞
✌
✕
✒ ✬
✪
✎
✒
✙
✳
✦
✍
✳
✩
✒ ✍
A
✳
✍
✖
✼✒ ✌
✒ ✛
❃❄
✎
✏
❄
7.
✌
✲
✔
✈
✘
✫
✱
✐☛
✎
✒ ✬
❁
✵
❞
✢ ✒
✔ ✍
✩
✪
543
✠
✡
✲
✲
✹
✐
✬
✬✥
✒
✎✓
6.
✖
✭ ✒ ✥
✒ ✪
✗
✟
✞
✄
✝
✒ ✖✩
✦✫
✴
✦✌
✕
✖
✢ ✔
✧ ✐✥ ★
✆
✄
✝
✸
✏
✘
✴ ✺
✦
✐☛
✖
✘
✖✥
✦
☎
✒ ✣✑
✌
✎
✘
✒ ✦❆
❴
✎✓
✢
✖
✬✌
✒ ❆
❜
✖
❇
✐
✒
❵ ✒
✬✒
✍
✘
A
✓✔
✘ ✾
✒ ✒
✐✍
❞
✱ ✲ ✲
✏✑
✒
km
✜
✒
❛
✥
✍
❋
❈
✖✥
✜
✬
B
❋
✩
✪
✛
✮
✖
✹
✐
✛
✐
✒ ✛
✔
✎✓
D, E
✍
✖✔
✏
✘ ✒
✦❆
✕
✒ ✬
✪
✲
✲ ✲
✶
F
✖✎
✘
▲
❣
❤
✕
✒ ✬
✪
✶
❙
❲
❩
✓
✣
✤
✍
✕
km
❳
❦
❭
P
❬
✔
✽✒ ✬
✩
✪
❈
❪
B
D
7
3
E
6
4
F
3
2
❧
❬
✖✥
✦
✖✥
✜
✬
A
❫
❩ ❬
/
✽✒ ✬❀
✒
✲ ✲
✲
❈
❞ ✲ ✲ ✲
✖✥
✜
✬
❴
✣
✤
✍
❄
✔
✒ ✛
✯
✏✑
✒
✕
✒
✕
✒ ✬
✪
✺
✐✖
✘
❋
✎
✬
✔ ✓
✩
✪
✖
✹
✔ ✎✓
✱ ✲ ✲
✖✥ ✜
✬
●
✏❝
✛
✍
❂
✒
✎✓
✩
✪
✫
✖
❇ ✐
✒
✍
✽
✍
✐✍
✜
✒
❛
✥
✍
✐
✛
✐
✒ ✛
✍
544
① ✁✂✄
❀☎ ✆✝✞✟✠ ☎✡☛ ☞✌
✍☞✎✏☎✞
✑❀❀ ✍✒☞ ✟ ✓✔ ☞✕✖✎
✍✎
✍✒☞ ✟ ☞✌✕✝✠✆ ✗✖✎
✓ ✘✍❀✝
☎✙ ✚ ✛✝✟ ✌✗☞✜☛
☞✌ ✏✙❢✢✗ ✣✝✍✤☞ ✟✥ ❀✝✠✜✞✝ ✣✍✞✝✦✥ ✜✝☛✚ ☞✜✢✢✠ ✍☞✎✏☎✞ ✑❀❀ ✌✝ ✧❀✤✞✟✆✗✌✎★✝ ☎✝✠ ✜✝☛✩ ✧❀✤✞✟✆
✑❀❀
8.
❖❀✝
☎✙✩
P
☛✌ ✍❢✕ ✱✪✍✝✫✌ ✣✍✞✠ ✬✝✭ ✆✠ ✮ ✫✝✠ ✍✒✌✝✎ ✏✠❢ ✯✝✝✫✝✠ ✮
✬✒✝❝
✮
✣✝✙✎
Q
✬✒✝❝
✮
✌✝ ✱✍❀✝✠ ✭ ✌✎ ✢✌✟✝
☎✙❣ ☞✆✰★✝ ✏✠❢ ✍✒✪ ❀✠✌ ✲✝✙ ✕✠ ✆✠ ✮ ✞✝✦✖✳✝✜
✠
✞✚ ✍❢✝✴✍❢✝✠☞✎✌ ✣✵✕✚ ✍✝✠ ✖✝✶✝ ✣✝✙ ✎ ❖✕✝✠ ✎✗✞ ✌✗ ✆✝✷✝✝ ✸
✮
✢✝✎★✝✗ ✆✠ ✮ ☞✫❀✝ ✭❀✝ ☎✙❣ ✍✎✗✹✝★✝ ✢✮✏❢
✆✠❡
✠ ✟ ✫✠ ✟✠ ☎✙ ☞✌ ✬✝✭ ✌✝✠ ✌✆ ✢✠ ✌✆ ✺✻✔
✣✵✕✚
✢✠
✌✆
kg
✺✈✔
✌✆
✣✝✙✎
✍✝✠✖✝✶✝
❖✕✝✠✎✗✞
✣☞✼✌
✌✗
✢✠
✣☞✼✌
kg
kg
✍❢✝✴✍❢✝✠☞✎✌
kg
✽✓✔
✌✗
✣✝✏✶❀✌✟✝ ☎✙❣
❀☞✫
✱✪✍✝✫✌
♣✝☎✟✝
☎✙
✞✝✦✖✳ ✝✠✜✞
✬✝✭
✟✲✝✝✚
✌✗
✏✠❢
☞✕☛
✍✒ ✪❀✠✌
☞✆✕✝✦✥
☞✆✰★✝
☞✞✵✞✟✆
✆✝✷✝✝
✏✠❢
✜✝✞✠
✏✝✕✗
✞✝✦✖✳ ✝✠✜✞
✲✝✙ ✕✝✠✮
☞✌✟✞✠
✾✿ ❁❂
P
♣✝☞☎☛✩
☞✆✕✝✦✥
✌✎✞✝
✜✝✞✠
✏✝✕✗
3.5
1
2
3
1.5
1.5
2
✣✵✕
✍✝✠✖✝✶✝
❖✕✝✠✎✗✞
Q
❇✿❄❉
❈
3
✞✝✦✖✳✝✜
✠
✞
✱✍✎✝✠ ❖✟ ✍✒✶✞ ♠ ✍✎ ❊❀✝✞ ✫✗☞✜☛❣ ❀☞✫ ✱✪✍✝✫✌ ✬✝✭ ✆✠ ✮ ☞✆✕✝✦✥ ✜✝✞✠ ✏✝✕✗ ✞✝✦✖✳✝✜
✠
✞ ✌✗ ✆✝✷✝✝
✣☞✼✌✟✆✗✌✎★✝
✌✝
✜✝✞✠
10.
☎✝✠✞✝
✧❀✤✞ ✟✆✗✌✎★✝
✌✝
❃❄❅❆❄
❇✿❄❉
❈
9.
✱✍❀✝✠ ✭
✌✝
✆✝✷✝✝
☎✙ ✩
❖❀✝
kg
✍❢✝✴✍❢✝✠ ☞✎✌
✌✗
☛✌
✟✲✝✝
✏✝✕✗
♣✝☎✟✝
✞✝✦✖✳✝✠✜✞
☞✯✝✕✝✙✞✝
✱✍✕r✼
A
✢✮✢✝✼✞✝✠ ✮
✭✡☞ ❝♥ ❀✝✠✮ ✢✠ ✣☞✼✌
✣✝✙✎
✢✠
✌✗
✭✡☞❝♥ ❀✝✠ ✮ ✌✗
✣✝✼✗
✍✒✌✝✎
✌✗
✭✡☞ ❝♥ ❀✝✠ ✮
✱✪✍✝✫✞
✍ ✒ ✪ ❀✠ ✌
✭ ✡☞ ❝ ♥ ❀✝
✣ ☞ ✼ ✌✟✆✗✌✎★✝
✍✎
☎✙❣
☞✆✕✟✝
✴✟✎
☞✕☛
✌✗
☎✙
B
✟✗✞
✓✺
✍✒✪❀✠✌
✏✠❢
✲✝✙ ✕✝✠✮
✌✝✠
☞✆✕✝❀✝
♣✝☞☎☛✩
✜✝✞✝
☞✆✕✝✦✥
☎✙ ✩
✭✡☞ ❝♥ ❀✝✠✮ ✌✝
✍✒✌✝✎
✭✡✞✠
✣ ✝✙ ✎
☞✞✆✝✥★✝
✢☞✵✆☞✕✟
☞✌
✌✗
A
✣☞✟☞✎❖✟
✏✠❢
Rs
☞✌✟✞✠
❖❀✝
✍✒✌✝✎
✦✢✏✠❢
■✆✶✝❏
✌✎✞✠ ✏✠❢
✫✝✠
✆✝✷✝✝
♣✝☞☎☛ ✣✝✙✎
✍✒✌✝✎
B
B
✏✠❢
☞✆✰★✝
✢✮✏❢
✠ ✟
✞☎✗✮ ☎✝✠✞✝
✏✠❢
✟✝✠
✣☞✼✌✟✆
✌✗
✏✮❢ ✍✞✗✚
☎✙
✢✠
✍✒✌✝✎
✐✔✔
Rs
☞✌✟✞✠
✱✪✍✝✫✞
✭✡☞ ❝♥ ❀✝✠ ✮ ✌✗
✌✗
✞✭
✓✐
✞✭✝✠✮ ✌✝
✌✝
☎✙❣
✌✎✟✗
✴✟✎
✍✒☞✟
✣☞✼✌
✭✡☞❝♥ ❀✝✠ ✮ ✌✝
✣☞✼✌
✕✝ ❑✝
✢✝●✟✝☞☎✌
☎✙❣
✆✝❋✌✖
✍✎✗✹✝★✝✝✠ ✮
✢●✟✝☎
✓✺✔✔
✢✠ ✣☞✼✌ ✆✝❍✭
✱✪✍✝✫✞
❀☞✫
✌✆ ✝ ✟✗
✱✪✍✝✫✞
✴✟✎
✫✤ ✢✎✠
A
✣✝✙ ✎
✕✝ ❑✝
✌✝
✌✮✍✞✗
☎✙ ✚
A
✌✎✞✝
♣✝☞☎☛❣
❥ ✁✂
✄
☎
✆
✄
✝
✟
✞
✄
✝
545
✠
✡
❧
☛ ☞
☛ ✌ ✍
☛
✱✏
✑
✒
✓
✔
✕ ✏
✖
✕
✗
✘
✙
✕
✗
✚
✛
✜✚
✢
✣
✕
✤✥
✜✚
✢
✣
✕
✥
✒
✦
✕
✘
✏
✧
★
✩
✑
✕
✪
✘
✤
✫
✘
✑
✓
✒
✔
✕
✏
✖
✫✬
✛
✤
✫
✘
✧
✭
✮✯
✚
✚
✕ ✛
✎
✰
✲
✳
✓
✴
✏
✯
✚
✣
✕
✵
✶
✷
✣
✛ ✯
✚
✏
✕
✘
✸
✕ ✯
✏
✑
✛
✘
✸
✳
✖
✗
✓✯
✜
✪
✣
✥
✥
✕
✘
✶
✓✏
✩
✑
✐
✽
✕
✯
✘
✑
❣
✏
✕ ✘
✥
✕
✪
✘
❣
✏
✑✯
✘
✑
✓
✒
✔
✕ ✏
❀
✥
✕
✪
✘
✩
✑
✕
✪
✘
✏
✕ ✘
✹
✺
✻
✯
✖✫
✬
✛
✏
✕
✘
✲
✕
✑
✒
✼
✜
✪
✯
✭
✮
✳
✓
✪
✲
✪
✥
✪
✒
✜
✘
✏
❁
✕ ✾
✘
✼
✣
✖
✫✬
✛
✰
✜✚
✾
✏
✑✽
✕
✕
✪
✘
✑
✓
✒
✔
✕
✏
✓
✛ ✽
✕ ✕
✣
★
✏
✩
✑
✦
✿
✣
✤✑
✕
✘
✏
✥
✯
✘
✲
✕ ✾
✸
✥
✪
✒
✳
✓
✦
✛ ✏
✕
✘
❀
✏
✏
✥
✯
✘
✏
✥
✯
✘
✥
✪
✒
✼
✕
✑
✒
✐
✽
✕
✯
✘
✑
✥
✪
✒
✵
✤✫♦
✚✥
❂
✤✖
✽
✕
★
✑✒
✓✔
✕
✏
✖✗
✕ ✘
✙
✕
✗
✚
✛
✜✚✢
✣
✕ ✱❃
✓✛ ❄
✛ ✓✬ ✓
✔ ✕
✯
✥
✪
✒
❅
✎
(i)
✲
✸
✳
✕
(ii)
✥
✕ ✑
✹
♠
✜
✪
✜✚✢
✣
✕
✪
✾
✜✚✢
✣
✕
✸
✳
✖✓ ✑✤ ✥
✛
✜
✪
✳
✕
✾
✸
✳
✛
❀
✜
✪
❆
✖✕
(iii)
✜
✪
✾
✿✣
✤ ✑✕ ✘
✪
✾
✜✚✢
✣
✕
✲
✕ ✘
✪
x
✳
✕ ✑
✒
✐
✽
✕ ✘
✯
✑
✿✣
✤ ✑✕ ✘
✕ ✪
✘
✎
❣
✑✘
✔ ✕
✾
✣
✖✗
❣
✕
✘
❉
✜
✜
✪
✙
✯
✙
✗
✕ ✚✛
✜✚✢
✣
✕
❊
✕
✘
✲
✕
✤
✫
✘
✏
✕
❀
✪
✯
❅
❉
✜✜
✪
✙✯
❊✕
✤
✫
✘
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✏
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y
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♠
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Z = ax + by
✳
✓
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✣
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✕ ✛
✏
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♠
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❪ ❭
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✖❆ ✕ ✪
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✕ ✓
(1)
✴
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❆ ❣
546
① ✁✂✄
(2)
✐☎ ✆✝✞ ✟
✟✠✞ ✡☛✝
②✡
✕☛✘✙✚
(3)
☞✌✍✎ ✜ ✠✞ ✢
✐✏
✐✏
✑✒✞ ✓✝
❀✎❀✢ ✩✗
✜✠✤ ✏
M
✜✠✤ ✏
(i)
✜✐✘✏✌✬
✑✒✞✓✝ ✐✔✕✡
✜✣✯✗ ✕
✐✔✕✡
(ii)
✑✒✞✓✝
❣✤
M
✟✠
❀✎ ❀✢✩✗
✟✠
✥✝✦✡✗❞
m
✟✠
❞✠✡
M
✧❞✓✠★
❞✠✡
✖✠✗
✟☛✘✙✚✛
✗r✠✠
m
❞✠✡
❣✤✢ ✛
✧❞✓✠★ ✑✒✞ ✓✝ ✐✔✕✡ ✮✞ ✔ ✜✘✣✟✗❞ ✗r✠✠
❣✢✤ ✛
❞✠✡
✪✠✞✫ ✠
Z = ax + by
✐✔✕✡
✜✘✣✟✗❞
✝✘✍ ❀✎ ❀✢✩✗ ✪✠✞✫ ✠ ✐✘✏✌✬ ❣✤✭ ✗✠✞
✥✝✦✡✗❞
✝✘✍
☞✌✍✎
✜✘✣✟✗❞
✪✠✞✫ ✠
✮✞ ✔
✜✘✣✟✗❞
✐✔✕✡
✗✌
❞✠✡
ax + by > M
✝✘✍
✑✈✠✝✘✡✱✲
☞✌✍✎
✡❣☛✢
✮✞ ✔
♦✠✏✠ ✘✡✣✠✯✘ ✏✗
✏✰✠✗✠
❣✤ ✛
✜✥✝r✠✠
✰✠✎ ✕✠
✑✒✞ ✓✝
❣✤ ✛
✡❣☛✢
✥✝✦ ✡✗❞
✟✠
✟✠✞ ②✯
❀✠r✠
❣✤
❞✠✡
m
❞✠✡
❣✤
ax + by < m
✝✘✍
♦✠✏✠
✰✠✎ ✕✠
✘✡✣✠✯✘ ✏✗
✜✣✯✗ ✕ ✜✠✤✏ ❀✎❀✢ ✩✗ ✪✠✞✫ ✠ ❞✞✢ ✟✠✞ ②✯ ☞✌✍✎ ✑✈✠✝✘✡✱✲ ✡❣☛✢ ❣✤ ✛ ✜✥✝r✠✠ ✑✒✞ ✓✝ ✐✔✕✡ ✟✠
✟✠✞ ②✯
✝✘✍
✳
✥✝✦✡✗❞
❀✎❀✢ ✩✗
✏✞ ✰✠✠✰✠✢ ❥
✮✞ ✔
❣✤ ✛
✡❣☛✢
✪✠✞✫ ✠ ✮✞ ✔ ✍✠✞ ✟✠✞ ✡☛✝ ☞✌✍✎ ✜✠✞✢
✜✘✣✟✗❞
✮❣☛
❞✠✡
✥✝✦✡✗❞
✝✠
✘✟❀☛
✈✠☛
☞✌✍✎
❞✠✡
✐✏
✐☎✍ ✠✡
✈✠☛
②✱✴✗❞ ❞✠✡ ✚✟ ❣☛ ✐☎✟✠✏
✟✠
✑❀☛
✐☎✟✠✏
✶✷✸✹✺✻✸✼✽
✘♦✗☛✝
✘✮ ✓✮
✥✝✦✡✗❞
❯✝✝
✏✤ ✘✰✠✟
②❀
✐✏
✐☎✠✞✩☎ ✠❞✡
✜❞✞✘ ✏✟☛
✙✌
✜✘✣✟✗❞
✮✞ ✔
✪✠✞✫ ✠
✜r✠✯✓✠✠❅✫✠☛
✐☎ ✠✞✩☎✠❞✡
❞✞✢ ✭
✝✎✬
✟✠✞
❞✞✢
❀✢ ❢ ✠✕✡
✝✎✬
❣✠✘✡
✐☎ r✠❞
✐❣✎❄❢✞ ✭
F.L.Hitch Cock
✐✘✏✮❣✡❜❀❞❅✝✠
✮✞ ✔
✟✠
✟☛
✍✠✞✡✠✞✢
②✱✴✗❞
✟✠
✝✠✞ ✙ ✡✠
✐☎ ✠✞✩☎✠❞✡
❀✦✫ ✠✐✠✗
❞✞✢
✡✞ ❈❉❊❈
✡✠❞
②✡
✗✌
☞✌✍✎ ✜✠✞✢
❣✕
✟✠✞
✘❞✕✠✡✞
✮✠✕✞
❣✤ ✛
✸✾✿❁❂✻❃
✏✤✘ ✰✠✟
✐☎ ✠✞✩☎✠❞✡
❣✤
✟✏✗✞
✟✠ ❣✤ ✜r✠✠✯✗✵ ✍✠✞✡✠✞✢
❀✞
✌✡☛✭
✘✮✘✣
❆❀☛
✙✠✡✠
✩✝✠✛
✟✠✞
L.Kantoro Vich
✗r✠✠
✜✘❅✗✆✮
✩✘❇✠✗✖
✘✟✚✛ ✍✠✞ ✡✠✞✢
✓✠✫✠✎✜✠✞ ✢
✘✙❀❀✞
❞✞ ✢
✘✟
✜✠②✯✛
✡✞ ❅✮✗✢✫ ✠ ❆✐
❀✡✵
❈❉❊❋
❞✞✢
❀✞
✟✠✝✯ ✘✟✝✠✛
✜✢ ✩✞☎ ✙
✜r✠✯✓✠✠❅✫✠☛
G.Stigler
✡✞ ✏✤✘ ✰✠✟ ✐☎✠✞✩☎✠❞✡ ❀❞❅✝✠✭ ✮✞ ✔ ✜✢✗✩✯✗ ②✱✴✗❞ ✜✠❣✠✏ ❀✢✌✢ ✣☛ ❀❞❅✝✠ ✟✠ ✮❇✠✯✡ ✘✟✝✠✛
❀✡✵ ❈❉❊❧
❞✞✢
❣✤✭
✟✠
✘✮✘✣
✏✤ ✘✰✠✟
❀✎ ●✠✮
G.B. Dantzig
✘✍✝✠
✙✠✞
✡✞
✚✟
✍✪✠✗✠ ✐✦❇✠✯ ✘✮✘✣
✏✤✘ ✰✠✟ ✐☎ ✠✞✩☎✠❞✡
❀❞❅✝✠✜✠✞✢
✟✠✞
✙✠✞
✘❀✢ ✐ ✕✞ ✉❀ ✘✮✘✣ ✮✞ ✔
❀☛✘❞✗ ✐☎✧❞✠✞ ✢ ❞✞✢
✡✠❞
❣✕ ✟✏✡✞
❀✞
✟☛
✐☎✘ ❀✬
❀✓✠✉✗
❣✤ ✛
✐☎ ✠✞✩☎✠❞✡
✜❞✞✘ ✏✟☛
✘✮✘✣
✩✘❇✠✗✝
✐✏
✐☎ ✠✏✘✈✠✟
✜r✠✯✓✠✠❅✫✠☛
✟✠✝✯
✟✏✡✞
T.C.Koopmans
✮✞✔
✟✠✞
✟✠✏❇✠
✜r✠✯
❀✡✵
❈❉❧❋
✓✠✠❅✫✠ ❞✞✢
❞✞✢
L.Katorovich
✜✠✤✏
✡✠✞✌✞✕ ✐✎ ✏❅✟✠✏ ✐☎✍ ✠✡ ✘✟✝✠
✩✝✠✛ ✐✘✏✟✕✡ ✗r✠✠ ✜✠✮✓✝✟ ❀✠❍■ ✴✮✞ ✝✏ ✮✞ ✔ ✜✠✩❞✡ ✮✞ ✔ ❀✠r✠ ✟②✯ ✪✠✞✫ ✠✠✞✢ ✟☛ ✙✘✴✕ ❀❞❅✝✠✜✠✞✢
❞✞✢
✏✤✘ ✰✠✟
✐☎ ✠✞✩☎✠❞✡
✐☎✘✮✘✣
✮✞ ✔
✜✡✎✐☎✝✠✞ ✩
❞✞✢
✑❡✠✏✠✞✗ ✏
—
❑
✮❏✘✬
—
❣✠✞
✏❣☛
❣✤ ✛
❖P◗❘◗ 13
❙❚❯❱❲❳❨❯ Probability
The Theory of probabilities is simply the science of logic
quantitatively treated – C.S. PEIRCE
❍✁✂✄☎✆✁ (Introduction)
✐✝✞✟ ✠✡ ✠☛☞☞✌☞✟✍ ✎✟✍ ✝✎✏✟ ✐✑☞✒✓✠✔☞ ✠☞✟ ✒✠✕✡ ✓☞✖✒✗ ✘✙✠ ✐✚✡☛☞✛☞
✠✡ ❞☞✜✏☞✌☞✟✍ ✢✟✣ ❞☞✒✜✔ ✝☞✟✏✟ ✠✡ ✌✒✏✒✤✥✔✔☞ ✠✡ ✎☞✐ ✢✟✣ ✦✐ ✎✟✍ ✐✧★☞
❋☞☞✩ ✝✎✏✟ ✦✕✡ ✪✒✛☞✔✫ ✬✭✬✏✭ ✠☞✮✯✎☞✟✪✑☞✟✰ ✱✲✳✴✵✶✲✳✷✸✹ ✺☞✚☞
✐✑✒✔✐☞✒✖✔ ✌✒✻☞✪✗✒✝✔✡✓ ✖✗✒✼✜✠☞✟✛☞ ✠☞ ✽✐✓☞✟✪ ✒✠✓☞ ❋☞☞ ✌☞✮✚ ✐✑☞✒✓✠✔☞
✠☞✟ ✐✚✡☛☞✛☞ ✢✟✣ ✐✒✚✛☞☞✎☞✟✍ ✐✚ ✐✒✚✻☞☞✒✼☞✔ ✐✣✞✏ ✢✟✣ ✦✐ ✎✟✍ ✒✏✦✒✐✔
✒✠✓☞ ❋☞☞✩ ✝✎✏✟ ✕✎✕✍✻☞ ❢✓ ✐✒✚✛☞☞✎☞✍✟ ✠✡ ✖✤☞☞ ✎✟✍ ✐✑☞✒✓✠✔☞ ✢✟✣
✌✒✻☞✪✗✒✝✔✡✓ ✖✗✒✼✜✠☞✟✛☞ ✌☞✮✚ ✈✞☞✒✕✠✞ ✒✕✾☞✔✍ (classical theory)
✎✟✍ ✕✎✠☛☞✔☞ ✻☞✡ ❡❋☞☞✒✐✔ ✠✡ ❋☞✡✩ ✿✕ ✕✎✠☛☞✔☞ ✢✟✣ ✌☞❀☞✚ ✐✚ ✝✎✏✟
✌✕✍✔✔ ✐✑✒✔✖✤☞❁ ✕✎✒✼✜ ✠✡ ❞☞✜✏☞✌☞✍✟ ✠✡ ✐✑☞✒✓✠✔☞ ✫☞✔ ✠✡ ❋☞✡✩ ✝✎✏✟
✐✑☞✒✓✠✔☞ ✢✟✣ ✓☞✟✪ ✒✏✓✎ ✠☞ ✻☞✡ ✌❂✓✓✏ ✒✠✓☞ ✝✮✩ ✿✕ ✌❂✓☞✓ ✎✟✍
✝✎ ✒✠✕✡ ❞☞✜✏☞ ✠✡ ✕✐✑✒✔✰✍❀ ✐✑☞✒✓✠✔☞ (conditional probability)
✢✟✣ ✰☞✚✟ ✎✟✍ ✒✢✥☞✚ ✠✚✟✍✪✟♦ ❃✰✒✠ ✒✠✕✡ ✌❄✓ ❞☞✜✏☞ ✢✟✣ ❞☞✒✜✔ ✝☞✏✟ ✟ ✠✡ ✕❅✥✏☞ ✝✎☞✚✟ ✐☞✕ ✝☞✟♦ ✔❋☞☞ ✿✕
✎✝❆✢✐❅✛☞❁ ✌✢❀☞✚✛☞☞ ✠✡ ✕✝☞✓✔☞ ✕✟ ✰✟❃✶✐✑✎✟✓ (Bayes' theorem), ✐✑☞✒✓✠✔☞ ✠☞ ✪❇✛☞✏ ✒✏✓✎ ✔❋☞☞
❡✢✔✍▲☞ ❞☞✜✏☞✌☞✟✍ ✢✟✣ ✰☞✚✟ ✎✟✍ ✕✎❈✍✟✪✟✩ ✝✎ ✓☞✖✒✗ ✘✙✠ ✥✚ (random variable) ✌☞✮✚ ✿✕✢✟✣ ✐✑☞✒✓✠✔☞ ✰✍✜✏
✠✡ ✎✝❆✢✐❅✛☞❁ ✌✢❀☞✚✛☞☞ ✠☞✟ ✻☞✡ ✕✎❈✍✟✪✟ ✔❋☞☞ ✒✠✕✡ ✐✑☞✒✓✠✔☞ ✰✍✜✏ ✢✟✣ ✎☞❂✓ (mean) ✢ ✐✑✕✚✛☞ ✢✟✣
✰☞✚✟ ✎✟✍ ✻☞✡ ✐✧★✟✍✪✟✩ ✌❂✓☞✓ ✢✟✣ ✌✍✒✔✎ ✌✏❇✻☞ ✪ ✎✟✍ ✝✎ ✬✠ ✎✝❆✢✐❅✛☞❁ ✌✕✍✔✔ ✐✑☞✒✓✠✔☞ ✰✍✜✏ ✱discrete
probability distribution✹ ✢✟✣ ✰☞✚✟ ✎✟✍ ✐✧★✍✟✪✟ ✒❃✕✟ ✒✺✐✖ ✰✍ ✜✏ ✠✝☞ ❃☞✔☞ ✝✮✩ ✿✕ ✌❂✓☞✓ ✎✟✍ ✝✎ ✬✟✕✟
✐✚✡☛☞✛☞ ✞✟✍✪✟ ✒❃✏✢✟✣ ✐✒✚✛☞☞✎ ✕✎✕✍✻☞☞❢✓ ✝☞✔✟ ✟ ✝✮✍♦ ❃✰ ✔✠ ✒✠ ✌❄✓❋☞☞ ✏ ✠✝☞ ✪✓☞ ✝☞✟✩
13.2 ❧❉❊✄●■❏❑ ❉❊✁✄▼✆●✁ (Conditional Probability)
✌✻☞✡ ✔✠ ✝✎✏✟ ✒✠✕✡ ❞☞✜✏☞ ✠✡ ✐✑☞✒✓✠✔☞ ✫☞✔ ✠✚✏✟ ✐✚ ✥✥☞❁ ✠✡ ✝✮✩ ✓✒✖ ✝✎✟✍ ✒✠✕✡ ✐✑✒✔✖✤☞❁ ✕✎✒✼✜
✠✡ ✖☞✟ ❞☞✜✏☞✬◆ ✖✡ ✪✿❁ ✝☞✟✍♦ ✔☞✟ ✈✓☞ ✒✠✕✡ ✬✠ ❞☞✜✏☞ ✢✟✣ ❞☞✒✜✔ ✝☞✏✟ ✟ ✠✡ ✕❅✥✏☞ ✠☞ ✐✑✻☞☞✢ ✖❅✕✚✡ ❞☞✜✏☞
13.1
Pierre de Fermat
(1601-1665)
548
① ✁✂✄
❞☎ ✆✝✞✟✠❞✡✞
✟❢✔✗✘✙
✆✟☛✧✞✞✪
✑✞✒✓
❞✞
✆☛ ✆☞✌ ✡✞ ✍✎✏ ✑✞✒✓ ✒✔
✑✈
✆✝ ✟✡✢✕✞✱
✔✪✔✩✫✞✞✬✠
✔✪✟✲✳
✟✜✓ ✓❞
✠✞✢✣ ✟✤✥❞
✆☛☎✦✞✧✞
✆☛
✟✗★✞☛ ❞☛✘✩
✍✎✭
✩
(fair)
✮✠✞✯✠
✡☎✖
✆✝✕✖ ✗✘✙ ✚✛✞☛ ✗✘✙
✟✔✰❞✞✘✩
❞✞✘
✚✥✞✜✖✘
✗✘✙
✆☛☎✦✞✧✞
✆☛
✟✗★✞☛
❞☎✟❢✓✭
✒✔
✆☛☎✦✞✧✞
✍✎✴
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
1
8
✰✠✞✘✩✟❞ ✟✔✰✗✘ ✙ ✮✠✞✯✠ ✍✎✩❉ ✒✔✟✜✓ ✍✪ ✆✝✟✡✢✕✞✱ ✔✪✟✲✳ ✗✘✙ ✆✝✵ ✠✘❞ ✆✝ ✟✡✢✕✞✱ ✶✈✢✷ ❞☎ ✆✝✞✟✠❞✡✞
❞☛
✆✳
✔❞✡✘
✍✎✩✭
✆✝ ✢✟✕✞✱ ✡
✪✞✖
✍✞✘✖✞✺
✜☎✟❢✓
❞✞✘
E
❄✞✳✖✞
✟✖✐✟✆✡
✸✮✠✹✖✡✪
❞☛✡✘
✢✞✘
✟★✡
✆✝❞ ✳
✍✞✘✖✞✺
✑✞✎☛
F
❄✞✳✖✞
✸✆✍✜✘
✟✖✟✢✱ ✲✳
✟✔✰✗✘✙
✆☛
✍✎✭
✩
✡✈
E = {HHH, HHT, HTH, THH}
✑✞✎☛
F = {THH, THT, TTH, TTT}
✒✔✟✜✓
P(E) = P ({HHH}) + P ({HHT}) + P ({HTH}) + P ({THH})
✑✞✎☛
1 1 1 1 1
(
?)
8 8 8 8 2
P(F) = P ({THH}) + P ({THT}) + P ({TTH}) + P ({TTT})
=
✰✠✞✘✩
1 1 1 1 1
8 8 8 8 2
F = {THH}
=
✔✞❧✞
E
✍☎
P(E
✒✔✟✜✓
✑✈
❄✞✟✳✡ ✍✷ ✒✱ ✍✎❉
E
❞☎
✻
✪✞✖
✡✈
✆✝✞✟✠❞✡✞
F) = P({THH}) =
✻
✜☎✟❢✓ ✍✪✘✩ ✟✢✠✞
❄✞✳✖✞
✾✞✡
E
✗✘✙
✟✔✰✗✘✙ ✆☛ ✆✳ ✖✍☎✩ ✍✎✭ ❄✞✳✖✞
F
✈✖
✼✠✞
✍✎✭
✑✮✠
✍✎ ✟❞
✆✝✞ ✟✠❞✡✞ ✰✠✞
❞☎
❞☛✖✘
✼✠✞
1
8
E
✕✞✿✢✞✘ ✩ ✪✘ ❉
✩
✟✜✓
✚✖
✆✍✜✘
✟✔✰✗✘✙
F
❄✞✟✳✡ ✍✞✘✖✘ ❞☎ ✔✹★✖✞
✍✎✏
✆✝✟✡✢✕✞✱
✗✘✙
✶✈✢✷ ✑✞✘✩
✆☛
✆☛
✆✳
✟✗★✞☛
✖✍☎✩
✗✘✙ ✟✜✓ ✒✔ ✔✹★✖✞ ✔✘ ✆✝✟✡✢✕✞✱ ✔✪✟✲✳
✒✔
✑✟✡✟☛✰✡
✔✹★✖✞
✖✘
✍✪✘✩
✗✞❀✡✗
✆✝❞ ✳ ✍✞✘✡✞
S
✪✘✩
✆☛
✍✎
✑❧✞✞✱ ✡✽
✠✍
F
✟✖✟✕★✡ ✍✎ ✟❞
❢✞✓✼✞
✟❞✠✞
❄✞✳✖✞
✟❢✖✪✩✘
✆✍✜✘
✔✘ ❄✞✳❞☛ ✒✔❞✞ ✚✆✔✪✷✤ ★✠
✠✍
✈✡✞✠✞
✍✎
✟❞
✍✞✜✞✡
❞✞✘
✓❞ ✓✘✔ ✘ ✖✓ ✠✞✢✣ ✟✤✥❞ ✆☛☎✦✞✧✞ ✗✘✙ ✐✆ ✪✘✩ ✔✪❁✖✞ ★✞✟✍✓ ✟❢✔❞✞ ✆✝✟✡✢✕✞✱ ✔✪✟✲✳ ✗✘✙✗✜ ✚✖ ✆✟☛✧✞✞✪✞✘✩
❞✞
✑✈
✠✞
✔✪✷✤ ★✠
✍✎
❢✞✘
✟❞
F
❄✞✳✖✞
❞✞
F
✗✍
✗✘✙
✑✖✷ ✗✹✙✜
✆✝✟✡✢✕✞✱
F
❞✞✘
✆✝✟✡✢✕✞✱
F
❞✞
❄✞✟✳✡
❢✞✘
✶✈✢✷
✔✪✟✲✳
✍✞✘✖✞
✍✎✭
E
✪✞✖✡✘
✟✢✠✞
✼✠✞
✗✘✙
✍✷ ✓
✍✞✘✖✘
✫✞☎
✑✖✷ ✗✹✙✜
❄✞✳✖✞
✆☛
E
E
❞☎
❞☎
✍✎❂
THH
1
=
4
1
=
4
✆✝✞✟✠❞✡✞
✆✝✞ ✟✠❞✡✞
✍✎✭
✑✡✴
✐
E
❄ ✝ ✞✟
✝
❣
✚
✕
✝
✑
✛
❞✠
✛
✈
✤
✦
✝ ✝ ✏
✕
✖
✗
✖
✡ ☛
✤
✛
✮
✍❞
E
✦
✝ ✝ ✏
✭
✕
F
F
❢
✟
☞✌
❞
✝
✛
✣
✕
✓
✖
✍
✓
❞✠
✙
✝ ✏
✧
❞
✝
P (E|F)
❞
✧
✤
✤
✑
✎
✧
☛❞
✏
♦
✢ ✚
✛
✮
✘
✉
✍
✏ ✝
❆❇ ❇
❈
✑
F
✑
❊
✔
F)
✰ ◗
✱
✤
✯
✝
✑
✍✟
❢
✟
☛
☞✌
☞✌
✝ ✍
✰ ◗
✱
✩
✝
✓
✔
☞✌
E
✝
✍
✎
13.2.1
❖P
❘ ❙
✯
①❥
♠
❇
✕
✝ ✟
✑
✫
✓
✑
✍
❞
❚
❯❱
❲
E
F
✈
✏
✑
✝
✝
❞✠
✓
✑
✈
✏
✝ ✝
☛
✪
✲✳
q
❞
✏ ✝ ✗
q
✰
✴
◗
②
✰
✴
◗
②
✘
✒
✍
✧★
✑
✵ ✶
③
✵ ✶
③
r
♥
r
♥
✷
❦ ❩
✷
✠
✙
✝ ✏
✕
✖
✍
❞
✙
✝ ✏
E
✚
✟
✧
✙
✝ ✏
✶
❦ ❩
✶
❝
♥
❛
❝
♥
❛
q
✲
❦ ✱
❛
✝
✚
✧
✎
✧✝ ✓
✑
❞
✠
✕
✖
✍
❞
★
✪
✫
✕
✗
✓
✖
F
✈
✏
✝ ✝
✕
✖
✍
❞
F
❄ ✝ ✞✟
✝
F
❄
✝ ✞✟ ✝
❄ ✝ ✍ ✞✏
✛
✧★
✑
☛
✝ ✬
✑
✧
✎
✧
❣
✚
❄ ✝ ✍ ✞✏
q
✲
❦ ✱
❛
✸
❤
✸
❤
❧
❬
❛
❧
❬
❛
✹
✹
✕
✝
✑
❞✠
✕
✖
❞
✝
✑
❦
❦
✍
✫
✕
✖
❞
✝
✑
✍✟
✘
✝
☛❞
✏
✝
❞ ☛✠
P(F)
✘
✒
❢
✟
✎
✝
☞✌
☞
✗
❚
❳
❪ ❫
❴
❞ ☛✠
☞✌
✍✏
❵❜
❡
✝
✤
✜
✎
✝
☛✑
✍✧
✩
✝ ✝ ✍
✘
✏
❞
✟
✜
✑
☞
✢ ✻
✑
✝ ✏ ✑
❀
... (1)
0
✛
✈
☛✑
✤
✦
✝ ✝ ✏
✜✩
❞
✝
●❍ ❞
☞✍
F
☞
❀
✺
✝ ✝ ✍
✧★
✑
☞✌
✝ ✍✎
✘
✒
❞
✏
✝
P(F)
✍❞
✍
✟
☞✌
❢
✟ ✍
◆
(
❉
✎
✝
✓
✑
?)
✜
✝ ✏
✮
❞
☛❞
✏
✢ ✣
■
✝ ❏✝
❁
✍
✏
✤
✝
✻
❞✠
✫
✍
✯
☛
✍
✑
✕
✓
✖
✺
✢
✞
☛✑
▲✝
✝ ✏
☛✪
☛
✒
✍
✓
✔
✏
✝
▼
☛✑
☞✌
✝
✏
✯
✍
✺
✞
S
✕
0
✢
❞✠
✝ ✑
❄ ✝ ✞✟ ✝
✉❑
✕
✓
✖
P(S|F) =
P (S F)
P (F)
P (F)
1
P (F)
✑
❄ ✝ ✞✟
✝ ✉❑
✮
✕
✝ ✏
✑
✠
(Properties of conditional probability)
❳
☛
✯
✕
✕
✖
✜✠
✍
❋
E
✜
✟ ✝
☛
✓
P(E F)
P(F)
✢
✍
✫
✮
✝ ✝
✿
✎
✻
✧★
✻
☛✑
✢ ✣
✍
❞
✛
✞
P (E F)
,
P (F)
P
❘ ❳ ❙
❨
❭
F
✩
✝
1 P (S|F) = P (F|F) = 1
✯
✕
✠
❞
✏ ✝
P(E|F) =
✝
✟
✎
✲✳
✺
✍
❞
✝
✯
✏
✎
✍
❄ ✝ ✍ ✞✏
❞
n (E F)
n (F)
❣
✧★
✑
✘
✒
✍
✕
✓
✖
F
☞✌
✍✏
❞
☛
☞✌
✍✏
✒
❇ ❇
✕
✗
✓
✖
✛
✧★
✑
✜
✕
❞✕
✏ ✑
✕
✓
✖
✜
❞
✝
✢
❂❃
❅
❞
✏ ✝
E
❄✝
✞✟ ✝
✯
✏
✎
✥
✒
✍
✏ ✒
✓
(E
✼
✽ ✾
❞✠
✍
✝
✍
✕
✓
✖
✘
✝ ✑
n(E F)
n(S)
P(E|F) =
n(F)
n(S)
✟ ✝ ✑
✞
☞✌
✥
✝ ✝ ✏ ✑
✝
☛
☞✌
✢ ✣
✕
✢ ✣
✝
✔
✥
❞
✜
✝
✑
☞✌
✍✏
=
✒
✧✝
✜
☛✑
☛
☞✌
✍✏ ✒
✓
✜
✝
✢
✣
✑
★
P(E|F) =
✛
⑥
✑
✛
✧★
✑
E
❄ ✝ ✞✟
✝
✜
✍✟
❞
✝
P (E|F)
✑
✧
✯
✏
❞
✏
✝
1
4
❞✠
✍✘
✉
✕
✓
✗
✖
☞✌
✝ ✍✎
✜
✝
✛
✈
✑
✡ ☛
P(E|F) =
✟ ✝ ✞
✑
✕
✝ ✏
✑
❞✠
549
✁
✂✄
☎✆✁
✖
✕
✗
✓
✖
550
① ✁✂✄
❧☎✆☎
P(F|F) =
✝✞
P (F)
1
P (F)
P(S|F) = P(F|F) = 1
✈✟✠
✡☛☞✌ ✍
✝✥
P (F F)
P (F)
A
❀✎✏
P(F)
✎✙
P[(A
✩
✎❢✔☎✚ ✗ ☎
✈☎✥✑
★
0,
B
✒✓ ✎✟✏✔☎✕ ❧✖✎✗✘
S
✙✞
✙☎✚ ✛✕ ✏☎✚ ✜☎✘✢☎✣✤ ✝✥✦ ✈☎✥✑
❧✚ ✬
✣✙ ✈✧❀ ✜☎✘✢☎
✛❧
✒✓ ✙☎✑
✎✢❀✖
✻☎✑☎✼
✟r
B)|F)] =P(A|F) + P(B|F) – P[(A
✫✒
F
❀✎✏
A
✈☎✥✑
B
✒✑✭✒✑
✪
✈✒❢✮✞✕
B)|F]
✜☎✘✢☎✣✤
✟☎✚
✝☎✚ ✬
✦
P[(A B)|F)] = P(A|F) + P(B|F)
✩
✝✖
✮☎✢✟✚
✝✥✦
✎✙
P[(A B)|F)] =
✩
=
P[(A B)
P (F)
✯
✰
F]
P[(A F) (B F)]
P (F)
✱
✳❧✖✴✵✶❀☎✚ ✦
❢✚✷
✲
✱
❧❢✕✎✢✗✸
✒✑
❧✎✹✖✺✢
❢✚✷
r✦✘ ✢
=
P (A F) + P (B F) – P (A B
P (F)
=
P (A F) P (B F) P[(A B)
P(F)
P(F)
P(F)
✰
✰
✰
✰
✰
✽
✰
✰
F)
✰
F]
✾
= P(A|F) + P(B|F) – P(A B|F)
✪
✮r
A
✟✆☎☎
B
✒✑✭✒✑
✈✒❢✮✞✕
P[(A
P[(A
✿
✈✟✠
✮r
❁
✡☛☞✌
❃✴❄ ☎
❅
A
✟✆☎☎
B
✒✑✭✒✑
✪
✩
✟☎✚
B)|F)] = 0
B)|F)] = P(A|F) + P(B|F)
✈✒❢✮✞✕
✜☎✘✢☎✣✤
✝☎✚ ✦
✟☎✚
P(A
✩
B) = P(A|F) + P(B|F)
P (E |F) = 1 – P(E|F)
❂
❧✚
✝✖✚ ✦
❆☎✟
✝✥
P (S|F) = 1
✎✙
P [(E
✿
✩
E )|F)] = 1
❂
❉❀☎✚ ✎
✦ ✙
P (E|F) + P (E |F) = 1
✿
❂
P (E |F) = 1
✈✟✠
✈☎✛✣
✝☎✚ ✦
❂
✈r
✙✴❇
❈✏☎✝✑❄☎
✺✚ ✦❊
❉❀☎✚✦✎✙
❾
P (E|F)
E
✟✆☎☎
E
❂ ✒✑✭✒✑
S=E
✈✒❢✮✞✕
✩
✜☎✘✢☎✣✤
E
❂
✝✥✦
✐
♠ ✝
✞ ✟✠
✡
✞
☛
❀
✕
✍
✟
❣
✘
✙
r
✒
✎
✚
♠ ✝
✞ ✟✠
✡
✞
✘
✖
✓
✏
✎
✩
✱☞
r
✍
✒
✌
✍
✘
✟
✍
❣
✙
✒
✍
✚
②
✔
✒
✙
✒
✍
✛
b
☞✕
✖
✚
✍ ✏
✢
✌
✍
✒
✪
✒
✛
② ✧
★
✪
✒
✗
✎
✣
✤✥
✒
✣
✤✥
✒
✛
✘
② ✧
✓
★
✍
g
✢
☞✌
❀
❑
✍ r
✍
✒
✒
✓
✔
✒
☞
✓
✍
✘
✫
❀
✍
✍
✚
✣
✤✥
☞
❀
✓r ✍
✓
✍
✒
r
✍
P (A|B)
✒
❑
✍
r
✓✔☞✕
✖✗
☞
✒
✙
✒
✛
✓
✦
✒
✙
✓
✦
✖
✓
✍
✙
✣
✤✥
✘
☞
✚
✮
✒
✛
✭
✎
✱✬
② ✧
✓
★
✔
4
,
13
B) =
✑
4
9
✘
✘
✓✍ ✒
4
13
9
13
✛
✍
✢
✢
❀
P (A
✈
✍ ✏
✎
P (A B)
P ( B)
P (A|B) =
☞
✓
✛
✜
② ✧
★
✓✍
7
9
, P (B) =
13
13
P (A) =
☞✌
551
✁
✂✄
☎✆✁
r
✓ ✏r
✒
✎
✱
✗
✏✔
✘
✛
✱
✍
✯
✍
✓✍
☞
r
✰
✌
✱✬
✍
✲
☞
✳
✦
✙
✎
✴✵
✘
✶
S = {(b,b), (g,b), (b,g), (g,g)}
✍
✙
② ✔
E
☞
✕
✖
✚
r ✷
✍ ✍
F
Ø
✍
✙✲
E:
✸✚ ☞
② ☞✹
✍ r
✌
✍ ✒
✍
✚
✒
✒
✛
✒
✣
E
✈
✣
✙
✶
♠ ✝
✞ ✟✠
✡
✞
✼
✖
✓
✫
✣
② ✍ ❀
✍
❂❀
✍
✗
✙
✏
✱
✹
❀
✍
✟
✏
✱
✍
✙
❆
✒
✣
② ✔
✹
❀
✍
❆
☞
✓
✒
✦
✍
✯
✍
✰
✌
✛
✓✍
✒
☞
r
✓✍
✌
☞
✚
✓✍
✖
✓
✶
✛
②
✧
✓
★
✍
✎
✘
✻
❫
✒
✦
✦
✙
✽ ✾
❀
✍
✌
❅
❀
✍
☞
✦✛
✖
✎
✘
✶
r
✓✏
✍ ✒
✧
✓ ✏r ✒
✱
✎
✘
✛
✹
✍
✒
❂
❂❀
✍
✗
❀
☞✌
❀
✘
✒
✗
✫
❀
✛
✍
✍
✍
☞
❀
✓r ✍
❃ ❄
✒
✈
✦✛
P(A|B)
✎
✈
✍ ✏
✎
✘
✻
❑
✍ r
✒
☞
✓
☞
✓✍
②
❅
✔
r ✏
✤
✒
✘
❂
✖
✓
✍
✚
✧
✳
✭
✎
✘
B
✺
✍
✓ ✏✚
✍
✍
✵
✗
✎
✘
S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
B = {4, 6, 8, 10}
✈
✍ ✏
✎
✘
✱✬
✹
❀
✍
✦
✙
✒
✘
✙
✓✔
✚
✏
✱
✖
❑
✍
r
✘
✦
✙
✓✍
✏
✘
✛
✓✍ ② ✍
✪
✒
✢
❂
☞
② ✹
✍
✰
✚
✳
☞
✍
✍ ✓
❁
✱✿
✤
❢
✓✍ ② ✒
☞
✴✵
r ✓
✚
☞
✳
✒
1
4
✦
✧
✚
✮
✍
✲
✑
✎
✘
1
3
A = {2, 4, 6, 8, 10},
A
✒
✳
✙
F)=
✹
❀
✍
❜
✍
✵
✎
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r ✍ ✒
✘
✻
✣
✓
✦
1
4
3
4
✳
✺
✍
✧
★
✔
✱✬
✍ ✍
r
✲
✎
✒
✑
✳
✖
✓
✩
✎
A
✣
✓✍
✌
F = {(b,b), (g,b), (b,g)}
✈
✍ ✏
✎
✧
✦
✘
☞✕
✖
✒
✘
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P (E
✈
✍ ✏
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☞
❇✓
r
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✏
✎
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P (E F)
P ( F)
✒
✙
✦
✏
✔
❜ ✦
✒
✦
✫
❜
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❣
3
4
P(E|F) =
☞
② ✖
❜ ✦
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✓✍
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F = {(b,b)}
✑
P(F) =
✈
r
✧
★
✚
✢
✒
✛
E = {(b,b)}
r
②
✣
✤✥
✣
✤✥
✍ ✈
✍ ✒
✵
✍
❫
✺
✍
✚
❫
F:
☞
☞
✶
B = {4, 5, 6, 7, 8, 9, 10}
✚
☞
❫
✓✍ ② ✒
✚
❂
✖
✓✍
✧
✳
552
① ✁✂✄
P(A) =
✈☎
♠✟✠✡☛☞✠
✌
%
XII
✔✕✖ ✩✕ ✗✘
✍②✏✏
✱✍
✓✬✭✙ ✍✮✏✯
✔✕✖
✎❡✭r ✏
✡❣ ✔✏★ ✓✢✙✧✱
✹✺★ ✏
✻✮✏
✎✏✑✒✏✏✓✏
✤✥
E
✙✚✛✏✜✏✢✣
✍②✏✏
✮✙✵
✔✕✖
✗✘✘✘
XII
✮✤
✲✏r
4
7
✙✚✛✏✜✏✢✣
✎❡✭r✢
✔✕✖
✤✥
✤✥✰
✖
✙✍
4
10
P (A B)
✆✝❙✞
4
10
7
10
P (A B)
P(A|B) =
P ( B)
r☎
✤✥✖✦
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✳✮✏
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♣✮✳r
✍❀r✕
✓✬✭✙ ✍✮✏✯
✻✮✏
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XII
P (E|F)
♠✟✠✡☛☞✠
❁
✎✙❀✐✏✏✙❆✏r
✙✍✮✏
A:
✱✍
✾r✢✩❀✢
✾✎✤✓✢
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✡❣
B
✍✏
✎✏✩✕
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❂✸✏✓
❂✸✏✓
❄✏✙✽r
✎✴✙r✵✒✏✣
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✔✕✖
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✩✖ ❫✮✏
✩✖ ❫✮✏
✤✏✕★ ✏
✔✕✖
✪
❈
✎✴ ✍✽
✈✏✥❀
F
✍❀★✏
❄✏✽★✏ ✾✮✏✵✶✷✸✮✏
✤✥✰
r✜✏✏
B
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✙✵✮✏
✻✮✏
✤✥ ✦
r✏✕
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A
✎❀
✍✢
✩✖❫✮✏
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✎✴✍✽
✲✏r
✤✏✕★ ✏✿
✍✢✙✧✱✰
B = {(6,5,1), (6,5,2), (6,5,3), (6,5,4), (6,5,5), (6,5,6)}
✈☎✦
A=
(1,1,4) (1,2,4) ... (1,6,4) (2,1,4) (2,2,4) ... (2,6,4)
(3,1,4) (3,2,4) ... (3,6,4) (4,1,4) (4,2,4) ...(4,6,4)
(5,1,4) (5,2,4) ... (5,6,4) (6,1,4) (6,2,4) ...(6,6,4)
✈✏✥❀
✈☎
r☎
✻✮✏ ✙✚✛✏✜✏✢✣
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A
●
B = {(6,5,4)}
P(B) =
P(A|B) =
6
216
✈✏✥❀
P(A
P (A B)
P (B)
●
1
216
6
216
B) =
1
6
1
216
✪✫✘
✤✥✼
✳✮✏✕✖
A
✤✥ ✙✍
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0.043 (
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430
43
0.43
P ( E F) =
1000
1000
P (E F) 0.043
0.1
P(E|F) =
P ( F)
0.43
r☎
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✖
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P(F) =
✈☎
B:
5
7
, P ( B) =
10
10
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✩
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✈
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S = {(H,H), (H,T), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6)}
❏❑
▲
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5
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P (E F)
P (F)
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5
11
P(F) =
36
36
F = {(2,4), (4,2)}
P(E|F) =
✫
✍
✺
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P(E) =
✳
✏
✯
✪
✍
✏
✙
✍ ✓
✚
E = {(4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (1,4), (2,4), (3,4), (5,4), (6,4)}
F = {(1,5), (2,4), (3,3), (4,2), (5,1)}
✯
✙
✍
✗
✏
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553
✁
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❄✍
✜
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554
① ✁✂✄
(H,H)
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t☎✆✝
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✪✯
✪✰
✪✱✩ ✰ ☎✲✳✆
✳✆✧✯
✪✆✴✵
☎✲✐✥✺
(H,H), (H,T), (T,1), (T,2),
1 1 1 1
, , , ,
(T,3) (T,4), (T,5), (T,6)
4 4 12 12
1 1 1 1
, , ,
12 12 12 12
✳✦✈
✼✆✧ ★✫✩
✻
✽✆✰✷✆✳✆✵ ❞
✩✬
✾✼ ✤✆✈
✪✱ ✆★✹✩✦✆ ★✷✿✆✥ ★✯✦ ✩✬ t✆ ✴✩✦✬ ☎✧❀ t✧✴ ✆ ★✩
★❢❁✆
❂❃❄❅
✴✵
F
✼✆✷ ✫✵❞
❈✪✆✴✵
✪✯
❆✪❇✰
❫
☎✧✺
❱❲❳❨❩❬❭
✴✵
❍■❏ ✬
✴❞✸✹✆
✪✱✩ ✰
☎✆✵ ✷✆●
✩✆✵
✣✤✆✆✥ ✦✵
E = {(T,5), T,6)}
☎✧✺
❞
✳✆✧✯
E
❑
F = {(T,5), (T,6)}
P(F) = P({(H,T)}) + P ({(T,1)}) + P ({(T,2)}) + P ({(T,3)}) +
✳❍
P (E
✳✆✧✯
❑
P ({(T,4)}) + P({(T,5)}) + P({(T,6)})
1 1 1 1 1 1 1 3
=
4 12 12 12 12 12 12 4
1 1 1
F) = P ({(T,5)}) + P ({(T,6)}) =
12 12 6
1
6
3
4
P (E F)
P(E|F) =
P (F)
✳✦✈
2
9
▲▼◆❖P◗❘❙
E
✹ ★✣
P (E
✹★✣
F
F) = 0.2
✳✆ ✧ ✯
❑
2. P(A|B)
3.
✽✆✰✷✆
F = {(H,T), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6)}
✦❍
1.
E
✽✆✰✷✆ ❈❉✹❊✷✦✼ ❋✩ ✪✰ ✪✱✩✰ ☎✆✵✷✆● ✳✆✧✯
13.2
✐✴
❀
❚✆✦
✦✆✵
✩✬★t❋❀
✪✱ ✩✆ ✯
✩✬
P (E|F)
✹★✣
4. P(A
❯
B)
❑
B)
❚✆✦
✩✬★t❋
✳✆✧✯
✳✆✧✯
☎✧❞
P (F|E)
P(B) = 0.5
P(A) = 0.8, P(B) = 0.5
(i) P(A
✽ ✆ ✰✷✆ ❋✝
13.1
✳✆✧✯
P (E) = 0.6, P (F) = 0.3
★✩
❚✆✦
✩✬★t❋✺
P (A
❑
P (B|A) = 0.4
(ii) P(A|B)
✹★✣
❅
P(A) = P(B) =
B) = 0.32
❚✆✦
✩✬★t❋
(iii) P(A
5
13
✳✆✧✯
❯
P A|B =
B)
2
5
✳ ✆ ✧✯
✐
5.
5
6
, P(B) =
P(A
B)
11
11
(i) P(A B)
(ii) P(A|B)
P(A) =
❀
✝✞
✈
✟ ✡
✠
7
11
☛
r ✟
☞
✌
❢ ✓❢
✝
6.
✔
✕
✝
✖✗
✘
❢
✙
✝
❢
✱
✜
☞
✢
✚
✝
✝
✭
✍
r ✬
✟
☞
✑
✍
✝
✣
✟ ✡
✖✗
✟
☞
✞ ✟
☞
✣
✟ ✡
✖✡
✤ ✥ ✟
✡
✍
✚✪
✪
✪
✖
✵
✚
✟
✪
✟
☞
✖
✟
☞
✖✗
✡
✚
11.
✪
✽
✟
✟
r
✧
✟
☞
✟
❯
❀
✟
✿
❀
✔
✜
☞
✢
✝
✚✪
✝
✔
✝
(iii) P (E
❢
✬
✝✍
✪
✬☞
✦
13.
❀
✟
✣
✧
✠
✖✗
✑
✍
✝
❢
✍✝❉
✈
✹
✏
❯
✬
r
✑
✍
✝
r
✖
✮
r ✬
✧
r
(i)
✪
✕
✞ ✟
☞
✖✗
✘
❢
✶
F:
✠
✚
✑
✍
✩
✍
✟
☞
✟
☞
❞
✯
✝
✝
✖✗
r
✖✡
✱
✩
✜
☞
✢
✮
✝
❢
✍
✖
✗
r
✮
✍
✧
✟
☞
r ✟
✧
✠
✪
✧
✎
✧
✟
r
☞
✟
✧
✠
✳
✴
✜
✟
❀
✟
✞ ✹
✸
✥
✖
✝
✟
☞
❀
✟
✟
✼
r
✟
✬
✪
❀
✪
✟ ✺
☞
✻
✧
★
✠
✕
✬☞
✪
✟ ✺
☞
✻
✧
✠
✔
✍
✟
☞
✤
✥
✟
✟
✟
✦
✴
✧
✖✗
✟
☞
✟
✖✡
✈
✷
✟
✧
✚✪
❁
✟
☞
✖✗
✍
✎
✕
✟
✝
✪
✝r ✣
❀
✍r
✟
✌
✟
r
✍✎✝
✏
✑
❀
✝
✞
❀
✧
❀
✍r
✟
✌
✟
r
✍✎✝
✏
✑
❀
✝
✞
❀
✧
✖✗
✭
✚
❢
✟
✖✗
✭
✠
✚
☞
❀
✮
✠
✧
❢
✍
✟
✪
✝r ✣
❑
✧
✧
✮
✖✗
✍✎
✮
❀
✟
☞
✦
✖✗
✧
★
✠
✚
☞
✾
✍
❀
✟
❢
❀
✟
✦
☞
✍
❀
❢
✧
✕
✵
F:
✖✡
✦
✮
✍
✖✷
r ✟
❑
✟
✖✗
✟ ✟
✪
✟
✟
✛
✖✟
✚
✌
✟
☞
✜✟
✟
✰
✟
✝
❑
☞
✍✬
✧
✠
E = {1,3,5}, F = {2,3},
✪
✟ ✈
✟
☞
✈
✟ ✡
✠
G = {2,3,4,5}
✏
✑
★
✪
✟
☞
P (E
✣
✹
✩
✣
(ii) P (E|G)
✹
☞
✥ ✟
☞
✔
❄
❅ ❅
✚
❆
✺
✻
✍
✎
❇
✖
✜
✍❋
✧
✟
☞
☞
✔
✟
❀
❊
✷
✝
❀
✟
✎❀
❢
✺
✍
✻
✎
❢
✺
✍
✻
✎
✹
✜
☞
✢
✣
✧
✟
✔
✣
✴
❅ ❅
✔
✺
✍
✻
✜
☞
✢
✩
✟
P (G|E)
✈
✟ ✡
✠
F|G)
✍✟
✪
✟
☞
✮
☞
✒
✩
☞
✚
✣
✍✎✝
✔
☞
✞
✟
☞
✚
✟
r
✈
✟ ✡
✠
❢
✖✟
✚
✟
✍
✝
☞
❢
✶
✠
✧
F
✠
P (F|E)
✟ ✡
✠
☞
❃
✟
✩
✝
❢
r
☞
✤ ✥
✪
☞
✝✍
✘
✖✡
★
✠
✧
✙
✍
✖✟
✚
✔
F|G)
☛
✩
✞
✟
☞
✪
✟
☞
★
✠
✪
❀
✟ ✈
✟
☞
✟
r
✔
☞
✟
✕
✝
✔ ✪
✟
✟
☞
✧
✠
✟ ✈
✟
☞
✟
✍
(i) P (E|F)
12.
✔
✤ ✥
❢
❯
❀
✫
✮
✔
☞
❢ ✓❢
✑
✪
✟
☞
✩
✖✟
✚
☞
r
✝✍
✟
✟
★
✟
✶
✧
✕
✖✟
✚
✑
✍
✞ ✟
☞
✈
✝✭
✍r ✬
✟
☞
r ✟
✧
✍
☞
❀
✔
✌
✟
r
✔
r
✍
❀
✘
✟
✽
✝✍
✟
☞
r ✟
✕
✔
✧
✧
✖
✬
✟ ✺
✻
✟
✟
✟
✧
✖✗
✰
✲
✬☞
✑
✍
✖✗
✦
✪
✟
✖✡
✡
✟
r
✖
❢
✎
✮
✧
✟
✬
✔
✟
☞
❀
✖✗
✕
✖✡
✔
(b)
✔
✧
✪
✤ ✥ ✟
✡
☞
✈
✟ ✠
✡
✦
✧
✎
✟
☞
✝
✝
✟
❢
❀
✟
✚
☞
✟
✖
✮
✮
✤ ✥ ✟
✑
✍
✌
★
✠
✔
r ✎
✖✷
✚
✏
✑
✖
F:
✍
✔
✟
r
✍
✩
✖
✤
✥
❢
✍
✍✟
F:
✩
✜
☞
✢
✖
✮
✡✜
✟
✝
✟
✧
F:
✖✡
✱
✖✟
✵
❀
✟
✞ ✟
☞
✍✟
☞
❞ ✯
✎
✝
✦
r
✚
✑
✍
✔
✑
✍
(a)
✖✡
✟
✝
✞ ✟
☞
✔
✧
✍
✎✝
✔
✍
✡
✎
✖
✟
r
✩
✚
r
✎
❑
✌
✖
✮
✟
☞
☞
✑
✍
✤ ✥ ✟
✤ ✥ ✟
r ✬
✖✟
✚
E:
✟ ✡
✪
✱
✍
✑
✍
F:
10.
❀
✫
✈
✝
(ii) E :
E:
✣
✔
✡
✎
✏
✑
✔
r ✎
❢
❯
(i) E :
9.
✟
☞
✚
(iii) E :
✞ ✟
☞
✍
r
✎
(ii) E :
8.
r
✍
✚
✑
✍
P(E|F)
✛
☞
(i) E :
7.
✚
✟
r
✍✎
✝
(iii) P(B|A)
✒
✝
✟
r
555
✁
✂✄
☎✆✁
✚
❆
✈
❀
✧
✜
☞
✢
✪
✟ ✡
✜
☞
✈
✟
✢
❢
✟
✚✪
✬
❡
❂
✟
✟
✟
✝
❀
✍r ✟
✩
r ✬
✈
✟
❑
❀
✧
✠
✚
❀
✝
✞
✝✍
✖
✎
✝
✡
✜✟ ✡
✖✗
✭
❢
❯
❀
✫
✚
✍
✚
✟
✝r
✣
(ii)
✚
✟ ✡
✟
☞
✖✗
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✖✗
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✠
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✘
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✧
✶
✠
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❇
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✝
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✠
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✝
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✜
☞
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✖✗
❀
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✟ ✡
✜
☞
✢
556
① ✁✂✄
❞☎✆✝
✞✟✠ ✝✡☛☞
❞✡
✞✟✠ ✝
❞✤
✈✡✌✡✝
14.
✒✎
15.
☎✓✒✡ ✍✒✡ ✎✏
✒✡☛✍
✯
✖❞
✞✡✌☛
✎✏
✢✡☛
✎✡☛ ✝☛
❞✤
❞✡☛
✞✡✌☛
✌☞✍✎
✟
✎✏ ✑ ✒☎✓
✥✎✛✦☎✔❞✧✞✤✒
✔☛✕
✎✡☛✝☛
✌☞ ✍✎
✟
❞✤
✞✟✡☎✒❞✢✡
✔☛✕
✞✛ ✝❣
✰✡✢
✞✟✡✪✢
✞✟ ✠✝ ✒✡✓✗ ✘✙✒✡
★✒✡
✚✛ ✝✡ ✜✡✢✡
✎✏ ✣
✢✡☛
✖❞
✎✡☛✍✤✩
✌☞✫✒✡✖✬ ☎✭✡✮✝✦☎✭✡✮✝ ✎✏ ✑
☞
✓✡☛✝✡☛ ☞ ✌☞✫✒✡✈✡☛☞ ❞✡
❞✤☎✜✖✑
✞❀✤✱✡✲✡
✞☛☞ ✕✔☛ ✕
☞
✌☛ ✖❞
✞✟ ✡☎✒❞✢✡
☎❞ ✓✡☛ ✞✡✌✡☛ ☞ ❞✡☛ ✞☛✕
☞ ❞✝☛ ✞❀
✞☛☞✕❞✝☛
❞✡☛
✞✟✠ ✝✡☛☞
✈✡✏❀
✞❀
✒☎✓
☎✔✚✡❀
❞✡☛✵✶
❞✤☎✜✖✑ ✒☎✓
✈✮✒
✌☞✫✒✡
✞✡✌☛
✞✟❞✳
✞❀
✎✡☛
✞✟❞✳
✢✡☛
✖❞
✌☞✫✒✡
✴
☎✌★✔☛✕
❞✡
✍✛ ✲ ✡✜
✷✙✡✸☛✑
☞
❞✡☛
❄✡✳✝✡ ✹✮✒✺ ✝✢✻ ✖❞ ✞✡✌☛ ✞❀ ✌☞✫✒✡ ✴ ✞✟❞✳ ✎✡☛✝✡✼ ☎✓✒✡ ✍✒✡ ✎✏ ✢✡☛ ❄✡✳✝✡ ✹☎✌★✔☛✕ ✞❀ ✞✳ ✞✟❞✳
✎✡☛ ✝☛✼
☎✝❢✝☎✸☎✫✡✢
16.
✒☎✓
✌✞✟☎✢✥☞✽
❞✤
✞✟✠✝✡☛ ☞
✻☛ ☞
✞✟✾✒☛❞
✌☛
✌✎✤
B
✈✡✏ ❀
(A) A
✓✡☛
✷✿✡❀
✚✛ ✝✑
☞☛
P (A|B)
✢✥
✎✏❣
(C)
✵✌
❄✡✳✝✡✖✬
B
❂
❞✤☎✜✖✑
1
2
(B)
A
✒☎✓
✻☛☞
1
, P(B) = 0
2
P (A) =
(A) 0
17.
✰✡✢
✞✟✡☎✒❞✢✡
✞✟❞✡❀
✎✏ ☞
(D) 1
✝✎✤☞
P (A|B) = P (B|A)
☎❞
(B) A = B
✞☎❀✭✡✡☎✐✡✢
(C) A
❃
B=
0
❁
✢✥
❅
(D) P (A) = P(B)
13.3
✻✡✝
E
F
❃
✔☛ ✕
❆❇❈❉❊❋●❈
✸✤☎✜✖
F
❋❈
E
☎❞
❄✡☎✳✢
E
✎✡☛✝☛
✔☛✕
✥✡✓
✓✺ ✌❀✡
✞✿✡✡
✖❞
✢▼✡✡
❞✡☛
✞✟✡✒❣ ✎✻☛☞ ✌✒✛ ☞ ★✢ ❄✡✳✝✡
✖❞
❉❑❊▲
F
F
✢▼✡✡
✓✡☛ ✝✡☛☞ ❄✡✳✝✡✈✡☛☞
✒✛ ✍✞✢❖
❍■❏❈❑
(Multiplication Theorem on Probability)
✞✟☎✢✓✠✡✶
✎✻
P(E|F)
❞✡
✷✞✒✡☛ ✍
✓✠✡✡✶✢ ✡
EF
✜✡✝✢☛
⑥✡❀✡
✎✏☞
✓✠✡✡✶✢☛
☎❞
✎✏ ☞
✎✏ ✑
✔☛✕
✞☎❀✲✡✡✻
P(E
✜✏ ✌✡
❄✡✳✝✡
✈✡✏ ❀
P(E|F) =
✷✞❀✡☛★✢
✎✏ ☞
✵✌☛
✓✡☛
❄✡✳✝✡✖✬
❄✡✳✝✡
E
❃
F
EF
❞✡☛
✭✡✤
☎✸✫✡✡
✎✏ ☞ ✑
E
◆✞✐✳✢✒✡
❃
✜✡✢✡
F
✌✻✛ ✘✚✒
❄✡✳✝✡✈✡☛☞
E
✢▼✡✡
✎✏✑
❞✤ ✞✟✡☎✒❞✢✡ ✰✡✢ ❞❀✝☛ ❞✤ ✈✡✔✠✒❞✢✡ ✎✡☛✢✤ ✎✏ ✑ ✷✓✡✎❀✲✡ ✔☛ ✕ ☎✸✖✣
☎✝❞✡✸✝☛
❞❀✢☛
❞✤
✔☛✕ ❄✡☎✳✢ ✎✡☛✝☛ ❞✡☛ ✓✠✡✡✶ ✢✡ ✎✏ ✑ ✈✮✒ ✠✡♦✓✡☛ ☞ ✻☛☞
✞❀✤✱✡✲✡
✻☛☞
✞✟ ✡☎✒❞✢✡ ✰✡✢ ❞❀✝☛ ✻☛☞ ✵✘✙✛❞ ✎✡☛ ✌❞✢☛ ✎✏ ✑
☞
❄✡✳✝✡
✞✟ ✡☎✒❞✢✡
S
✌✻☎✐✳
☎❞
✝✤✚☛
✔☛ ✕
☎✓✖
F
☎✝❢✝☎✸☎✫✡✢
✎✻
EF
☎✻P
✞✟❞✡❀
✹✖❞
✥✡✓✠✡✡✎
✈✡✏ ❀
✖❞
❀✡✝✤✼
❞✤
❞✤ ✞✟✡☎✒❞✢✡ ✰✡✢ ❞❀✝☛ ✔☛✕ ☎✸✖ ✎✻ ✌✞✟☎✢✥☞✽
☎✓✫✡✡✒✡
✜✡✝☛
❄✡✳✝✡
✍✒✡
✞❀
✌☛
✎✏ ✑
❄✡✳✝✡
✰✡✢
❞❀✢☛
E
❞✤
✌✞✟ ☎✢✥☞✽
✞✟ ✡ ☎✒❞✢✡
❞✡☛
✎✏ ✑
☞
P (E F)
, P (F) 0
P (F)
✌☛
✎✻
☎✸✫✡
✌❞✢☛
✎✏ ☞
☎❞
❃
F) = P (F) . P (E|F)
... (1)
✐
❣
✝
✞
❣
✟
✠ ✡
☛
✠ ☞ ✌ ✍
❣
✎
✏
✑✒
P (F E)
, P (E) 0
P (E)
P(F|E) =
P (E F)
P (E)
F) = P(E) . P(F|E)
P(F|E) =
✞
✠
✈
P(E
✕
✌
✓
557
✁
✂✄
☎✆✁
(
E
❉
✞
✠ ✑
✏
✍
✒
F=F
✓
E)
✔
... (2)
✈
✭
✖
✗
✠ ✎
✘
✭
✙
✗
✒
✠
P(E
♠
✍
✑
✝
✍
✍
✛✍
❣
✝✏
✍
✜✢
✠ ✣
✌
✦
✌
✜✑ ✘✥✠
✠ ✝
❣
✠ ✌
✍
✠
❣
✎
✑✒
F) = P(E) P(F|E) = P(F). P(E|F)
✓
❉
✜✘✠
✚
✠ ☞
✒✠
✍
✧★
✥
✜✢
✠ ✑✞
✒✌ ✠
✒✠
☛
t
✩
✠
☞
✑☞
✞
✪
✝
✒
❣
✌
✍
✈
❣
✏
✎
P(E)
✑✒
✫ ✬
✬
✠
0
✤
✠ ✘
✎
♠ ✮
✒
P(F)
✈
✤
0
✪
✠ ❣
✘✥✠
✚
✏
✍
✹
✯ ✰
✱ ✲✳
✴
✱
✵
✬
✶
✒
✈
❣
✎
✏
✧✍
✮
✠ ✎
✘
✜
❣
✚ ✡
❧
✛
✲
✍
✜✢
✷
✠
✝
♥
✺
✛
✘
✍
✏
✍
✈
✖
✒✠
✚
✡
✒
✍
✒✠
E
❀
✑✒
❁
✠
✑
☞ ✒✠
✚
☞
✛
✝
✍
✛✍
❣
✚ ✡
✒✠
✚
✡
❣
✝✏
P(E F)
✍
✞
❂
✮
✠
❣
✡
✑
✧✍
✮
✏
✍
✮
❣
✏
✎
✧✍
✮
✠
✍
✬
✏
♥s
✒
✽
✛
☞
✍
♥s
✑☞
✒
✚
☞
✍
✒✠
❀
✍
❣
✚ ✡
✘
✻
✠
✡
☛
✠ ✌ ✡
❣
✎
✑✒
✜
❣
✒
✒
✡
✚
✍
✝✠ ☞
❣
✎
✚
☞
F
✈
☞ ✠
✚ ✡
✠ ✘
✎
✍
✒
✡
✑✒
✒
✜✢
✠ ✑✞
✒✌ ✠
✧✍
✮
✛
✘
✡
✚
✠
✝✏
✍
✾
✞
✠
✦
✮
✺
☛
✠ ✌ ✡
✶
✚ ✡
✑
☛
❉
✑☞
✒
✏
❁
✠
✑☞ ✒
✠
✒✠
✚ ✡
❣
✎
♥s
✏
✩
✍
✑
☞ ✒
✚
☞ ✍
✪
❑
✠
✌
✑☞ ✒✠ ✚
✚ ✡
✬
✠
✬
❣
✏
✎
✒ ✘☞ ✠
❣
✎
✝✏
✍
✒✠ ✚ ✡
✏
✑☞ ✒
✠
✚
☞ ✠
✑☞ ✒✠
✚
✝
✏
✍
✒✠ ✚ ✡
10
15
)=
✧✍
✮
✞
✠
t
✮
✧✍
✠ ☞
✍
✠ ✏
✩
✍
❣
✡
✏
✮
✌
✠
✮
✍
✪
✠ ✜
✧
✞
✠
✪
✏
✧✍
✮
✜
t
❣
✚ ✍
❣
✎
P (EF)
✠
P(E) = P (
✈
✛
✠
✜
✞
✏
✓
❣
✎
✛
✏
✟
✠
✠
✧✍
✮
✪
☞ ✠
✚
☞ ✠
✦
✜
s
✮
✛
✜✍
♥
✑☞ ✒✠
✼
✏
✝✠ ☞
✠
✸
✠
✎
✘
s
✍
✧✍
✮
✞
✡
✚
✮
✏
✿
✒
✒
✈
❂
✏
✑☞ ✒
✚
✡
❣
✎
❃
❄
❀
✠ ✠ ✌
E
❁
✠
☞
✠
❀
❁
✠ ✑
✫
★
❃
✌
✽
❣
❣
✎
✹
✈
✶
t
✒✚
✧✍
✮
✠
✝✏
✍
❅
✒✠ ✚ ✡
✈
✏
✸
s
✮
✠ ✘
✎
✧✍
✮
✛
✜✍
✧
✫ ❃
✏
✘
✧✍
✮
☛
t
✑
✒
E
☛
t
✈
❂
❃
✜
❣
✚ ✡
❀
✒✠
✒✠
✒✠
✚
✠
❣
✠ ☞
✍
✠
❁
✠ ✑
❣
✠ ☞
✍
✠
❑
✠ ✌
P(F|E) =
❄
✈
❣
✝✏
✍
❑
✠ ✌
✫
❣
✏
✎
♥s
★
❆
✬✽
✺
✈
❣
✎
✮
✛
✑✚
♥
✠ ✘
✎
☞ ❣
✡
✏
✧✍
✮
✛
✘
✡
✏
F
s
♥
✍
✒✠
✚
✚ ✡
❣
✠ ☞
✍
✍
✒
✡
✜✢
✠ ✑✞
✒✌ ✠
❇
✒✠
✛
✜✢
✑
✌ t
✏
✜✢
✠ ✑✞
✒✌ ✠
❣
✎
✪
✌
✠
✠
✌
♥s
t
✽
✏
✪
❣
✜✢
✠ ✑✞
✒✌ ✠
❣
✎
9
14
✧★
✥✠
✍
☞
✑☞
✞
✝
❈ ✠
✘✠
❣
✝
✏
✍
✜✢
✠ ✣
✌
❣
✠ ✌
✍
✠
❣
✎
P(E F) = P(E) P(F|E) = P(E) . P(F|E) . P(G|EF)
✓
=
❊ ❋
●
❍
●
■❏
▲▼
◆
❋
❖ P
❋
■
❋
● ◗
10 9 3
15 14 7
❘
● ❙
❏
❚
❯
❱❲
❋
❏
❳
▼
❨
❋
▼
❋
❩❬
❭
❋
P
❏
P
❳
❪
✮
✞
✑
❀
✒✡
❁
✠
E, F
✈
✠ ✘
✎
G
✬
✮
✒
✜✢
✑✌
✶
❃
✠
❫❁
✛
✝✑
✬
☞
✠
❞
❣
✎
✏
✌ ✠ ✍
P(E F G) = P(E) P(F|E) P(G|E F) = P(E) P(F|E) P(G|EF)
✓
✓
✫
✓
♥s
✛✡
✜
✢
✒
✠ ✘
✜✢
✠
✑
✞
✪
❣
✎
✒
✌
✠
✧
✥✠
★
✍
☞
♠
✮
✑
☞
❵
☞ ✑
✚
✑
✻
✠ ✌
♥
❜
✑☞
✞
✝
❀
✠
❣
✘✥✠
✌
✡
☞
✠
❁
☞ ✠
✒✠
✈
✠ ✏
✑
♥
✍
s
✍
❴
✌
✠
✘
✠ ✘
✈
✞
✠
✬
✑✚
❇
✑
❀
✒
♥s
✜✢
✠ ✑✞
✒✌ ✠
✍
❁
✠
✈
☞ ✠
♥
s
✠ ✏
✍
✍
✬
✑
✚
✧★
✥✠
✟
✠
✮
☞
✑☞
✞
✝
✒✠
✡
❫❁
❛
✠ ✌
✏
✑
✒✞
✠
❜
✜✢
☛
✠
✛
✒✌
✠
★
✌ ✌
✪
✒ ✘✌
✠
❣
✎
558
① ✁✂✄
♠☎✆✝✞✟✆ ✠ ✺✡ ☛☞✌✌✍✎ ✏✑ ✒✓✔✑ ✕✖✗ ☛✍✘
✎ ✙✑ ✚✛✜ ✚✢✣✢✑ ✤✍✎ ✥✍ ✦✏ ✧✍✘ ★✌✩ ✦✏ ✕✑✪ ☛☞✌✍ ✫★✪✌ ☛✬✫✕✭✮✌✌✫☛✕
✫✏✦
✫✪✏✌❢✍
✝❣
✤✌✪ ❢✍ ✎ ✫✏
✗✱✶
✏✌✍
✚✦✯
✷✳✲✕
☛✗❢✍
K
✩✌✍✍
☛☞✌✌✍ ✎ ✏✌
★✌✩✰✌✌✗
✒✌✱✖
✗✱✯
✎
✏✖✕✍
✭☛✸✙✕✳✌
P (KKA)
✗✤✍✎
P(K|K)
✳✗
❑✌✕
★✌✩✰✌✌✗ ✗✌✍✪✍ ✏✑ ☛✬✌✫✳✏✕✌ ✏✌✍
✗✌✍✪✍
☛✖
✫✏
✵☛✗❢✍
P(A|KK)
✗✱
✩✌✍
✫✏
✕✑✥✖✍ ✫✪✏✌❢✍
★✌✩✰✌✌✗
☛✗❢✍
✗✑
☛✬ ✌✫✳✏✕✌
✚✻✽✌✪
✧✍✘
A
✾✌✖✌
✫✪✳✤
✗✤✍✎
☛✬✌✿✕
✗✱✴
☛✬✌✫✳✏✕✌
❄✌✙✪✌ ✵✫✪✏✌❢✌ ✚✳✌ ☛☞✌✌ ✛✲✏✌
✫✪✏✌❢✌
✚✳✌
(52
❾
☛☞✌✌
★✌✩✰✌✌✗
1) = 51
☛☞✌✍
✚✦ ☛☞✌✍ ✏✌ ✛✲✏✌ ✗✌✍ ✪✍ ✏✑ ✥☛✬✫✕★✎r
✫✪✏✌❢✍
✲✳✌
✗✱✯
✏✖✪✌
✹✻✏✍
✐✌
✗✱ ✎✯
✒★
✚✢✣ ✢✑
✗✱✶
☛✖
✩❀✥✖✍
☛☞✌✍
✏✌
✗✱✎ ✫✐✪✤✍ ✎ ✕✑✪ ★✌✩✰✌✌✗ ✗✱
✗✌✍✕✌
✗✱
✤✍ ✎
✺✼
☛✬✌✫✳✏✕✌ ✗✱
☛☞✌✍
✖✗
✚✦
✐★ ✫✏
✗✤✍ ✎
✗✱✎
4
50
P(A|KK) = P (A|K K)
✛✥✫❢✦
✗✌✍ ✪✍ ✏✑
3
51
P(K|K) =
✒✎✕✕✈
❑✌✕
✩✰✌✌✜✕✌ ✗✱✯ ✒★ ✚✢✣✢✑ ✤✍✎
✛✥✫❢✦
❑✌✕
✛✲✏✌
4
52
P(K) =
✗✑
✏✌
❄✌✙✪✌ ✵✫✪✏✌❢✌ ✚✳✌ ☛☞✌✌ ★✌✩✰✌✌✗ ✗✱ ✶ ✏✌✍ ✒✌✱ ✖
✒★
✥✌✮✌
✕✑✥✖✍
✫✏
P(KKA) = P(K) P(K|K) P(A|KK)
4 3 4
2
52 51 50 5525
=
13.4
✺✡
▲❁❂❃ ❅❆
☛☞✌✌✍ ✎
❄✌✙✪✌
✗✱✶
✏✑
✏✌✍
✒✌✱ ✖
❇❆❈❉❆❊❋
✚✢✣ ✢✑
✤✍ ✎
✥✤✥✎■ ✌✌✷✳
✵✫✪✏✌❢✌
(Independent Events)
✥✍
✦✏
✤✌✪✌
✚✳✌
☛☞✌✌
✗✱✯
✚✳✌
☛☞✌✌
✦✏
✫✪✏✌❢✪✍
✳✫✩
✛✲✏✌
E
✗✱✶
✧✍ ✘
✗✑
✵
E
✒✌✱ ✖
F
✶
❄✌✙✪✌
✵✫✪✏✌❢✌
P(E
✏✌✍
◗
F) =
❏✤✰✌✈
✷✳✲✕
13 1
52 4
☛☞✌✌
☛✖
✫✹✢▼✑
✏✖✕✍
❖P❦❦
✏✌
✲✳✌✍ ✎✫✏
P(E) =
E
☛✬✌✫✳✏✕✌
1
= P (E|F),
4
☛✖
✏✌✍✛✜
✗✤
☛✬■ ✌✌✧
✏✗
✪✗✑✎
✗✱ ◆
✎
✏✑✫✐✦
☛✬❍ ✳✍ ✏
✫✐✥✤✍ ✎
✵✫✪✏✌❢✌
✚✳✌
☛☞✌✌
✤✌✱✫❢✏
✫✹✢▼ ✑
✏✌
✕✌✍
4 1
52 13
P(F)
✛✲✏✌
✗✱✶
✏✌✍
✷✳✲✕
✏✖✕✑
✗✱◆
✛✥✫❢✦
1
52
✥✏✕✍
✢✌❢✌
✫✧✹✌✖
❄✌✙✪✌✒✌✍✎
P (E F)
P(E|F) =
P (F)
✒✕✈
✏✑
✚✳✌
F
✕✮✌✌
P(E) =
✥✌✮✌
☛✖✑●✌✽✌
✗✱✯
✗✱✎
✫✏
❄✌✙✪✌
1
52
1
13
F
1
4
✧✍✘
❄✌✫✙✕
✗✌✍ ✪✍
✏✑
✥❀✹ ✪✌
✪✍
❄✌✙✪✌
✐
❣
✝
✞
✟
✠
❣
✡
☛
☞
✌✍
☛
✎ ✏
❣
✑
✒✓
1
= P(F|E)
13
P(F) =
✔
✕ ✖
✘
✌
✍
☛
✒
✠
✓✏
☛
E
✖
✈
✏
✌✤
✏
r
✥
✓
☛ ✞
✦
F
☛ ☛
✌✍
✡
❣
☞
✌✍
☛
✒
✠
✓✏ ☛
✓
☞
✙
✌✍
✓☛ ✤
✓
☞
✺✻ ✻
✼
✻ ✻
✌✍
✡
☛
✈
✙
☛
✞
✽
✒
✓
E
✚✕
☛
☛
❣
✜
☛
❜✪
❣
✑
✒
✓
✒
✓
✖
☛ ✞
✟
☛
✈
❣
☞
✟
✧ ☛ ★ ✏
☞
✫
▲ ✬ ✭
✮
✯
E
☛ ✞
✟
✏
r
✤
✡
☛ ☛
☛ ☛
✝
✥✛
✈
❀
✌✍
☛
✒
P(E)
❂
✌
✒
✠
✓✏
☛
✰
✱
F
☛
☛
❣
✰
✲ ✳
♥
E
✈
✞
✟
✔
✞
F
☛
✑
✤
✕
❄❅
✈
✕
☛
✒
P(F)
☛
✑
✤
✖
❇
✏
✥
✾
✙
✏
✿
✟
❉
☛ ✞
✜✣
✢
✕
✞
✕
✓
☞
☛
✙
✞
F
✚✕
☛
☛
✓
☞
❜
✓
✥✛
✚✕
☛
☛
✙
✚
☛
✒
✞
✕
✏
❣
☛ ✞
✜✢
✞
✣
✕
✓
☞
♥ ✜
✢
☛
✙
✤
☞
✚✕
☛
☛
✰
✴
✵
✶
✓
❣
✾
✙
✏
✿
✟
☛
✏
✞
✚
❣
✑
☛
✕
☛
♥
☛ ❜✪
t
❀
❣
☛ ✟
✞
❋ ✌✠
☛
❄
✞
❈ ❉
❃
✠
✞
❣
☛ ✞
✟
✠
✒
0
❁
P(F)
✓
☞
✕
☛ ✤
✑
P(E)
✓
☞
t
❀
❣
0
❁
✥
✗
☛
✈
☛
✠
✓
❣
✩
✑
✞
❣
✝
✙
☛ ✞
❣
✝
✟
✞
✌✍
☛
✎ ✏
❣
☛ ✏
✞
☛
❣
✑
✒
✓
F) = P(E). P(F)
❆
♥
✞
... (1)
✜
✏ ☛ ✞
✜
✞
❞
✏
✞
F) = P(E) . P(F|E)
❆
❇
☛ ❜
✪
✥✛
❊
✢
☛
✓
❣
✞
✚✕
☛
✗
✓☛
✩
✟
✜
✠
✝
P(E
✈
❣
✩
✑
✥
✓☛ ✞
P(E
✠
✒
✕
✏
✕
☛ ☛
✓☛
✞
✚✕
☛
✜
✦
✚
☛
✒
✙
☞
P(E| F) = P(E)
✏
✙
✞
✩
✑
P(F| E) = P(F)
✈
✥✛
✚✕
☛
♥
✷✸
✹
✑
✚✕
☛
✥
✓☛
✦
✞
✙
❣
✧ ☛ ★ ☛
✙
✌
✍
✓
☛ ✤
✌✤
✜
✦
✟
✜
✦
✘
☛ ☛ ✏
☛
P(F)
✕
☛ ☛
✘
✓
☞
♥ ✗
1
52 1
1 13
4
P (E F)
P(E)
P(F|E) =
✌
559
✁
✂✄
☎✆✁
✚✕
☛
... (2)
✥
☛
✈
☛ ✟
✞
✓
☞
✾
✕
✏
✿
✟
☛ ✏
☛
✓☛ ✞
✒
✜
●✕
✒
★
✒❍ ☛ ✏
✏
✤
❣
❂
✞
✡
☛
☞
✌
✒
✤
✡
☛
☛
✒
☛ ✏
✜
✓✤
✓✏
✞
❣
✑
✩
✟
E
✕
✷✸
✹
E
✺✻ ✻
✼
✈
☛
✑
✤
✻ ✻
F
■
✝
☛
★
✞
✟
✥
✾
✙
✏
✿
✟
☛
✈
☛
✑
✤
F
✜
✒
✓
✚✕
☛
P
◗
❘
✲
✠
☛
❢
❏❑
✒
▼
✓
✌✤
☞
✥✛
☛
❅
☛
♥
✗
✞
✌✍
✒
✏
✘
✜
☛
❂
✚
♥
✝✒
✓
☞
✙
☛ ✞
✚✕
☛
◆
☛
❜✪
❣
✑
✟
✏ ☛
✞
♥
☛ ❜✪
❣
☛ ✏
✞
☞
❣
✑
✟
✠
✒
P(E
❖
♥
☞
F) = P(E) P(F)
❆
❙
✰
1.
♥
✙
☛ ✞
✚✕
☛
P(E
2.
☛
✈
☛
✞
✟
F)
❆
E
❁
❱
✓
✡
☛
☞
✙
✏
✟
✿
☛
✌✥
t
☞
✈
✌✥
t
☞
☛
☞
✓☛ ✞
✾
☛
✙
✙
☛ ✞
✟
✶
✓
☞
☛
✞
✟
✈
✌
✒
✤
✡
✓
☞
✌
✒
✚✕
☛
☛
✑
✤
☛ ☛
✙
☛
☛
✫
✌✤
✾
✤
✡
☛ ☛
☛ ☛
☛ ✞
✟
✝
✞
✟
✓☛
✦
✞
✌✤
✈
✙
✫
☛ ✟
✞
✓
☞
✚✕
☛
☛
☞
✌
✒
✙
☞
✤
❅
❣
✑
♥
✟
✚✕
☛
✥
✠
✒
☛ ✟
✞
✥
✞
✥✛
☛
✈
✞
☛ ☛
✝
✌✍
☛
✒
✠
✓✏
☛ ✶
✞
✾
✕
✏
✿
✟
☛ ✟
✞
✶
✥✘
☛
✞
♥
✓
✌
✝
✞
✟
✌
☛
✝
✍
✝
✟
✞
✓
☞
❄✦
❄✦
✟
❣
☛ ✞
✌
✑
✜
❣
✩
✑
✦
❣
✈
r
☛
❣
✑
✞
☛ ✞
♥
☛
☛ ✏
❯
t
❣
✑
❳
✒
✓
✏
✓
❲
✈
✒
✔
✓✏
☛
t
❀
✥✛
✜
❣
☞
☛ ✟
✞
✠
✒
☛ ✏ ☛
❣
✩
✑
✘
✓
☞
✕
☛
✒
❣
♥
✡
✘
❫
✘
☛
✣
❀
☞
❫
✌
✥✛
☛
✈
✜
✡
✌✥
t
✏ ✞
✥✛
☛
✈
✘
☛
✈
◆
✓
❣
✚✕
☛
❂
☛
✞
✟
✶
❉
✘
☛
✈
✚✕
☛
✈
dependent
✚✕
☛
❂
☛
✈
☛
✘
✙
✏
✿
✟
❇
✌✤☛
✒❚ ✏
✚✕
☛
✘
✈
☛
☛
✥
✓
✡
✥
✫
✾
F
P(E) . P(F)
✏
r
✏
✒
✤
✥
✾
☛
✌✤
◆
✏
✙
✏
✿
✟
✫
✌✤
✾
✌✤
✾
✌✤
✚✕
☛
☛
✈
☛ ✟
✞
✝
✟
✞
560
① ✁✂✄
✐☎✆✝✞✞✟
✠✞✡☛
☞✞✌
✍✞✎
✠✏✑✎
✦✐✆✤✐✆
✜✐✡❫✌☛ ✙✞✚✛✞★✩✪
✍✒✔
✓
✕☎✖
✐✗✘ ✕✎✏
✠✟✞✛✞✫✞✌☛
✛✍✌✓
✙✞✚✛✞
✜☎✆✢✑
✍✒✣
✤✐✥✚✑✕✞
✦✤✡✑✓✧✞
✙✞✚✛✞★✩✪
✜✞✒✆
✍✒✣
✓
✖♥✠ ✆✎ ✬✞✭✖✞✎✓ ✟✎✔
✓
✕☎✖ ✖✞✎ ★✎✠ ✌ ✤✡✑✓ ✧✞ ✙✞✚✛✞★✩ ✙✞✚✑✌ ✍✒✓ ☎❫✛✏✌ ✐✗☎✕✏✑✞ ✬✞♥✮ ✕✎✑ ✆ ✍✒✔ ✑✞✎ ✡✍ ✐✆✤✐✆
✜✐✡❫✌☛ ✛✍✌✓ ✍✞✎ ✠✏✑✌ ✍✒✣
✓
☎✡✈✞✎✟✑✯ ✕☎✖ ✖✞✎ ✬✞♥✮ ✕✎✑✆ ✐✗✞ ☎✕✏✑✞ ✡✞✈✌ ✐✆✤✐✆ ✜✐✡❫✌☛ ✙✞✚✛✞★✩
✙✞✚✑✌
3.
✍✒✔
✓
✑✞✎
✡✍
✖✞✎ ✕✞✖✰ ☎✱✲✏
✤✡✑✓ ✧✞
✐✆✌✳✞✝✞
✛✍✌✓
✍✞✎
✠✏✑✌
✍✒✣
✓
✤✡✑✓ ✧✞ ✏✍✈✞✑✎ ✍✒✔
✓
F
✐✍✈✎ ✐✆✌✳✞✝✞ ✠✎ ✑✫✞✞
✕☎✖ ✐✗✘ ✕✎✏ ✙✞✚✛✞
✖♥✠ ✆✎ ✐✆✌✳✞✝✞ ✠✎ ✠✓✷ ☎
✓ ✸✑
✕✴✵✟
E
✜✞✒✆
4.
✷✆✞✷✆
✍✒✣
✜✫✞✞☛✑❣
✑✌✛
✍✞✎✑✌
✍✒✔
✓
☎❫✛✏✞
✐☎✆✏✈✛
✖✞✎✛✞✎✓
✐✆✌✳✞✝✞✞✎✓
P(E F) = P(E) . P(F)
A, B
C
P(A B) = P(A)
P(A C) = P(A)
P(B C) = P(B)
P(A B C) = P(A)
✜✞✒✆
✏✞✎
✤✡✑✓✧✞
❫✞✑✞
✏✍✞
✺
✺
✺
✜✞✒✆
✺
✕☎✖ ❀✐✆✞✎✢✑ ✟✎✓ ✠✎ ✏✟ ✠✎ ✏✟
✍✒✪✔ ✏✞✎
E
✜✞✒✆
✾❉
❂❃
E
F
❫✞✑✞
★✏
✐✞✠✎ ✏✞✎
★✏
❀✲✞✈✞
✷✞✆
❫✞✑✞
✤✡✑✓ ✧✞
✍✒✓
✻✠
☎✏
✐✆✌✳✞✝✞
✏✞
✐✗☎✑✖✬✞☛
P(E) =
✤✐✥✚✑✕✞
P(E
♠✼✽✾✿❁✽
❀✲✞✈ ✐✆
✍✒✓✣
❂❂
✕☎✖
✤✡✑✓✧✞
★✏
A
✠☞✞✌
F
✜✞✒✆
❆❑
✺
✜✛☎☞✞✛✑
✐✗✞❄✑
B
✡✎✶
✜✈♦✹✜✈♦
☎✏✕✞
❫✞✑✞
✕☎✖
✜✞✒✆
✡✎✶✡✈
✕☎✖
✦✐✞✠✎
✐✆
✐✗✞❄✑
✠✓❅ ✕✞
❆
✏✞
✜✐✡✘✕☛
✠✎ ☎✛❇☎✐✑ ☎✏✕✞ ❫✞★ ✑✞✎ ✷✑✞★✩ ✢✕✞ ✙✞✚✛✞★✩
P (A)
S = {1, 2, 3, 4, 5, 6}
E F = {6}
✍✒✯
✺
❊❋❙●
P (E
F)
1
6
✍✒✣
✓
❍
unbiased
B
■
✍✞✎✛✞✪ ✜✞✒✆
✤✡✞✑✓✧✕
✟✞✒☎✈✏
✜✞✒✆
2 1
3 1
, P (F)
6 3
6 2
F) = P(E) . P(F)
✙✞✚✛✞★✩
☎✡✥✞✟ ✠✓❅ ✕✞
✙✞✚✛✞✜✞✎✓
✾❉
F
✍✒
✍✒✣ ✙✞✚✛✞
✠✟☎✥✚
✑✷
✜✞✒✆
✐✆
✡✎✶ ♦✴✝✞✛✐✶✈
✍✒✓❈
E = {3, 6}, F = {2, 4, 6}
E
✜✞✒✆
✍✒ ✣
✜✷
✜✑✯
✜✞✸✞✆
★✏ ✠✞✫✞ ✙✞☎✚✑
★✏ ☞✞✌ ✬✞✑☛ ✠✘✕ ✛✍✌✓ ✍✞✎✑ ✌ ✍✒ ✑✞✎ ✖✌ ♦✻☛ ✙✞✚✛✞✜✞✎✓ ✏✞✎ ✤✡✑✓✧✞
✠✎ ✜✞✒✆ ✦✐✞✠✎ ✐✆ ✐✗✞❄✑ ✠✓❅✕✞ ✠✟ ✍✒✪✔ ✏✞✎
❫✞✛✑✎
✍✟
✡✎✶
✑✫✞✞
E
P(B)
P(C)
P(C)
P(B) P(C)
✺
♠✼✽✾✿❁✽
☎✈★✔ ❫✍✞✩
✺
✙✞✚✛✞✜✞✎✓
✛✍✌✓ ✏✍✞
✡✎✶
✡✎✶
E
F
P(E)
P(F)
✍✒✔
✓
✙✞✚✛✞✜✞✎✓
✍✞✎✛✎ ✏✌ ✐✗✞ ☎✕✏✑✞✔ ❫✷ ✖✞✎✛✞✎✓ ✐✆✌✳✞✝✞ ✠✓✐✮✛ ☎✏★ ❫✞★✩✔ ✐✗✞☎✕✏✑✞
✡✎✶
F
✏✞
✙✞✚✛✞✜✞✎✓
❀✲✞✈✞
✙✞✚✛✞ ✦☎❏✑✌✕ ❀✲✞✈
✐✆✌✳✞✝✞
✏✞✎
18 1
36 2
✐✞✠✎ ✏✞✎ ✖✞✎ ✷✞✆
✏✌☎❫★✣
✠✟✠✓☞✞✞▲✕
✜✞✒✆
P (B)
✟✞✛
✈✎✓
18 1
36 2
✑✞✎
♦✕✞✣ ✟✞✛ ✈✎✓
✐✆ ☎✡✥✞✟ ✠✓❅✕✞
A
✙✞✚✛✞
✐✗✞❄✑
✦✐✍✈✌
✍✞✎✛✞✪ ✖✬✞✞☛✑✎
✐
❧✝ ✞
✝
P(A
✟
✠
✡
B) = P (
♥
A
✈
✚ ✢
P(A
✝
✝
✣
✤
♠ ✪
✫ ✬✭
✮
✫
F
✈
✝
✣
✤
✱✤
✚
✬
❣
☛
B
✈
✯
✰
▲
✓
✚
✚
✠☞
✵
✦✝ ✜
☞
✟
✣
✌
✩
✖
✗✤
✠✼
❞
✝
❄
✦
✒ ❧r
✝
✜
☞
✱✝ ✌
☛
✝
✧★
♥
✝ ☛
✱✝ ☛
✍
✎
✒✳
✚
(E,F), (E,G)
✱✝
✗✘
✒✚ ♥ ✾
✝
✿
✒ ✓ ✔✝
✑
❧✕
✌
✖
✝
✗✘
✝
✙
✚
✟
✝
☞
☛
✝
)
✟
✣
✩
✌
✝ ✏
✝
✲✖
✝
✟
✣
✩
✈
☞ ✚
✑
✌
☛
B) = P(A) . P(B)
✶
✖
✹
✑
✝
✽✝
✌
✥✝
✡
✑
1 1 1
2 2 4
P(A) . P(B) =
▲
✗✔
✜
✚
✖
✍
✎ ✝ ✏ ✝ ✌
☛
9 1
36 4
=
✈
✛
✝
☞
☛
✝ ✌
☛
561
✁
✂✄
☎✆✁
✗✘
✝ ✙
✚
✈
✝
✣
✤
❧✑✒
✔✜
✟
✝
☞
☛
✝
✴
(F,G)
✟
✣
✝
✣
✤
✑
✌
☛
✑
✝
☞
G
✱
✝
☞
✣
E
✏ ✌
☛
✦
✝ ✜
☞
✈
✦
❡
✝ ✜
☞
✝
❄
✱
✝
☞
✣
☛
❄
✚
✠☞
✒
✳
✚
✖
✝
✚
✠☞
✗✜
✗✘
✝ ✙
✚
✷
✒
❧
✝
▲
✟
✝
☞
☛
✝
✴
✸
✱
✚ ✑
✓
✚ ✥
✌
✝
♥
✟
✣
✝
☛
✺
✌
✗✜
✱✝
☞
✣
✗✘
✝ ✙
✚
✟
❡
✱✝
☞
✣
✝
☞
☛
✝
✴
❧
☛
✱✝ ☛
✒☞
✗✤✝
✒✻
✚
✒
✟
✣
✗
✚
✺
✌
✢
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
▲
✗✔
✜
✚
✖
E = {HHH, TTT}, F= {HHH, HHT, HTH, THH}
✝
G = {HHT, HTH, THH, HTT, THT, TTH, TTT}
✈
✝
✣
✤
❧✝ ✞
✝
E
✟
✠
P(E) =
❜
❧✒
✏
✧
P(E
❧✝ ✞
✝
F = {HHH}, E
✡
✡
F) =
P(E) . P(F) =
✟
✠
2
8
1
, P (F)
4
1
, P (E
8
P(E
P(F
✈
✝
✣
✤
✏
✧
♠
✪
✫
✬❣
✦
✬
✭
✮
✫
r
✖
✝
✜
☞
✝
✧★
✯
❅
✝ ☛
✌
✒✱
(E
✒❧
E
✈
❢
✚ ✞
✝ ✝
✝
✣
✤
F)
✓
✚ ✌
✥
✝
✒✱
❇
▲
1
, P(G)
2
1
, P(F
8
G)
✡
G)
✡
✓
✚ ✥
✌
✝
✟
✣
✟
✣
✌
❃
✖
✒♥
G = { HHT, HTH, THH}
✡
7
8
3
8
G)
❀
1 7 7
2 8 16
P(F) P(G)
❁❥
❙
F) = P(E) . P(F)
✡
❂
❂
P(E) . P(G)
P(F) . P(G)
✛
▲
✱
✠✒❃
✧
F
4
8
G)
P(E
✚ ✢
❧✒
G = {TTT}, F
1 1 1
1 7 7
, P(E) P(G)
4 2 8
4 8 32
✈
❜
✡
✱
✠
E
✦
✈
✝
✣
✤
✝ ✜
☞
F
✝
✧★
♥
✝ ☛
(F
▲
✈
✓
✚ ✌
✝
✣
✤
✥✝
✦
G)
✈
✝ ✜
☞ ✝
✧★
✝
✣
✤
✟
✣
✌
(E
✚
✝ ☛
✈
✝
✣
✤
E
G)
✈
✝
✣
✤
✗✤✝
✒✻ ✚
F
❆
❍
✝
✠
▲
✟
✣
✌
✩
✓
✚
✥
✌
✝
✟
✝
✲✠
✌
☛
✩
❜
❧✒ ✏
✧
P(E
✡
F) = P(E) . P(F)
... (1)
562
① ✁✂✄
❢☎✆✝
13.3,
♦✞✟
♦✞✠✡☛✝☞✞✌ ✝ ✍✞ ✎✏
E = (E
E
❉✎✝✞❢
✛ ✖
✗
F
E
☛✝✕☞
P(E
✎✝
✘ ✒☞✑✒☞
✗
✘
❾
= P(E)
(E
✗
☛✒♦✐✙✚
F) + P(E
P(E
❾
= P(E) [1
E
✏✕ ❢✖
✢
F
✗
F ) = P(E)
✗
F)
✗
P(E) = P(E
❜✍❢✤✥
✑✒✓✔
✗
F
✗
E
✗
F
✘ ✒☞✑✒☞
☛✒♦✐✙✚ ✏✕✛ ☛✝✕☞
✍✝✜✝ ✏✙
F)
✘
✏✕✣
✛
F)
✗
✘
F)
P(E) . P(F) (1)
✦
☛✝✕☞
✍✞
P(F]
◗❘❙❚❯❱❲
13.3
= P(E) . P(F )
✘
☛✈✧
E
☛✝✕☞
F
✘ ✑♦✈✛✆✝
✘
✈✜✝✝
✘
✈✜✝✝
F
F
✸✹
❾
✘
✶❣
P(A
✒✲✖ ✝☞
✎✏
✑♦✈✛✆✝
✘
✎❢✳
☛✝✕☞
✳✴✝✝✚✎✝
✐✝
✍✖✈✝
✏✕
❢✖
✎❢✳
B
❡✞✛ ✍✞
✏✕ ✛
✑♦✈✛✆✝
A
= 1 P(A ) P(B )
♠✵✰✶✷✯✰
✏✕✛✩
❜✍✙
✫✬✭✮✯✰✱
✪
(a) E
(b) E
▲✝✔✠✝✥★
✏✕✛✩
B
✑♦✈✛✆✝
A
▲✝✔✠✝✥★ ✏✕✛ ✈✝✞
✎✝
✺✎✻✠✈❡
✥✖
♦✞✟ ✏✝✞✠✞ ✖✙ ✒✲✝❢✎✖✈✝
✘
✎✝
B
❡✞ ✛
✺✎✻✠✈❡
✍✞
✥✖
✖✝
✏✝✞✠ ✝
) = P(A
✢
B)
= P(A) + P(B) P(A B)
= P(A) + P(B) P(A) P(B)
= P(A) + P(B) [1 P(A)]
= P(A) + P(B) . P(A )
= 1 P(A ) + P(B) P(A )
= 1 P(A ) [1 P(B)]
❾
✗
❾
❾
✘
❾
✘
❾
✘
✘
❾
= 1 P(A ) P (B )
✦
✘
13.2
✼✽✾✿❀❁❂❃
3
5
1
5
✘
1.
✎❢✳
2.
❅❆ ✒❇✝✝✞✛ ✖✙ ✥✖ ❈❊❋❊ ✙ ❡✞✛ ✍✞ ✎✝✳●❍■✎✝ ❢❏✠✝ ✒✲❢✈✑✜✝✝❢✒✈ ❢✖✥ ❈✥ ✳✝✞ ✒❇✝✞ ❢✠✖✝✤✞ ❈✥✩ ✳✝✞✠ ✝✞✛
P(A)
✒❇✝✝✞✛
3.
✍✛✈☞✝✞✛
♦✞✟
♦✞✟
✖✝✤✞
✥✖
✣
☞✛❈
P (B)
✖✝
❢❊❧❏✞
✏✝✞✠ ✞
✖✝
✏❖ ✥ ❢✠✖✝✤ ✖☞ ❢✖✎✝
☛✝✕☞
✖✙
A
✈✜✝✝
✒✲✝❢✎✖✈✝
❢✠☞✙❑✝▼✝
◆✍❡✞✛
B
✑♦✈✛ ✆✝ ▲✝✔✠✝✥★ ✏✕✛ ✈✝✞
❄✝✈
✍✞
P (A
✗
B)
❄✝✈ ✖✙❢✐✥✩
✖✙❢✐✥✩
✈✙✠
✍✛✈☞✝✞✛
✖✝✞
✎✝✳● ❍■✎✝
❢❏✠✝
✒✲❢✈✑✜✝✝❢✒✈
❢✖✥
✐✝✈✝ ✏✕ ✩ ✎❢✳ ✈✙✠✝✞✛ ❢✠✖✝✤✞ ❈✥ ✍✛✈☞✞ ☛❍■✞ ✏✝✞ ✛ ✈✝✞ ❢❊❧❏✞ ✖✝✞ ❢❏P✙
♦✞ ✟
✐
❢✝
✞
✟
✠✡✠☛
☞✌
★
❢✑
❢✍
✎
✩
✥✙
✚
✤✙
✏
✑
✏
✌ ✏
✪✫
✦
✔
✒
✓
✬
✙
✔
✕
✎
✖
✏ ✏
✔
✟
✠✡✠☞
☛
✭
✠
✌
✍
✗
✮
✏
✗
✏ ✣
✤
✚
✌
✗✙
✘ ✙
✌
✙
✒
✛
✚
✓
✞
✍
❢✜
✢
✣
✏
✯
✒
✚
✓
✠☞
✙
❢✣
❢
✑
✤✥✙
★
✡
✠✙
☞
❢
✝
✞
✟
✠✡✠☞
☛
✌
✒
✏
✙
563
✁
✂✄
☎✆✁
✚
✦ ✧
✤
✌
✚
✗✙
✰✱
✙
✍
✡
✒
✚
✓
✲
✏ ❢
✎
✍✌ ✏
✏ ✌
✍
✡❢
✑
✞
✛
4.
✳
✞
✍
✰✱
✕
✎
✏
✴
✎
❢
✤
✒
✏
✙
✌
✏
✒
✓
✼
✸
5.
❢✑
✞
❢✍
✰✏
✞
✍
6.
✥
✝
✙
✚
✎
7. A
✏
E
✔
p
✌
✖
✝
✙
✚
F
✎
✸
✘ ✏
✙
✸
✚
✔
✏
✏
✌
❞
✔
(i) P (A
B
✏
✗
✓
✷
✘ ✡
✷
❄
✘ ✡
✸
✏
P(A
❇
10.
✹★
❣
❢✑
✞
✎
❀
✌
✚
❢
✘
✸
✏
✔
✏
✝ ✙
P(A
❇
11. A
✚
✚
B
✏
✗
✓
✔
✔
✎
✍✏
✙
✍✌ ✏
✒
✚
✓
✸
❆
✹★
✏
★
✒
✏
✙
✌
✡
✒
✼
✓
✍✏
✙
❢
✺
✏
✴
❢
✤
✰
✠☞
✙
✻
✗
✰✌
❢
✌
★
✽
❢
✍
✗
✌
✙
✒
✛
✚
✓
❢
✗✡ ✾
✏
✿
✏
✚
✷
✒
✗✙
✸
✗
✤
✹★
✏
✭
✚
✺
✏
✙
❢✝
✷✞
✰✏
✏
✙
✒
✛
✚
✓
✭
✤
✚
❣
✤
✷
✎
✏
✝ ✏ ✝
✗
✶
✤✙
✭
✚
✤✙
❢✝
✫
✍✏
✙
✷
✏
✡
✏ ✝ ✏
❄
✮
❣
✒
✼
✓
✍✏
✙
✚
✓
P(E)
3
, P (F)
5
P A =
1
, P (A
2
❢✍
✹★
✏
❞
✒
✚
✓
✰
✏
✞
❞
✌
✖
✮
✞✤
✙
✡
✒
✏ ✏
✰
✗✟
✰
✗
✔
3
10
P (E
✔
✏
✗
✓
✑
✒
✏
✑
✡
✔
✒
✛
✚
✓
✏
✗
✓
3
5
B) =
❃
❄
✠
P(A) = 0.3
1
5
F) =
❂
✌
✣
❅
ii
✸
❆
✌
✖
✹★
✏
❞
❀
✏
✞
P(B) = 0.4.
P(B) = p.
✏
✏
✟
✠
✌
✚
✏
✒
✛
✚
✓
✌
✣
B)
❃
✲
✏
✌
✍
✡
❢✑
✞✛
1
1
, P (B) =
2
4
P(A)
❞
✚
✓
✏ ✌
✔
P(A
✏
✗
✓
❂
1
8
B) =
✌
✣
✍
✡❢✑
✞✛
❀
✟
✠
✌
✚
✒
✡
❞
✷
✏
✞
✘ ✡
✹
★
✏
1
.
4
❆
✚
✹★
✸
✏
✎
✞
✒
A
✏
✔
❄
❣
❞
✒
✓
✚
✑
✒
✏
✓
✚
✔
✏
B
✏
✗
✓
✓
1
2
P(A) =
❞
✏
✴
B)
✗
❀
✟
✠
✌ ✚
✸
✏
✹★
✏
✌ ✖
❞
✏
✞
(iv) P(A
✔
✔
B
✏
✗
✓
B
✏
✗
✓
✑
✏ ✌
✏
✒
✓
✌
✏
✙
✍
✥
✤✙
✔
✏
✓
✗
❁
✌ ✏
✙
❇
★
✒
✡
❆
✚
❄
✥✙
✚
✍
✏
✙
❣
✵
★
✏
✡
✒
✡
)
✚
★
✍✏
✥
❈
✏ ✝
✏
7
12
P(B) =
✏
P(A) = 0.3, P (B) = 0.6
(ii) P(A
✣
✏
✗
✏
✒
✚
✓
✶ ✫
✌
✡
✸
❁
✒
✡
✏
✗
✓
✒
✑
✒
✏
i
B)
✏
✏ ✌
✍
✡
✷✙
✏
✙
✍
✥
✞✍
✣
✏
✗
❢
✠
✏ ✌
✭
✏
✥
✤
✚
✲
✏
✎
✰✱
✏
✏
✍
✡❢✑
✞✛
❉
★
✌
✒
✏
✙
✙
✍
✡
✞
✍
✣
✏
♥
✷✙
✏
✙
❢✑
✞✛
✴
✘
✚
✝ ✏ ✝
✒
✚
✏
✞
❇
★
✏
✏
✗
✓
✝
✙
✚
❁
✏
❞
✹★
❆
B
✏
✏
✲
✏ ❢
✎
✘
✠✌ ✚
✥
✲
★
✤✙
✰✱
13.
B
✏
✗
✓
✒
✡✚
✸
✏
✰✏
✍
✏
❀
(iii) P(A
✞
✔
B
✏
✓
✗
✟
✠✌ ✚
(i) P(A
12.
✌ ✖
★
✒
✡
✔
✏
✗
✓
A
★
✥
✔
❀
✒
✓
❅
✏
❇
★
✚
✏
(ii) P (A
B
★
✒
✡
✎
(iv) P B | A
A
❞
✏
✞
✚
✓
✏
✛
✹
✍
4, 5, 6
B
❄
❣
(iii) P(A|B)
9.
✒
✏
✗
✓
✍
✏
✗
✰✱
✏
★
✏
✒
✼
✓
✟
B)
❂
✬
✎
✰
✱
❣
✤
❞
✟
✠
✚
A
★
✎
?
✒
✚
✓
✏
✞
✍
✡
✌ ✚
✔
B
✲
✏
✤
✷
✏ ✝
✏
❀
✟
✠
✤✥
✏
✗
✓
✍
✏
✙
✭
✗
✤✙
✎
✏
✞
✹★
✏
✶ ✫
✤✙
✰
B
✗
✚
✹
★
✏
✰✏
✤✙
✏
✗
✓
A
✏
✏
✌ ✚
✰✏
✌
✭
✤
❀
✟
✠
✞✙
✤✡
A
★
✏
✺
✏
✒
✓
✛
✚
F
✏ ✏
★
✥
✌
✙
✌
✖
✏
✏
B
✏
✗
✓
✍✏
✥
✹★
✏
★
✏
✷
✴
E
★
✏
✔
✝ ✏
✝
✝
✙
✚
✍
✗
✺
❢
✏
❞
✸
✏
✔
✹★
✏
✏
✞
✰✌
✴
8.
✗
✽
❢
✥
✞✍
✏
✗
✓
✹★
★
✏
✛
★
✏
✗
✓
B
A
1, 2, 3
A
✔
✏
✰
✤✙
✷✎
❢
✔
✸
✹
✍
✍
✡
✵
✍✏
✘ ✙
(iii)
★
✤
✤✙
❢✣
✰✱
✚
✒
✓
✚
✌ ✏
✙
✏ ❢
✎
✰✱
✏
✰✌
❢✌
✟
✖
✏ ✏
❢
✲
✍✌ ✏
✏
✌
✍
✡
❢✑
✞
★
❢✍
✞
(i)
●
✞✍
✍✏ ✝
✡
✌
✖
✏
✏
✘ ✤✗✡
✝ ✏ ✝
✒
✏
✙
✛
❢
★
✘
✏
✙
✴
✍✏
✝
✡
✑
✏
✌
✡
✷✙
✏
✙
✚
✘
✚
✙
✚
✝ ✏ ✝
✒
✏
✙
✚
✒
✛
✓
(ii)
✣
✏
♥
✤
✥✙
✚
✦ ❊
✍✏ ✝
✡
✰✱
✖
✔
✏
✗
✓
❋
●
✏
✥
✍✏ ✝
✡
✞✠✚
✘ ✤✗✡
✝ ✏ ✝
564
14.
① ✁✂✄
1
3
✖✎✒ ❣
✍✆✣
☎✙✆❞✱
(i)
☛✠
☛✠✤
☛☞✌✍✟
✖✗
☎✏✘✠
♠✘☞✠✒
✖✗
☛✠
✖✟✠
✈✟✎✏
☛☞✌✍✟
✖✗
☎✏
☎✟
✕✚✍✟☛
☎✏✑✠
✖✎✤
✒
✑✟✠
✕✚✟✆✍☎✑✟
✥✟✑
✖✎
❞✟✑✙
✑✦✍✑✢
☎✟✠✧★
✱☎
✆☎✘
✣✞✟✟✈✟✠✒
☞✠✒
E
❡✟✷✘✟✱✛
F
✈✟✎ ✏
✌✝✑✒ ✓✟
(i) E : ‘
✆✘☎✟✗✟
✮✍✟
✕✫✟✟
✖✬ ✝✬r☞
F:‘
✆✘☎✟✗✟
✮✍✟
✕✫✟✟
✧❢☎✟
✖✎
(ii) E : ‘
✆✘☎✟✗✟
✮✍✟
✕✫✟✟
☎✟✗✠
✏✒✮
F:‘
✆✘☎✟✗✟
✮✍✟
✕✫✟✟
✱☎
✸✟✣✞✟✟✖
(iii) E : ‘
✆✘☎✟✗✟
✮✍✟
✕✫✟✟
✱☎
✸✟✣✞✟✟✖
F:‘
✆✘☎✟✗✟
✮✍✟
✕✫✟✟
✱☎
✸✠✮☞
✱☎ ✴✟✓✟✟✝✟☛
✱☎
✴✟✓✟✟
(a)
✕✚✟✆✍☎✑✟
(b)
✍✆✣
(c)
✍✆✣
☎✙
☞✠✒
☎✟✠
✝✖
60%
✥✟✑
✺✖✣✙
✝✖
☎✟
✕✚✟✆✍☎✑✟
✱☎
☎✟
✗✠✑✟
✖✎ ❣
☎✟
✥✟✑
(A) A
✈✟✎✏
B
B
✕✏✌✕✏
(C) P(A) = P(B)
✖✎
?
’
’
☎✟
✖✎
✖✎
’
’
✍✟
✍✟
✱☎
40%
☎✟✤
✱☎
✸✠✮☞
✮✬ ✗✟☞
✈✒✮ ✻
✚ ✠ ✟✙
✖✎
✖✎
’
’
☎✟ ✈✟✎✏
20%
✣✟✠✘✟✠✒ ✈✶✟✸✟✏ ✕♥✼ ✑✠ ✖✎✒ ❣
✖✎ ❣
✘
✕♥✼ ✑✙
✑✟✠
✖✎
✺✖✣✙
✈✟✎ ✏
✑✟✠ ♠☛✝✠r
✘
✖✙
✈✒✮✻
✠✚ ✟✙
✈✒✮ ✻
✠✚ ✟✙
☎✟
✈✶✟✸✟✏
✕♥✼✑✙
✖✎ ❣
☎✟
✈✶✟✸✟✏
❀✟✙
✕♥✼ ✘✠
✝✟✗✙
✖✟✠✘✠
✕♥✼ ✑✙
✖✎
✑✟✠ ♠☛✝✠r
✺✖✣✙
☎✟
✈✶✟✸✟✏
❀✟✙
✕♥✼ ✘✠
✝✟✗✙
✖✟✠✘✠
☎✙✆❞✱❣
❞✟✠ ✯✼✟
✈✟✎ ✏
✝✖
✈✶✟✸✟✏
♠✴✟✗✟
✆✘✵✘✆✗✆✶✟✑
A
✆☎
☎✟
✖✎✒
☎✙✆❞✱❣
(B)
❡✟✷✘✟✈✟✠✒
❞✟✑✟
✈✶✟✸✟✏
✥✟✑
(A) 0
✣✟✠
✽✬ ✘✟
☎✙✆❞✱
✈✒ ✮✚✻
✠ ✟✙
✕✚✟✆✍☎✑✟
✕✟☛✟✠✒
✆✝✹✟✦✟✙★ ✺✖✣✙
✍✟✣✲✳ ✴✍✟
✕✚✟✆✍☎✑✟
✍✆✣
☎✙
18.
✔✕
1
2
⑥✟✏✟ ✌✝✑✒ ✓✟ ✔✕ ☛✠ ✖✗ ☎✏✘✠ ☎✙ ✕✚✟✆✍☎✑✟✱✛ ✜☞✞✟✢
✆☎
☎✙
17.
✌✝✑✒ ✓✟
B
✈✟✎ ✏
✑✟✞✟ ✝✠r ✩✪ ✕✫✟✟✠✒ ☎✙ ✱☎ ☛✬✆☞✆✭✑ ✮✯✰ ✯✙ ☛✠ ✱☎ ✕✫✟✟ ✍✟✣✲✳✴✍✟ ✆✘☎✟✗✟ ❞✟✑✟ ✖✎ ❣ ✆✘✵✘✆✗✆✶✟✑
☞✠✒
16.
✣✟✠ ✘✟✠✒✤
☛☞✌✍✟
(ii)
15.
A
✱☎ ✆✝✞✟✠✡✟ ☛☞✌✍✟ ☎✟✠
☎✟✠
☞✠✒
☛✠
❞✟✑✟
❢✍✟
✖✎
✕✚✾ ✍✠ ☎
(C)
✕✏✌✕✏
✖✎✒
✕✟☛✠
✕✏
☛☞
✈❀✟✟✿✍
☛✒✶ ✍✟
✕✚✟❁✑
✖✎ ❂
1
3
✈✕✝❞✙★
✑✟✠
✌✝✑✒ ✓✟
☎✖✑✠
1
12
✖✎ ✤
✒
(D)
1
36
✍✆✣
(B) P(A B ) = [1–P(A)][1–P(B)]
❃
❃
(D) P(A) + P(B) = 1
☎✏✘✠
✐
13.5
❡✍ ✎
II
❝
✝
✞
✟ ✠
✡☛
☞
✏ ✑✒✓
✔
(Bayes' Theorem)
✝
✌
✒
✕
565
✁
✂✄
☎✆✁
✖
✍
✗
✘
✍ ✏
✙
I
✗
✈
✍
II
✚
✙
✒
✖
✔
❢
✔
✛
✜
✢
✙
I
✘
✍ ✏
✗
✍
✰
❡✜
✗
✣
✤✥
s
✗
✖
✈
✍ ✙
✚
♥
✏ ✍
✏
❢✖
✜
✗
✜
✜
✗
✛
✜
✙
✢
✈
✍
✚
✙
✘
✍ ✏
✙
✍
✰
❡
✗
✜
✦
✒
✕✤
✑
✤
✥
s
✗
✖
✔
✕
✈
✍
✘
✍
✏
✙
✗
✙
✚
✧
✕✍ ✗
✏ ✍ ✏
✭
✮
✎
✎
✗
❢✖
✜
✗
✜
✕
✑
✜
✗
✛
✜
✙
✢
✒
✕
✤
✑
✔
✕
1
2
✥✯
✍ ✒★
✕✬
✍
✘
✍ ✏
✙
❑
✍ ✬
✕
✗
❡
✚
✜
✗
✤✗
✤
✕
✔
✕
✬
✗
✛
✜
✙
❢✖
✜
✗
★
✍
★
✍ ✖
✒
✕✤
✑
✪
✩
✫ ★
✍
✒✎
✕✍ ✏
✑
✒
✱
✲
✍
✳
✗
✍
✘
✍
✏
✙
✗
✓
✍
✴
❡✍ ✎
✬
✑
✛
✙
✏
✜
✗
✢
✛
❡
I)
✘
✍
✏
✙
✍
✰
❡✜
✗
✤✗
✔
✕
✲
✍ ✸
✖ ✍ ✗
✜
❢
❡✜
★
✍
✛
✍
✤
✕
✬ ✗
✕
✍
✚
✜
❢
✗
✒✱✲
✍ ✳
✗
✍
✛
❡
✗
✒
✕
✒
✕
✛
✙
✜
✥
✤
✑
❢✜
✖
✗
✒
✕
✬
✍
❢✜
✛
✚
✜
❢
✙
✙
✎
①
✒
✕
✽
✍
✖
✛
✍
✗
✎ ✗
✤✗
✤
✑
★
✛
✍
❞
✺ ✤✱
s
✗
✤
❡
★
✍
✗
✍
✗
✤
✥
s
✗
✖
✚
❢
✜
✕✑
✘
✍ ✏
✙
✗
✗
✘
✍ ✏
✙
✍
✱
✍ ✏ ✑
✍
✕
✍
✤
❡✍
✍
✎
✍
II
①
✗
★
❊
✱
s
✗
✎
✍ ❡
✖
✜
✗
✕✍ ✗
✗
✒
✎
✕✍ ✏ ✎ ✗
✴
❡✍
✱
s
✗
✛
❡✜
✗
✭
❢
✎
✎
✮
✎
✗
❑
✍ ✎
✺
✻
✏ ✜
✗
✕✑
✛
✢
✙
✥✯
✥✯
✤
✥✯
✒✬
✽
✜
✥✯
✍
✒
★
✕
✬
✍
✓
✍
✎
✍
✓
✍
✎
II)
①
✤✗
❢
✒
✛
✙
✓
✍
✗
✥
✎
✕✑
❑
✍
★
✍
✬
✶
✍ ✑
✕✚
✛
❡
✺ ✤
❢
✓
✍
✎
✽
❂
✍
✗
✍
✽
✍
✺
✻
✛
❋
✩
★
✍ ✥
✗
✚
✍ ✜
✎ ✗
✥✯
✘
✍
✎
13.5.1
✱
✥
✯
❡✍
✎
✍
❏
▲
✈
✍ ✜
✤✗
✥❈
✚
✍
(a) Ei
▼❖
◆
P
◗ ❘
❙ ❚
✱
✻
✈
✍
❣
❤
✺
✔
✔
✕
✥
✒✚
✶
✍
✍ ✳✍ ✍
✈
✍
✚
✙
✱
s
✮
✫
❣
❦
❦
❦
✤✚✗
✲
✍
✸
✖
✍ ✜
✗
✥
✯
❡✜
✹
✗
✬ ❧
✈
✤
✜
★
✖
✍
✛
✚
❆
✍
✍
▲
❙
✪
✮
✭
❖
En= S
★
✕✍
✗
✥✯
✒✬
✖
✓
✎
✬ ❧
✛
✙
S,
✒
✳
✿
✲
✍ ✻
✎
✍
E1,
✔
✾
✤
❡❢✯
, ... En
2
❤
✛
✙
✬ ✘
✍
✍
✱
s
✗
✥✯
✤
❡
✒
✏
✒✬
✖ ✲
✍
✻
✒
✕
★
✍
● ❍ ■ ♥
✥✯
✍
S
✒
✳
✎
✕
✑
❴
✤
❡
✥✯
✍
✒★
✕
✛
❡✜
✚
✜
✒
✶
✍ ✕
✛
❡
✖ ♠
✗
✍ ✬ ✗
✛
✜
✙
✒
✕
✕✍
✺
✗
✻
✍
❜
✼
★
✍ ✜
✚✗
♠ ✍
E
✗
✒
✕
✱
s
✗
✤
✜
❢
✬
❴
E =
♣
13.3,
✒
✭
✕✍ ✗
✺
✻
✮
✪
✭
✤✗
❵
✈
✍
✛
❡
✢
✙
♣
✛
❡
13.5.2
✍
❜
✖ ✍
♣
❴
E
E = S.
✎
✍
✿
E
✚
✙
✈
✍
✗
✍
✎
✍
✔
✿
❣
✕ ✚✎
✑
✒
★
✕
❴
✤
✜
✥
✍ ✻
♣
✥✯
✍
❯
r ▼
t ✉
❙
✗
❢
✛
✙
✬
✍
❑
✍
✕✚
✺
✻
❢✜
✓
✽
✬
✖
✗
✒
✕
✕
✚
✎ ✗
✍
✗
✥✯
✕✍
✚
✍
✽
✎
✍
★
✍
❢
★
✍
✤
❇
✒✲
✍ ✬
✍
❉
❈
✛
✮
✈
✍
✘
✍ ✍ ✢
✽
✗
✍ ①
✥✯
❡
★
✗
✥
✒✚
✍ ✍ ❡✍ ✜
✗
✥
✚
✒✱
✭
✍
✚
✕✑
✒✓
✔✢
❴
✒★
✕
✬
✍
✒
✱
✶
✍
✍
✓
✎
✕✍
✗
✒✎
✒
✥
✬
✕
✚✬
✍
✛
✙
★
✒
✖
★
✒
✖
❪
✲
✍ ✷
✱
s
✗
✒✱
✶
✍
✍
✓
✎
✕✍
✗
✒✎
✤
✍ ✎
✑
✛
✜
✹
✙
✬
✍
❴
♣
❴
♣
✕✑
✥✯
❡
★
✗
✈
✍
✤✗
✥✯
{E
✗
♣
✤
✥
✜
✕✍
✗
✒
✥
✬
✕
✚✬
✑
✛
✜
✙
❪
★
✬
✗
✚
✛
✢
✙
✚
✙
✤
✕✑
✥❈
✚
✕
✍
❆
❜
E
✎
✍
♣
✿
✥✯
✒✬
✖
✲
✍ ✻
✤
❡
S
✒
✳
✿
✕✍
♣
✾
❴
q
✗
✍
✕
✚
✤
✕
✬ ✗
✛
✜
✙
✒
✕
E
★
✒
✖
✤
❡
✍ ✻
✥✯
F, E
❴
✪
✮
✭
✒✬
✖
★
✲
✍ ✻
✒✤
✕
✚✗
❢
✜
❴
E
S
F}
♣
❣
✕
✍
F
✤
❡
✕
✍
✔
✕
✪
✮
✭
✔
✕
✒
✱
✶
✍
✍
E
★
✈
✍
✕✍
✒✱
✶
✍ ✍ ✓
✓
✎
✛
✚
✙
F
✔
✕
✎
✒
✕
✤
✑
✒✱
✶
✍ ✍
✛
✙
✈
✍
✥✯
✒✬
✖
✓
✎
✚
✙
✲
✍ ✻
✛
✢
✙
✤
❡
✪
✮
✭
★
✢
✙
✢
✗
❀
❚
▼
❙
◆
❖
✇
▲
P
❙
▲
{E 1, E 2,...,E n}
E 1, E 2 ,...,E n
✏
✎ ✍
✿
✬
✥✯
✍
✎ ✱
s
✗
❁✍
❜
★
✱
s
✗
S
✒
✳
✬
✍
❁
❈
✎
✒✎
✕✍
✏ ✑
✬
❈
❈
❁
✍
✗
❑
✍
✒
✖ ★
✍
✔
✿
{E
F, E F, E F }
F , E F, E
F, E
F}
✤
❡
❡
✒
✬
✏ ✍
❡
✗
✒★
✕✬
✍
✛
★
✬
✘
✍ ✍
i = 1, 2, ..., n
✿
✛
✙
✜
✹
❉✍
✈
✽
❑
✍
★
✈
✷
(Partition of a sample space)
❨ ❩
❙ ❙ ❬
❭
❁✍
✱✗
✎ ①
✈
✍
{E
✬
✍
✛
❡✜
✗
✢
✙
j, i, j = 1, 2, 3, ...n
❛
❣
✍
✮
✼
✬
❳
✤
❡
❋
★
✗
✕
❜
❲
❆
✤
❡
★
✒
✖
✥✯
✍
✛
❡✜
✛
✜
❃
✿
❥
❈
★
✮
♦
❡
✱
s
✗
❵
(c) P(Ei) > 0,
✖
❯❱
❖
Ej = , i
❴
(b) E1
✒
✱
✶
✕✑
❆
✬
★
✒
✖
✬ ✗
❁
E1, E2 ... En
✗
✿
✱✗
✛
✜
✹
✙
✤
✕
❆
✕
✍
✬
✕✚
❃
❁✍
❜
✙
✬
✬ ✗
✒★
✕
❇
❡
❑
✍
✤
✕
✵
✥✯
✍
✬ ❑
★
✍ ❢
✗
✬
✍
reverse
❁✍
✱
s
✗
✒★
✕
✒✎
✕
✍ ✏ ✑
✴
❆
✬
✍
✥✯
✍
✬
✍
✼
✒✬
✏
✍ ❡
✗
✒✤
❅
✕✑
✒
★
✕
✒
✕
❀
✎
✤✗
✥✯
✍
✏ ✗
✘
✍ ✏
✙
✍
✕✑
✛
✢
✙
❃
✱✗
s
✒
✎
✕✍ ✏ ✎ ✗
✿
❅
①
✥✯
❡
❢
✒✎
✕✍ ✏
✑
✘
✍ ✏
✙
✕
✍
✵
❢✖
✜
✗
✤✗
✒✱✲
✍ ✳
✗
✍
✛
❡✜
❄
❫
✽
✏ ✜
✾
❜
✕✑
✍ ✎
✒✱
✲
✍ ✗
✳✍
✕✍
✗
✖
✴
❡
✜
✕
✑
✥
✥
✯
②
✒
▼◆
❱
③
✇
✬ ✖
✲
✍
✻
(Theorem of Total Probability)
✤
❡
✒
✳
✿
✯
✍
✒
★
✕
✬ ✍
✲
✍
✷
❈
★
✗
✍
④
✚
S,
✛
✙
✢
✕
✍
❡
✔
✍
✎
✕
✏
✒
✑
✒
✱
✓
✶
✍
✔
✍
✓
✎
A
✥
✛
✯
✒
✙
✬
✈
✖
✲
✍
✍
✻
✙
✚
✤
❡
❡
✍
✒
✎
✳
✏
✗
✜
✱
✿
✒
✗
s
✕
✤
✜
❢
✥
✬
✯
❋
★
✗
✔
✕
✕
566
① ✁✂✄
❄ ☎✆✝ ☎
✞ ✟✠
✡☛
,
P(A) = P(E1) P(A|E1) + P(E2) P(A|E2) + ... + P(En) P(A|En)
n
P(E j ) P (A | E j )
=
j 1
♠☞☞✌✍✎
❢✏✑☎ ✒✑☎ ✞✟ ❢✓
E1, E2, ..., En
S = E1
Ei
✈☎✟✫
✞✘❣✰
✱☎✡
❢✓
✞✟
✬
Ej =
❢✓✗✲
E2
✩
i
✭ ✮
✩ ✪✪✪ ✩
S
✓☎ ❞✓ ❢✚✛☎☎✜✝ ✞✟
(
❢✢✣☎ ✤✥✦✧
)
❜✗❢★❞✠
En ... (1)
j, i, j = 1, 2, ...., n
✯
A,
❄☎✆✝☎
✐✔❢ ✡✏✕☎✖ ✗✘❢✙✆
✚❣♦
A=A S
= A (E1
= (A E1)
❢★❞
✬
✬
✬
✗☎❧☎
❢★❞
✞✲
A
✈✗✰ ✑✴✸✡
Ei,
✬
A
✈☎✟✫
i
❜✗❢★❞
✞✟
Ej,
✬
= P(A
P(A
✈☛
✐✔ ☎❢✑✓✡☎
✚❣♦
✬
✒✴ ✹☎✝
✩
E2 ... En)
(A E2)
Ø✘✕☎✳
E1)
✗✘✴✵ ✢✑☎❣
Ej
✈☎✟ ✫
✚❣♦
❢★❞
✩
A
✬
E1) + P(A
✬
E2) + ... + P(A
❢✝✑✘
✞✟✰
✜☎✝✡❣
✞✘
P (Ei)
✩
✶✐✗✘✴✵ ✢✑
Ei
E2)
✸✑☎❣✰❢✓
.....
✚❣♦
✬
✩
Ei) = P(Ei) P(A|Ei)
✺☎✫☎
Ei
13.4
En)
✬
(A
✬
✬
▼◆❖P◗❘❙
(A
✩ ✪✪✪✩
✬
j, i, j = 1, 2 ..., n
✯
P(A) = P[(A
❜✗❢★❞
✩
(A
✈☎✟ ✫
✞✟✰
A Ej
✬
i
✜☎❣
✛☎✲
✷
j,
✈✗✰✑✴ ✸✡
✚❣♦
✞✟✰❍
En)]
✬
✬
En)
0 i = 1,2,..., n
✯
✮
❢✓
❜✗❢★❞
P(A) = P(E1) P(A|E1) + P(E2) P(A|E2) + ... + P(En) P(A|En)
✑☎
P(A) =
n
P(E j ) P (A | E j )
j 1
♠✻✎✼✽✾✎
❆✦❇❈
✞✟ ❍
✿❀
❢✓✗✲
✞❃❅ ✡☎★
✝
❁✑❢ ✸✡
✞☎❣ ✝ ❣
✝❣
✓✲
❞✓
✡❧☎☎
❢✝✘☎✖✹ ☎
✞❃❅ ✡☎★
✓☎✑✖
✞☎❣ ✝❣
✓☎
✓✲
❂❣✓ ☎
❢ ★✑☎
❢❉❧☎❢ ✡✑☎❣✰
✘❣✰
✞✟ ❍
✞❃❅ ✡☎★
❢ ✝✘☎✖ ✹☎
✓☎✑✖
✞☎❣ ✝ ❣
✚❣♦
✓✲
✐✔☎❢✑✓✡☎
✗✘✑☎✝✴✗☎✫
✐❊✹☎✖
✞☎❣ ✝❣ ✓✲ ✐✔ ☎❢✑✓✡☎❞❋ Ø✘✕☎✳ ❆✦●❆ ✡❧☎☎ ❆✦✥■ ✞✟✰ ❍ ❢✝✘☎✖ ✹☎ ✓☎✑✖ ✚❣♦ ✗✘✑☎✝✴✗☎✫ ✐❊ ✹ ☎✖ ✞☎❣✝❣ ✓✲ ✐✔ ☎❢✑✓✡☎
✱☎✡
✼❏
✓✲
✓✲❢✜❞❍
✘☎✝
❄☎✆✝☎
★✲❢✜❞
✓☎❣
B
❢✓
❡❢✝✘☎✖ ✹☎
✺☎✫☎
❢✝⑥❢✐✡
✓☎✑✖
✚❣♦
❢✓✑☎
✗✘✑☎✝✴ ✗☎✫
✜☎✡☎
P(B) = 0.65, P(
✞❃❅ ✡☎★
✞✟❍
✞✘❣✰
✝✞✲✰
✐❊ ✹☎✖
❣
✞☎❣✝❑
P(A)
✓✲
✱☎✡
) = P(B ) = 1
▲
❄☎✆✝☎
✓✫✝☎
❾
✞✟❍
A
✓☎❣
✞✘❣✰
P(B) = 1
✈☎✟✫
✱☎✡
❾
❡✞❃❅✡☎★
✞✟
❣
✞☎❣✝❑
❢✓
0.65 = 0.35
P(A | B) = 0.32, P(A | B ) = 0.80
▲
✸✑☎❣✰❢ ✓
❄☎✆✝☎❞❋
B
✈☎✟ ✫
B
▲
✗✘❢✙✆
✗✘✴✵ ✢✑
✚❣♦
❢✚✛☎☎✜✝
✞✟✰
❜✗❢★❞
= P(B) . P(A | B) + P(B ) P(A | B )
▲
▲
✗✰ ✐❊ ✹☎✖
✐✔☎❢✑✓✡☎
✐✔✘❣✑
✺☎✫☎
✐
567
✁
✂✄
☎✆✁
= 0.65 × 0.32 + 0.35 × 0.8
= 0.208 + 0.28 = 0.488
✈
✝ ✞
✈
✤
✟ ✠ ✡☛ ☞
✌☛
✕
✡
✤
✖
❝✬
✭
✮ ✯
✰✱
✲
✥
✍☛
✎
☛
✦
☞
✏✡
✎
✓
✘
✡
✖
✎
☛
✠ ✑
✍☛
✏☛
✒
✓
✘
✍
✧
✓✔
✌☛
☞
☛
✠
✕
✍
✒✖
★
(Bayes' Theorem)
✬
✳
☛
✠
✖
✖
✩
✖
✝
✧
❀
✎
✟
✟♦✽
☛ ☛
En = S
✠
✈
✍☛
A
☛
✒
✢
✟✠
✡☛
☞
✌☛
✍
✒✝
✗
✷
✍☛
✖
✪ ☞
✵
✖
✏
✗
✕
★
✢
✈
✧
✶
☛
✓
✘
☛
✟✎
☛ ☛
✪ ✏
✖
✟
✍
✝ ☛
✏✫
✍
✒✖
★
E1, E2 ,..., En
E1, E2 ,...., En
✹
✠
☛
✕
✢
✕
✟
✩
✣
✖
✓
✘
☛
✟✎
✍
✝
☛
✿
✑
✡
✝
✞
✷✸
✠
✈
❞
☛
✔
,i
n
✶
☛
✹
☛
✕
★
✢
❀ ✺
☛
✖
✟
✍
✓
✟
✘
✝
✏
✎
★
✑
✝
✕
★
✢
✈
✻
☛ ☞
☛
✒
✢
S
✶
✏✡
✟
E1 E2 ,...,
✴
✎
✺
✏✍
✗
✣
✢
✵
☛
☛
✝
☞
✾
P (E i ) P (A|E i )
P(Ei|A) =
✙ ✚✛ ✜ ✜
✈
✟ ✒✴
✝
✹
♦
✖
✼
✍
✗
❁
❁
❁
❂
✎
✖
✝
✒
✕
✢
✝
☛
✖
1, 2, 3, ..., n
P(E j ) P (A | E j )
j 1
♠
✰✰❃
❄
✮
❣
✕
✡
✖
★
☛
✝
✕
✢
✟
✍
P(A E i )
P(A)
P(Ei|A) =
P (Ei ) P (A|E i )
P (A)
=
(
✓
✘
☛
✟✎
P (E i ) P(A|E i )
=
(
n
✏
★
✍
✝
✓✔
✌☛ ☞
☛
♦
✖
✼
✓
✘
☛
✟✎
✩✑
✍
✝
✌☛
✠
✟✠
✎
✡
✏
✖
❅
♦
✖
✼
✟
✠ ✎
✡
✏
✖
❅
☛
P(E j ) P(A|E j )
j 1
❢
❆
❇❈❉
❊ ❋
●
✤
✵
✠
☛
✈
P(Ei)
☛
✦
✓
✘
✡
✖
✎
♦✖
✼
✈
E1, E2, ... En
✶
☛
✖
✥
☛
✖
★
✍☛
✖
P(Ei A)
✓
✟✒
✍☛
✖
✍ ▲✓
✠
✓
✟✒
✍
▲✓
✠
✓
✟
✒
✥
☛
✓
✘
✡✖
✎
✷
♦
✖
✼
✷
✟♦✽
✵
Ei
☛
✵
✕
✗
✍
✝ ✗
✠
✍ ▲✓
✠
✓✔
♦
✍
☞
☛
☛ ☛
✝
✕
☛
✢
❅
✈
✍☛
✕
☛
✖
✝
✗
✕
✝ ✞
✵
▼
✈
✍
✗
✧
✴
✓
✒
☛
✖
▼
a priori
❅
❍
✍☛
✗
✠
✓
✘
☛
✟✎
✍
✝
☛
▼
✍
✒✝ ✖
☛ ☛
✝
☞
✾
✏✔
✶
☛
✟
❑
✗
✍☛
❅
✍
✕
✝ ✖
✓
✘
☛
✟✎
✍
✝
Ei
✕
★
✢
✏
✖
✡
★
✖
✟
✍✏
✗
✻
☛
✍
✕
☛
✈
☛
✖
★
✵
✍
☛
✖
✠
✷
♦✼
✖
♦
✟
♦
☛
✗
✶
☛
✺
✕
✽
✵
Ei
☛
✖
✩
✍
✕
✓
✘
☛
✟
✎
✕
✗
☛
✝
✖
✕
✣
★
✢
✍
✝ ☛
✏✓
✟
✘
✝ ✤
★
✍
✕
✝ ✖
☛
✕
✣
✢
✎
☛
✟
★
✖
✻
❙
✓
✥
☛
✦
✓
✘
✡
✖
✎
✕
☛
✖
✠
☛
✟
✟♦✟♦◆
✶
✟
✍
✎
☛
✩
✎
☛
✕
◆
✵
✠ ☛
✏
✖
☛ ☛
✝
☞
✾
✍
✝
Ei
❀ ✺
✓
✟
✘
✝
✻
☛ ☞
✏✡✎
✡
★
✖
❍
✍
✈
☛
✒
✢
♦✼
✖
♦
✷
✝ ✗
✕
✍☛
✒
S
✶
✏✡
✟
✢
✈
☛
✒
✢
✵
✍
✏
✖
✈
✟
◆✍
✠ ✕
✗★
❀
✍
☛
✣
★
✢
✷
▼
✈
✧
☛
✟✎
✷
✍
✶
☛
✓
✘
✌☛ ❅
✍
✗
✓
✘
☛
✟
✎
✍
✝ ☛
✖
✶
☛
✹
✝ ☛
✕
✢
✤
✟
✍
❀
✝
✍
✗
✕
✍
✎
☛
✩
✎
☛
✕
✣
✢
❑
✤
✖
✢
✷
✡★
✖
✶
☛
✕
★
✴
☛
✝
Ei
✍
✒✝ ✖
✣
✢
✹
✏✔
❍
✡
★
✖
☛
❅
✪ ✏
✟
✓
✎
✕
★
a posteriori
✍
☛
❘☛
✝
❍
☛
♦
❘☛
✌☛
☛
★
✖
✟ ✠ ✡☛ ✌
☞
☛
✢
❏
❀
☛
▼
✗✠
☛
✒
✺
☛
✝
❍
✷
✠
✍☛
■
✟
hypotheses
☛
❑ ❖
✍
✗
✶
☛
✟
A
✶
☛
❍
✠ ✟
◗
✍☛
✒
❑
✏
✟✠
✹
✍
✍
✍☛
✖
✡
★
✖
❍
✍
✗
P
✤
✖
☛
✖
✩
✷
✸
✍☛
✖
Ei
☛
✠ ✓
✑
✘
✎
✣
✢
✓
✟ ✒✟ ❙
✧
☛
✟
✝ ✎
☛
★
✖
✡
★
✖
●
✓
✎
☛
✟
✖
✩
✝ ☛
✕
✣
✢
✪
✠
✡
★
✖
✏
✖
♦✼
✑
❚
✍☛
✖
✟✠
❍
✠ ✟
■
✟
❑
❀
☛
✝
☛
✕
✒
✌☛ ☛
★
✖
✡
★
✖
568
① ✁✂✄
♠☎✆✝✞✟✆
❡☞✓
✜
✎☛✎
✝❣
✎☛✎
✏✓✙
✌☛✍ ✎☞
✎☛✎
✠✡
✏✓✙
♥☛☞
✢
✈☛✍ ✏
✗✘
I
✌☛✍ ✎☞
❞✤★
✗☛
✗✘
✙☞✓♥
II
✈☛✍✏
✗☛✎✘
✩✣
✒✍✔
I
✙☞♥
✓ ✓☞
✛☛②
✒✍✔
✓
✗✘
✒☛☞★☛
✗☛☞
❢★✗✎★☞
❢♥✑
✗✘
❢✗✣✘
✫✤☛
✣☛✌☛
E1
✣☞
✈☛✍✏
I
P(A|E2) = P(
II
II
= P(E2|A),
✈✛
✗✘
✒✍
✌☛✍ ✎☞
❡ ☞✓
✣☞
✙ ☞✓ ♥
✗☛☞
✣☞
❡☞✓
✒✍
II
✣☞
✎☛✎
✣☞
❡☞✓
♦☞✯
✈☛✍✏
✖
✗☛✎✘
✤☛♥✥✦✧✤☛
✤✒
❢✗
✌☛✍✎☞
A
✕
✌☛✍ ✎☞
✎☛✎
✣☞
❡☞✓
❢ ★ ✗☛ ✎★ ☞
✛☞❝☛✴✬✭ ❡✤
☞
❡☞✓
✙☞♥
✓
II
✌☛✍ ✎☞
❞✤★
❢★❧❢✬②
✑✗
E2
✗☛☞
✗✏②☞
♥☛☞★☛☞✓
✠✶
❢✣✫♦☞✯
②✘★
✣☛☞★ ☞
✈❢r☛✷★
♦☞✯
✒✍✲
✏✓ ✙
✎☛✎
✗✘
✏✓ ✙
✙☞♥
✓
✗✘
r☛✘
✣☛☞★☞
✝❣
❡☛★
✒✍✔
✗☛
✎☞✓
✤❢♥
✒✘
②✛
✣☛☞★☞
II
I, II
✈☛✍ ✏
♥☛☞★☛☞✓
❡☞✓
III
❢♥✑
❢✣✫♦☞✯
✗☛
✒✍ ✲
②☛☞ ✩✣
✙✑
❞☛✺♥✘
✛☛②
✈☛✍✏
E3
Ø❡✿☛❀
❢✸✹✛☞
I, II
✙✩✪
❡☛★
✎✘❢✚✑✔
❡☛★
)=
3
7
❡☛★
✎☞✓
A
✰☛✱★☛
II
✌☛✍ ✎☞
✒✍
✚☛☞
❢✗
✎✘❢✚✑
❢✗
✒✍ ✮
✒✍✔
✓
❢★✗☛✎★☛
✚✛
)=
✤✒
❢✗
5
11
✳☛ ②
✒✍
❢✗
♦✒
✎☛ ✎
✏ ✓✙
1 5
35
2 11
1 3 1 5 68
2 7 2 11
✒✍ ✓
✚✒☛✺
♦☞✯
✒✍✓
✬✭✻ ✤☞✗
✈☛✍✏
❡☞✓
❢✸✹✛☞
♥☛☞
III
❢✣✫♦☞✯
❡☞✓
✑✗
✒✍ ✔
✓
❢✸✹✛☞
✣☛☞★ ☞
✈☛✍✏
❄❢★✗☛✎☛
✙✤☛
✗✘
✫✤☛
P(A|E1) = P(
❢✸✹✛☞
I
P(A|E2) = P(
❢✸✹✛☞
II
P(A|E3) = P(
❢✸✹✛☞
III
✣☞
✈☛✍✏
III
✣☛☞★ ☞
✗☛
❢✣✫✗☛
✣☛☞★ ☞
✣☞
✣☞
✗☛
✣☛☞★ ☞
✣☛☞★☞
❢✣✫✗☛
✗☛
✗☛
I
❡☞✓
✑✗
❞✼★ ②☛ ✒✍ ✈☛✍ ✏ ✽✣❡☞✓ ✣☞ ✤☛♥✥✦ ✧✤☛ ✑✗ ❢✣✫✗☛
✬✭☛ ❢✤✗②☛
✒✍
❢✗
❢✸✹✛☞
♦☞✯
❞✤★
✗☛☞
✗☛☞
♥✿☛☛✪ ②☛
❢★❧❢✬②
1
3
P(E1) = P(E2) = P(E3) =
✒✘
✙✩✪
❡☞✓ ♥✾✣✏☛
✒✍ ✮
E1, E2
②✛
✣☛✌☛
❢✣✫✗☛
❢★✗☛✎✘
❢✗
❢★✗☛✎✘
❢★✗☛✎★☛
✙☞♥
✓
✬ ✭ ☛ ❢ ✤ ✗ ②☛ ✲
✗✘
❞☛✺♥✘ ✗☛ ❢✣✫✗☛ ✒✍✔ ✑✗ ♣✤❢✫② ✤☛♥✥✦✧✤☛ ✑✗ ❢✸✹✛☛
❢★✗☛✎②☛
✚✛
✵☛✏☛
❢✸✹✛☞
❢✸✹✛☞
✙☞✓♥
✒✍✓
✣☞
P(E 2 ) P(A | E 2 )
P(E2|A) =
P(E1 ) P (A | E1 ) + P(E 2 ) P(A | E 2 )
♠☎✆✝✞✟✆
✙☞♥
✓ ✓☞
1
2
P(A|E1) = P(
✌☛✍ ✎☞
✑✗
✈☛✍ ✏
✰☛✱★☛
✌☛✍ ✎☞
✒✘
I
✌☛✍✎☞
✬✭☛❢✤✗②☛
P(E1) = P(E2) =
②✛
✒✍ ✔
✓
✑✗
✒✍ ❁
❢★✗✎★☛
❢✣✫✗☛
❢✣✫✗☛
)=
✒✍✔
2
=1
2
)=0
❢★✗✎★☛
❢★✗✎★☛
)=
1
2
✗✏②☞
✒✍ ✓
❢✣✫✗☛
✐
✈
✝
✞✟ ✠
✝
✡
☛✡
☞
✌ ✍
✎✏✑
✞ ✎✒
✓
✑
✔
✑ ✕
✎✑
✡
✖
✡
=
= P(E1|A)
❢
✞
✢
✈
✝
✝
✡
✣
✑
✖
✓
✑
✓
✑
✗
✑
✡
✖
✡
✓✕
✘✙
✑
✞
✚
✓✛ ✑
I
✜✚
✑
✑
✎✑
✡
✖
✡
✓
✑
✞
✎✒
✓✑
✞
✟ ✠
✝
✡
✎✡
✗
✑
✡
✖
✡
✓✕
✘✙
✑
✞
✚
✓✛ ✑
✤
✘✙
☛✡
✚
✑ ✏✑
P(E1 ) P(A|E1 )
P(E1 ) PA|E1 ) + P(E 2 ) P(A|E 2 ) + P(E 3 ) P (A|E 3 )
P(E1|A) =
1
1
3
=
1
1
1 1
1
0
3
3
3 2
2
3
❢
♠ ✥
✦ ✧★
✩
✦
569
✁
✂✄
☎✆✁
✪
✫
☛
✑
✖
✡
❢
☞
✞✓
❡
✓
❡✬
✭
✈
✑
✮
✭
✯
✰✕
✭
✘
✏✕
✱
✑ ✲✑
✓
✕
✞
✰✳
✰ ✎✖ ✕ ✚
✛
✑
✞
✖
✴
✖
✞
✞✵ ✑
✛
✘✙
✓
✑ ✏
✎
✡
✞
✖
✞
✌
✶
✯
✷
✜
✓✕
✮
✯
✗
❞
✸
✹
❡
✬
✭✈
✑
✮ ✭
✯
✰
✕
✭
✘
✑
✡
✺
✕
✞
✷
✰
✻
✚
✞
✒
✛ ✚
✼
☛✡
☞
✎
✱
✑
☛
✗
❞
✸
❡
✬
✭✈
✑
✮ ✭
✯
✰
✕
✭
✜
✈
✑
✮ ✭
✯
✰✕
❝
✝
✖
✡
✽✑
✰
✝
✾
✖ ✎
☞
✵
✚
✑
✛ ✑
✛ ✑
✗
✎
☛✡
☞
✑ ❞
✏
❂
✎
✓
✑
✘
✏✕
✱
✑ ✲✑
✞
✓
✺
✗
❞
✞✓
✰
✗
❇
☛
✑ ✖
✒
✛ ✚
✒
✛
E
✡
☞
✰
✑
❁
✬
✎✑
❄
✑
✑
✮ ✭
✯
✗
✰✕
✭
E
✕
✘✶✷✛ ✚
✑
✏✕
✱
❁
❈
✼
✘
✰
✏✕
✞
✱
✑
✛
✒
✛
❡
✬
✭✈
✬
✜
✖
✡
☛✡
{E, E }
☛☞
✡
❡
✘
✏
✏✑
✡
✬
✭✈
✰✕
✭
99%
✑
☞
✡
✙
✑ ✲✑
☞
✈
✑ ✏
❞
✞
✕
✘✛ ✑
✛
❡✬
✭✈
✑
✮ ✭
✯
✰
✕
✭
✘✑
✡
✕
✞✷
✺
❞
✖
✑ ✖
✡
✑
✗
❞
✚
✑ ✖
✕
❡
✬
✭
✹
❡
✾
✗
❢ ✜
✘✛ ✑
❢
✜✑
✎
✗
❢
✰
✡
✛
☛✡
☛☞
✡
✎
✡
❡✓
✰
✝
✛
✑
✛ ✑
✿
❖
❀
✚
✞
✒
✛
✚
✑ ✌
✗
❁
✚
✑
✬
✸
❞
❡✓
✹
✖
✑
✑
✛ ✑
✗
✰
❃
✑ ✖ ✕
❡✬
✭
✈
✑
✮ ✭
✯
❣
✘
☞
✘
✑
✡
✑
✡
✰✕
✭
✰✑
✓
✕
❂ ✘
✞
❄
✑
✞✛
✝
✛
✑
✛ ✑
✗
✸
❞
✒
✚
✑
✘✙
✑
✞
✚
✓✛ ✑
✰
☛☞
✡
❡✬
✭✈
✑
✮ ✭
✯
✰✕
✭
✘
✑
✡
✗
✑
✡
✖
✡
✓
✕
✑
✷
✖ ✑
✓
✑
✡
✌ ✳
✑ ✑
✛
✯
✡
❡✬
✭
✈
✑
✮
✭
✯
✰✕
✭
✘
✑
✡
✰
✗
✗
☞
✸
❞
✗
✎
✔
✑ ✕
☛☞
✡
✑
✡
✖
✡
✓
✕
✑
✷
P(E|A)
❃
✖ ✑
✑
✛
✰
✖
✗
✑
✡
✖
✡
✓
✕
✑
✷
✖
✑
✓
✑
✡
✒
✛
✚
✑
☞
✡
✓
✌ ✳
✑ ✑
✛
✯
✻
✚
✞
✈
✑
❞
✏
✏✖
✑
✗
✺
✻
✚
✞
✒
✛
✰
✡
✸
❞
♣
✕
✞✷
✺
☛☞
✡
A
♣
✕
✞✷
✹
✰
✡
✑
❆
❞
♣
✰
✹
✚
✗
✹
✛
✻
✖ ✎
☞
✵
✰❅
✼
✰
✡
✕
✞
✷
✒
✛
✕
✞
✷
✰
✡
✑
✗
✸
❞
✹
✘✙
✞
✛
✌ ✳
✑ ✯
✎
☛✞ ✶✷
✓
✑
❡✓
✞
✰
✔
✑ ✑
✖
✗
✸
❞
✗
☛☞
✡
❃
✑
✛
✗
0.1
0.001
100
P(E) = 0.1%
P(E ) = 1 – P (E) = 0.999
❈
P(A| E) = P
✺
❣
✹
✚
✞
✒
✛
✼
✰
✑
✓
✑
✘
✏✕
✱
✑
✲✑
✹
✛
✰
☛✡
☞
❡
✬
✭✈
✑
✮ ✯
✭✰✕
✭
P(A|E ) = P
✘
✑
✡
✕
✞✷
✰
✗
❞
❅
☛☞
✡
❡✬
✭✈
= 90% =
✑
✮ ✭
✯
✰✕
✭
✘✑
✡
✹
✕
✞✷
✰
✌ ✳
✑ ✑
✯
✖
✑
✜✚
✝
✞
✓
✞
✌ ✚
✑
✑
✗
❞
✞
✓
✰
✗
9
0.9
10
❉
✈
✑ ❞
✏
✺
❣
✹
✚
✞
✒
✛
✓
✼
✰
✗
✰✑
✛ ✰
✑
✘
✏✕
✱
✑
✲✑
☛☞
✡
❡
✬
✭✈
✹
☛✡
☞
❡
✬
✭✈
❞
✼
✞
✑
✮ ✭
✯
✰✕
✭
✻
✚
✞✒
✛
✚
✞
❈
✏✕
✱
✖
✡
✜
✑ ✖
✡
✺
❡
✘
✻
✚
✞✒
✛ ✚
✑
✮
✭
✯
✑
✾
❡
✜✼
✚
✞
✺
❡
✑ ✲✑
✞
10%
❢ ✜✑
✘✛
❢
✰
✡
✺
✑
✑ ✲✑
✹
✛
✜
✖
✡
✏✕
✱
✺
✘
✹
❡✬
✭✈
✑
☞
✡
✼
✚
✞
❢
✧
✚
✘
✻
✹
✈
❡
✺
1%
✝
✞✓
0.1%
✹
✞
❞
✞
✚
✞
90%
❢
✰
✡
✰
✛
☞
✹
✡
✞✷
✹
✟ ✕
✎
✡
✑
✡
☞
✑
✮ ✭
✯
✰✕
✭
✘
✑
✡
✕
✞✷
✰
✖
✗
✕
☞
✗
❅
❞
✑
✮ ✭
✯
✰✕
✭
✘
✑
✡
✹
✕
✞✷
= 1% = 0.01
✰
✌ ✳
✑
✑ ✖
✯
✑
✝
✜✚
✞
✓
✞
✌
✚
✑
✑
✗
❞
✞✓
❞
570
① ✁✂✄
✈☎
☎✆✝ ✞✟✠✡☛✆☞
✌✞✍✞
P(E) P(A|E)
P(E) P(A|E) + P(E ) P (A | E )
P(E|A) =
=
0.001 0.9
90
= 0.083
0.001 0.9 0.999 0.01 1089
✈✓✔ ✕✖ ☞✞✗✘✙✚☞✞ ✛✜ ✢✆ ✏✕ ✣☞✤✥✓ ✦✆✧ ✦✞★✓✦
✤✖
❢✞✓
✱✲
♠✸✹✺✻✼✹
✤✖
✽✾
25%, 35%
✑✞✞✏
❀✞✍✞☎
✕✛✪✈✞✫✬✪✦✮✪
✳✴✖✞
✕✖ ☎✞✆✿✰ ☎✢✞✢✆ ✦✆✧
40%
✈✞✲✍
✭❃✞✜ ✤✰✠❍ ✶✞✬✒
✱✲✩ ❣
☎✢✞✓✮
✱✲ ✩ ❣ ☎✞✆✿✰✞✆✩
☛✞✢
B1 :
☎✞✆✿✰
☛❂✞✮✢
A
✌✞✍✞
☎✢✞☞✞
✏☞✞
✱✲
B2 :
☎✞✆✿✰
☛❂✞✮✢
B
✌✞✍✞
☎✢✞☞✞
✏☞✞
✱✲
B3 :
☎✞✆✿✰
☛❂✞✮✢
C
✌✞✍✞
☎✢✞☞✞
✏☞✞
✱✲
★✠▲✰
✠✡ ✖✞✍
✱✲✔
❡✞✰✢✞
E,
✱✲
E
B1, B2, B3
✤✖ ❡✞✰✢✞✕❉
☎✞✆✿✰
❡✞✰✢✞✈✞✆✩
❀✞✍✞☎
✱✲ ❣
B1
B2
☞✞
✥☞✞
☞✞
B3
✠✞✆✯ ✮✤✰✦
☛✆✩
☛❂✞✮✢✞✆✩
✳♦✠✞✗✢
✤✢❊✢
✠✍★✠✍
✦✆ ✧
✴✞❋✞
0.083
✱✲✷
☛❂✞✮✢✆ ✩ ✭☞✩ ❃✞✒
✠✡✞✤☞✖✓✞
B1, B2, B3
❡✞✰✢✞✕❉
✫✢
✦✜✧✎
✺❈
✤✖
✫✴✖✮
✦✆ ✧
❀✞✍✞☎
✤✎☞✞
✱✲ ❣
✖✞✍❀❁ ✞✞✢✆
✦✱
✠✞☞✞
✯✞✓✞
☎✞✆✿✰
✠✍✮✵✞✶✞
☛✆✩ ✕✛✪✈✞✫✬✪✦✮✪
✱✲
✦✆✧
☛✆ ✩ ✴✆
✤✖
✠✡✖✞✍
☞✱
P(E|B1) =
✤✢✤☛✬✓
✈✞✲✍
✖✞
☎✞✆✿✰
C
✦✜✧✎ ✳♦✠✞✗✢ ✖✞ ❄☛❂✞✔
❝✷
❄☛❂✞✔
☞✞✗✘✙✚☞✞
☛❂✞✮✢
B
❅✷
✈✞✲✍
✤✢✖✞✎✞ ✯✞✓✞
✌✞✍✞
☎✢✞☞✞
✱✞✆ ✓✮
✠✤✍✠❍ ✶✞✬
✱✲ ❣
✤✗☞✞
✈✞✲✍
✱✲ ❣
☛✞✢
✤✎☞✞
☎✆✝ ✞✟✠✡☛✆☞
✱☛✆✩
❢✞✓
☎✞✆ ✿✰ ✦✆✧ ❀✞✍✞☎ ✱✞✆✢✆ ✖✮ ✠✡✞✤☞✖✓✞ ✯☎ ✤✖ ✤✗☞✞ ✱✞✆ ✤✖ ✦✱ ☛❂✞✮✢
P(B2|E) =
=
✱✲
✱✲❇
E
✤✢❊✢
P(B3) = 0.40
P(E|B2) = 0.04, P(E|B3) = 0.02
✌✞✍✞
✱✲ ✈✞✲✍
✱✲ ✔
✱✲
✠✡✖✞✍
✠✡✤✓❂✞✓
✏☞✞
✤✖ ❡✞✰✢✞
= 5% = 0.05
✫✴✮
❆
✱✲✔
P(B1) = 25% = 0.25, P (B2) = 0.35
✠✜ ✢✔
✖✮ ✠✡✞✤☞✖✓✞ ✯☎
✱✲ ❣
✕✖ ☎✞✆✿✰
✈✠✦✯✮✬ ✈✞✲✍
❡✞✤✰✓
✠✞✆✯ ✮✤✰✦ ✱✞✆✢✆
A, B
✳♦✠✞✗✢
✭✎✏✑✞✏✒
✤✖
P(B2 ) P (E|B2 )
P(B1 ) P (E|B1 ) + P (B2 ) P(E|B2 ) + P (B3 ) P (E+|B3 )
0.35 0.04
0.25 0.05 0.35 0.04 0.40 0.02
0.0140 28
0.0345 69
B
✌✞✍✞
✐
♠ ✝
✞ ✟✠
✡
✞
✯
✪
✛
✜
✛
✜
✥
✔
✦
✒
✔
✰
✫
✮✗
✲
✪✔
✣
✟❣
✙
✲
✌
✺
✎ ✙
✲
✒
✔
✰
✙ ✲
✮✌✪
✙
✔
✱✌
✳
✪
✥
✛
✓
✩
✭ ✧
✚
✎
☛
☞
✖
✖
✒✓
✴
✥✵
✦ ✒✓
✳
✪
✚
✶
✣✷
✎ ✮
✤ ✖
✮
✍ ✎ ✏
✑
✫
✎
✫
✫
✎
✥
✎
✛
✙
✪✔
✪✔
✬
✎ ✭
✮✌
✱
✌
✮✌
✪
✴
✥
✦
✵
✒✓
✫
✌✭ ✎
❧
✱
✌✎ ✔
✚
✙
✔
✖
✮❧
✱
✒✓
✶
✕✖
✫
✚
✎
✭ ✎
✚
✎
✌
❡
✍ ✎ ✑
✏
✖
✓
✎ ✔
✥
✦
✔
✣
✌
✎ ✔
✥
✎ ✛
✛
✜
✓
✭ ✎
✔
✯
✪
✗ ✘
✔
✎ ✙
✔
✙
✪✔
✯
✪
✔
✗ ✔
✚
✎
✥
✦
✔
✓
✚
✎
✙
✔
✗
✔
✓
✙
✛
✖
✩
✙
✎
✛
✜
✌
✪✔
✖
✢
✖
✙
✔
✛
✜
✲
✴
✥
✵
✦ ✒✓
✫
✎
✮✌✪
✖
✚
✶
✫
P(E|T1) =
✓
✎ ✔
✕✖
✗ ✔
✖
✓✖
✣✷
✌✎ ✓
✙
✛
✖
✩
✥
✔
✦
✫
✛
✎ ✸
✗ ✔
✓
✪✔
✚
✙
★
✧
✎
✥
✎ ✩
✔
✪✔
✹
✺✻
✎ ✼
✥
✛
✗ ✔
✓
✫
✛
✬
✎ ✭
3 1 1
, ,
10 5 10
1 1
, ,
4 3
✹
✺✻
✎
✪✔
✚
E
T1, T2, T3,
2
P (T4 )
5
✎ ✙
✔
✿
✎ ✓
✎
✚
✎ ✙
✔
✌
✖
❁
✎
✒✙
✎ ✱✸
✹
✈
✌
✺✻
✎
❙
❥
❦
✥
✦
✔
✒
✔
✰
✙
✛
✜
❀
✽
✮
✌
2
5
1
12
❦
✾
✼
✎ ✫
✎
✲
✭ ✎
✔
✖
❁
✎
✒✙
✎
✛
✜
✢
✫
✮
✗
✚
✎ ✜
✼
✍ ✎ ✑
✏
T4
✓
✯
✪
✓
✎
✚
✎ ✙
✔
✣
✓
✗ ✔
✓
✪✔
✣✛
✸
✧
✙
✔
✌✖
✣✷
✎ ✮
✒✓
✛
✎
✔
(
✮✗
✥
✦
✔
✒
✔
✰
✙
✫
✌✭ ✎
✥
✦
✔
✲
✒
✔
✰
✙
✲
✭ ✎
✔
✫
✎
✛
✜
)
1
4
=
▼
✎
1
1
, P (E|T3) =
, P(E|T4) = 0,
3
12
, P(E|T2) =
✛
✙
✒✓
✚
✣
✎
✷
✮✫
✌
✭ ✎
✱✸
✫
✮✗
❝
✍ ✎ ✑
✏
✥
✔
✦
✢
3
1
1
, P (T2 ) , P (T3 )
10
5
10
P(T1) =
❜
✪
✥
✎ ✛
✌✖
✢
❝
✳
✪
✤
✔
✣✷
✎ ✮✫
✌✭
✎ ✱✸
✚
✎
✛
✎ ✭
✔
✣✛
571
✁
✂✄
☎✆✁
❝
✑
✫
✎ ✩
✔
✮
✌
✚
✶
✫
✥
✎ ✛
✙
✎
✓
✎
✚
✎
✙
✔
✣
✓
✯ ✪✔
✎
✔
✭ ✖✢
❝
✚
✳
✳
✔
❂
✎ ❃
✣✷
P(T1|E) =
✺
✔
✫
✎
✓
✎
❝
✍ ✎ ✏
✑
✒✓
❝
✎
✓
✎
✗ ✔
✓
✎ ✖
✒
✣✷
✎ ✮
✫
✌✭ ✎
❅
♠ ✝
✞ ✟✠
✡
✞
✌✎ ✔
✯ ❞
✎
✮✌
✟
✣✎ ✪✔
❣
✪✩
✘
✪
✩
✘
✺
✎
✫
✎
✫
✎
✙
❊
❊
✣
✱✌
✛
✜
✓
✛
✜
✿
✙
✛
✚
✎ ✜
✚
✎
✤ ✖
✮
✛
✓
✒
✔
✰
✙
✎ ✓
✎
✚
✎
✙
✔
✓
✖
✚
✎
✳
✭
✥
✎
✙
✔
✭
✜
✖
✣✷
✎ ✮
E
✒✙
✎
✌✖
✥
✦
✔
✪✘
✩
✳
✎
✛
✜
✫
✎
✓
✔
✺
✩
✔
✮✌
✥
✎
❡
✢
1
4
1
10
1
12
❁
✎
P(S1) =
✒✙
✎
✪✩
3 120 1
40 18 2
2
0
5
✥
✛
✜
✣
✺
✔
✩
✮✌
✓
✚
✎
❊
✛
✜
✙
✔
✥
✛
❈
✺
✩
✔
✥
✎ ✤ ✖
✪✔
✪✘
✩
✫
✎
❉
✳
✎
❊
✛
✜
✓
✢
✪
✫
❜
✪
✳
✎ ✔
✌✖
✤
✭ ✎
✣✷
✎
✮
✛
✜
✢
✥
✛
✫
✌✭
✎
✱✌
✬
✎
✭
✣✎ ✪✔
✌
✖
✮
❧
✱
✢
❝
✫
✮✑
✺
✎
✬
✎
✭
✯
✪
✴
✭
❆
✲
✛
✜
✫
✌✭ ✎
✢
✤
✎
✭ ✎
✤
✖
✮
✌
❁
✎
✌
❇
✫
✮
✑
✙
✔
❧
✱
✌
✖
✩
✣
❆
☛
✤
✭ ✎
✎ ✙
✔
3
10
=
3 1 1 1
10 4 5 3
1
2
❄
✚
★
✚
P(T1 ) P (E|T1 )
P(T1 ) P (E|T1 ) + P (T2 ) P(E|T2 ) + P (T3 ) P(E|T3 )+ P (T4 )P (E|T4 )
=
✚
✭ ✼
✪✔
✭
✙
✛
✩
✜
✘ ✫
✎
✎ ✓
✎
✤ ✖
✮
✢
✣✎ ✪✔
❧
✱
✮
✌
✌✎ ✔
S1
✲
✯
❞
✎ ✤
✣✎ ✪✔
✣
✌
✓
✓
✫
✛
✪✩
✘ ✫
✎
✳
✭ ✎
❊
✙
✔
✚
✎ ✙
✔
✭
✳
❊
✚
✎ ✙
✔
✌
✖
❁
✎
✒✙
✎
✌
✖
✣✷
✎ ✮
✫
✌✭ ✎
=
1
6
✌
✖
✌✖
✮✌
❁
✎
✯ ✪
✒✙
✎
✣
✚
✎ ✜
✓
✓
✚
✎
S2
✙
✔
✥
✎
✣✎ ✪✔
✤ ✖
✣
✓
572
① ✁✂✄
P(S2) =
P(E|S1) =
❧☎✆ ✝✞
✟
✡✛✦
✍☛
=
P(E|S2) =
❧☎✆ ✝✞
❖✝✓✕✔
✥✌ ☎
✢✞✤✔✢
✟
✘✌✈✞★✑✒✥✌✝
P(S1|E) =
✠✡☛☎
☞✞✠✌
✍☛
✎✞✏✠✞
✘✔✞✠✌ ✑✗ ✓✍ ✑✞❧✌ ✓✍
✍☛
✑✒✞✓✝✍✔✞
5
6
=
❧☎✆ ✝✞ ✟ ☞✞✙✚ ✡✛ ✜✘✓✍ ✑✞❧✌ ✑✗ ☞✞✠✌
❧✧✝
✖✞✗✞
✘✞✌ ✣✠✌
✍☛
✝✡ ✘✔✞✠✌ ✑✗ ✓✍ ✑✞❧✌ ✑✗
✡✛✦
✍☛
❧☎✆✝✞
✟
3
4
=
✑✒✞✓✝✍✔✞
☞✞✙✚ ✡✛
✜✘✓✍ ✑✞❧✌ ✑✗ ☞✞✠✌
❖✝✓✕✔
❧✧✝
✖✞✗✞
✠✡☛☎
✘✞✌ ✣✠✌
✍☛
✝✡
✖✞✗✞
✘✔✞✠✌
✍☛
✑✒✞✓✝✍✔✞
✓✍
❧☎✆✝✞
3
8
✑✒✞✓✝✍✔✞
✟
✑✒✍✏
✡✩✙✚ ✡✛✦
✜✘
✵✍
✍✣✶✞
✠✞✌✏
✍✗✠✌
✍✣✶✞
2.
✥✌ ☎
✡✞✌✠✌
✍☛
✵✍
❞✞✛ ✣✌
✥✌ ☎ ✷
✣✞✣ ☞✞✛ ✗ ✷
✢✌ ✉ ✘✞✹
✗✆✞
✹☛
✑✩ ✠✪
✥✌☎
✿
✓✍
3.
✣✞✣
✡✛ ✭
✡✛
✣✞✣ ☞✞✛ ✗
✙❧
✠✡☛☎
❧✌
30%
✗✡✔✌
☞✞✛ ✗
✡✛ ☎✭
✙❧
✘✞✔
✥✌ ☎ ✗✆✞
✔❞✞✞
✿
✍✣✶✞
✘✞✔
✝✞✹✺ ✻✼✝✞
✵✍ ✸✌ ☎✹
✓✠✍✞✣☛
✜✞✔☛ ✡✛ ✭ ✑✩ ✠✪ ✓✠✍✞✣✌ ✸✵
✹☛
✥✌ ☎
❧✌
✍✞✣☛
☎
✸✌✹
✵✍
✸✌ ☎✹✌☎☎ ✡✛ ☎ ☞✞✛✗
✵✍ ✍✞✌ ✝✞✹✺✻✼✝✞ ❁✩✠✞
✍☛
✕✝✞
✑✾ ✢✢
✚ ✔☛✚
✢✬✞✚
✼✞❅✞✞✢✞❧ ✥✌ ☎ ✠
✍☛
✡✛
1 24 3
8 8 8
✓✠✍✞✣☛
✗☎✸
✜✞✔☛ ✡✛ ✦
✍☛ ✽
✜✞✔☛
✡✛ ✭
✥✌ ☎
✣✞✣
✕✝✞
✙❧✍✞
✗☎✸
☞✓✔✓✗✕✔ ✸✌☎✹✌ ☎
✹✾ ❧✗☛
✑✒✞✓✝✍✔✞
✢✌✉ ✑✓✗❡✞✞✥
✗✡✠✌
✢✞✣✌
✵✍
☞❀✝
❞✞✛ ✣✌
✽
☎
✜✞✔✞ ✡✛ ☞✞✛ ✗ ❂❧✥✌ ☎ ✵✍ ✸✌✹
✡✛
✓✍
☎
✸✌✹
60%
❧✾✓❁✔
✍✗✔✌
✼✞❅✞✞✌ ☎ ✥✌☎ ❧✌
✑✡✣✌
❞✞✛ ✣✌
❧✌
☞✞✛ ✗
☎ ✌☎☎
✸✌✹
✓✠✍✞✣☛
✼✞❅✞✞✢✞❧ ✥✌ ☎ ✗✡✔✌ ✡✛ ☎ ☞✞✛ ✗
✡✛ ☎ ✓✍
20%
✼✞❅✞✞✢✞❧
✼✞❅✞✞✌ ☎ ✠✌
✑✒✞✓✝✍✔✞
✡✛
✓✍
✢✡
✼✞❅✞
✼✞❅✞✞✢✞❧
✥✌
✟
✓✠✍✞✣☛
✥✌ ☎
✸✙✚
✍☛
✣✞✣
✍✞✣☛
✗✡✠✌
☎ ✌☎
✸✌✹
✜✞✔☛ ✡✛ ✜✞✌
✡✛ ❣
40%
✢✞✣✌
✼✞❅✞✞✢✞❧
✼✞❅✞✞✌ ☎ ✥✌ ☎
A
★✸✒✌ ❆ ✓✣✝✞✭ ✢✬✞✚ ✢✌ ✉ ☞☎✔
✥✌ ☎ ✥✡✞✓✢❄✞✣✝ ✢✌ ✉ ✵✍ ✼✞❅✞ ✍✞✌ ✝✞✹✺ ✻✼✝✞ ❁✩✠✞ ✸✝✞ ☞✞✛✗ ✝✡ ✑✞✝✞ ✸✝✞ ✓✍ ❂❧✌
✡✛ ✭
✟
13.3
✝✡ ❃✞✔ ✡✛ ✓✍ ✵✍ ✥✡✞✓✢❄✞✣✝ ✢✌ ✉ ✼✞❅✞✞✌ ☎ ✥✌☎ ❧✌
✥✌☎
3
4
5 1
6 4
❧☎✆✝✞
✡✛ ❣
✕✝✞
✡✛ ☎✭ ✹✞✌✠✞✌ ☎ ❞✞✛ ✣✞✌ ☎ ✥✌ ☎ ❧✌
✍✞✣☛ ✸✌✹✌ ☎ ✡✛ ✭
☎
✍✣✶✞
✜✞✔☛
✑✒ ✞✓✝✍✔✞
✥✌☎
✢✞✤✔✢
✡✛ ✭
✐✮✯✰✱✲✳✴
1.
3 1
4 4
1
✑✒✞✓✝✍✔✞
✖✞✗✞
❖✝✓✕✔
☞✫✞☛✬✏
✢✞✣☛
✑✒ ✞✓✝✍✔✞
1
P(S1 ) P(E|S1 )
6
=
=
1 3
P(S1 ) P(E|S1 ) + P(S2 ) P (E|S2 )
6 4
☞✔✪
✢✞✣☛
✑✒✞✓✝✍✔✞
✖✞✗✞
✠✡☛☎
✟
❖✝✓✕✔
=
☞✘
✝✡
❖✝✓✕✔ ✖✞✗✞
✥✌ ☎
✢✞✤✔✢
❧☎✆✝✞
✗✡✠✌
✢✞✣✞
✡✛ ❣
A-
✸✒✌ ❆ ✓✥✣✞
✐
4.
✱✝
✞
✟
✠
✡☛
✝
✈
☞✌✍
✌✎
✏
✑
✦ ✧
✑
✠
✘
✒
✑
★
✒
✌✎
✒ ✡✢
✝
✣
✒
✌✎
✒ ✡✢
✝
✣
✒
1
4
1
4
✣
✒
✟
✘✒
✑
★
✟
5.
✟
✣
✘✒ ✑
✣
✒
✡✝
✡☛✏
✒
✧✍
✕
✒
✗
✕
✒
✩
✗
✘
✗
✙
✩
✧
✓
6.
%
✕
✒
✩
✒ ✖ ✷
✸❣
✢
✗
✖
✟
✒
✣
✗
✒
7.
✥
✡✝
✡✖ ✱
✠
✕✣ ✍
✰
❃
✟
✕
☛
✙
✣
✗
✒
✈
✍
✜
✗
✙
❄
✘
✕
✗
✙
✝
✱✝
✍
✖ ✠
✗
✕
✝✒
✈
✟
✥
✣
✒
✟
✥
✈
✒
✕
✥
✢
✒
☛
✟
✦ ✧
✑ ✘
✠
✦ ✧
✑
✠
✘✒ ✑
✤
✒ ✑
✒
✑
✒ ✑ ✗
✝
✍
✝
✍
✩
✒
✑
✗
✌✕
✟
✡✝
✝✒ ✗
✜
❣ ✒
✫
✒
✌✎
✏
✑
✝✒
✍
✓
✓ ✔
✒ ✕
✪
✗
✕
✕
✒
✴
✤
✒
✡✝
✳
✍
✩
✗
✕
✥
✔
✒
✕
✖ ✑
✗
✤
✒ ✑
✣
✒
✗
✟
✥
❝✢
✡
✕
✢
✡✖
✞
✣
✟
✥
✡✝
✳
✟
✥
✤
✞
✭
✣
✦
✣
✲
✕
✖ ✒
✭
✛
✒
✧✎
✳
✝✣
✒
✝
✍
✕
✣
✒
✟
✝
✒
✗
✤
✒
✘
✈
✙
✗
✙
✗
✢
✕
✗
✡
✹
✒ ✣
✑
★
✢
✒
✟
✍
✟
✡
✩
✥
✢
✢
✡✖
✒
✖
✤
✒ ✣
✒
✩
✥
✖ r
✡
✝
✸
✷
❣ ✢
✢
✡✖
✡
✟
✥
✒
✩
✭
✕
✒
✡
✕
✑ ✗
✤
✑
✪
✙
✗
✰
☛✗
✌✕
✡
✓
✘✖
✠
✒ ✢
✭
✢
✒
❝✢
✡
✘
✢
✟
✮
✗
✟
✥
✹
✒ ✑
✣
✱✝
✲
✌✎
✝
✟
✒
✟
✥
✡
✤
✩
✭
✗
✴
✣
✕
✒
✩
✡
✙
✗
✣
✧
✗
✈
✝✒
✣ ✍
✑
✒
✙
✗
✌✕
✧
✩
✟
✥
✘
✩
✍
✑
✠
✒
✡✝
★
✝
✒
✩
✭
✟
✥
✣
✒ ✢
✒
✩
✭
✑
✡
✧
✞
★
✣
✈
✕
✒
✟
✰
✒
✝
✰
✜
✣
☛
✰
✯
✣
★
✒
☛✒
✩
✤
✒
✖ ✒ ✑
✗
✒
✣
✍
✡
✘
✙
✗
✪
☛
✗
✝✒ ✗
✭
✗
✣
✒
✗
✢
✒
✌✎
✒ ✡✢
✝
✣
✒
✮
✟
✦
✒
✥
✲
✰
✝✒ ✙
✗
✻ ❡
❡ ❡
✾
✣
✒
✱
✝✒ ✕
✿
✟
✒ ✗
✤
✒
✣
✒
❝✢
✡
✕
✥
✼ ❡
❡ ❡
✪
✕
✥
✪✴
☛
r
✝✒
✰
✙
✗
✝✒
✞
✪
✟
✥
✕
✦
✒
★
❡ ✵
✶
✳
☛✗
✰
✝
✽
❂
✒
✣
✴
✒
✈
✭
✓
✙
✗
❡ ✵❡
✩
✟
✥
✈
✝
✒
❁
❡ ✵❡ ✶
★
✣
✦
✒
✲
✘✏
✒
✧✎
✳
✑ ✒
✈
✩
✳
☛
✗
✌✎
✒ ✡✢
✝✍
☛✗
✒
✰
✕
✒
✪
✗
✝✒
✌✎
✒ ✡✢
✝
✜
✍
✰
✯
✩
✭
r
✝
✩
✞
✒
99%
✰
✯
✣
✬
✢
✩
✒
✡
✴
✒
3
4
✣
✒
✕
✞
✍
✘✒ ☛
✷
✍
✘✒
✭
✣
❝✢
✡
✣
✢
✒
✙
✗
✦
✒
✝
✟
✒
✑
✗
✗
✝
✍
✌✎
✒
✡✢
✝✣
✒
✟
✥
A
B
60%
A
40%
B
1%
✦ ✧✍
✈
✱✝
✝ ✒ ✕❅
✒ ✒ ✑ ✗
✘
✗
✙
✒
✖ ✒
✕
✥
✗
✘✏
✒
✍
✑
✘✏
✒
✈
✒
✍
✑
✕
✥
❆
✒ ✥
✕
✘✏
✒
✈
✍
✑
✩
✒ ✥
✕
✝
✒
♠
✓
✓
✟
B
✕
✝
✍
✌✎
✒ ✡
✢
✢
✒ ✖
✝
✦
✗
✸
✷
❣ ✢
✒
✣
✒
✡✑ ✝✒
✱
✝
✡✑
☛
✤
✍
✣
✑
✗
✝
✍
✡✑ ✖ ✏
✗
✒ ✝
✌✎
✒ ✡✢
✝
✥
✣
✒ ✗
✱
✝
✑ ✱
✬ ✩
✓
✣
✝
✍
☛
✗
✕
✒
✌✎
✒
✕❋
✹
✒
✒
✬
✍
✡
✤
✱
✡✝
✝✒ ✗
✩
✭
✟
✥
✣
✒ ✗
☛
✟
✱✝
✡
✡✖
✣
✒
✦
✬
✩
✥
✓
✕
✒
✞
✈
✗
✟
✥
✡✝
✣
✡✕
✣
✝✒
✬ ✩
✟
✒ ✗
✣
✒
✱✝
☛
A
✘✏
✒ ✍
✑
✞
✑ ✒
✡
✢
☛
2%
✝✒
✗
❇
✘✏
✒
✒
A
✪
✣
✠
✌
✒ ✖ ✑
✦
✕
✗
✳
✗
❆
✓
✭
✡
✦
☛
✠
♠
✌
✒
✖ ✑
✲
✒
✪
☛
✪
✣
✒
❆
☛✠
❅
✣
✠
✟
✦
✌✣
✒
✍
✑
❈
✤
✒
⑥
✒
✣
✒
✕
✒
✟
✥
✞
✑
✗
✳
✘
✗
✙
✳
✛
✒ ✒ ✑
✌
✒
✑
✗
✝✍
★
✣ ✛
✒ ✒
❡ ✵
✻
✟
✌✎
✡
✬ ✩
✙
✥
✣
✪
★
✌
❊✒
✜
✈
☛✗
✘
✙
✗
✟
✟
✒ ✑
✗
✝✣
✒
❡ ✵
★
✟
✗
✝
✍
✌✎
✒ ✡✢
✝
✣
✒
❡ ✵
✬
✩
✥
●
✌✟
✣
✡
✕
✍
✌✎
✒ ✡
✢
✝
✣
✒
✣ ✛
✒ ✒
✢
✡✖
✌✟
✩
✒
✕
✥
✖ r
✦
✣
✢
✡✖
✖
r
✣
✝
✍
✡✤
✱
✡✝
✖
✦
✒
✖
✤
✍
✣ ✣
✒
✖
✤
✍✣ ✣
✒
❆
✒
✦
✕✗
✦
✕
✒
❀
✝
✩
✗
✭
✡
✟
✥
✦
✙
✥
✑ ✢
✒
✓
✟
✥
✣
✒ ✗
✩
✌
✒ ✖ ✑
✖
r
✕✗
✦
✖
★
✢
✒
✦
✘✒ ✑
✪
✢
☛
❡
✵✼
❁
✌✎
✒
✡✢
✡✝
✢
✤
✒
✰
✗
✈
✌✎
✒ ✕❋✹
✒
✧
⑥
✒
✕♥
✒
✪
✌
✒ ✖
✙
✒
✿
✘✏
✒
✩✧✣
✞
✒
✡
✝✢
✳
✜
✘❉
✙
✾
✣
✒
✱
❆
✟
✕
✒
✟
✥
✦
✗
✯
✗
✡
☛☛
✟
✒ ✗
✪
☛
✜
★
⑥
✒
✧
✬
✍
✪
✘
✌r
☛
✧✍ ✮
✢
✒
✧
✖
✕
✒
✞
✦
✗
✩
✙
✥
★
✒
✩
✗
✍
✑
❅
✌
✒ ✖ ✑
✭
✟
✒ ✑
✗
✗
✖ ✒
✘✏
✒
★
✗
✈
✝✒
10.
✌✎
✒ ✡✢
✝
✓ ✔
✒
✮
✢
✒
9.
✔
✒
✧
✪
☛
✓ ❣ ✒
✒
☛
❞
✗
✕
✝
✒
✣
✦
✗
☛✒
✙
✗
✍
✭
✱
✣
✯
✒
✝
✟
✥
✭
✗
✈
✑ ✒
✓
✝✣
✒
0.5%
✣
✒
✪
☛
✟
✥
✓
✺
❡
❡ ❡
✴
✒
✗
✝
✒
✟
✥
✡
✡
✟
✥
✡
✦
✣
✑
✈
✗
✦
✗
✣
✠
❝✢
✩
✕
✥
✩
✝
✒
✭
8.
❞
✖ ✠
✦
✒
☛
✩
✭
✳
✌
✑
✒
✈
✒
✌✎
✏
✭
✓
✱✝
✟
✒
✢
✒
✡
✍
✣
★
✧
✗
✙
☛
✌✎
✒
✡✢
✧✎
✳
✢
✡✖
✌✎
✝
✖ ✒ ✗
✑
✒
✥
✢
✒ ✑
✩
✟
✙
✥
✗
✬
✒
✡
✖ ✢
❥
✝
✩
✒
✱
✑
✒
★
✝
✥
✴
✞
✒
✒
✍
✘✒
✟
✗
✧
✗
☛✟
✞
✌✎
✏
✑
✢
✡
✑ ✖ ✒ ✑
✣
✗
✒
✗
✪
✱✝
✍
✓ ✔
✒ ✕
✧
✗
✣
✒
✮
✟
✥
✧
✟
✒
✢
✒
✪
☛✗
✪
✰
✟
✝
✍
✟
✍
✕
✒
✣
✤
✒
✑ ✑
✭
✜
✧✎
✳
✰
✢
✓ ✔
✒ ✕
★
✓
✪
☛
✗
✩
✗
✩
✗
✗
75%
✣
✕✌
✒
✧
✒
✗
✡
✰
✡
✡
✦
✩
✭
✣ ✍
✑
✫
✒
✡☛✚
✒ ✛
✒ ✍✜
✪
☛
✴
✣
❡ ✵✶
✣
✟
✣
✧✦
✡✑ ✖ ✒
✑
❣
✒
✧✎
✳
✗
✱✝
✩
✗
✪
☛
✙
✗
✭
✞
✒
✑
✕
✒ ✗
✧
✓
✡
✝
✗
✓
✘
✪
✙
✗
✧
✒
✩
✗
✓
✬ ✩
✥
❢
✍
✡✝
✦
✥
✟
✟
✥
✩
✡✝
✖ ✑
✗
✗
✩
✙
✗
✩
✒
✓ ✔
✒ ✕
✦
✥
❀
✢
✝✒
573
✁
✂✄
☎✆✁
✛
✒ ✒
✦
✜
✩
❉ ✝
❍
✍
✱✝
✌
✒
✒
✦
✓ ❣
✒
✪
☛✗
★
✣
✍
✦
✝
✒
✗
✣
✍
✑
✞
✒
✕
✓ ❣ ✒
✟
✥
✩
✢
✡
✖
✈
✣ ✍
✟
✥
✒ ✥
✓
❂
✩
✗
✢
✒
✰
✕
❇
✡
✼
✝
✍
✩❅
✣
✒ ❈
✙
✗
✝
✍
✙
✢
✒
✙
❅
✢
✒
✌✎
✒ ■ ✣
✴
✑ ✒
✗
✟
✒
✣
✗
✍
★
✝
✕✣
✍
✟
✥
✢
✡✖
574
① ✁✂✄
♠☎✆ ✝✞
✟✞ ✠ ✡☛ ☞ ✌✍ ☎✎ ✏✡☛ ✑✒☛✓✔ ✕☛✆ ✔✍
✡✕ ❀☛✆ ✦ ✌✢✔✍
✕✖✞
11.
✔☛✆
♠☎✗✆ ✛
♠✣☛✤✆
✮✘
✑☛☎✆
✑✢
A, B
✘✌ ✱✡☛✗☎☛✙✡✌ ✙❀✰☛✲✔☛ ✗✆✛ ✑☛☎
✕✖✫
✌☛✡✲
☎✰✡
✌☛
20%
t☛❀✆
12. 52
✕✖✎ ✫
❣☛✢☛
✙★✔
(A)
14.
✡✙✬
✌✍
✏☛☛✆
✮✘
4
5
A
✕✖
B
✑✷☛☛
Ø✰❡☛✴
✤✮☛✔☛
✕✖✞
5%
B
✗✪✛✤
♠❧✑☛✙✬✔
☎✰✡
✕✖
✔☛✆
t☛✔☛ ✕✖✫
✌✍
4
5
✑✒☛✙✡✌✔☛
✺✑
✑✒☛✙✡✌✔☛
✕✖✫
✰✆✎
✘✌
✙★✔
✕✖✎
✚✡☛
A
✽
B
♠✣☛✤☛
✕☛✆ ❀✆
✌✍
1
5
P(B)
✔r☛☛
✑✷☛✆
✰✆✎
✙✌☎✍
✗✪✛ ✣
✬☛✆
✑✢✍❲☛❳☛☛✆✎
✑☛☎☛✆✎
☎✌✔✆
✘✌
✕✖
✌☛✆
✕✰
✙✗ ❡☛✆ ✸☛
✑✆✎✛ ✌❀✆
✑✒✙✔✬❡☛✲
✏☛✢☛✜
A
30%
❣☛✢☛
C
✔r☛☛
♠❧✑☛✙✬✔
✗✪✛✤
✙✌✘
t☛✔✆
t☛✆ ②✹ ✦ ✗✆ ✛
✕✖✎
t☛✔☛
②❀✰✆✎
✕✖
✑✒☛✙✡✌✔☛
✔r☛☛
0
✔☛✆
✌☛✆ ②✲
☎✆
✜✔☛✔☛
✕✖
✙✌
✕✖ ✴
2
5
(D)
✾
A
✰✆✎
✙❀✿❀
☎✆
✌☛✖❀
✭✍✌
✕✖ ✴
❀✕✍✎
(Random Variables and its Probability
✙❀✰☛✲ ❳☛
✑ ✙✢ ❳ ☛ ☛✰
✑✙✢❳☛☛✰☛✆✎
✗✆✛
A 1%
✗✆✛
✜☛✢✆
✰✆✎
② ❱✣✪ ✌
✗✆✛
✑✕✤✆
❀ ✕✍✎
✕✍
r ☛✆
☎✍✏☛
★✪✗✛
✆
❩ ✌✔ ✪
✕✖✎
②❀
②❀
✑✢✍❲☛❳☛☛✆✎
✑ ✙✢ ❳ ☛☛✰ ☛✆✎
☎✆
r☛✆✫
②❱✣✪✌
♠❀✗✆✛
✕✖ ✯
✑✢✍❲☛❳☛
✑✢
✰✆✎
✙✗★☛✢
✌✢✆✎ ✫
✬☛✆ ❀☛✆✎
✑☛☎☛✆ ✎
✕✰
✑✢
✑✒✌✦
✙★✔☛✆ ✎
✌✍
☎✎ ✏✡☛✥☛✆ ✎
✗✆✛
✡☛✆✮
✰✆✎
②❱✣✪ ✌
✕☛✆
✕✖✎✫
✙☎✚✗✆✛
✑✢✍❲☛❳☛
❀
♠❀✗✆✛
✥☛✖✢
✌☛✆
✟❭ ✗❢✔✪✥☛✆✎ ✗✆✛
✗✆✛
✥☛✖✢
✰✆✎
☎✎ ✏✡☛
✌☛
✙❀✌☛✤✆
(D)
✥ ✙❨ ✌✔ ✢
✚✡☛
✕✖✯
✙☎✚✌☛
(C)
✙✌
✬☛✆
✑✷☛☛✆✎ ☎✆
✑✒✌✦
1
2
✻☛✦❀☛✘✼
❡☛✆ ✸☛
P(A)
✑✢✍❲☛❳☛☛✆✎
✕✖ ✥☛✖ ✢
✏☛✢☛✜ ☎☛✰✮✒✍ ♠❧✑☛✙✬✔
②☎✆
(C) P(A|B)
❁
✑✒ ☛✙✡✌✔☛
7%
✥☛✖ ✢
(B) P(A|B) < P(A)
☎✎ ✜✎✙❨✔
(ii)
(iii)
✌✍
P (B)
P (A)
☎✆
(i)
✕☛✆ ❀✆
✑✒☛✓✔
(A) P(A | B)
✡☛✬❯ ✙❱✣✌
✥☛②✘
☞
✰❡☛✍❀ ✥☛✳ ✑✢✆ ✦✢ ✕✖✎✫ ✑✒r ☛✰ ✥☛✳ ✑✢✆ ✦✢
❂❃❄❅❆❇❈❉ ❊❋ ●❃❍❋ ■❏❑▲▼ ◆❖ ❃❆❂❉P❃ ◗❘❙❚
✰✆✎
✡☛
C
C
✔r☛☛
13.6
Distribution)
✕✰✞
✠
✏☛✢☛✜ ☎☛✰✮✒✍
✏☛☛✆
✕☛✆❀✆
✗☛❢✔✙✗✌
✘✆ ☎✍
♠✣☛✤✔✍
?
(B)
✥☛✖✢
✟✞
✥☛✖✢
✌☛
✡✙✬ ✘✌
②✹✦
✌✍
✌✍
✕✪✥☛✫
☎✰✡
☎✆ ✘✌
✑✷☛✆
✜☛✆✤❀✆
✑✒✬✙❡☛✲✔
✕✖✫
✚✡☛
✮✵✶ ✵✍
☎❧✡
✗✪ ✛✤
✤✮☛✔☛
✑✒☛✙✡✌✔☛
✔☛❡☛☛✆ ✎
✑✷☛✆
13. A
✌✍
A
✑✢
✝✞
B
50%
☎☛✰✮✒✍ ♠❧✑☛✙✬✔ ✌✢✔☛ ✕✎✖ ✔r☛☛ ✥☛✳ ✑✢✆✦✢
✌✢✔☛
✌☛✆ ✘✌ ✜☛✢
✑✢ ✧✙★✔✩ ✡☛ ✧✑✦✩ ✑✒☛✓✔ ✕✪ ✥☛✫ ✡✙✬ ♠☎✆ ✭✍✌ ✘✌ ✙★✔ ✑✒☛✓✔ ✕☛✆✔ ☛
✕✖ ✙✌ ♠☎
❣☛✢☛
✕✖ ✔☛✆ ✗✕ ✘✌ ✙☎✚✗✆✛
✌✍
✰✆✎
❬❭
✜☛✢
✘✌ ❫✆✢
✕✰☛✢✍
✏☛✢☛✜
✥☛✖✢
♠✣☛✤❀✆
☎✆ ✞
✰✆✎
✙t☎✰✆✎
❪✙★
☞
✭✍✌
✗❢✔✪ ✥ ☛✆ ✎
❴
✗❢✔✪✥☛✆✎
✕✰☛✢✍
✏☛✢☛✜
✗✆ ✛
✗✆ ✛
❪✙★
✕✖✞
☞ ✗❢✔✪ ✥☛✆✎
✑✒✙✔✬❡☛✲
✙✌☎✍
✰✆✎
✙✗❡☛✆ ✸☛
✏☛✢☛✜
☎✎ ✏✡☛
✌☛✆
❵✘✌
✗❢✔✪ ✥☛✆✎
✥❀✪ Ø✰
✰✆✎
✰✆✎ ✫
✕☛✆
✌✆
✌✍
☎✌✔✍
✕✖✫
✜☛✬ ✘✌❛
☎✎✏✡☛
✰✆ ✎
✙❀✌☛✤❀✆
✕☛✆
☎✌✔✍
✐
♠ ✝✞
✣
✖
✈
♦
☛ ✚
✣
✒
✫
✥✬
✣
✌
✤
X
✟
✠
✡
☛
✈
✫
✥
❧
✞
✝✏
✚
☛ ✧ ✸
✖
✎
✝✏
X,
✓
✘
✖
✌
✑
✞
✙✒
✝
✘
✖ ✌
☛ ✖
✘
✪
✜
✘
✤
✫ ✛
✜
✚✢
✘
✖ ✌
☛
✖
☞✌
✘
✜
✍
✘
☞✖ ✓
✌
✘
✪
✜
✮ ✎
✚
✮
✎
✚
✭
✖ ✖ ✚ ★✖ ☛
✒✔
✝✖
✝✓✔ ✕
✖ ✗✖
✫
✙
✫
✙
✞
✘
✎
✙✒
✣
✌
✤
✙
✒
✮ ✎
✌
♦
☛
✚
✭
✖ ✖ ★✖ ✖
✝✏
✖ ☛
✍
✙✒
✣
✖
✒✓
✝✚ ✓
✎
✒✖
✘
✪
✜
✷
✝✏
✯
✓
✘
✖ ✌
☛ ✖
✮ ✎
✑
✞
✒
✌
✘
✪
✜
✞
✖ ✧ ✚
✱
✲✳
✒
✞
✖
✧ ✚
✱
✣
✒
✎
✒☛
✌
✶
✒
✖
♦
☞✓✗✖
☞
✘
✜
✢
✖ ✌
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codomain
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S = {HH, HT, TH, TT}
X,
✞
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X (HH) = 2, X (HT) = 1, X (TH) = 1, X (TT) = 0.
✙✒
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Y (HH) =2, Y (HT) = 0, Y (TH) = 0, Y (TT) =
S X
Y
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2.
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●
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Rs 1.50
X
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✍
✝✓
✝✚ ✓✭
✖ ✖
✚ ★✖ ☛
✘
✪
✜
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✖ ✗✖
✒✖
✝✏
✚☛ ✧ ✸
✖
✠
✎
☞✚ ★✩
✘
✵
✜
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
✘
✪
✜
✮ ✎
✚
✫
✙
✠
576
① ✁✂✄
r☎
X (HHH) = Rs (2 3) = Rs 6
X (HHT) = X (HTH) = X (THH) = Rs (2 2 1 1.50) = Rs 2.50
X (HTT) = X (THT) = X (TTH) = Rs (1 2 2 1.50) =
Re 1
X (TTT) = Rs (3 1.50) = Rs 4.50
✆
❾
✆
❾
✆
✈✞✟✠
✝
X
{ 1, 2.50,
✘✞
✎✔❢
❢✘
✝
♠✱✲✳✴✵✲
✠✹ ✻
✙✞✚✪
✝
✈✎✮r✗❀
★✞✙
✫✚ ✬
☎✞✛
✿✧✚
✥❥✙ ✤
✸✞✟ ✔✚
★✚ ✹
✳❣
★✞✙
✔✹✚
r☎
✥✦✎r✛✜✞✢
✫✚ ✬ ✥✦✭❀✚ ✘ ✈✫❀✫ ✫✚ ✬
✥✠
❢✘
✥✬✔✙
✔✞✔ ✻✚✹ ✛ ☛✟✹✣ ❀✞✛✼ ✽✾❀✞
❢✘
✻✚✹ ✛ ✎✙✘✞✔✗ ✻✯✢ ✈✞✟✠ ✿✧✘✞
✥✦✎r✛✜✞✢
✕✞✔✞
✎✙✘✞✔✞✚✹ ★✚ ✹ ✧✥✬✔r✞ ✘✗ ✧✹ ✓❀✞ ✘✞✚ ✛✜✞✞✢ r✞
★✞✙✞
☛✟✣ ✈r✤ ✥✦✎r✛✜✞✢ ✧★✎✩✪
✧★✎✩✪
☛✟ ✎✰✧✘✞
✥✎✠✧✠
☛✟✤
4.50, 6}
❢✘ ✸✞✟✔✚ ★✚ ✹ ✺ ✧✥✖✚ ✬✛ ✈✞✟ ✠ ♥
✶✷
✘✠✙✚
✧✥✬✔r✞
X
✯✧✎✔❢
☛✟✒
✝
✆
✝
✡
✎✏✑✒ ✎✓✞✔✞✕✖✗ ✘✗ ☛✞✎✙ ✘✞✚ ✛✜✞✞✢ ✠☛✞
❀☛✞☞ ✌✍✞
✆
✻❀✞
✎✘
✯✧
✻❀✞✣
☛✟ r✞✚ ✒
X
✥✦✎❁❀✞
✘✞✚
✥❥✙ ✤
✎✘❀✞
✻❀✞✣
❀✎✛
X
✛✞✚
✘✞ ✎✫✫✠✍✞ ✛✚✹ ✒ ✰☛✞☞ ❢✘ ✔✞✔ ✻✚ ✹ ✛ ✘✞ ✎✙✘✔✙✞
☛✟ ✣
✸✞✟✔✚
✧★✎✩✪
✠✓✞✗
★✚ ✹
✻✚✹ ✛✞✚✹
w1, w2, r
✘✞✚
❧❀❂r
✧✚
✘✠r✚
☛✟✹✣
☛✟✤
S = {w1 w1, w1 w2, w2 w2, w2 w1, w1 r, w2 r, r w1, r w2, r r}
X=
=
X ({w1, w1}) = X ({w1 w2}) = X ({ w2 w2}) = X ({w2 w1}) = 0
X ({w1, r}) = X ({w2 r}) = X ({ rw1}) = X ({rw2}) = 1
X ({rr}) = 2
0, 1 2
✈☎
✻✚ ✹ ✛✞✚ ✹
✔✞✔
✯✧✎✔❢
✘✗
✧✹ ✓❀✞
✧✥✬✔r✞
✘✗
✧✹✓❀✞
✈✞✟✠
X
✈r✤
❢✘
❀✞✛✼✎ ✽✾✘
13.6.1
variable)
❃❄
✈✞✯❢
✛✧
✥✦ ✭❀✚ ✘
❅❆❇❈❉❊❋❄
✥✎✠✫✞✠
✘✞
●❍
✏❥✙✞✫
✧✛❘❀
❄■
❀✞
❏❑❆❉ ❅❄▲❆
❢✘
✧✚
★✞✙
★✞✙
✘✞✚
✔✚✹
✧✘r✞
☛✟✣
(Probability distribution of a random
▼◆❖P
✥✎✠✫✞✠
☛✞✚✣
✔✚
✯✧
✎✘
✥✦✘✞✠
✥✎✠✫✞✠✞✚ ✹
✏❥✙✙✚
✫✚ ✬
✥✠✗◗✞✍✞
f1, f2 ...f10
★✚✹
✥✠
❁★✜✞✤
✎✫✏✞✠
✘✠✚✹
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3, 4, 3, 2, 5,
☛✟✹✣
✏❥✙✚
X
☛✟
❀✞✛✼✎ ✽✾✘
✰✞✚
✧★✧✹ ✐✞✞❧❀
✈✞✯❢ ❢✘ ✥✎✠✫✞✠ ✘✞✚
❢✘
☛✟
f1, f2 ... f10
✥✎✠✫✞✠✞✚ ✹
4, 3, 6, 4, 5
✏✠
✏✠
✫ ✿✧✫✚✬ ✧✛❘❀✞✚✹ ✘✗ ✧✹ ✓❀✞ ✘✞✚ ✙✞✚✪
✎✰✧✚
✎✙❙✙
✥✦✘✞✠
✧✚
✥✎✠✐✞✞✎✩✞r
✎✘❀✞
✘✠✒
X
✻❀✞
☛✟ ✤
✧✚ ❧❀❂r ✘✗✎✰❢✣ ❘✥✩✪r❀✞
X (f1) = 3, X (f2) = 4, X (f3) = 3, X (f4) = 2, X (f5) = 5,
X (f6) = 4, X (f7) = 3, X (f8) = 6, X (f9) = 4, X (f10) = 5
✈r✤ ✺✒
✈☎
f3, f7
★✚✹
❚✒
❯✒
X
✘✞
✧✚
❱✒
❲
★✞✙
✎✘✧✗
★✚✹
✺
✧✚
X
☛✞✚✻✞
✥✎✠✫✞✠
✰☎✎✘
✘✞✚
X = 4,
✘✞✚ ✯✢
✏❥✙✞
✰☎
✐✞✗
★✞✙
✥✎✠✫✞✠
✰✞❢✣
✥✎✠✫✞✠
✔✚
f4
✯✧✗
✧✘r✞
✘✞✚
✏❥✙✞
☛✟
✻❀✞
☛✞✚✣
X
✘✞
✥✦✘✞✠
f2, f6
❀✞
f9
✘✞✚
✏❥✙✞
✰✞❢✻✞
★✞✙
3
☛✞✚
✧✘r✞
☛✟
✰☎
f1,
✐
X = 5,
X = 6,
✈
☛
✠
✒
♣
✓
✔
✕
✌
✟❞
✕
✎ ☞
✔
☛
✎
✤
✞✖
☛
✟
❀
❞
X
✥
✦
✈
✔
✕
✕
❞☛
✣
✟
☛
✟
❞
✞✖
✔
f1, f2,
✔
✤
☛
❧
☛
X
✥
✦
✈
✔
✕
✣
✟
✢
✘
✩
✒
✤
✘
✥
☛ ✠
✞
☛
☛
✟❀
❞
✘
☞
✌
✎
✍
☛
✌
✎
✍
☛
t
☛
✏
❞
☛
✌
✎
✍
☛
t
t
☛
✏ ✑☛
✑☛
✚
☛
✎ ☛
✛
☛
☛
✔
✜
❀
✢
✘
✒
✣
✟
✏
✞✟
✠✡ ☛
✠
f4
✡
☞
♦
✌
✎
✍
☞
t
☛
✎
☞
1
10
☛
✔
❣
✒
✤
✚
✟❞
✤
✤
☛
✏❞
✞✟
✠✡ ☛ ✠
❞☛
☞
✌
✎
✍
✎ ☞
✥
❞
✞✖
☛
✟
❀
❞
☛
✔
❣
✒
✤
☛
☞
✎ ☞
✔
✕
✞✖
❞
☛
☞
☛
☞
1
10
✞✖
☛
✟
❀
❞
3
10
=
✥
❞
☛
3
10
P (X = 3) =
✔
✙
☞
✞✖
✔
☛
✎
✥
★ ☛
✠✡ ☛ ✠
✥
❞
❀
☛
✕
❞☛
✞✟
✤
☛
☞
✎ ☞
f7
3
P ({f1, f2, f3}) =
10
✘
❞
❞
✘✕✘✙
❀
❞
☞
P(X = 2) =
✒
f8
✞✟ ✠✡ ☛ ✠
f10
☛
✒
✧
✔
✙
☞
t
✝
❀
✔
❣
☛
✎
✥
★ ☛
f5
✞✟ ✠✡ ☛ ✠
✗
✒
1
10
✥
❞
☛
t
✝
577
✁
✂✄
☎✆✁
✔
✙
☞
✒
✟❞
3
2
P(X = 4) = P({f2, f6, f9}) = , P(X = 5) = P({f5, f10}) =
10
10
1
P(X = 6) = P({f8}) =
10
✈
☛ ✒
✠
✢ ✘
❜
✞✖
❞
✔
✜
✪
✫
✒
❞
☛
☞
❀
✛
☛
☛ ✠
❞
❞
✥
✦
✶
✶
❀
☛
✸
✘
✔
✥
☛
☞
✟t
✤
✪ ✫
❀
☛
✤
✞✖
☛
✟
❀
❞
X
✌
✠
❞
✘
❞
✙
☛
✡
☞
♦
❞
☛
✟❀
❞
✝
☛
✟
❀
❞
✘✙
✥
✑✥
✞✖
☛
✟
❀
❞
✙
☛
✈
✣
☛
☞
❞
☛
☞
✟
✥
★
☛ ☛
t
☛
☛
✒
✰
✎
✙
☛
✤
❞
✔
❣
✙
☞
✥
✞✖
☛
✯
✙
☛
✤
❞
✮
✘
☛
❧
✔
✥
✎
✥
✞✖
X
✌
✠
✌
✠
✯
✝
✤
❞
✬✭
✟
✬✭
✟
✥
❞
❞
❀
☛
✪ ✫
☞
✬✭
✟
✟❞
✘✕✙
☛
X
✌
✠
✪ ✫
✏❞
✵✶ ✶
✷
✟✡✡✠
✬✭
✟
❀
☛
✞❞
✲✳
✴
☛
✝
❞
✯
☛
☞
✟✎
✘
✎
✞✖
✘✙
✎
❞☛ ✠
✙
★ ❀
☛
✈
✤
☛
☞
✱
☞
✞✟
✰
❞
✟✎
✠✚
☛ ☛
✟
✣
✎
✟
✥
✥
☛
✟
❞
✥
✟★ ☛
❜☛
✞✖
❀
✣ ✤
☛
t
☛
✔
❣
☛
✒
✹
☛
✺
✟
✎ ❞
☛ ❀
✔
☛
✒
X
P(X)
:
:
x1
p1
...
...
x2
p2
xn
pn
n
✔
t
pi 0,
✓
☛
pi = 1, i = 1, 2, ..., n
i 1
✥
✡
☛
✻
✪ ✫
❀
☛
✘
✙
✟✡
❞
X
✬✭
✟
❞
✌
✳
✠
✲
❢
❁
✓
★
❀
☛
✏
✕
❞
☛
✪
✶
❂
❀
✟
❃
x1, x2, ... xn
xi
✪
✫
❀
☛
✔
☛
✎
xi
☛
✎
☞
☞
✪ ✫
❀
☛
❞
❞
✌
✠
✤
❞
✬✭
✟
✬
✭
✟
✌
✠
✥
✞✖
☛
✟
❀
❞
X,
X
✘
✚
✙
✡
♦
☞
✔
☛
✥
✒
✈
✢
❞
☛
❞
☛
☞
❧
☛ ☛
✽
✘✙
✾
✾
✚
✽
✕
☛
✡
✺
✔
✼❀
✒
✈
☛
✠
✒
pi (i = 1,2, ... n)
P (X=xi) = pi
✕♣
☛ ✡
✹
✕
♣
☛
✎
✔
✼
❀
✒
✥
☛
☞
❞
❧
☛
✎
X = xi
✥
✪
✞✖
✟
✘✕
❄
☛ ✽
✱
✯
✟
✿
✭
✡
☞
♦
✔
☛
☞
✡
✍
♦
✥ ✤
❅
✪
✝
✹
✍
✔
✥
✒
✈
❧
☛ ☛
✽
✙
✺
✈
☛
☞
✾
✣
✡
☞
♦
✟
✔
✤
✘✗
✏
P(X = xi)
✔
❀
❇
0
❣
☛
☞
✥
☛
✔
❣
✒
✥
✦
✈
X
❞☛
xi
✕
♣
✼
✣
❀
✎
☞
☞
✤
❞
✥
✞✖
☛
✟
❀
❞
✘✪
☛
❡
♣
✡
✒
❄
☛
❆
❀
☞
☛
✠
578
① ✁✂✄
❧☎✆☎
t☎✌☎
X
✝✞
✝✣✤
✥❧☞✏✑
♠✩✪✫✬✭✪
✮✯
t☎✌✟
☞✎✢☎✏✟
♦✟ ✠
❧✡☎✞
☞✢❧✞
✌☎✕☎
✝✣✤
☛
♦✟ ✠
❧☛✡☎☎☞♦✌
✒✓☎ ☞✦✢✌☎
52
✥❢✢☎✟☛
✒✐☎☎✟ ☛
✍☎✎☎✟ ☛
✚☛✘ ✎
✢✞
❧☛✶✦☎
✢✞
♦✟ ✠
♦✟ ✠
✑✢
✢☎
☞✏✑
☞✏✑
✒✓ ☞✌✔✕☎✖
❧✡☎✞
✚☛✘ ✎
✦☎
✵♦✌☛▲☎
2
✝✣✤
❢✦☎✟ ☛ ☞✢
✒✐☎☎✟ ☛ ✢☎✟ ✒✓☞✌✵✆☎☎✒✎☎
X
✔☎✟
❧✡☎✞
✙✚✔✛ ✜☎✟ ☛
✦☎✟ ✧
✢☎
✑✢
✳✐☎✴☎✟ ✐☎✴
✒✐☎✟
✢☎
✝☎✟ ✎☎
❧✍☎♦✟✕☎
✝☎✟
★☎☞✝✑✤
✒✓☞✌✵✆☎☎✒✎☎
♦✟ ✠
❧☎✆☎
✢✞☞t✑✤
✵✒✗✘✌✦☎
❧✟ ☞✎✺☞✒✌ ✢✴✌✟ ✝✣✤
☛
☞✎✢☎✏☎
✧✦☎
✝✣ ✥❧☞✏✑ ✔☎✟ ✎☎✟ ☛ ✒✐☎☎✟ ☛ ✢☎
X
✢☎ ✍☎✎
☞✎✢☎✏✎☎
✝✣✤
☛
✒✴✞✻☎✼☎
P(X = 0) = P(
= P(
✥❧☞✏✑
♦✟ ✠ ❧☎✆☎
❧✟
✷☎✌
✫❣ ✥❢✢☎✟☛ ✢✞ ❧☛✶ ✦☎ ✑✢ ✦☎✔❜ ☞✸✹✢ ★✴ ✝✣✤ ✥❧✢☎✟ ✝✍
0, 1,
♦✟ ✠
✒✓ ☎☞✦✢✌☎✜☎✟ ☛
✧✱✲✱ ✞
❧✛ ☞✍☞✰✌
✒✓☎ ☞✦✢✌☎
❧✍☞✗✘
✥❢✢☎
✎✝✞☛
✥❢✢☎
✎✝✞☛
✜☎✣✴
✥❢✢☎
✎✝✞☛
)×P(
)
✥❢✢☎
48 48 144
52 52 169
P(X = 1) = P(
= P(
) . P(
)
✎✝✞☛
=
✜☎✣✴
✥❢✢☎
✜☎✣ ✴ ✥❢✢☎
✎✝✞☛
✥❢✢☎
✥❢✢☎
✎✝✞☛
✥❢✢☎
✜✆☎♦☎
)+P(
✥❢✢☎
4 48 48 4 24
52 52 52 52 169
P(X = 2) = P(
) = P(
✎✝✞☛
✜☎✣ ✴
).P(
✥❢✢☎
✎✝✞☛
)
)
✥❢✢☎
=
✥❢✢☎
✜☎✣✴
=
✜✌✈
✜✡☎✞✗✘
✒✓ ☎☞✦✢✌☎
♠✩✪✫✬✭✪
✮✾
✒☎❧☎✟☛
✒✓ ☎☞✦✢✌☎
✚☛✘ ✎
✷☎✌
✥❢✢☎
✍☎✎
✏✞☞t✑
✝✣
☞✢
✥❢✢☎
✑✢
X
0
1
2
P(X)
144
169
24
169
1
169
t☎✟✱✟ ✿
✢☎✟
✌✞✎
✚☎✴
✳✹☎✏✎✟
✒✴
✢✞☞t✑✤
☞✢
☞❀✢☎✟ ☛
✢✞
❧☛✶✦☎
✜☎✣✴
✵✒✗✘
P(
✽
✢☎
✦☎
✍☎✎
✑✢
☞❀✢
✒✓☎ ❃✌
✝☎✟✎✟
✑✢
☞❀✢
✒✓☎ ❃✌
✎
✢✞
✝☎✟✎✟
)
✝✣✈
X
(1,1) , (2,2), (3,3), (4,4), (5,5),
(6,6)
X
0, 1, 2,
3
6 1
36 6
1 5
1
6 6
✫❣
)
4 4
1
52 52 169
✚☛✘ ✎
♦✟ ✠
✜☎✣✴ ✥❢✢☎
✒✓☎ ☞✦✢✌☎
✢✞
✒✓ ☎☞✦✢✌☎
✝✣✤
☞✎✺☞✒✌
✢✴✌☎
❧☛✡☎♦
☞❀✢
✝✣✤
✝✣✤
☛
☞❀✢☎✟ ☛
❁
doublets
❂
✢✞
❧☛✶ ✦☎
✢☎
✐
579
✁
✂✄
☎✆✁
✈
✝
P(X = 0) = P(
✱✞
✟
✠ ✡
P(X = 1) = P(
✱✞
☛☞
✞
☛☞
✞
✌ ✍
✡✎
✠
✈
✠
✏
✑
✒
5 5 5 125
6 6 6 216
)
)=
☛
☞
✞
✓
✌
✍
✡✎
1 52
3
6 62
1 5 5 5 1 5 5 5 1
=
6 6 6 6 6 6 6 6 6
P(X = 2) = P (
)
✠
✒
☛☞
✞
✠
✓
✈
✱✞
✏
☛☞
✞
✌
75
216
✍
✡✎
✑
1 1 5 1 5 1 5 1 1
6 6 6 6 6 6 6 6 6
1
5
15
=3 2
6
216
6
1 1 1
1
P(X = 3) = P (
)
6 6 6 216
=
✡✌
☛☞
✞
r
✔
✈
r
X
✞✠
✟
✠ ✡ ✥✕
✖✗
✠
☛
✘
✞
✈
✖✗
✠
☛
✘
✞
❧✜
✢
✣ ✤
✦
✠
r
✠ ✎
✈
n
✞✠
✘
✓
★ ✖
♠ ✬
✭✮
✯
✣
t
✠
r
✠
✑
✠
✩
✓
✖✗
✠
☛
✘
✞
r
✰
✲
✚ ✎
✣
✍
✫
✏
✠
r
❡✠ ✌
✸♦
☛
✚ ☛✛
✠
✍
✎
r
✔
✏
X
0
1
2
3
P(X)
125
216
75
216
15
216
1
216
✠
125 75 15 1 216
1
216
216
✎
✕✌
✪✍
✡
☛
✞
✏
✪
✡
✘
✠
✳
☛
✴✵
✞
✶
✌
✷
✒
x
❡✠ ✌
✓
✍
✫
✝
✓
X
☛
✌ ✙✌
✧
✓
=
✔
✕
✎
✌
✝
1
pi = 125 75 15
216 216 216 216
i 1
✈
r
✠
r
✚ ✌
✓
✞✡
✓
☛
✸✹
✠ ✚
✘
✓
✖✗
✠
☛
✘
✞
✠
✡
☛
✌ ✙✌ ☛
✚
☛✛
✠
r
P(X = x) =
❡✠ ✌
✠
❞
✞✡
r
☛t
✱
❡✎
r ✑✍
kx
k (5 x)
✍
②
✓
❂
❂
❃
❃
❂
✖✺
✠
✻
✼
✽
✞
✓
✪
r
0
✞✠
✸✪
✒
0.1
(a) k
☛
✧✱
♥
❃
k
t
✍
✠ ❀
✏
x
♥
x 1
♥
❄
❅
❂
❋
✱✞
0
❂
x 3
❦ ❦
✾✠
✎
✕✠ ✎
✓
2
❦
❂
❦
4
✞✠
✓
✸
✠
✓
❁
r
X
☛
✸
✞
✪
✓
✿
✠ ✠ ✘
✽
✠
✒
✪
✎
✛
✘
✠
✍
✏
✔
580
① ✁✂✄
(b)
❜☎
✆✝✞
✟✠
❯☛✒ ✓✞✔
X
❣✥
✟✝
✕✝✖
✡☛✝
✗✝✘✙✖
☞✌ ✝✍☛✟✞✝
☞✚✛ ✞✖
✆✘✙ ✓
☞✌ ✝✍☛✟✞✝
✘
✎✏ ✜
✎✏
✍✟
✕✝✖
✞✢☛✞✣
✓✠❞✖
✑✝☞
✦☛✝
✍✕☛✝
✗✝✘✙✖
☞✚✛ ✞✖
✎✏ ✘ ✜
✕✝✖
✑✍✤✟✞✔
✗✝✘✙✖
☞✚✛ ✞✖
✘
✎✏ ✜
✎✏ ✣
X
0
1
2
3
4
P(X)
0.1
k
2k
2k
k
n
(a)
✎✔✖ ✘
✧✝✞
✎✏
pi 1
✍✟
i 1
0.1 + k + 2k + 2k + k = 1
k = 0.15
) = P (X 2)
= P (X = 2) + P (X = 3) + P (X = 4)
= 2k + 2k + k = 5k = 5 × 0.15 = 0.75
) = P (X = 2)
= 2k = 2 × 0.15 = 0.3
) = P (X 2)
= P (X = 0) + P(X = 1) + P(X = 2)
= 0.1 + k + 2k = 0.1 + 3k = 0.1 + 3 × 0.15 = 0.55
❜☎✍★✩
✪
(b) P(
❯☛✒ ✓✞✔
✕✝✖
✗✝✘✙✖
☞✚✛ ✞✖
✎✏ ✘
P(
✞✢☛✞✣
✕✝✖
✗✝✘✙✖
☞✚✛✞✖
✎✏ ✘
P(
✑✍✤✟✞✔
✑✝☞
✑✝☞
✑✝☞
13.6.2
✆✎❝ ✞
✎✝✖ ✞✝
☎✠
☎✔✸☛✝✑✝✖✘
✎✏ ❄
✔✖ ✘
✑✢✝✝✖ ❊
✐❋●❍■■❏■■
✎✏✘❉
✎✏
❑
✴✵
✔✖✘
❞✼ ✟✠
✍❅☎✖
✎✝✖ ✞✖
✆✎❝★✟
❜✓
❀✭✮✯✰✱✲✳
❜☎
✍✟
✔✝✓
✕✝✖
✳✭
✔✖✘
✩✟
✬
❞✼
✆✘✙ ✓ ☎✖
✔✝❈☛
✎✔
✍✟☎✠
X
✎✏ ✘
(Mean of a random variable)
✶✭✷❀
☞✌ ✝✍☛✟✞✝
☛✎
☞✚✛ ✞✖
☛✝✕✹✍ ✺✻✟
✍✟☎✠
✟✽✝✝
★✖ ✘
✗✝✘✙✖
✫
✟✖
✍✟☎✠
★✽✝✾✝
✧✝✞ ✟✼ ☎✟✞✖
☞✼
☛✝✕✹✍✺✻✟
☛✝✕✹✍ ✺✻✟
❞❞✝❂
❞✼
❞✼
✖
✟✼✖✘✦❉
❃✖ ❆
✎✏
✎✏ ✘ ✩✖☎✠
✔✝❈☛
✔❈☛✔✝✓
✍❅☎❃✖ ❆
✟✝✖
☛✝
✩✟★
✎✠
✕❁✝✝❂ ✓✝
☎✖
❃✝✘✻✓✠☛
❃❝❆✻ ☎✘ ✿☛✝✩❇ ✔✝❈☛❄ ✔✝❈☛✟
✑❃✍✸✢✝✍✞
✑✝✏☎ ✞
☎✘▲ ✝✝✍❃✞
☎✘✿☛✝
✔✝✓
☛✝
✔✝✓
❃✖ ✘❆✕✌✠ ☛
✟✝✖
✹ ✞
☞✌ ❃✍
❜✘✍✦✞
✔✝☞
✟✠
✎✏❉
✟✼✞✝
x1, x2, x3, ..., xn
❃
▼✔❁✝✣
✟✠
n
p1, p2, p3, ..., pn
☞✌✝✍☛✟✞✝
x
✑✢✝✝❂ ✞✈
☎✘✦ ✞
☞✌✝✍☛✟✞✝
☛✝✕✹✍ ✺✻✟
E (X)
X,
✟✝ ✔✝❈☛❄ ❞✼
☎✖
❧☛✡✞
☎✖
▲✝✝✍✼✞
❞✼
X
✟✼✞✖
✎✏❉
✟✝ ✔✝❈☛❄ ✍❅☎✖
◆
,
☎✖
❧☛✡✞ ✟✼✞✖
❃✖ ❆ ☎✘▲ ✝✝✍❃✞ ✔✝✓✝✖ ✘ ✟✝ ▲✝✝✍✼✞ ✑✝✏☎ ✞ ✎✝✖ ✞✝
❃✖❆
✎✏ ✘❄
✦☛✝
✔✝❈☛
✎✏ ❄ ❅✆ ☞✌ ♦ ☛✖✟
X
✟✝✖
✟✠
☞✌ ♦ ☛✝❁✝✝
P
Expectation
◗
✑✞✣
E(X) =
➭
=
i 1
❁✝❘✕✝✖ ✘
✔✖ ✘
✎✝✖ ✞✠
✎✏❉
✔✝✓ ✟✝✖
❖☎✟✠
✎✝✖❉
n
✑❯☛
xi pi
☎✘✿☛✝
i 1
✍✟☛✝
✎✏ ✘❉
X
xi pi = x p + x p + ... + x p
1 1
2 2
n n
▲✝✠
✟✎✞✖
✎✏✘❄
✍❅☎✖
✐
❀
✝ ✞
✟
✠ ✡☛
☞
✌
✤
✥
♦
✔
✕
✢
X
✍
☞
✦
✝ ✛
☞✝
♠ ✩
✪ ✫✬
✭
✪
❀
✝
✔
✢
✝
❞
✦
✧
✝
✎
❀
✖
✙
✶
❀
✝
✣
✏✑
✒
❀
✰
✝
✔
✙
✝
✛
☞
✫
✝ ✓
✝ ✝
X
✱
✔
✙
✠
☞
✏
✝ ✖✝
✔
✙
✝
❀
✝
✔
✢
✠
♦
✔
✕
✲
❀
✝
✲
☞
❜
✖
(xi, yi)
❀
✏
✍✘
✈
✣
✝
✦
✧
★
✝
✚ ✝
X
✔
✝ ✞
✟
✠ ✡☛
☞
❇
❍ ❍
☞
❀
✝
✝
✏
✑
✠✚
✰
✏
❍
❞✔
✌
✝
✘
✖
✗
✙
✝ ✝ ✠♦
✚
❞
✝
✛ ✝
✙
✔
☞
✝
✜ ✛
☞✘
✖
✙
✢
✚
✏✑
✝
✠
❀
☞
✚
✝
✣
✝
✙
✔
✙
✠
✍
✞ ✓
✠✺
✻
✼ ✽
✦
✧
✝
✖
☞✚
❞
✝ ✛
✣
❞
✲
✦
✝
❅
♦
✔
✕
✦
✝
✔
✖
❞
✲
✶
✝ ✝
X
✝ ✹
✝ ✝
✚
✹
✰
☞
✝
✔
✜
☛
✝
✲
✝
✦
✧
✝
✝
❞
✣
❪
✝ ✍
✝
✎
❀
❀
✝
✏✑
✒
❀
✝
✓
✝
✝
✷
✝
✚
❀
☞✘
✰
✠
✝ ✞
✟
✠ ✡☛
☞
✾
✝ ✻
✛ ✝ ✣
✝
✔
✙
✖✔
✠✛
✠
❞
✚
✹
✦
✧
✣
✝ ✍
❆
❇
❪
✝ ✖✝
✔
✙
✏
✍
✏✑
✝ ✵
✚
✖
✙
✶
❀
✝
✣
✝
✙
✔
☞
✝
❀
❪
✲
✠
✝
✔
✢
❪
❈ ❪
✼
✏
✝ ✖✝
✙
✔
1
36
P(X = 3) = P({(1, 2), (2, 1)})
2
36
P(X = 4) = P({(1, 3), (2, 2), (3, 1)})
3
36
P(X = 5) = P({(1, 4), (2, 3), (3, 2), (4, 1)})
4
36
P(X = 6) = P({(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)})
5
36
P(X = 7) = P({(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)})
P(X = 8) = P({(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)})
P(X = 9) = P({(3, 6), (4,5), (5,4), (6,3)})
P(X = 10) = P({(4, 6), (5, 5), (6, 4)})
P(X = 11) = P({(5, 6), (6, 5)})
1
36
2
36
3
36
4
36
✦
✿
✔
✏
✍
✤
✙
❁
5
36
❉
❪
❪
✽
✦
✧
P(X = 12) = P({(6, 6)
✍
✠
❞
✚
❀
❂
❞
yi = 1, 2, 3, 4, 5, 6.
✧
✣
✝
✖
☞✚ ✝
P(X = 2) = P({(1, 1)})
✌
✠
✦
✤
☞
xi = 1, 2, 3, 4, 5, 6
❄
✝
✏
X
✧
✝
✚ ✝
✲
✱
★
☞
✧
✝
❃
♦
✔
✕
❪
❍ ■
✸
✖✗
✳ ✴
✝
✔
✥
❣
♦
✔
✕
★
✰
✝
✵
✚
✝
✝
✔
✚ ✝
✮
✯
❞
✏
✑
❀
581
✁
✂✄
☎✆✁
6
36
❊
❪
❋ ❪
●
❪
582
X
① ✁✂✄
❞☎
✆✝ ☎✞✟❞✠☎
X
✟☎
P(X)
✡☛ ☞✌
✍✎ ✏
2
3
4
5
6
7
8
9
10
11
12
1
36
2
36
3
36
4
36
5
36
6
36
5
36
4
36
3
36
2
36
1
36
xi
✟☎
pi
n
❜✑✞✒✓
➭
= E(X) =
✘
xi pi
✕
2
✖
i ✔1
5
36
6
✙☎✚
13.6.3
✆☎✑☎✚☛
✆✚☛✜❞✌✚
❀✩✪✫✬✭✮✯
✟☎✙✶ ✞✷✸❞
✞✛✞❢☎❁✌
✛✚✜
✹✢
❞☎
✆✝☎✞✟❞✠☎
✞✌❄✌✞✒✞✣☎✠
✡☛ ☞✌☎✚☛
✰✱
✥☎✦✟
✡☛ ☞✌
✥✚☛
✗
✖
6
36
7
✗
✆✢
✑☛ ✣✟☎✈☎✚☛
✆✝❞☞
✯✩
✹✢
✛✚✜
✥☎✌☎✚☛
✟☎✙✶ ✞✷✸❞
✛☎✒✚
✞✙✣☎☎✟☎
✤✟☎
✗
✖
4
3
2
1
10
11
12
36
36
36
36
9
42 40 36 30
36
✟☎✚✤
❞☎
✥☎✦✟
✧
22 12
✍✎★
✥✚☛
✹✢☎✚☛
✞✛✹✢✻☎
✛✚✜
✛✚✜
✥☎✦✟
✡☎✢✚
✑✥☎✌
✥✚☛
✍☎✚
❞☎✚❜✼
✑✽✹ ✌☎
✑❞✠✚
✍✎ ❂
☛
X
1
2
3
4
P(X)
1
8
2
8
3
8
2
8
–1
0
4
5
6
P(Y)
1
8
2
8
3
8
1
8
1
8
▲✆❅☞✠✟☎
E(X) 1
1
2
3
2 22
2
3
4
2.75
8
8
8
8 8
✈☎✎✢
E(Y)
❢☎✾
X
✈☎✎✢
✈☎✑☎✌✾
Y
✑✚
✈✒✤❆✈✒✤
✆✝✚✞❍☎✠
✞❞✟☎
1
✍✎☛
✟❇✞✆
❃☎
✌✍✾☛
❃✎ ✑☎
✙✚✠☎
✞❞
✍✎★
X
✑☎✿☎
✍✾
Y
✛✚ ✜
✞✌❊✆✻☎
✑✚
✈☎✎✢
✍✎★
Y
✹✢
=7
(Variance of a random variable)
✲✳✴✱✵✩
✺✑
✛✚ ✜
✖
5
36
8
2 6 12 20 30
=
✈✠✏
1
2
3
4
3
4
5
36
36
36
36
1
2
4
1
0
3
5
8
8
8
8
✺✌✛✚ ✜
✑❞✠☎
✍✎
✥☎✦✟
✑✥☎✌
❋✈☎✛✶ ✜✞✠
✍✎ ☛
●■❏❑▼★
✟✍
6
1
8
22
2.75
8
❜✌
✹✢☎✚☛
✛✚✜
✞✹❈☎☎❉✥❞
✐
❘❙
X
Y
❞✝ ✞
✟
✝ ✌
❧✞
✔
❞
✟
✠ ✡
✢
✝
✑
✈
✑
❣
✑
❞ ☛☞ ✞
✒☞ ✞
✌✍
✞
❧✝
✓
✎✛
✎
✠
✏
✔
❞
✜
✑
✒✓
✞
✒
✓
✞
✣
✤
✥
❧✜
✣✫
❞
✝
☛
✔
✝
✕ ✎
✖
✗✘
❞
✝
✑
✦
✧
✔
✝
✕ ✖
✎✗
✘ ❞
✙
☛
✌✍
✞
✒✭
✬
❚
❱
❯
✎❞
✔
✝
✓
✞
13.5
❲
❳
✙
☛
✟
✝
❞
✤
★
✒
✓
✞
✎
✚
✛
✝
☛
✌✞
✍
✒✝ ☞
✝
✓
✞
✒
✓
✞
✎
✒✓
✞
✎✚
✛
✌
✙
☛✪
✝
✝
☛
✔
✝ ✌
✝
✎
❞✜
✚
✛
❧✜ ✒✝
✝
☛
✝ ✌
✢
❞
❞
X
p(x1), p(x2), ..., p(xn)
✒✝
☞
✝
✎
✲✳ ✳
✴
✳ ✳
✵
✔
❞✢ ✝ ✟
✝
✓
✞
✒✝ ☞
✣❞ ✝
☛
❧✞
✠
✓
✞
➭
✣✎ ☛✶
✠ ✜
✎
✮
✏
✏
❞
= E (X), X
✝ ✝
✎✼
✝
✢
✎❞
✔
✝
❞
✝
✒✝ ✸
✔
✝
✕ ✎
✖
✗
✘
❞
✝
✌
❞
✝
✞
✣✫
❧
☛✪
✝
❧✞
✮
✝
✢ ✝
✑
✙
☛
✑
✎✮
❧✌ ✍
✞
✒
❧
✓
✤
✌✍
✞
❧✝ ♦
✝
✝
✣
✝
✮
✝
❧
❞
✎
✌
✷
✔
X
✑
❞
✝
✣✫
✒✝ ☞
❧
☛✪
✝ ✝
✎
✌
✢
✒
✭
✬
✒✝
✣
✑
✜
✣✫
✢
✝
❞✜
❧
☛✪
✝
✑
x1, x2 ...xn
✔
✶
✑
✤
✥
✤
✫
✝
✥
2
var (X)
✔
x
✝
⑥
✝
☛
✝
✎
☞
✹
✎✣
✽
✤
2
= Var (X) =
2
= E (X
x
✔
✝
❧✒✢
❀
✭
x
✔
✢ ✾
)
) 2 p ( xi )
( xi
i 1
2
n
✝
✞
➼
✪
❞✝ ✞
✒
❧
✓
✛ ✔
✔
✝
✕ ✎
✖
❄ ❅
✳
✝
☛
x
✝
✿
✗
✘
❞
❆ ❇ ❈
❉
✰
✮
✓
✑
❞
❈
✱
✝ ☞ ✢
✞
X
✙
☛
✳
✝
✒
✎
✌
✙
✠
☞
❁
❊
❋
✯
✝
☞
❞
●
✱
❍
✳
■
❈
✳
❏
❑
✱
Var (X) =
=
) 2 p ( xi )
( xi
i 1
standard deviation
❞
✑
✢ ✞
❂
❈
▲▼
❃
✳
❋◆
❖
✳
✎
❞
✤
n
( xi
Var(X) =
P
) 2 p ( xi )
i 1
n
( xi 2
=
2
◗
2 xi ) p ( xi )
◗
i 1
n
n
n
2
2
( xi p ( xi )
=
i 1
◗
i 1
❧
✓
✡✢
✓
n
✑
✒✝
✣
✍
✥
✤
✯✰
✱
❃
✜
✌✞
✩ ✧
✤
✥
✣✫
583
✁
✂✄
☎✆✁
p ( xi )
2 xi p ( xi )
◗
i 1
✑
✓
✤
✥
✢
✺
❞
✝ ✞
✎
☞
✻
☞
584
① ✁✂✄
n
n
( xi 2 p ( xi )
=
☎
n
2
p ( xi ) 2
i 1
i 1
n
i 1
n
xi 2 p ( xi )
=
xi p ( xi )
☎
✆
2
2
✆
2
n
p (xi ) =1
❉✝ ❦✞
❛s ❞
i 1
✈❦✟❥ ✆
=
i =1
xi P( xi )
i 1
n
( xi 2 p ( xi )
=
☎
2
i 1
n
( xi p ( xi ))
Var(X) =
❀✠
2
n
2
xi p( xi )
i 1
i 1
n
2
2
Var(X) = E(X ) [E(X)]
❀✠
❪
✡☛✠☞
2
E(X ) =
✍
xi 2 p( xi )
i ✌1
♠✎✏✑✒✓✑
✏❣
✔✕
✜✦✬✐✠✪✠
✲✠✘ ❡✣✤
✱✖
✖✠
X,
✗✘✙✚✠✘✛
✜✧✙✛✮✯✠✰
✜✠✢✣ ✜✦
✜✠✢✣
✢✲✙✳✴
✜✧✖ ✴
✖✠✣
☛✵
✜✣✥
✤ ✖✘✣
✜✦
✜✧ ✠★✛
✢✤ ✩❀✠✗✠✣✤
✢✤ ✩❀✠ ✖✠✣ ✶❀✷✛
✖✦✛✠ ☛✵✭ ✛✸
X
1
6
❀✠
6
✢✠❧✠
☛✬
P(1) = P (2) = P (3) = P (4) = P (5) = P (6) =
❜✢✙❡✱
X
✖✠
❡✣
✢✖✛✠
✜✧ ✠✙❀✖✛✠
✜✧✢✦✪✠
✫✠✛
✖✬✙✡✱✭
S = {1, 2, 3, 4, 5, 6}
4, 5,
✲✠✘
✖✠
✱✖
❀✠✮✹✙ ✺✻✖
✼✦
☛✵
☛✵✭
✸✤ ✴✘
☛✵ ✽
X
1
2
3
4
5
6
P(X)
1
6
1
6
1
6
1
6
1
6
1
6
n
xi p ( xi )
E(X) =
✗✸
i 1
=1
✢✠❧✠
✗✛✽
☛✬
1
1
1
1
1
1
2
3
4
5
6
6
6
6
6
6
6
21
6
1 2 1 2 1 2 1 2 1 2 1
2
3
4
5
6
6
6
6
6
6
6
2
2
Var(X) = E (X ) – (E(X))
E(X2) = 12
91
=
6
21
6
2
91 441
6 36
35
12
91
6
✡✠✣
1, 2, 3,
✐
♠ ✝
✞ ✟✠
✡
✞
✱
☛
☞
r
✌ ✍
✌
✲
✳
✔
✣r
r
✒
✓
✴
✵
✌
✌
✔✰
✌
✎
✑
✏
✷
❑
✌
✎✏
✑
✗✘
✣
✔✕
✌ ✌ ✖
✏
✳
✌
✙✗
✫✌
✗✘
✙✗
✶
✚
✌ ✛ ✘✜
✚
✌ ✌ ✣
✢
✷
✌
✣✰
✗
✌
✛
✏
✸
✺
✹
✌ r
✏
✵
❁
✪
✌
✑
✖
✤
✘
✥✦ ✧
✯
✌
✬ ✍
✌
✌
✥
★
✩
★ ✘
✻
✵
✌
✖
✏
✗
✘
✫✖
✪
✖
✏
✼
✵
✽
✌
✗
✌
✪
✌
✰
✛
✘
✣
✿
✗
✙
✸
✹
✮
✸
✣
✗
✬
✌ ✏
✔✕
✌ ✏
0,1
✷
✌ ✏
✣
✰ ✗
✌ ✛ ✰
✏
2
✵
✌
P(X = 0) = P(
✈
✯
✪
✖
✏
✯
✌ ✬
✍
✌ ✌
✻
✵
✌ ✖
✏
✗✘
✫✖
✌
✗
✌
✏
✰
✛ ✏
✫
=
✯
✌ ✬
✸
✍
✌ ✌
✰
✘
✖
)
✸
✗
✯
✈
✌ ✮
✵
✔✱
✌ ✣
✸
✹
✗r
✌
✯
✖
✤
)
52
C2
C2
✸
✙
✗
✯
✌ ✬ ✍
✌
✌
✰
✌ ✖
✏
✯
✌ ✬
✍
✌ ✌
C2
C2
❧
✸
✹
✺
❀
r
✗
✮
✸
✰
✘
✖
)
C148 C1
52
C2
4 3
1
52 51 221
52
✸
✬ ✌
✏
✗
✌
✵
✫✏
4 48 2 32
52 51 221
P(X = 2) = P (
✹
✈
✌
✮
❄
✰
X
0
1
2
P(X)
188
221
32
221
1
221
n
✼
✵
xi p ( xi )
X = E(X) =
i 1
188
32
1
34
1
2
221
221
221 221
n
2
188
32
1
36
xi p ( xi ) = 02
12
22
E(X2) =
221
221
221 221
i 1
2
2
Var(X) = E(X ) – [E(X)]
= 0
✳
✫
✌
✌
✸
✘
✈
✯
36
34
–
=
221 221
✦
✫✣ ✛
✙
✪
✌ ✰
✗
✜
✣
✎
x
=
r
✏
✖
48!
2!(48 2)! 48 47 188
52!
52 51 221
2!(52 2)!
✹
✌ ✬ ✍
✌
✌
4
✪
✌
✣✯
✰
✌
✿
✎
✗r
✌
✸
✙
✈
✯
✭
✕
✌ ✮✌ ✕
✏
✌ ✮
✸
✺
✹
✪
✌
✗
✌
✏
✦
✧
P(X = 1) = P(
X
✔✕
✌ ✏
✱
✔
✫
✮✾
✌
X
4
❄
✬
✌ ✏
✛ ✰
✺
48
✈
r
✫✏
✙
❂❃
✌ ✬ ✣
✔✏
✸
✖
✷
✟❣
r
585
✁
✂✄
☎✆✁
Var(X)
2
6800
(221)2
6800
0.37
(221)
✴
✛ ✥✚
✌ ✥✶
X
✙
✗
586
① ✁✂✄
13.4
✐☎✆✝✞✟✠✡
1.
❝☛☞✌✍
✬✜❣
✎✏
✮✕✑☞
✎✑✒✑✎✓✎✔☞☛
✯✰☞★
(i)
✏☞★✱☞
X
✕✖ ☞✎✗✏☛☞
✢✎✬☛
0
P(X) 0.4
(ii)
X
0
P(X) 0.1
(iii)
Y
(iv)
P(Z)
2.
✏✭
5.
✍✏
✗☞✣✤ ✎✥✦✏
✧★
✩✚✪
✎✓✍
✢✘✫ ☞✩
0.4
0.2
1
2
0.5
0.2 – 0.1
3
0
1
0.1
0.2
4
0.3
2
1
0
–1
0.3
0.2
0.4
0.1
0.05
✺
✏☞✚
✸✗✹☛
✯✦☞✓☞
❝☞★
✩✚✪
✕✖ ☞✎✗✏☛☞
✢✘✔✗☞ ✮☞✜★
✷☞☛☞
✍✏
✎✢✹✩✚✪
✏✭
(ii)
☛✭✑
✎✢✹✏☞✚ ✘
✏☞✚
✍✏
(iii)
✍✏
✎✢✹✩✚✪
✏✭
✧☞★
✍✏
✕☞✢☞
(ii)
❫❀
✯✦☞✓☞✚ ✘
✢☞r☞
✩✚✪
X
✍✏
✛✚✘
❝☞★
✯✦☞✓✑✚
✕★
✢✕✪✓☛☞
✢✚
❝❁❂✭
✢✘✔✗☞❃
✏☞✚
✍✏
✕★
✢✘✔✗☞
✺
✕✖✏✙
✬✜♦
✛✚✘ ✮✘☛★
✛✻✼✗
✢✘ ✫☞☞✎✩☛
✎✧☛☞✚✘
✣☞✚
❫✕☞✢✚
✩✚✪
✹✗☞
✛☞✑
✹✗☞
✸✗✹☛
✏☞✚
✹✗☞
✬✜
X
✗☞✣✤✎✥✦✏
✏★☛☞
✬✜❢
✏☞✓✭
✧★
✬✜♦
✷❝ ✍✏
?
✏✭✎✷✍✾
✛✚✘
✯✦☞✓☞✚✘
✢✘✫☞☞✎✩☛
✕✙☞✚✘ ✏✭ ✢✘✔✗☞
X
✬✜❣
✽☞☛
❝✘✙ ✑
(i)
✣☞✚
X
✬✜❣
✏★☛☞
✎✧☛☞✚ ✘ ✏✭
✑✬✭✘
✎✓✎✔☞✍❣
3
✏☞✚
X
✎✑✒✑✎✓✎✔☞☛
(i)
6.
✢✘✔✗☞
✛☞✑ ✓✭✎✷✍
✎✢✹✩✚✪
4.
✢✚
✍✏ ✏✓✲☞ ✛✚✘ ✳ ✓☞✓ ✮☞✜★ ✴ ✏☞✓✭ ✵✚✣
✘
✬✜❣
✘
✣☞✚ ✵✚✣
✘
✗☞✣✤✥✦✗☞ ✎✑✏☞✓✭ ✵✌✶❣ ✛☞✑ ✓✭✎✷✍
✵✚✣
✘ ☞✚✘
3.
✏☞✜ ✑
2
0.6
Z
✛✚ ✘
1
–1
P(Y)
❝✘✙ ✑☞✚✘
✏✭
✯✦☞✓✑✚
❝☞★
✎✧☛☞✚✘
✏✭
✏☞✚
✏✭
✛☞✑☞
✍✏
✏☞
✕★
✕✙☞✚✘
✢✘✔✗☞
✢✘✔✗☞
✢✕✪✓☛☞
✬☞✚ ✑☞❃
✢✘✔✗☞
✏☞
✵✗☞
✏✭
✢✘ ✔✗☞
✏☞
✏☞
✕✖ ☞✎✗✏☛☞
❝✘✙ ✑
✽☞☛
✏✭✎✷✍
✷✬☞✿
✬✜❣
✢✕✪✓☛☞
✛☞✑☞
✵✗☞
✬✜❣
❄❅ ❝✼❝☞✚✘ ✩✚✪ ✍✏ ❆✚★ ✢✚❢ ✎✷✢✛✚ ✺ ❝✼❝ ✔☞★☞❝ ✬✜✘ ❀ ❝✼❝☞✚✘ ✏☞ ✍✏ ✑✛✻✑☞ ❇✕✖ ✎☛✣✲☞✶ ❈ ✗☞✣✤✥✦✗☞
✎❝✑☞ ✕✖ ✎☛❉r☞☞✕✑☞ ✩✚✪ ✎✑✏☞✓☞ ✷☞☛☞ ✬✜❣ ✔☞★☞❝ ❝✼❝☞✚✘ ✏✭ ✢✘✔✗☞ ✏☞ ✕✖ ☞✎✗✏☛☞ ❝✘✙ ✑ ✽☞☛ ✏✭✎✷✍❣
7.
✍✏
✎✢✹✏☞ ✢✛✢✩✶ ✗ ✢✘☛❊ ✎✓☛
✑✬✭✘ ✬✜ ✎✷✢✛✚✘ ✎✧☛ ✕✖ ✏✙ ✬☞✚ ✑✚ ✏✭ ✢✘✫☞☞✩✑☞ ✕✙ ✕✖✏✙ ✬☞✚✑✚ ✏✭
✢✘✫ ☞☞✩✑☞ ✏✭ ☛✭✑ ✵❊ ✑✭ ✬✜❣ ✗✎✣ ✎✢✹✏☞ ✣☞✚ ❝☞★ ✯✦☞✓☞ ✷☞☛☞ ✬✜ ☛☞✚ ✕✙☞✚✘ ✏✭ ✢✘ ✔✗☞ ✏☞ ✕✖ ☞✎✗✏☛☞
❝✘✙ ✑
✽☞☛
✏✭✎✷✍❣
✐
8.
✱✝
✞
✟ ✠ ✡
☛ ☞✌ ✝
X
❑
✟
✑
X
✍
✎
✝
✟
✒
✔
✓
✕
3
✕ ✖
✍
4
5
✝
✟
X
✍
✎
✞
☛
❑
✟ ✑
✺
✗
✓
✞
✟ ✻
✟ ✟
✝
✟
✽
✟
✬
✗
✞
❑
✟ ✑
✼
❞✏
❞✏
✟ ☛
✞
✝
✑ ✟
❞
❢
✕
✾
❞
13.
✓
❆
✟
✗
✓
✬
✗
✱
✤
❞✏
✠ ✟
✝
❊
❏ ✥
✗
✓
✗
❞✟
✓
15.
✓
✗
✝
✟
✬✟
✕
✝✞
✟
✙
✟ ✗
✘
❢
☛
✕
✕ ☛
16.
✣
✛
✑
✟
✞
✟ ✛
✗
✤
✱✗
✗
✗
✓
☛
✜
✗
✴
✌ ✟
(A) 1
✕
✝
✟
✓
✗
✓
✝
✖
✞
✟ ✠
✡
☞✌ ✞
✟
✞
✲
✑
✓
✞
✟
✝
✟
✬✟ ✶
✤
✩
✗
✓
✞
❑
✟ ✑
✞
✟
✝
✖
☛✜
◆
✞
✟
✝
✟
✗
✙
✚
✱✛
X
✥
✞
✲
✓
✗
❄
✼
☛✒
✝
✎
✑ ✟
❞✎
❞✏
✟ ✵
✑
✓
✕
✟
❞✏
☛
✑
E(X)
✙
✚
✛
✤
✩
✝
✕ ✗
✕
❑
✟
✑
✑
✝
✎✑ ✟
❅
✙
✚
✓
✳✗
✞
✟ ✘
✗
✷
✟ ✟ ❞✕
❑
✟ ✑
✑ ✟
✗
✝
✟ ✗
✧
★
✍
✕ ✖
✝
✖☛✜
X
✢
✞
✟ ❆
✟ ✗
✤
☛✜
✕
✝
✖
❆
✟
✝
✖
✤
✘
X
✣
✛
✬✟ ✕
✓
✗
✱✛
◆
✗
✞
✲
✑
☛
✝✞
✟
✘
✞
✟
✙
✚
✛
❊
■
✥
✛
✓
✱✝
✍
✝
✟
✕
✗
✘
✱
X
✱✛
✗
☛
✝
❆
✟
✚
✎
✗
✝
✳
✗
❞✏
✝
✖
✞
✟
✠ ☞
✡
✌
✞
✟
✞
✟
❞✎
❆
✟
✬✟ ✶
✍
✬
✓
✗
✝
❑
✟ ✑
✴
❞
❞✏
❞✏
✟
✵ ✑
✓
❊ ▼
✥
✘
✘
✞
✟
☛✝
❅
❞✏
▲ ❖
✥
✳
✞
✟
✴
✝
✟
✳
✌
✟
☛
✟ ✟
✘
✞
✟
✛
(B) 2
▲
❖
✥
✤
✞
☛✠
✴
✬✟ ✕ ✝
☛
✳
✍
✳
✝
✟
✷
✍
✕ ✗
✜
☛
✝✞
✟
✟ ✖
%
✤
✠
❆
✟ ✎
✚
❄
✝
❑
✟ ✑
✕
✗
❞✏
✤
❆
✕ ✬✟ ✠
✗
✕
☛
✝✞
✟
✱✛
❄
✠
✾
✞
✟
✓
✗
✕
✗
☛
✳
✎✟ ✗
✾
✑ ✟
✍
✕
✓
✗
❞✎
✳
✝
✟
✙
✟
✗
✑ ✟ ✗
☛✳
✎✟
✗
☛
X=1
✝✞
✟
✣
☛
✞
✟
✱
✛
✛
✓
✗
▲
❆
✯
✞
✑
✖✕
❞✎
✣
❆
✟ ✎
✚
✱✝
❞✢
●
✝
✓
✗
✝
✟
✬✟
✶
✞
❞✎
✙
✚
(C) 5
(D)
✗
X
✍
✎
✝
✖
☛✜
❄
✞
✟
✕
❙
✕
✷
✑
✟
✢
✷
✟ ✎
❆
✟
❊
◗
✥
✳
✗
✞
✟ ✠ ☛
✡
☞✌
❯ ❖
✟ ✠
✗
✕
❄
❞
✏
✝
✗
✣
✎❁✟
❆
✕ ✬
❆
✟ ✎
✚
✕ ✗
✺
✞
✽
✞
✟
❊
P
✥
✣
✩
✝
✟ ✗
✷
✑ ✟
❊
✝
✟
X
✥
✕
✟
✝
✖☛
✜
✢
✣
✑ ✖
✕
❊
❏ ✥
❍✟
✍
✕ ✟
❃
✞
❄
❞✏
✞
☛✠
❱
✙
✖
✢
☛
▲ ❊
✥
✤
✞
✤
✜
✒
var (X)
✝
✟
✎
✥
✝
✟
✖
❊
■ ✥
✷
✗
✷
✌
✟
✥
✘
❊ ●
✥
❍✟
✕ ✗
✤
✓
✗
❊
❏
✥
✧ ✤
✌
✟
❄
✞
✟
✱✛
✞
❍✟
✙
✚
✝
✖☛✜
✠
✝
✟
✞
✟
✺
✞
✝
✖☛✜
✤✩
✗
✝
✖
❃
✱
✝
✟ ✗
☛✳
✍
❆
✟ ✎
✚
❑
✟
✑
✣
✝
✟
✓
✗
✤
✩
✌
✲
✦
✞
✟
✷
✣
✤
✗
☛✍
✑ ✟
✷
✞
☛
❞✏
✝
✟
✱✛
◆
✞
✟
✤
✗
✤
✥
❞✟
✓
❄
✠
X=0
E(X)
✬
✗
★
✤
✟ ✑
❞
✝
✙
✚
✓
✙
✚
%
❖
✗
✓
✱
✝
✧
★
✙
✟
❦ ❦
✝
✖☛✜
❞✏
✟ ✵
✑
X,
✞
☛✠
✷
❏
✩
☛
✠
✟
✓
✗
✳
❘✟
✒
✔
✓
✕
✬
✜
x 2
✪ ♥
❑
✟
✑
✢
✟
✬✟ ✕
✞
✝
✑ ✟
✝
❞✎
✤
✩
✗
❇
✖
❈
▲ ❖
✤
☛
✓
✗
✤
✒
✚
✓
✗
❍✟
❊ ▼
❚
✱✝
✬
✌ ✟
✟ ✟ ✳
✕ ✟
❞✏
✟ ☛
✦
✙
✚
✣
✓
✤❙
✝
✟
✓
✗
✒
❆
✟ ✚
✎
✝
✖
✥
✗
x 1
✈
✫
❀
❋
2)
✮
✤
✩
✗
❆
✟ ✚
✎
✬
✤
✝
✟ ✎
x 0
✪ ♥
✪ ♥
❀
❞✏
✤
✩
✟
✛
✤
✝
✟
❊ ●
✟ ✟
❊
P
✥
✞
✟
❁✟
✟ ✎
❉
✱✝
✘
✤
✖
☛✜
✝
✟
✴
✌ ✟
✟
✤
✞
✟
3k
0
✣
✴ ✌
✟
❂
✟
✟
✣
✬✟ ✕
X
14.
✗
✓
❀
✕
✱
✛
✿
❁
✕
❀
2k
2), P(X
✭
☛✕
(iv) P(0 < X < 3)
k
✱
✑ ✖✕
✹
✬✑
✝
✖☛
✜
✌
✝
✖
☛✜
✝
✖
✤
✩
✠ ✟ ✕
✗
✟
7
✣
✳✗
✷
✸
✟
6
✛
(iii) P(X > 6)
P(x)
✢✣
✝
✟
✢
✲
✤
❞✏
12.
✬✟
✕
✤
❞✟
✙
✚
✱
(a) k
(b) P (X < 2), P (X
✗
✞
✟
2k 2k 3k k 2 2k 2 7k 2 +k
P( x)
✠ ✟
✘
k
(ii) P(X < 3)
✯
✞
✟ ✰
✞
✟
P(X) 0
✞
✟ ✠ ✡
☛ ☞✌ ✝
✱✝
☛✠
2
k
10.
11.
✗
1
✝
✖☛✜
✱✝
✞
✝
✑ ✟
0
(i) k
9.
❞✏
✟ ☛
587
✁
✂✄
☎✆✁
8
3
✣
✩
☛
✥
✟ ✟
✘
✞
✟
✙
✚
588
① ✁✂✄
17.
❡☎✆ ✝✞✟✠✡ ☛☎☞☎ ✌✞ ✡✌
✑✜❞✓☎
✌✞
✌✧☛☎
37
221
(A)
13.7
✘✤✌ ✦
✍✎✏ ✎✞ ✑✒ ✓☎✔✕✖✗✓☎ ✔☎✒ ✘✙☎✒
✚✛✢
☛★
E(X)
✹✺✻✼✽✾✿
✠☎☛✒ ✚✛✢
✜
❡☎✆ ✝✞✟✠✡
X
✥✣✌☎✒ ✜
✚✛✩
❡☎✆
5
13
(B)
1
13
(C)
❝✪✫✬✭✮✯ ✰✪✯✱✬✲✬ ✳✬✭✪ ✴✵✰✶ ❝✷✸✫
13.7.1
✌☎
✟✆✌☎✝✒
2
13
(D)
(Bernoulli Trails and Binomial Distribution)
❀✺✿❁✼❂✼
✈✆✒✌ ✘✤✓☎✒ ✍☎✒ ✜ ✌✞ ✘✤❃✕❄✟☛ ✟❅✘✟✧❆☎☎❡✞ ✚☎✒ ☛✞ ✚✛✢ ❇✔☎✚✧❆☎☎❈☎❉ ❇✗☎✝☎ ✍✓☎ ✟✑✣✌☎ ✡✌ ❊✟❋☛● ✓☎ ✡✌ ❊✘✦●
✔☞☎☎❉☛☎
✚✛♥
✟✌✑✞
✘✤☞ ✆
✓☎ ❊✆✚✞✜ ✟✆✌✝☎ ✚✛♥
✟✌
✘✤☎❢☛
✘✟✧❆☎☎❡☎✒ ✜
✟✝✡♥ ✡✌
❇✙☎✧
✌☎
✟✆❆☎❉✓ ❊✚☎❍ ●
✡✌
❡✒ ✜ ✑✒
❊✚☎❍ ●
✡✌
✓☎
❊✆✚✞✜ ●
✓☎ ❊✆✚✞✜● ✚✛
❊✑✘❄✝☛☎●
✌☎✒
✟✑✣❃✒ ❄ ✌☎✒ ❇✗☎✝✆✒ ✘✧
✚☎✒
✈☎✛✧
✑✌☛☎
✈☎✟✔✢
✔❑✑✧✒
✚✛♥
✥✑
✌☎✒
✈✜✎✒
✡✌
✘✤✌ ☎✧
★✖❋☎
✑✒
✌✞
✟❏❈☎✟☛✓☎✒ ✜
❊✈✑✘❄✝☛☎● ✌✚☎
❊✟✆✌✝
❡✒ ✜
✡✒ ✑ ☎
✠☎☛☎ ✚✛✢
❋■✌ ☎
✚✛●
✘✤❋✝✆ ✚✛
❇✔☎✚✧❆☎
❃✒ ❄
❊✟❋☛● ✈☎✆✒ ✌☎✒ ✑✘❄✝☛☎ ❡☎✆☎ ✠☎✡ ☛☎✒ ❊✘✦● ✈☎✆✒ ✌☎✒ ✈✑✘❄✝☛☎
✌✚☎ ✠☎✡✍☎✢
✘✤✐✓✒✌ ★☎✧♥ ✠★ ✚❡ ✡✌
✚✛♥
✜
☛★ ✚❡ ✥✑✒ ✡✌
☛☎✒
✘✧✞❣☎❆☎☎✒✜ ✌✞
✈✑✘❄✝☛☎✢
❃✒ ❄
✘✤✌☎✧
❏❃☛✜❖ ☎
✘✤✐✓✒ ✌
r
✡✌
✘✧✞❣☎❆☎
trial
✥✆❡✒ ✜ ✑✒
✘✧✞❣☎❆☎ ✌☎
✘✟✧❆☎☎❡
✚☎✒ ✍✞
❡✒ ✜ ✑✘❄✝☛☎
❃✒ ❄❃✝
✌✚✝☎☛✒
✟✑✣✌☎ ❡☎✆ ✝✞✟✠✡♥ ❋☎✧ ★☎✧ ❇✗☎✝☎ ✠☎✡
◆ ✌✚☛✒ ✚✛✢
✜
✓✟✔ ✡✌
✈☎✛✧
✘✧✞❣☎❆☎
✟✠✆❃✒ ❄
✘✧✞❣☎❆☎♥
★✧✆☎✛✝✞
✘✧✞❣☎❆☎ ▼
✑✜❞✓☎
✟✌✑✞
✟✑✣✌☎ ❇✗☎✝☛✒ ✚✛ ✜ ✓☎ ✡✌ ✘☎✑☎ ❇✗☎✝☛✒ ✚✛✜ ✓☎ ✌☎✒ ✥❉ ✈▲✓ ✘✤✓☎✒✍ ✌✧☛✒
✔☎✒
✘✤✐✓✒ ✌
❃✒❄
✘✟✧❆☎☎❡
✔❑✑✧✒
✟✌✑✞
✘✧✞❣☎❆☎ ❃✒ ❄
✈✑✘❄✝☛☎◆ ✌✞
▼✓☎
✘✟✧❆☎☎❡
✚☎✒ ☛✒
✚✛✜
✠☎✒
☛❈✓☛✩
✘✤☎✓✩
✔☎✒
✘✟✧❆☎☎❡
✘✤☎✟✓✌☛☎✡❍
✑✒
✈❋✧
❊✑✘❄✝☛☎●
✈❈☎☎❉☛✏
✚☎✒ ✜ ✍✒
❏❃☛✜❖ ☎ ✚☎✒ ☛☎ ✚✛✢
✚☎✒ ☛✞
✚✛✢
✓☎
✥✑
✘✤✌ ☎✧
❃✒ ❄
✌✚✝☎☛✒
✚✛♥
✜
✥✑
❊✈✑✘❄✝☛☎●
✓☎
✑✘❄✝☛☎
✚✛✢
✜
P◗❘❙❚❚❯❚❚ ❱ ✡✌ ✓☎✔✕ ✟✖✗✌ ✘✤✓☎✒ ✍ ❃✒ ❄ ✘✧✞❣☎❆☎☎✒ ✜ ✌☎✒ ★✧✆☎✛✝✞ ✘✧✞❣☎❆☎ ✌✚☛✒ ✚✛✜ ✓✟✔ ❃✒ ✟✆❲✆✟✝✟❞☎☛ ☞☎☛☎✒❳
✌☎✒
✑✜☛■❨ ✦
(i)
✘✧✞❣☎❆☎☎✒ ✜
(ii)
✘✧✞❣☎❆☎
(iii)
✘✤✐ ✓✒✌
(iv)
✟✌✑✞
❇✔☎✚✧❆☎
✘✤ ✐✓✒ ✌
✚✛✩
✜
✌✧☛✒
✌✞
✑✜ ❞✓☎
❏❃☛✜❖ ☎ ✚☎✒ ✆✒
❃✒ ❄
☛❈✓☛✩
✘✟✧❆☎☎❡
✌✞
✘✤ ☎✟✓✌☛☎
❃✒ ❄
✟✝✡
❇✗☎✝✒✜ ❏❃☛✜❖ ☎
☛✌
✟✝❞☎✞
✍✥❉ ✚✛
❋☎✟✚✡
✚☎✒ ✆✞
❋☎✟✚✡
✡✌
✘☎✑✒
✔☎✒
✌☎✒
✘✧✞❣☎❆☎ ✌☎ ✘✟✧❆☎☎❡ ✑✘❄✝☛☎
❇✙☎✧☎✒ ✙☎✧
❧
▼✘✟✧✟❡☛◆
✘✧✞❣☎❆☎
✘✤✌ ✦ ✚☎✒ ✆☎◆ ✚✛ ✈☎✛✧ ✑❬☎✞
✑✒
✟✆✟☞❋☛
✚✞
✘✟✧❆☎☎❡
✘✤✐✓✒✌
♠❩
✘✧✞❣☎❆☎
★☎✧
❋☎✟✚✡♥
✚☎✒ ✆✒
❡✒ ✜
❇✗☎✝✆☎♥
✑❡☎✆
♠❩
▼❡☎✆ ✝✒ ✜ ✑❡ ✑✜❞✓☎
☛☎✒
✚☎✒ ☛✒
p=
1
2
✚✛✢
✜
✓✟✔
✘☎✑☎
✑✘❄✝☛☎
✌✞
▲✓☎❭✓
✈☎✛✧
✚✛
✧✚✆✞
✈☎✛✧
✌✞
1
2
✟❏❈☎✟☛
✚✛♥
✟✠✑❡✒ ✜
✓☎ ✈✑✘❄✝☛☎ ▼✟❃❨☎❡ ✑✜❞✓☎
✡✌ ✑❡☎✆ ✚✛✢
✥✑❃✒❄
q=1–p=
✈✑✘❄✝☛☎
✘✧✞❣☎❆☎☎✒ ✜
✘✤✌ ✦ ✚☎✒ ✆☎◆
(p)
✓☎
❋☎✟✚✡
★✧✆☎✛✝✞
♠❩ ❇✗☎✝☎✒ ❡✒ ✜ ✑✘❄✝☛☎ ✌✞ ✘✤ ☎✟✓✌☛☎
✘✤✓☎✒ ✍
✑✘❄✝☛☎
✗✩
✘❄✝✌☎✒ ✜
✈✑✘❄✝☛☎
✟✆✩✑▲✔✒✚ ✘☎✑✒ ✌✞
✘✧
✗✩
✌✞
✑✜❞✓☎✡❍
❪
✘✤☎✟✓✌☛☎ ✚✛✢
✐
♠ ✝
✞ ✟
✠
✡
✞
☛
☞
✼
✌
✍ ✌
❢
✧
✓
✕
✖
✗
✘
(i)
(ii)
✧★
✓✍
✮✳
✌ ★
✖
✵
✶
✧✭
✮✳
✮
✵
✶
✒
✑✔
✓
✍
✧✓
✌
✔
✙
✍
✮
✍
✬
✑★
✍
✕✲
✍
✍ ✖
✌
✏
✔
✌
✖
✯
✑✔
✚
✓
✰
✍
✓✌
✛
✍
✱
✍
✲
✗
✏
✜
✱
✍
★
✗
✖
✢✖
✣ ✤
✍
✲
✔✗
✑
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✕✖
✗
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✓✍ ✌ ✔
✮✳
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✧
✘
✕✩ ✫
✪
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✭
✍ ✩
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✖
✧★
✓✍
✌
✙
✖
✬
✍
✘
✕✖
✘
✗
✓✍ ✖
✫
✲
★
✗
✖
✱
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✭
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✘
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✰
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✲
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✍ ✍ ✧
✑
✏
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✖
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✍ ✍ ✧
✧✭
✎
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589
✁
✂✄
☎✆✁
✧✓
✱
✍
✕✲
✍
✍
✫
✖
❣
✟
(i)
✮
✯
✑✔
✰
✍
✮✳
✲
✍ ✍
✖
✗
✓
✵
✶
✧✭
✔
✢✗
✷
✮
✍ ✍ ✧
✮
✍
✧✓
✻
✍
✮
✯
✑✔
✖
✧✓
✰
✍
✢
✾
✍
✔
✕✲
✍
✥
✦
✯
✑✔
✖
✗
✖
✭
✍
✖
✗
✘
✮
✏
✗
✘
✍
✓
✍
✯
✑✔
13.7.2
❂
✖
✗
✓✍
✧
✢❊
✧
★
✓
✖
✗
✜
❃
❄❅
✖
✧✬
★
✍
✵
✶
✧✭
✌
★
✍ ✺
✖
✗
✢
S
★
✌
✖
✗
❆❇
✜✍
✣ ✥
✍
★
✱
✵
✮▲❁
✏
✗
✫
✌
★ ✖
✌ ✔
✧
✢✔
✸
✢
✜✍ ★
✜✍
★
✌
✖
✗
✌
✍ ✌
✖
✕
✖
✧
✘
✗
✮✳
✏
✎
✭ ✦
✕
✗
✘
✖
✍ ✖
✗
✓✍
✻
✚
✧
✿
✓✍
❢
★
✍
✖
✧
★
✓✍ ✌ ★ ✖
✵
★
✓
✶
✧✭
✮✽
✙✖
✌ ★
✍
✺
✮
✍ ✍
✮✱
✙✖
✧
★
✓✍
7
16
✓✭
✍
✔
✮✳
✱
✿
✕✲
❀
✏
✍
✭ ✍
✮
✘ ✢✑✖
✢✜✍ ★
★
✖
✬
✓
❢
★
✍
✮
✌
✖
✯
✑✔
✱
✓✭ ✍
✍
✜
✱
✿
✔✗
6
15
✗
✖
✲
✏
✯
✑✔
✰
✍
✍
✘
✔
✓
✌
✛
✍
✜
✗
✖
★
✦
7
16
p=
✲
✍ ✧
✓✭
✍
✶
✙✖
✢
✍
✍
✧★
✓✍
✱
✰
✍
✌ ★
✍
✮❢
✍
✜
✗
✖
✢
✬
✑★
✍
✌
✏
✔
✶
✌
✭
✍
✱
✮
✎
✍
✯
✑✔
✮✳
✴
✲
✑
✮❁
✌
✭
✍
✑
✏
✩
✢
✮▲❁
✬
✑★
✍
✌
✏
✔
✯
✑✔
✍
✰
✍
✌
✍
✌
✲
✭
✮
✍
✍ ✍ ✭
✪
t
✭ ✑
✰
✍
✵
✎
✢
✾
✱
✍
★
✍ ✔
✱
✔
✗
✗
✏
✫
✸
✧
✙
✹
✍
✱
✺
✑
✱
✍ ✭
✖
✖
✓
✔
✧
✻
✚
✮✳
✴
✲
✗
✏
✫
✧
✻
✢
✜
✮
✓
✖
✯
✑✔
✗
✖
✰
✍
✮
✓
✖
✯
✑✔
✰
✍
✮✧
✍
✓✍
✮❢
✍
✜
✗
✖
✢
✮
❢
✑✰
✍ ✍
✜
✢
✮❢
✌
✭ ✍
✎
✍
✑
✏
✎
✢
✌
✭ ✍
❋
✌
✭ ✍
✓✍ ✖
✜✛
✍ ✦
✫
✧✓
✙
✧✾
✍
✮✑✔
✯
✜
✥
✦
❢
★
✰
✍
✮❢
✍
✍
✗
✖
✜
✗
✖
✚
✓
✢
❢
✌
✭
✍
✱
✭ ✑✔ ✙ ✖
●
✙✖
✧✙
✧✾
✍
❀
✏
✗
✻
✢
✏
✍
✧
✓
★ ✔
✹
✖
❍
★
✭
✑
✔
✓
✍
✲
✢
✹
✔
✬
❏
✧✓
✗
✖
✓✍ ✖
✢
✎
✍
✑
✏
✹
✍ ✑
✎
✢
✓ ✑★
✖
✜
✗
✖
✩
✥
✓
✽
✱
✍
✕✲
✍
❋
✮❢
✌
✭ ✍ ✚❜
■
✍ ✭
✦
✏
SFFFFF, FSFFFF, FFSFFF, FFFSFF, FFFFSF, FFFFFS
6!
4! 2!
n
n
✘
✍ ✖
✸
✱
✏
✎
✭ ✦
✮
✍ ✕
✖
✮❢
✑
✌
✍
✲
✍ ✧
✲
●
✥
✦
✮✳
✩
✓
✙✖
✎
✢
✓✔
✧✻
✚
✍
✗
✖
✲
✮❢
✍
✲
✭
✓✍
✮✳
✌
✭
✍
✮
✍ ✧
✱
★
✍
✘
✗
✖
(Binomial Distribution)
❈ ❉
❞
✮
✓
✢
✜
✍ ✍
✔
❢
✙
✖
✧✹
✭ ✺
F
✎
✍ ✏
✑
✗
✮✳
✌
✭ ✍
✲
✸
✜✍
✓
❢
✙
✖
✕
✱
✍ ✍ ✖
✮❢
✍ ✍
❢
✚
✓
✻
✬
✫
✰
✍
✖
✰
✍
✮✳
✕
❢
✫
✏
✱
✍
✕
★
✧✛
✹
✭ ✺
✮❢
✍
✮✳
✻
✬
✸
✧
✱
✍
✮
✏
✑
✧ ✜✭
✲
✭
✱
(ii)
✱
✧
✲
✱
✜✹
✌
✭ ✍ ✚❜
✜
✗
✖
✍
✖
✱
✢
✓✭
✔
✫
✏
✗
✩ ★
✢✾
✍
✔
❑
❋
✲
✜✹
❀
✍
✖
✗
✓
✔
✮
❢
✢
✹
❍
✔
✬
★
✍
★
✍
✲
✔
✌
✬
✗
✍
✓✍
✓
✑★
✍
✲
✢
✗
✷
✌
✬
✗
✮✳
✍
✓
✔
✍
✎
✍
✏
✑
✲
✍ ✧
✌ ✖
✎
✍
✮
✮✳
✓
✔
✯
✑✔
✏
✑
✢
✾
✗
✍
✙
✢
✖
✬
✷
❝
✍
✎
✍
✖
✗
✓
✮✳
✍
✓
✍
✍ ✧
✔
✫
✧
✌
✚
✢
✧
▼ ✿
◆
✿
❖❖❖
✿
✢
✓✭
✍
✮
✏
✫
✬
❧
✢
✑★
✍
✰
✍
✓
■
✘
✧
p
✗
✔
✓✍
✧
★
✜✍
✪
✌
✏
✔
✶
✭
✔
✬
★
✍ ★ ✖
✮
✲
✳
✍ ✍
q
✢✖
✓
✍
✱
✖
✌
✭
✖
✖
✧✓
✩
✍ ✍
✱
✍
✕✲
✹
✍
✻
✍
✢
✧
✻
✢
✜
✮
✍ ✕
✖
✢
✓✭ ✍
✸
✯
✑✔
✗
✖
✮
✓
✖
✍
✰
✍
✗
✖
✜
✓✍
✓
✗
✖
✢
✢
✻
✢✢✖
✣
✰
✍
✜
✗
✖
✕
✪
✗
✓
✔
✍
★
✍
✜
✢
✗
✖
✌
✕
★ ✖
❢
✘
t
✘
✛
✖
✙
✖
✧
✌
✚
✭
✔★
✮❢
✌
✭ ✍
▲❁
✧✭ ✘
✛
✍
✓✭ ✍
✌
✭ ✍ ✎
✍ ✖
✮
❢
✍
✲
✍ ✧
✮❢
✢✖
✲
✩
✔
✰
✧
✮✳
✍ ✺
✗
✖
✿
✏
✫
✏
✯
✑
✔
✮✳
✎
✍
✰
✍
✲
✍
✮✳
✴
✲
✗
✏
✮✳
✲
✫
✗
✏
✬
✱
✍ ✕
✖
✯
✑✔
✌
✭ ✍
✱
✢
✹
✥
✓
✮❢
✢
✱
✍ ✖
✚✓
❤
✿
✌
✚
✱
✪
❀
✙
✖
✩
❀
✎
✍ ✑
✏
✎
✢
✲
✢
✜✧
✓✍ ✭ ✔✪
lSSS, SSF, SFS, FSS, SFF, FSF, FFS, FFFq
✌
✭
✍
✰
✕✽
✍
★
✱
P
✲
✌
✭
✍
✲
✢✗
★
✖
✜✛
✍ ✦
✮❢
✔
✑★ ✖
✑✰
✍ ✍ ✜✍ ✖
❋
S
✓
✓
✲
✍ ✍ ✖
✗
✓✭ ✍
✚❜
✢
✓✍
❢
✍
✭
✰
✍
✲
✍ ✧
✌
✍
✮✧
✢
✜
✑
★
✍ ✏
✌
✔
✙
✍
❍
✲
✙
✍
✌ ★
✖
✖
✿
✍ ✕✍
✖
✲
✢
✜
✓✭ ✍
✱
✪
✲
✍ ✭
✬
✓✍
✢✗
✷
✲
✲
✍
✚✓
❁
✓
✭ ✍
✬
✗
❝
✍ ✘
✧
✥
✓
✹
◗
★
✧
★
X
■
✑
✱
✏
✎
✍ ✑
✏
✮✳
★
✧✌
✧
✷
✍
✭
✓✍
✮✳
✑
✢✖
✍
0, 1, 2,
❘
✭
✲
✍
3
✲
✧
✓
✱
✜✍ ★
✱
✍
✕✲
✍
✏
✏
✫
✌ ✖
✢
✓
✭ ✍
✮❢
✫
✏
✢
✌
✭ ✍ ✎
✍
✗
✖
590
① ✁✂✄
P(X = 0) = P(
)
= P({FFF}) = P(F) P(F) P(F)
❞☎✆✝✞
✟✠✡☛☞☎
✌✍✎✏
= q . q . q = q3
✭✑✒☎✆✏ ✓❞
✠✔✎✕☎✖☎
✗✘☞✏✙☎
✍✚✏ ✛
P(X = 1) = P(
)
= P({SFF, FSF, FFS})
= P({SFF}) + P({FSF}) + P({FFS})
= P(S) P(F) P(F) + P(F) P(S) P(F) + P(F) P(F) P(S)
= p.q.q + q.p.q + q.q.p = 3qp2
✱❞
✟✠✡☛☞☎
P(X = 2) = P (
)
= P({SSF, SFS, FSS})
= P({SSF}) + P({SFS}) + P({FSS})
= P(S) P(S) P(F) + P(S) P(F) P(S) + P(F) P(S) P(S)
= p.p.q. + p.q.p + q.p.p = 3qp2
♥☎✆
✟✠✡☛☞☎✱✜
P(X = 3) = P(
) = P ({SSS})
= P(S) . P(S) . P(S) = p3
✈☎✚✔
☞✎✌
✈☞✢
X
❞☎
✤✏ ✥✌
✠✣☎✓✒❞☞☎
✟✠✡☛☞☎✱✜
✍✚
X
P(X)
✟☎❧☎
✍✎
(q + p)3
❞☎
✓✦✠♥
0
q3
✓✘✗☞☎✔
1
3q 2p
✓✌✧✌✓☛✓★☎☞
3
2
3qp 2
3
p3
✍✚
q + 3q p + 3qp2 + p3
✌☎✆ ✥
✠✍☛✎✪
♥✐ ✟✔✎✪
✟☎❧☎
✉✚ ✟☎
❞✎✓✉✱
✓❞
✍✎
✓❞
✈☎✠✆✓✕☎☞
✈☞✢
✍✰
✫✪
✈☎✚✔
☞✎✟✔✎
✑✒☎✆ ✏ ✓❞
✩✪
✒✍
✓✌✷❞✷☎✞
✠✣☎ ✓✒❞☞☎✱✜
✉☎
✍✚✏✵
✝✟
✹✟✠✡☛☞☎✈☎✆ ✏
✗✠✷✥☞✒☎
✈✤
x
x
❞✎
✠♥
✍✚
✟✠✡☛☞☎✈☎✆✏
❞✎
✠✣☎ ✓✒❞☞☎✱✜
✯✰✲☎✢
(q + p)3
✘✆ ✡
✓✘✗☞☎✔
❞✎
✍✚✏ ✵
✓✉✟✟✆
✓✌❞☎☛
✈❧☎✞
✒✍
✓✌❞☛☞☎
(q + p)
✠✣☎✓✒❞☞☎
(S)
(S)
✈☎✚✔
❞☎✆
✺☎☞
❞✎
✘✆✡
✓✟t
❞✔☞✆
♥✲☎☎
(n–x)
✓❞
n-
✓✘✗☞☎✔
❞✎
✟❞☞✆
n
✠✓✔✖☎☎✰
✟✠✡☛☞☎✈☎✆✏
✟✠✡☛☞☎✱✜
✮
✍✚
✓❞
✟❣☎✎
✠✣☎ ✓✒❞☞☎✈☎✆ ✏
❞☎
✒☎✆ ✶
✫
✍✚
❧☎☎✵
❞✎
x
✒☎
✳☞✴ ❧☎✞
q+p=1
✟✠✡☛☞☎✈☎✆ ✏
✟❞☞✎
✬✪
2
✰✆✏
✍✚✏
❞✔✌✆
✤✔✌☎✚ ☛✎
✘✆ ✡
✠✣❧☎✰✪
✓☛✱
✍✰
✠✔✎✕☎✖☎☎✆ ✏
✓✦☞✎✒✪
n
✘☎☛✆
☞✸☞ ✎✒✪
✹✤✔✌☎✚ ☛✎
✠✣✒☎✆ ✶
✰✆✏
...n
✘✎✏
0, 1, 2 ...., n
✠♥
✟✆
✠✔✎✕☎✖☎☎✆✏
✘☎☛✆
✯✰✳✒
✍☎✆☞ ✎
✠✣☎♦☞
✠✣✒☎✆ ✶
✍✚✏ ✵
(n–x)
✈✟✠✡☛☞☎✱✜
✈✟✠✡☛☞☎✱✜
(F)
n!
(F), x !(n x)!
✍☎✆✏ ✶✎✵
☞✔✎❞☎✆✏
✟✆
✍✚ ✏✵
❞✎
✰✆✏
✐
❜ ✝
✞✟
✠
✡✟
☛☞
✌
✍
✟
✎
✏ ✑✒ ✓
✟
✔
✞✟
x
✠
✡☛
= P(x
n-
✫
✑✝
✕
✗
✥
✒
☛
✑✒ ✬
✕
✭
✕ ✕ ✟
▲
☛✰✲✏
✍
P (x
✕
n
✶
❜ ✡
☛☞
✎✕
✷
✫
✑
(q + p)n
✡☛
✑✝
✕
✗
✥ ✒
✞✟
✡☛
✈
✡☛
✔✥
✏ ✕ ❧
✘
✖
✳
✕
✕ ✏
✴
✵
❾
x)
✖
✡☛
). P[( n x)
✙
✔✥
✏
✕
✖
✕ ✟
✚
✛✜ ✢ ✣
❦ ✱
→
✠
✎
✒
✑✒ ✬
✕
✭
✕ ✕ ✟
✠
✈
✓
▲
✏ ✕
✑
✖
✕ ✟
✠
✎
✓
✕
☛☞
✕
☛☞
✕
✥
✟
x
❧✎
x
✸
✞✟
✠
✓
✟
✔
✡☛
✈✓
▲
✔
✥
✏ ✕ ✖
✕
✟
✼
✏
✎
✒
= px qn–x
❝
✩
❥
x
n
❀
✮
Cx p x q n
✏
✕
✠
✑
✎
✎
✒
(x + 1)
✒
✡
✓✒✠
✹
✠
✍
✕
✎
✒
☛☞
✕
☛♦
✡✎ ✏ ✒
❣
❣
✯
✗
✖
✏ ✪
✡☛
✔✥
✏
✕
✖
✕ ✟
✠
✎
✒
❣
✯
✗
✯
✗
✈
✍
✎✏
✕
X
✶
✕
x
, x = 0, 1, 2, ..., n, (q = 1 – p)
, (q + p)n
☛☞
✍
✕ ✟
✈
✍
✎✏ ✕
n!
p x qn
x !(n x)!
✈
✍
✎✏ ✕
) = nCx p x q n
✻
✕ ✑
✕
✒
☛☞
✕
]
✤
✦
✦
✧
✦
✦
★
(n x)
Cx p x q n
✺
☛♦ ✷
(n
✑
✗
❝
✩❥
✔✥
✏ ✕
n
)
☛
x
✠
✺
✓
✟
✔
✔✥
✏
✕ ❧✘
x
✠
✔
✥
✏ ✕
❧✘
✖
✕
✤
✦
✦
✧
✦
✦
★
P(x
✖
✏ ✪
✡☛
✠
P (S).P (S)...P(S) . P (F).P (F)...P(F)
=
✖
✏ ✪
✔
✥
✏ ✕ ✖
✕
✟
591
✁
✂✄
☎✆✁
✡✹
✠
✍
✕
✎✕
✫
✲
✠
✝
✫
✲
✠
✝
✼
✈ ✝ ❢✝ ✈ ✥
✈
✹ ✕ ✏
☛☞
✎✕
✑
✡✟
✈
✥
✹ ✕ ✕
X
✕
✡✎ ✏
✕
0
❣
✯
✗
1
2
...
P (X) nC 0 q n nC 1 q n–1 p 1 nC 2 q n–2 p 2
n
n
✼
♠
☛✍
✽
✾
✴
✏
☛☞
✕
✈
✍
✎✏
✕
✫
✲
✠
✝
✎✕
✟
✿
❁
❂❃
❄❅
❆ ❇
✎❣
✏
✟
❣
✗
✠
✈
✡✞✟
✠
...
x
n
C x q n–xp x
p,
✏ ✳
✕ ✕
n
Cn pn
n
✶
☛☞
✕ ❈
✥
❣
✠
✗
✾
✍
✕ ✟
✈
✠
✎
p
✏
✳
✕ ✕
✓
✟
✔
❉
✞✕ ✝
✈
♦ ❧
❣
✕ ✟
✝
✟
☛
✑
❣
✞
✡☛❡
✠
✭
✕
✴
☛☞
✕
✈
✍
✎✏ ✕
✫
✲
✠
✝
✕
✏
x
P (X = x)
P(x)
n
n–x x
P(x) = Cxq p , x = 0, 1, ..., n (q = 1 – p)
P(x)
nB(n, p)
✡☛
✔✥
✏ ✕ ✖
✕
✟
✠
✎
✒
☛☞
✕
✈
✍
✎✏ ✕
✎✕ ✟
✎
✑
✡✟
✡✎ ✏
✟
❊
✕
✒
❣
❋✍
✾
✠
✗
✯
✏
✎
✑
✏ ✟
❣
✗
✠
✖
✕
✑
✗
❜ ✡✟
✻
✡✟
☛☞
✕
✏
✎
✑
✏ ✟
❣
✯
✠
✗
✺
✎✕ ✟
❜
✡
✈
✫
❧✎
✡✟
☛♦
✑✝
✕
❋
✍
✫
✗
✥ ✒
✾
✏
✲
✠
☛
✎
✝
✎✕
✑✒ ✬
✕
✑
✏
✟
✭
✕ ✕ ✟
❣
✠
✗
❂
❍
●
✿
■
✠
✖
✕
✑
✗
❏
❑
❍
☛☞
✌
✍
✟
❂▼
◆
✎
☛
❇
✑✒ ✬
✕
✎❣
✏ ✟
✭
✕
✞✟
✠
❣
✡
✯
✠
✗
☛✔ ✥
✏ ✕
✎
✒
☛☞
✕
✈
✍
p,
✎
✏ ✕
✺
✓
✕
✥
✟
✈
☛♦
✫
✲
✠
✝
✎
✕ ✟
✯
❖
✖
✕ ❜
❧
P
✖
✫
◗❘
❙
❃
❍
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✽
✔
❚
✈
♦
❧✎
❲
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✕
❳ ❨
✍
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✓
✟
✔
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❖
✫
✕ ✑
♠
❉
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✕
✸
✍
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✏ ✕
✟
✪
❡
✈
❈
✪
❬
✝
▲
☛✰✲✏
✍
✈
✡✾
✥
✒✈
✕
✕ ✈
✍
✎✏
✕ ❧✘
✼
✕ ✏
✎
✒
✈
❧
✪
❖
✎✏
✞
✪
✈
❈
✏
✓
✟
✔
✎✕
✟
✫
✕
✑
✫
✕
✑
♠
✕
❳ ❨
✥
✝
✕
✫
✑✝
✕
✥
✗
✒
☛
✑✒
✬
✕ ✭
✕
❣
✕ ✟
✏
✟
❣
✼
✞✕
☛☞
✈
❈
✏
❖
X
✒
❖
✝
✏
✞
✖
✈
❧✎
✎
✏
◗
◆
✈
✝ ❢✝
❖
✒
✎
✍
✥
✟
✯
✠
❱
✍
❩
❱
✑✭
✕
❯
❍
(i)
(ii)
(iii)
♠ ♦ ✕
❣
X
❧
✫
✯
✲
✠
✝
n = 10
✖
✕ ✑
✗
p=
1
2
✺
✓
✕ ✥
✕
✈
☛♦
✫
✲
✠
✝
❣
✯
✗
✯
✠
✗
☛✑✒ ✬
✕
✭
✕ ✕ ✟
✠
✞✟
✠
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✠
✎
✒
✡
✹
✠
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✕
✎
✕ ✟
592
① ✁✂✄
P(X = x) = nC x q n–x p x
❜☎✆✝✞
1
1
, q=1–p=
2
2
n = 10, p
❀✟✠✡
P(X = x) =
❜☎✆✝✞
10
10 x
1
2
Cx
x
1
2
10
Cx
1
2
10
✈☛
(i) P(
(ii) P(
❇☞✌
) P (X = 6)
✍✎
✆✏✑
10
1
2
C6
10! 1
6! 4! 210
105
512
) = P(X 6)
= P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
❯❀✒✓ ✑✔
✍✎
10
=
✕
✆✏✑
10
1
2
C6
10
10!
6! 4!
=
(iii) P
10
✭✈✆✖✌✑✔
✍✎
1
2
C7
10
10
10!
7! 3!
✆✏✑✗
1
2
C8
10!
8! 2!
= P (X
10
10
10
1
2
C9
10!
9! 1!
10
1
2
C10
10
193
512
10! 1
10! 210
6)
✘
= P (X = 0) + P (X = 1) + P (X = 2) + P (X = 3)
+ P (X = 4) + P (X = 5) + P (X = 6)
10
1
2
=
+
10
10
C5
C1
10
1
2
10
10
1
2
10
C6
C2
10
10
1
2
C3
10
10
C4
1
2
10
10
1
2
848 53
1024 64
10%
1
2
=
♠✙✚✛✜✢✚ ✣✤
❜☎
☛✠✑
✌☞
❬✠✥✠☛ ✈✦✧✠★✦ ✩✠✝★ ✞✌ ✪★✥
✰✱✠✆❀✌✑✠
✛❣ ✔✠✓ ✝☞✆✸✞
X
✷✠✑
✌☞✆✸✞
✆✌
✫✬
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✩★✴
✆♦✰✹
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✞✌
❬✠✥✠☛
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❬✠✥✠☛ ✈✦✧✠★✦ ✌☞ ☎✦❬❀✠ ✌✠★ ✽❀✾✑ ✌✥✑✠ ✟✼✶ ✾❀✠★✆
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✆✓✌✠✝✠ ✵❀✠ ✟✼ ❜☎✆✝✞ ❀✟ ☛✥✓✠✼✝☞ ✰✥☞❢✠✿✠ ✟✼✶
✦ ✲✰❁❂✑❀✠
✩✠✝✠
✰✱ ✆✑✹✺✠✻
✟✼✶
X
✌✠ ☛✦❂✓
n = 10
✈✠✼✥
p 10%
10 1
100 10
✐
1 9
10 10
) = P(X 1) = 1 – P(X = 0)
q =1–p=1–
❜ ✝✞
✟ ✠
P(
✈
✡
❯
☛
☞
✌ ✍ ✎
✠
✏
✑ ✒ ✓
✒ ✡
✈
✕
✔
✒
✖
10
=1
✗✙
✘
✚
1.
✠✏
✍ ✒
✤
✱✒ ✝✤
✞
✌ r
✌
(i)
2.
✱✒
✝
✒
✝
✱
3.
✏
✒
✞
✟
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★
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✎
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✤
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✳
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13.5
✭
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★
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(ii)
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✒ ✓
✸
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✪
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✤
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✞
☛
✏
✍
✒
✡
✡
✒
✔
✤
✎✔
✤
✝✤
✈
✞
✏
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✝✤
✏✎
✠✏
❧
✞
✬ ✌
✒
✔
✤
✏
✤
✦
✱
☛
✒ ✶
✤
✼
✤
✲
▼
☛
☞
✒
✩
✒ ✤
★
✒ ✠
✶
✴
✤
✫
P
✓
✝✤
✹
✒ ✟
✪
✤
✩
✒ ✌
✤
❆
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✏
✎
❇
☛
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✤
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❑
✼
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▲
✠✏
▼
☛
☞
✻
✠
✏
(iv)
✻
✈
❜ ✝
✡
✒ ✬
☛
✡
✒ ✬
✺
◆
6.
✝
✑
✔
✳
✷
☛
✌
☞
✍ ✎
★
910
1010
1
✺
❯
☛
✵
✍
✏
10
9
10
C0
✛
✜✢
✣
✺
✞
✌
5.
✎✤
✔
☛
✒
✝
✱
✻
✱✮
✫
✪
✦
✧ ✒ ✟ ✒
❀
5%
❃
✤
✔
✡
✒
✵
✩
? (ii)
✲
✟ ✍ ✒ ✠✴
✏
✒ ✤
★
✒ ✍
❂
✍ ✈
✒
★
✒
✍ ✒
✺
☛
✍
✲
✟ ✍
✒
♦
✼
✤
✔
✠✏
4.
593
✁
✂✄
☎✆✁
✍
✏
✼✲
✤
✈
✏
✔
✒
✔
✤
✎
✔
✤
✝✤
✠✏
✈
✔
✏
✞
✟ ✑ ✒ ✒
✩
✫
✪
❂
✱❂
✌
✼
✒
✱
✝
✓
✑
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✤
✩
✠
✞
✌
✏
✒ ✟
✳
★
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✪
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❜ ✝✏ ✳
✵
☛
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✱✮
✒
✞☛
✏
✍ ✒
❇
✩
7.
✪
✞✏
✦
✌
✎✤
✔
✝✤
❖
✠✏
✝
✞ ✏ ✝✳
❋
✒
✳
✶
✬
✤
✔
❖
☛
●
✈
✝
✱
✓
✈
✏
✔
✌
✞
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❇
☛
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✏
✒
✓
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✤
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✱✮
❈
✌ ✒
✔
✤
✼
✒ ✟ ✳
✱
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❘
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✤
✤
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✏
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✏
✒
❁
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✓
✏✳✞
★
✠
✭
✝
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✌
✟
✔
✤
✞✏
✠✏
✞✼
✒
✻
☛
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✤
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❅
☛
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✱❅
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❙
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☛
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✏
✩
✒
✤
❖
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✏
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✤
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✩
✤
✳
✭
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✝
✦
❊
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✤
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❜
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✏✳
594
8.
① ✁✂✄
❡☎✆ ✝✞✟✠✡
♦☎✝☎
✍✟✚✛☎☎❡
(
✡☛
✥✏✱✤✟♦☛✦✍✞✙
✍✘☎✟✙☛✗☎
✎✜
10.
✏✑
✥☞ ✌ ✆ ✏✑✒
✎✓☎☎✔ ✡✕
✟☛
X=3
✈✟✖☛✗❡ ✍✘☎✟✙☛✗☎
✟☛
❧✣☎✞
P(xi), xi = 0,1,2,3,4,5,6
✍✚✞✧☎☎
✡☛
❡✜☞
★
✟♦✐☎✮☎✞✔
✍✘✓ ✆
✏✑
♦✜ ✢♦✝
✟✠✆❡✜☞
✈✆✱ ❡☎✆
❡✜☞
❧✜
✈✟✖☛✗❡
✍✘✩✙✜ ☛
♦✜ ✢
✗✞✆
✝✯☎
☛✚
✰☎✚
✏✑
)
❧☞✣ ☎☎✟♦✗ ✪✫☎✚ ✏✑✒
☞
✙☎
✈✟✖☛
✍✘✓ ✆☎✜☞
✬❧☛✞ ✭✙☎
♦✜ ✢
❧✏✞
✪✫☎✚
✡☛ ✲✙✟✭✗ ✡☛ ✝☎✳✌ ✚✞ ♦✜ ✢ ★✴ ✟✌☛✌ ✵☎✚✞✎✗☎ ✏✑ ✶ ✟✠❧❡✜☞ ✪❧♦✜✢ ✍✘✩ ✙✜ ☛ ❡✜☞ ✠✞✗✆✜ ☛✞ ✍✘☎✟✙☛✗☎
✏✑✒
❯✙✷ ✆✗❡
✬❧☛✞
✎☎✜
✥☎✚✶
✍☎❧✜
☛☎✜
✸
12.
✡☛
✍☎❧✜
☛☎✜
✹r
13.
✙✏
✏✑
✟☛
✭✙☎
✍✘☎✟✙☛✗☎
✡☛
✥☎✳✭ ❧
♦✜ ✢
❂☎✱ ✟✌✙✱✭✗
(A) 10
✥☎✚
✥☎✚
✟☛
✟♦✓☎✜❀ ☎
✏☎✜ ✆✜
☛✞
–1
☛✞
(A)
✹☎❂☎
☛✞
✍✘☎✟✙☛✗☎
5
C4
❃
❅
❇
(C)
✗✑ ✚☎☛
5
C1
✆
✍✚
✏☎✜✆✜
(a)
❃
❅
1
2
☛✞
✎☎✜
❯✙✷ ✆✗❡
✥☎✚
★
✻
✥☎✚
✈✟✖☛✗❡
☛✞
☛✞
✟✠❧❡✜☞
❇
✡☛
♦✏
✗✮✙✗r
✍✘☎✟✙☛✗☎
(B)
✟☛
✍✘☛☎✚
✍✘☛☎✚
☞
❉✴✴ ✥✦✥ ✏✑ ✒
✆
✍✚
✪✹☎✝✆✜
✬❧
✏✑
✡☛
✥☎✚
(b)
✟✆✟❡✔ ✗
❉✻
✈☎✆✜
❉✴ ❂☎✱ ✟✌✙✱✭ ✗
♦✜ ✢
✏✑✒
☞
✍✘☎✟✙☛✗☎
✈☎✆✜
✹r
♦✼✗✱✈☎✜☞
♦✼✗✱ ✈☎✜☞
☛✞
☛✞
☛✞
❧☞ ✵✙☎
✙☎✎✽✟✾✹☛
4
❄
❆
❈
1 4
5 5
1
5
❄
❅
❆
❇
❈
10%
❡✜☞
✍✘✟✗✎✓☎✔
★ ✥✦✥ ♦✜ ✢
✆❡✷ ✆✜
✡☛
✥☎✚
(c)
❡✜☞
❡✜☞
☛✞✟✠✡✒
✺☎✗
✵☎✚☎✥
❧✜
✿
☛✞✟✠✡✒
✏✑✒
✬❧☛✞
✵☎✚☎✥
❧✜ ✶ ✟☛❧✞ ✣☎✞
✏☎✜ ☞❁
✥✦✥
✏✑ r
5
❄
(C)
❆
❈
✍✘☎✟✙☛✗☎
❃
❅
❇
1
5
✏✑✒
9
10
✗✥
5
❄
(D)
❆
❈
★
✹☎❂☎☎✜☞
❡✜☞
❧✜
(B)
❃
❅
❇
4
5
4
❄
❆
❈
1
5
4
❃
✺☎✗
✍✘☎✟✙☛✗☎
✏✑ r
4
5
✗✮✙✗r
✝✜ ✯☎✒
✪✹☎✝✆✜
✟☛❧✞
✏✑
❡✜☞
✠✞✗
✬✆☎❡
✡☛
✺☎✗
✍✘☎✟✙☛✗☎
✭✙☎
11.
15.
✟❢✍✎
?
✎✜ ✯☎
1
100
14.
1
2
B 6,
☛☎ ✥☞✌ ✆
✏✑✒
: P(X = 3)
❧☞ ♦✜✢✗
9.
X
✟☛
(D)
✬✆❡✜☞
❧✜
☛☎✜✬✔
✆✏✞☞
❊
9
10
✹☎❂☎☎✜☞
☛✞
✗✑ ✚☎☛
✏☎✜✆✜
✐
❢ ✝❢ ✝✞
♠ ✍
✎ ✏✑
✒
✎
✓
✓
♣
✔ ✕
✖✗ ✘
✙
✔
✚
✛
✜
✚
✛
✕
✢✣ ✤
✛
✢✚
✥
✛
✚
✛
✖✤ ✦
✤
✧✔ ✕
★
✎
✎
✘
✙
✚
✰✔
✚
✭
✲
✬ ✔
✚
✱
✧✰
✣
❣
❇
✏
✜✔
✤
✸
✣✖❡
A:
E2 :
E4 :
❣
✔
✚
✖
✛
✰
✖✗
✘
✙
✫✰
✖✗
✔
✚
✛
✘
✙
✘
✙
✔
❈
✔
❈
✰✔
✚
✧
✔
✔
❏
✭
✣
2
2
III
1
2
3
1
IV
4
3
1
5
♣
✤
✔
✢✽
✔
✮
✔ ✕
✲
✖
❁
✰
✬
✕
✧
❂❃
✔
✭
✲
✖✰
✢✚
✥
✛
✜
✚
✛
✧
✚
✢✚
✥
✛
✰✔
✮
✰✔
✚
E4
✔ ✕
✲
✖
✤
✰
✖✤
♣
✤
✖
✗
✘
✙
✔
❈
✦
✤
❇
✰
✔
✰✔
✕
✖
✗
✔
✾
✿❀ ✽
✔
✧
✚
✖✤
✰✔
✣
✢✱
✖
✤
✰✔
✔
✧
✚
✖
✕
❉
✔ ✔
✖❊
✢✽
✔
✬
✪
✖
✥
❆
✔
✭
✲
❅
E1 :
✤ ✔
E3 :
❋✔
✯✤
✢✚
✥
✛
✰✔
❈
✪
✕✛
✢
✽
✭
✳
✲
?
✔
✫●
✭
✛
✲
✖
✗
✖
✗
✘
✙
✘
✙
✔
❈
✔
❈
I
✰
✔
♣
✤
✳
✔
❍
❁
III
✰✔
♣
✤
✔
✢✽
✔
♣
✤ ✔ ❍
❁
✭
■
✲
❁
1
4
3
2
1
, P (A|E2) = , P (A|E3) =
18
8
7
III
✰✔
♣
✤
✔ ❍
■
❡
✙
❁
✭
✽
✙
✚
❝
✔
❈
❇
✜
✚
❂
✔
✬
✭
✲
✖
✰
❑
✰✔
✮
✔ ✕
✲
✣
❅
P (A|E4) =
✢✚
✥
✛
✖✤
✰
✔
✣
❅
4
13
✢✱
✪
✭
)
✲
✧
✚
✽
P(E 3 ).P(A|E3 )
P(E1 )P(A|E1 ) P(E 2 ) P(A|E 2 ) + P(E 3 ) P (A|E 3 ) P(E 4 ) P(A|E 4 )
1
4
= 1 3 1 1
4 18 4 4
1
7
1 1 1 4
4 7 4 13
0.165
✰✔
✔
❅
✔
❍
= P(E3|A)
P(E3|A) =
✢✱
❁
✔
✥
✘
✙
✰
✣
❂
✔ ❍
♣
✤
III
❂
❁
IV
✢✚
✥
✛
❅
❅
✰✔
✫✰
❄
✽
II
✔
✙
✛
✖ ✯✬
✉
✼ ✸
2
✣
✮
✎
✸
2
P(A|E1) =
P(
✧
✚
✎
✸
P(E1) = P(E2) = P(E3) = P(E4)
✫
❅
✬ ✕✭
II
✽
✧
✖
✍
❧✹ ✻
✺
❅
✖✗
✫
6
✰✔
✽
✱
✎
✸
A, E1, E2, E3
✫
❅
✢
5
✔
✖
❂
✔
✔ ✫
✪
4
✔
✔
✥ ✩
3
✾
✿❀ ✽
✽
✛
I
✔
✥
✽
✜
✚
①
❛
❞
✖✗
✔
✣
✡
✑
✴
✵
✶
✷
✫
✰
✟✠✡ ☛
☞✌
595
✁
✂✄
☎✆✁
596
① ✁✂✄
♠☎✆✝✞✟✆
✝❣
✠✡
X
❡✎✜
✓✒✑
❢☛☞✌
♦✢
✍✎✏✑
✕✒✌✣ ❢✤✥❞
✦✧
❞✒
✢★
✓✒✔✕
❢✙✩❞✒
✙✒✑✗✜
✭✮✒✒✯ ✗✰
X
✢★✎
❞✒
✖✒✗
1
3
✈✫❙✬
4
P(X = x) = C x
❢❞
✍✎✏✑
xi
0
4
2
C1
C2
2
3
4
3
C0
2
3
4
1
2
3
4– x
4
2
3
3
2
2
3
C3
4
✢★ ✛
2
3
x
1
3
,x
0, 1, 2, 3, 4
xi P(xi)
P(xi)
4
1
3
q 1
1
3
B 4,
✍✎✏✑
is
✢★
❢✑✱✑❢❡❢✲✒✗
❞✘❢✙✚✛
☞✪✒ ❢✕❞✗✒
n = 4, p =
✕✢✒❀
✢✓
1
3
B 4,
C4
4
0
1
3
C1
2
3
C2
2
3
2
1
3
1
3
1
3
4
2
4
3
3
4
C3
4
2
2
3
4
4
3
C4
1
3
1
3
1
3
1
3
2
3
4
n
✭✍
✓✒✔✕
xi p ( xi )
( )=
➭
i 1
= 0
4
C1
2
3
3
1
3
4
2. C 2
2
3
2
1
3
2
23
22
2
1
2
6
3 4 4 4 4
4
4
3
3
3
3
32 48 24 4 108 4
=
81 3
34
= 4
4
3. C3
2
3
1
3
3
4
4. C 4
1
3
4
✐
♠ ✝
✞ ✟
✠
✡
✞
☛
☞
✱✌
✍✎
✏
✑ ✑ ✎
✒
✓
✑ ✔
✕
✖
✒
✗
✘
✙
✚
✛
✑
✒
✜ ✎
✌
✢
✣
✑
✤
✍
✙
3
4
✌
✥
✑
❣
✧
✦
♣
✕
❣
✌
★
✬ ✭✮ ✮
✗
✑ ✱
✍
✌
✗
✘
✙
✌
✑ ✒
✌
★
✩
✒
✌
★
n
✳
✟
★✑
✎
✗
✢✍✔
✱
✍
✌
✍
✎
✏
✑ ✑
✱✌
✎
✓
✒
✑ ✔
✓
✑
✪
✛
✑ ✒
✜ ✎
✒
✌
✢
✣
✑
✤
✍✙
✪
✫✑ ✒
✗
✢
❝
✗
✑
✥
✑
❣
✧
✦
✍
✎
✩
✒
✴
✩✜
❣
✒
n
✌
★
✯
✌✥
✑
♣
✓
✑
✩
✒
✍
✌
✥ ✎
✢
✰
✍
✓
✪
✫✑ ✒
✑ ✪
✫
✑ ✒
✗
✢
✪✎
✑ ✦
✗
✢
✲
✌
❣
✑ ✒
n
♣
✓
✑
597
✁
✂✄
☎✆✁
✗
✢
✗
✑
✎
✑
✓
✴
✣
✪
✢✘
✑ ✵✑
p=
❣
✦
✧
✣
✪
✢
✘
✑ ✵
✑
n
P (X x)
✥
✓
★
✒
✗
✘
✙
Cx qn
✛
✑
✒
✜
✎
x
✌
✢
px
n
✣
✑
✤
✍
✙
✌
✥
✑
n x
1
4
Cx
3
4
=
✴
✣
✤
✶
✙
✒
✌
q=
✯
✑
✪
✦
x
3
4
n
Cx
✗
✘
✙
✌
✑
✒
✎
✛
✑ ✜
✒
✎
✒
✌
✢
✣
✑
✤
✍✙
✌✥
✑
=
1
4
3x
4n
✯
✓
P(
✍
❯
✜
✙
✑
❣
✦
✷
✙
✎ ✥
★
✱
✌
✓
✑
✪
✗
✘
✙
P (x
✯
✈
✑
✑ ✸
✥
✹
❜ ✩
✍
✛
✑ ✒
) > 0.99
1) > 0.99
✺
1 – P (x = 0) > 0.99
✗
✱
✙
✑
1
n
4
C0
✙
✑
✯
✻
✼
✽
✴
✌
✑
✌
✑
✒
✾
1
0.01
4n
✌
✪✎
✒
✝
✞
✍
✎
✏
✑ ✑
✟
✠
✡
✞
✎
✒
✓
✑ ✔
✌
✑ ✒
A
☛
❂
❅
✣✪
❄
✣
✤
✑
✍✙
❁
✣
✤
✑
✌
✥
✑
★✑
❉
✑
✎
✌✪
✥
✎
✑
❊
✌
✑
✒
✑
✦
✪
B
❯
✌
✢
✗
S
❋
✙
✪
✢
✚
✓
✔
✢
✗ ✥
✑
✫✑ ✒
✗
✢
✗
✑
✎
✢
✑ ✪
✢
✩✒
✱
✌
✣
✑
✩
✒
✎
❣
✧
✦
❣
✑ ✫
✒
✢✧
✌
✑
✒
✥
✎
❣
✢
✗
✒
✥
✑
✧
✙
✍
A
✜
❃
❄
✑
✗ ✥
✒
❣
✦
✴
✔
✓
❆
❀
✣
✑
✩
✒
✌ ✪✥
✒
❧
✣
✤
✌
✑ ✒
✾
✗
✌
✑ ✒
✽
❣
❣
✧
✦
1
5
, P(F)
6
6
✴
✗
✢
✣✪
✴
✥
P(S)
✣
❣
★✑
♣
✌
★
✻
✥
❁
✕
✒
✖
❞
✎ ✥
★
✴
✓
✑
✌
✑ ✒
✩✣
✖
✯
P(A
✷
✙
✴
✑ ✒
0.01
✥ ✌
❇
✏
✑
✍
✌
❃ ✎
★✒
✩
✒
✌
✑ ❜
✒
✸
✱
✴
✌✪
✒
✌
✣
✑
✩
✒
❈
✥
✑
✒
❃ ✎ ✕
✖
✒
✔
✢
✥ ✎
✒
✌
✢
★✏
✑
❁
✌
✢✍✔
✱✧
✗
✢
✍✔
✱
✽
❣
✑ ✒
✯
✩
✒
❆
✥
✳
✟
✌
★
✥
✸
✹
... (1)
❞
✥
❁
♠
✈
✑
✑
✕
✑ ✗
✢
✯
1
4n
✯
1
100
0.01
n
✿❀
✩✥
1
0.99
4n
C0
4n >
✙
✑
✩
✍★
✜
✎
❃
❄
✑
✗
★✒
✔
✢
✥ ✎
✑
) = P(S) =
1
6
✑
✒
✎
✑
✯
✌
✑ ✒
✑ ✦
✪
F
✯
✻
✩✣
✖
✗ ✥
✑
❀
✣
✑
✩
✒
✣✪
❧
✣
✤
✌
✎
598
① ✁✂✄
A
✑✆ ✕
❞☎✆
✝✞✟✠✞
✌✟✐❡☞
P(A
✘✟✞
✌☎✓ ✠
❞☎
✒✓ ✕✗
✒☎✆ ✝✆
✝✞✟✠✞
✐❜ ❞☎✠
✘✟✞
❞☎
✌✍✟✠
✏✑☞✝☎
✑✆✕
✔✞✝✚☎
✐☎✛ ✜✍✞✕
❞☎
✡☛☎☞
✑✆ ✕
P(A
✌✈✢ ✌✝✣
✔✞✝✚☎
✔✎
A
✐✒☞✞
✡☛☎☞
✔✞✝✚☎
1
)=
6
5
6
2
4
5
6
) = P (FFFFS)
✑✆ ✕
✌☎✓ ✠
B
♥✖✟✠✞
✡☛☎☞
5
6
1
6
1
6
1
6
4
5
6
2
1
6
...
1
6
6
=
=
1 25
11
36
6 5
) = 1 – P(A
)=1
11 11
✔✞✝✚☎
✔✞✝✚☎
a + ar + ar2 + ... + arn–1 + ...,
✢✏♥
✒✓
5 5 1
) = P(FFS) = P(F) P(F) P(S) =
6 6 6
P(B
❢✤✥✦✧★✩
✝✎
✘✟✏☞✙
✡☛☎☞
P(A
✐❜ ❞☎✠
✡☛☎☞
r | < 1,
✔✒☎✛
✝✎
✘✟
✌✚✕✝
r✆ ✪☎✞
❞☎
✢☎✆✫
a .
1 r
✬
(
♥✆ ✏✭☎✙
❞✮☎☎
37
♠✳✴✵✶✷✴
✡✻✐☎✏♥✝
✍✲✝✱
XI
✎✚☎✝✞
❞✞
✢✏♥
✙❞
✒✓ ✗
❞✠✝✞
✒✓ ✗
✐☎✯✰✢ ✐✱✲ ✝❞
✢✏♥
✐✖✍▲
✑✸☎✞✚
✢✒
✌✚✱♦ ☎✍
❞☎
✟✑✱ ✏✜✝
✟✑✱✏✜✝
✢✒
A.1.3)
✹✕✫
✹✕✫
♥✸☎☎▲ ✝☎
✟✆
✲✺☎☎✏✐✝
✟✆
✒✓
✲✺☎☎✏✐✝
✏❞
✚✒✞✕
✑✸☎✞✚
❞✞
❞✞
✲✺☎☎✐✚
✔☎✝✞
✒✓
✝☎✆
✢✒
90%
✲✍✞❞☎✢▲
✝☎✆
✢✒
✑☎✼☎
40%
✔☎✝✞
✒✓
80%
✟✑✱✏✜✝
✒✓ ✗
✢✏♥
✲✺☎☎✐✚ ✍✆ ❡ ✎☎♥ ✑✸☎✞✚ ✽ ✲✍✞❞☎✢▲ ✍✲✝✱ ✡✻✐☎✏♥✝ ❞✠✝✞ ✒✓ ✝☎✆ ✑✸☎✞✚ ❞✞ ✟✑✱✏✜✝ ✹✕✫
❞✞
✵❣
✐❜☎✏✢❞✝☎
✑☎✚
✾☎✝
☞✞✏✔✙
✿☎❀✚☎
✌✎
❞☎✆
A
✙❞
B1
✐❜ ♥✏✸☎▲✝
✿☎❀✚☎
✒✓
✏✔✟✑✆ ✕
✙❞
✑✸☎✞✚
♥☎✆
✲✍✞❞☎✢▲
✍✲✝✱ ✌ ☎✆✕
❞☎
✟✒✞ ❞☎✢▲ ✐❜ ✪☎☎☞✞ ❞✞ ✿☎❀✚☎ ❞☎✆ ✐❜ ♥✏✸▲ ☎✝ ❞✠✝☎ ✒✓ ✌☎✓ ✠
❞✠✝☎
✒✓ ✗
P(B1) = 0.8, P(B2) = 0.2
P(A|B1) = 0.9 × 0.9
✘✟✏☞✙
✲✍✞❞☎✢▲
✏✚✏✸✜✝
✟✆ ✲✺☎☎✏✐✝ ✒☎✆ ✚✆
❞✞✏✔✙✗
✟☎✺☎ ✒✞ ✑☎✚ ☞✞✏✔✙
❞✞
✙❞
✍✲✝✱
P(B1|A) =
=
✌☎✓✠
P(A|B2) = 0.4 × 0.4
P (B1 ) P (A|B 1 )
P (B1 ) P (A|B 1 ) + P (B2 ) P (A|B 2 )
0.8 × 0.9 × 0.9
0.8 × 0.9 × 0.9 + 0.2 × 0.4 × 0.4
❁
648
680
❁
0.95
✡✻✐☎♥✚
B2
❞✠✝✞
✒✓✗
✫☞✝ ❞☎✢▲ ✐❜✪☎☎☞✞
✐
✈✝
✞
1. A
2.
B
(i) A,
✖
✗ ✘
✙
✦✢
✰ ✛✪
✩
❜ ✚
✛✜
✢✗
✙
✚
B
✯
✶
✷
✸
✰ ✗
✷
✠
✡
✣ ✗ ✤✥
✗ ✦✧
❧✱
✲✳
✬
✟
✞
☛☞
★
✘
✢✗
✛
✩
✷
★
✘
✟ ✌
✟ ✍ ☞
✎
P (A)
✪✢
✚
❞
✹
✈
✯
✍ ✏✍ ✏✌
☛
✑ ✒✓
✟ ✏✔
✕
0. P(B|A)
(ii) A B =
✫
❑
✗ ✬
★
✘
❧✲
✱
✳
599
✁
✂✄
☎✆✁
✢✭
✪
✴
✮
✦
✯
✪✰
✵
✩
✲✳
(i)
✰ ✗ ✷
✥
✗
✷
✩
✹
✗
✷
✲
✷
✩
✚
✷
✶
✷
✸
✢✗
✚
✷
✥
✗
✷
✩
✹
✦✢
✢
✹
❧
✗ ✷
✩
✭
✛✜
✗ ✪✯
✢
✗
✬
✗
✢✗
✲✳
✶
✷
✸
✢
✲✳
✢✭
✥
✷
❑
✗
✬
✢
✭
✪
✮
✦
✯
✪✰
✯
★
❑
✗ ✬
★
✘
✩
✪✢
✰ ✗ ✷
✥
✗ ✷
✩
✹
✗ ✷
✩
✲✳
✢
❧
✰ ✗ ✷
★
✗
✷
♥ ✺
✻
✢
❧
(ii)
✩
✳
✭
★
✗ ✷
✥
✷
★
✘
❡
♥
✺
✻
✢
✭
✛✜
✗
✪✯
✢
✬
✗
❑
✗
✬
✢✭
✪
✮
✦
✯
✪✰
✯
★
❑
✗ ✬
★
✘
✪
✢
✹
♥
✺
✻
✗
✹
✺
✻
✗
✲✳
★
✘
❡
♥ ✺
✻
3.
✢
✛
✥
✗
✢✭
✪
✮
✦
5%
✪✢
✼
❖
✯
✪✿
✬
✢✗
✷
✯
✗ ✰ ❀
✪
✷
✩
✪
✢
✛
4.
✗ ✥
✥
✗
✩
✖
✗
✘
✙
✷
✩
✚
✷
✯
✗
✰
❀
❧
✦✢
❉
✗
✪
▲
✢✗
✷
✖
✩
✪
✢
✬
✩
✢
✤
✩
❂
✦
✪
❍ ■
✭
✢
✚▼
✗
❍
✭
✛✙
❂
✷
✰
✩
✷
✩
❉
✗
✚
✷
★
✗ ✷
★
✘
✷
✩
✢
✚
✷
✸
✩
❏
✛✙
❅
✦✢
✚
✷
✖
✪
✩
✢
✹
✗
✪
✗
✰ ✗
✘
✗
✷
✺
✻
✢✗ ✷
✛✗ ✙
✩
✪
▲
❂
✰
✷
✩
▲
7.
✦✢
✥ ★
✭✩
❂
✰
✷
✩
✗ ✷
✛✗ ✚
✷
✢
✗ ✷
✦✢
✬
✥
★
✭✩
✷
★
✗ ✷
✩
✗
✙
✮
✗
★
✘
✬
'Y'
✛✙
✪
✗ ✷
✩
✚
✷
✩
✢
✚✩
✛✜
✪
✬
✯
✗ ✷
❂✭
★
✗ ✷
✥
✷
✢
✭
✛✜
✗ ✪✯
✢
❁
✗
❞
8.
✯
✪✰
✷
5
6
❂
✗
✛✗
✦
◗
✹
✗ ✙
✬
✗ ❡
✛✙
✶
✗ ✙
★
✘
❡
✿
✯
✗
★
✘
❃
✯
✬
✹
❜ ✚
✢
✭
✛✜
✗
✬
★
✗ ✷
✷
♥
★
✗
✥
❧
✶
✗
✷
★
❡
✩
✘
❜
✚
✢✭
✗
✷
❂
❈
✰
✗
✪★
✥
✷
✩
✛✜
✪
★
✗ ❄
✗
✚
✷
✿
✯
✗
✢
✗
♥
❂
✷
✰
✩
✗ ✷
✯
✗ ✰ ❀
✷
✛✜
✗ ✪
✯
✢✬
✗
★
✘
✪
✢
✩
✢
✙✥
✷
✛✙
❁ ✯
✗
❅
❆
✪
'X'
▲
✪
✥
✢✗
✭
✖
✪
✩
✢
✬
★
✘
❄
✗ ✗
✪✛
✬
✷
★
✗
❃
✷
✩
♥
✖
✗ ✙
✘
❉
✗ ✷
✗
❅ ■
✛✙
✾
✮
✗ ✬ ✭
★
✘
✖
✗ ✙
✘
✚
♥
✬
❢
✶
✗
❧
✛✙
✖
✪
✩
✢
✬
❞
✢✙
✪✰
✯
✗
✮
✗
✬
✗
★
✷
❡
✯
✪✰
❜
✚
✛✜
✢✗
✙
❧
✗
✬
✛✜
✗ ✪✯
✢
✬
✗ ✦✧
❑
✗
✬
✢
✭
✪
✮
✦
❡
②
✭✩
✖
✪
✩
✢
'Y'
▲
❂
✷
✰
✩
✗
✷
✬
★
✗ ✷
✖
✪
✩
✢
✩
✢
✭
✬
❡
★
✗ ✷
✚✩
✯
❡
✗ ✦
✧
✚
✗
✢✗
✗ ◆
✯
✗
✥
★
✗
✷
❡
✩
❧
▼
✗ ✭
❑
✗
✬
✢
✭
✪
✮
✦
❡
❧
✢✗ ✷
❅ ❆
✹
✗
✗
✦✧
✛✗ ✙
✢✙
✥ ✭
★
✘
❜ ✚
✢
✭
✛✜
✗ ✪✯
✢
✬
✗
✪✢
✶★
✛✜
❜ ✚
✢✭
✿
✯
✗
✯
✷
✢
P
✛✜
✗ ✪
✯
✢
✬
✗
★
✘
✪✢
✶★
❍
✚
✷
✢
✹
✗
❧
❂
✗ ●
✶
✗
✗
✖
✗ ✷
✩
✢✗
✷
✪
❂
✙✗
❇
❃
✬
✢
❁ ✗
✛✜
✗ ✪✯
✢
✗
✮
✗
✬
✗
★
✘
✮
✹
✬
✢
✪✢
✚
♥
✬
✗
❑
✗
✛✙
❞
✬
✢
✭
✪
✮
✦
✪
✢
✛✗ ✚
✷
✛✙
✢✗
✖
✢
✩
✬
✗
✬ ✭✥
✹
✗
✙
❈
✬ ✭✚
✙✗
✢✗
✖
✢
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✚
✷
❁
❞
✭
❙
★
✘
❡
✭
✛
✶
★
✗ ✷
✗
✾
✢
✗ ✷
✯
✗ ✰ ❀
❚
❁
✯
✗
✥
✗
✲
❂
✯
✗
★
✗
✷
✬
✗ ✷
❜ ✚
✢
✭
✿
✯
✗
✛✜
✗ ✪
✯
✢
✬
✗
★
✘
✪✢
✳
✱
✚
✶
❞
✗
✾
✷
❚
❧
✩
■
❯
✩
❂
✷
❃
♥
✦✢
✛✜
✯
✗
✷
❂
✶
✷
✸
✚
✛
✸
★
✗
✷
✥
✷
✢✗
✚
✯
✩
✗
✷
❂
♥
✪✢
✩
❘
✦✢
✩
❂
9.
✥
✷
♥
♥
❧
✬
✗
❈
✗
✙
✗ ✷
✾
❘
✹
✹
✗
♥
✳
✪
✥
★
✯
✚
✛
✷
✸
✰
❡
✖
✪
✩
✢✬
✭
✦✢
♥
❅ ❆
❉
✗
❂
✪
✜
★
✗
✷
★
✘
❡
✩
❡
✢
✚
✷
❂
✷
✰
✩
✢
❞
✛✜
✗
✥
✦
✢
✖
✻
✚
✛
✷
✸
✰
❇
✢✙
✹
✗
✢
✙✥
✷
✖
✪
❧
✢✙
✛✗ ✙
✛
②
♥
❊
✥
✳
✬
❇
❂
✗
✪✮
②
❇
✰ ✷
✚
✷
✳
'Y'
✶
✷
✸
✢
✗
♥
✬
✗ ✷
❧
✬
❄
✗ ✗
✗
✹
✗
♥
❇
✚
✷
✳
✢
✶
✷
✸
✲
✳
✗
❏
✩
✚
✷
✖
✪
✩
✢
❇
❧
▲
'X'
▲
✖
✪
✬
❧
✢ ✙✶
✷
✬
✭
✳
✢
✪
✚
✷
♥
✪
✩
❧
✢
❞
✮
✗
✖
✗ ✷
❧
✗ ●
✥
✢✗
✚
❇
♥
(i)
(ii)
(iii)
(iv) 'X'
'X'
✹
✯
✗
★
✗ ❄
✗
✖
✪
♥ ②
✩
✗
♥
❧
✥
✷
❡
❊
✪
❈
6.
✚✩
✰
✗ ✪★
✥
✷
♥
✥
✗
✷
❂
✷
✰
✩
✷
✭
②
✗
✷
❂
✷
★
✘
❖
✯
✪✿
♥
❊
✛✗ ❋✗ ●
✳
✚
✷
❜
✚
❧
'Y'.
▲
✳
★
✘
❡
✳
✱
♥
✪
✗
✻
✪★
❧
✽
✱
✗ ✖
✗
✷
❁
✯
✗
✲
✢
❂
✯
♥
90%
✪
✢
0.25%
✖
✗ ✙
✘
✱
✪★
❧
♥
✗ ✷
❂
✗
✷
✩
✳
✽
✾
✭
✪✮
✦
❧
♥
5.
✗ ✗ ✷
✱
✗ ✗
✷
✩
✽
✱
✾
❁ ✢
✲
♥
✛
✖
❂♥
✷
❁ ❱
✛✙
✭
❲
✗ ❳✗
✗ ✷
✚✶
✷
✸
✖
✚
✛
❞
✩
✷
❧
✩
✢
✚
✷
❧
✢
★
✗ ✷
✥
✷
♥
❨
❧
✸
✚
✛
✸
★
✗ ✷
♥
✩
❂
✷
❡
✚
✷
✰ ✗ ✷
❂✱
✥
✗
★
✘
❡
✛✜
✗
✪✯
✢
✬
✗
❑
✗
✬
✢
✭
✪✮
✦
600
10.
① ✁✂✄
✱☎ ✆✝✞✟✠ ✱☎ ✡✝☛☞✝ ✞✌✟✍✎✏ ☎☛✎ ✞☎✠✑✒ ✓☛✔ ✕✖☛✗✎ ✞☎ ☎✘ ✌✎ ☎✘ ✱☎ ✞✙✠ ☎✒ ✚✛ ☛✞✝☎✠☛
90%
11.
✱☎
❧✞✜☎
✌✎
✣☛✎✗
✘✎✤
✢☛✎
?
✞☎✌✒
✆✝✞✟✠
☎☛✎
✱☎
✡✝☛☞✝ ✚☛✌✎ ☎☛✎
✕✖☛✗✑✎ ✍✎✏
✓☛✥
✖✦
✢☛✎✑✎
✚✛☎ ✧
✚✔
✱☎
★✚✝☛ ✞✘✗✠☛ ✢✩ ❧☛✩✔ ❧✡✝ ☎☛✎✪✫ ✌✤✣ ✝☛ ✚✛☎ ✧ ✢☛✎✑✎ ✚✔ ✍✢ ✱☎ ★✚✝☛ ✢☛✔ ✬☛✠☛ ✢✩✭ ✱☎ ✆✝✞✟✠
✝✢ ✞✑❀☛✫✝ ✗✎✠☛ ✢✩✮ ✞☎ ✍✢ ✚☛✌✎ ☎☛✎ ✠✒✑ ✓☛✔ ✚✎✏
✤ ✍✎✏✯☛ ✗✎✞☎✑ ✬✓ ✰☛✒ ✖✦ ✚✛ ☛✲✠ ✢☛✎✯☛ ✍✢ ✣☛✎✗✑☛
✖☛✎◆✳
12.
✥✎✯☛✭
✘☛✑
✧❞☎ ✞◆✳ ✝☛✎✤ ☎☛
✧❞☎◆✳ ☛
✱☎
✞✑☎☛✗✎
✬✒✠✒✵✢☛✔✒
✢✘☛✔✎ ✚☛✌
✗✒✞✬✱
☎☛✗✒
✴☛✔☛
✕✌✍✎✏
A, B, C
✞✍✍✔❀☛
✞✑☎☛✗☛ ✬☛✠☛
✬☛✑✎
☎✒
✟✝☛
13.
✘☛✑
✞☎
✗✒✞✬✱
❏✝☛✑
✞✍☎❑✚
14.
15.
✘☛✑
✢☛✎✑ ✎ ☎✒
✙❞✑✎
✬☛
✪✗✎✟✧❙ ☛✼✞✑☎
✚✛ ☛✞✝☎✠☛✱❚
✸☛✠
P(A
❈❉❆❊❋●❍✾■ ❃ ❆✾
2
2
C
8
1
1
D
0
6
4
☎☛✎
✞✍✞✜
☎✘
✌✎
✞✥✗
✞✥✗
✱☎
✠✽☛☛
✥☛✩✔ ☛ ✚◆✳✑ ✎ ☎☛ ✌✤✝☛✎✯
✥☛✩✔ ☛
✚◆✳✑ ✎ ✍✎✏
✝✢ ✞✥✝☛
✙❞✑☛
✕✚✝☛✎✯
✚✛✷ ✝✎☎
✍✎✏
✌✰☛✒
❖✘☛✑
☎✒
✔☛✎✯✒
✬☛✑✎
✞☎✱
✍✎✏
✥☛✎
✯✝☛
✙❞✑ ✎
✬☛✑✎
✌✢☛✝☎
✢✩✦
) = 0.2
❧✍✎✏✗✎
❧✌✚✏✗
✢☛✎✑ ✎
☎✒
) = 0.15
❧✌✚✏✗
✢☛✎✑ ✎
☎✒
) = 0.15
B
✍✎✏
☎✒
✍✎✏
40%
30%
✥☛✩✔ ✎ ✌✎
✚✛ ☛✞✝☎✠☛
✶☛▼✡ ✝
☎✒
✝☛
✚✛ ☛✞✝☎✠☛
✞✑☎☛✝
☎✒
❧☛✩✔
✮
C
✌✎
✬☛✠☛
✢✩
✓☛✼✟✌
✢✩✭
✝✢ ✘☛✑
☎✘
☎✔
✞✗✝☛
✥✎✠☛
✢✩
✞✍☎❑✚☛✎✤ ✌✎ ✞☎✌✒ ✱☎
A
❧☛✩✔
✯✛ ✞✌✠
✸☛✠
✱☎
✗✒✞✬✱ ☎✒ ✌☛✔✞❀☛☎
✢☛✎✑ ✎
P(A
✕✚✔☛✎✟✠
✞✥✗
❧✍✝✍
❧✌✚✏✗
✍✎✏
✣☛✠✔✎ ☎☛✎
✢✩ ✞☎
✯✝☛
✍✎✏
P(B
B
✪✌✌✎
❧☛✩✔
✥✍☛
✞☎✝☛ ✬☛ ✌☎✠☛ ✢✩✭ ✞☎✌✒ ✰☛✒ ✌✘✝ ✔☛✎✯✒ ✪✑ ✥☛✎✑ ☛✎✤ ✘✎✤ ✌✎ ✞☎✌✒ ✱☎
✌☛✔✞❀☛☎
✱✌✎✓
✤ ✗✒
☎☛
☎☛
✝☛✥✺✻✖✝☛
☎☛
✓☛✼✟✌
✢✩ ✠✽☛☛
❅❃❄
6
✞✍✞✜
✢✩✤
❆❇
B
✟✝☛ ✚✛ ☛✞✝☎✠☛ ✢✩✤✭
✌☎✠✎
❴
3
✝☛✎✯
✍✎✏
A
✢☛✎ ✠☛✎ ✪✌✎ ✓☛✼✟✌
6
✝☛✎ ✯
✱☎
❂❃❄❡❅❡❅
✔☛✎✯✒
✙❞✑ ☛ ✬☛✠☛
✓☛✼✟✌
1
❏✝☛✑
☎☛✎✞✧
✧❞☎◆✳ ☛ ✗☛✗
✱☎
A
✔☛✎✞✯✝☛✎✤
▲
✝☛✥✺✻✖✝☛
❆✾②✾
☎✔✑✎ ✍☛✗✎
✝✞✥
✢✩
✌✎
☎✒ ✗☛✗✮ ✌✚✎✏ ✥ ❧☛✩✔
✢✩❢
✚✛ ☛✞✝☎✠☛
☎☛ ✙✝✑ ☎✔✠☛ ✢✩✭
❧☛✩✔
☎✒✞✬✱✭
✓✟✌✎ ✢✩✤ ✞✬✌✘✎✤ ✔✣☛✒ ✌✤ ✯✘✔✘✔
✠✔✒✍✎✏
✝✞✥
✸☛✠
●
❂✐ s
■♥
25%
✴☛✔☛ ✣☛✠✔✎ ☎☛✎
D
✚✛✷ ✝☛✶☛☛
☎✒
②✾②
✞☎✌✒
❧☛✎✔
✔☛✞✶☛
❧☛✩✔
✞✑✹✑
✢✩✭
❝✾✿❁❂
✯✪✫
✢☛✎ ✠☛✎ ✌☛✔✞❀☛☎
✢✩✭
✢✩✤✭
✢✩✭
✔☛✎✯✒
✴☛✔☛
☎✒✞✬✱✭
✍✎✏ ✚✛✷ ✝✎☎
1
2
B
✢☛✎ ✬☛✠☛
☎☛ ✙❞✑ ☛✍
❧✍✝✍
☎☛
✜✑☛✷✘☎
P✍✠✤ ◗ ☛ ❘✚
✌✎
)
✫ ✠✒✫
✚▼✍✍
✞✑✔✒❣☛❀☛
✴☛✔☛
✞✑✹✑
✐
r ✝
✞
✟
✠ ✡ ☛✡
☞✌
✝
✠✍
✎
r
✝
✏✑
(i) P(A
✈
✖
✗
✢
✣✞
✤
✎✝
✞
✥
❋
✝ ✜
✓
✠
✗
✘
✈
✖
✞
✞
✡ ✎✝
✗
✘
❋
✝ ✝
✦
✝
✞
✡
✞
✘
❋
✝
✜
✎✝
①
✝
✞
)
)
✎✓
✙
✓
✧
✞
★
✤
✞
✤
II
✘
✜
r
❋
✝ ✝
❋
✝ ✜
✝
✫
✝
✧
t ✭
✙
✎
✓
✙
☞
✣
✞
✘
✓
✚
✛
✝
✞
✘
r
✢
✙
☞
✘
✪
✜
✖
✘
✝
✙
✝
r
✈
✗✘
✖
✝
/B
♦✞
✘
✧
✞
✤
★
✔
✈
✘
✝
✗✘
♦✞
✘
❋
✝ ✜
✎✓
✠✔
✏✕
☞
(ii) P(A
16.
✒
✝
r
601
✁
✂✄
☎✆✁
✧
✞
★
✤
✤
✘
✣✞
✙
❋
✝ ✝ ✡ ✝
r
✤
✠
❡
r
✘
✠
✎
✍
✙
✝
❡✤
✧
✎✓
✝
✔
✝
r
✝
✤
✦
✈
✜
✘
✘
✝ ❡
✜
r
❡
r
✎✝
✎
✧
✞
★
✤
✎✓
✎✝
✧
✞
★
✤
❋
✝
✜
✙
✓
✘
✏
✘
❋
✝ ✝ ✡
✝
r
✤
✠
✩
✬
✝ ❡
✜
✫
✪
✜
✈
✝
✧✤
★
✞
✞
①
✤
✪
✤
✜
✏
✖
✝
✘
✞
✠✡ ✎✝
✘
✡ ☛✡
✠
17.
18.
19.
✖
✠❢
✝
r
☞✌
✮
✡ ✝
✞
✤
A
(A) A
✈
✍
✠
★
✝ ✜
❡
✳
✣✞
✙
✤
✝
✞
✡
✞
✎✓
☞✌
✝
✠✍
✎
r ✝
✒
✝
r
✚
✛
✓
B
B
✯ ✰
✝ ❡
✖
★
✝
✞
✏✞
♥
✓
✍
✠
★
✠
★
A
B
✈
✝ ✜
❡
✡
✝
♦
✱
✝
★
✠
✎
✡
☛✡
✣✞
r
✝ ❡
✜
A
✳
✬
✈
✴
✤
✞
✖
✙
✎✝ ✡
✜
✓
✴
✙
✪
✜
(B) P(A B) < P(A) . P(B)
(D) P(B|A) = P(B)
✵
✝
✞
P(A) + P(B) – P(A
(A) P(B|A) = 1
(C) P(B|A) = 0
P(A) 0
P(B/ A) = 1,
(C) B =
(D) A =
✲
✜
✖
✠
♥
✓
✑
✬
r
✖
✏
✞
✎✓
✠✔
✏✕
✙
✡
✝
✏
(B) B
P(A/B) > P(A),
(A) P(B|A) < P(B)
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✍
✎✝
✱
✝
✙
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✏
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✝ ❡
✜
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✬
r
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❧
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t
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1,
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P (E F)
P(F) , P ( F )
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❡
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♦
r
✧
t
✭
✽
P (E F|G) = P (E|G) + P (F|G) – P (E
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P (E
F) = P (F) (E|F), P (F) 0
E
F
P (E
F) = P (E) P (F)
P (E|F) = P (E), P ( F ) 0
P (F|E) = P (F), P(E) 0
✍
✙
★
✓
✟
✜
P (E | F) = 1 – P (E|F)
✵
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0
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r
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r
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602
☎
① ✁✂✄
❧✆✝✞✟✠✡
✝☛ ✠☞✌✍✎✠
✍✏
✝☛ ✑✒ ✌✓
S
E1, E2, ...En,
A
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E1, E2, ....En
S
E1, E2, ..., En
E1 E2 ... En = S
A
✑✠❡
{E1, E2, ...En)
✔✒✆
✝☛☞✎✐✕✠✡
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✍✠
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P(E i |A)
n
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✍✠✒
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n
2
x
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( xi
❆
) 2 p ( xi )
i 1
✌✠
❧✑✎❀ ❂ ✌✎✓
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2
x
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n
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x
( xi
❈
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603
✁
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✝
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604
① ✁✂✄
❡☎✆ ✝✞✟✠✡✞☛☞
✌✍✎✟✏
✑✞☞✒✓✟
✔✍✕✞✖✗✘
✙✡✍✞✚✗ ✒
✍✟ ✛✜✛✢
❡✟☛ ✠✣✕✞✔✤✞☞
✕✒
✥✞✒✦
✌✧★✟☛
✩✕
❡★✪✎✠✫✬✞✭
✠✣✞✔✝✕☞✞ ✙☛✐✍ ✮✔✯✠✰ ✙☛✐✍✱ ✕✒ ✲✞✞✟✓ ✕✞ ✳✟✝ ✑✞✒ ✓✞☞✞ ★✚✦ ✠✣✞✔✝✕☞✞ ✠✡ ✴✵✗✞ ✴✞✕✶✞✭✕ ✕✞✝✭
✮✴✙✣✞★❡ ❫✟ ❡✞✟✔✎✝✡ ✷✛✸✸✜ ✹ ✛✜✺✻✼ ✕✒ ✠✽ ✾☞✕ ✮✰ ❫✞✖✔✿✐❀✍ ✴✞✖✠ ✏ ❁✞☛✘✱ ❡✟☛ ✔✎❂❡✞✍ ★✚ ✔✓✘✟
✛✜✛❃ ❡✟ ☛ ✠✣✕✞✔✤✞☞ ✔✕✝✞
✮✙✟❄ ✞✹✠✣❡ ✟✝✱
✕✞✟
✲✞✵✞✟✗✤✞✞✾❬✞✒
✮✔✠✝✡✟
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✙✞✰
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✵✔✬✞☞●✞✟☛
✷✛❃❆✛✹✛❋✛❃✼
✕✡✍✟
✘✞❊❡✍
✔✕✝✞ ✴✞✚✡ ✛❃✛❆ ❡✟☛ ✩✕
✥✞✞✦ ✥✞✞✖❡ ✘ ✙✟❄ ✞ ✷✛✜❅❆✹✛✜✸✛✼ ✍✟ ✌✍✎✟ ✏ ✍✞❡ ✠✡ ✠✣✔✘❇ ✠✣❡ ✝
✟
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✩■✩✍■
✷✛❃❆✛✹✛❃❋✻✼❍
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✣ ✎
✟
✠✣✞✔✝✕☞✞
✷✛✜✻❋✹✛❃❆✜✼
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✤✞✟✙ ✒✤✞✟ ✎
✴✞✚✡
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✠✡
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❡✞✖ ✡✕✞✟✎
✷✛❋❅✢✹✛❋❃✜✼
✷✛❃✺✸✹✛❋❆❆✼❍
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✠✡
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✣ ✎
✟
✍✟ ✠✣✞✔✝✕☞✞ ✕✞ ✘❡✽ ❑❁✝ ✠✏✗✍ ✎✟✏ ❝✠ ❡✟☛ ✘✫❬ ✞✠✞☞ ✔✕✝✞✦ ✔✓✘✟ ✛❋✢✢
❡✟☛ ✠✣✕ ✞✔✤✞☞ ✠✽ ✾☞✕
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✓✞☞✞
✔✘❇✞☛ ☞✱ ❡✟☛ ✠✣✞✔✝✕☞✞
★✚✦
—
▲
—
✎✟✏ ✴✔✑✞✵☎ ✔★☞✒✝ ✰☎ ✔✶✐✕✞✟✬✞
♠ ✎✏ ✑ ✒
✏ ✓
✏
✐✁✂✄ ☎✆✝
1.
4.
1
cos 2 x
2
1
(ax b)3
3a
2.
5.
7.
x3
3
10.
x2
2
12.
2 2
x 2 x2 8 x C
7
14.
2 2
x
3
16.
2
x C
8.
log x
7
3
✞
✟ ✠
1
sin 3x
3
1
4 3x
e
cos 2 x
2
3
ax3
3
bx 2
2
1 2x
e
2
4 3x
e
x C
3
3.
6.
2 3 x
x e C
3
cx C 9.
2x C
11.
x2
2
13.
x3
3
15.
6 2
x
7
3
5
✡
✡
4
C
x
✡
x C
7
2 2
x C
5
5x
5
4 2
x
5
3
2x 2
C
3
x
3sin x + e
x
2 3
10 2
x 3cos x
x C
3
3
19. tan x – x + C
21. C
17.
C
18. tan x + sec x + C
20. 2 tan x – 3 sec x + C
22. A
☛
✐✁✂✄ ☎✆✝
1. log (1 + x2) + C
2.
4. cos (cos x) + C
5.
3
6.
2
(ax b) 2 C
3a
3
8.
11.
1
(log| x |)3 C
3. log 1+ log x C
3
1
cos 2(ax b) C
4a
☞
5
7.
1
(1 2 x 2 ) 2 C
9.
6
2
x 4( x 8) C
3
✞
✟
2
( x 2) 2
5
✌
3
✍
4
( x 2) 2 C
3
✌
✌
3
4 2
( x x 1) 2 C 10. 2log
3
x 1 C
606
✥❢✳❦r
7
12.
1 3
( x 1) 3
7
14.
(log x)1
1 m
1
17.
20.
2e
C
m
C
18. e tan
2x
1
C
18(2 3x3 ) 2
13.
1
log|9 4x2 | C 16.
8
15.
C
1
log (e 2 x e
2
1
x
1 2x
e
2
3
C
x
x
19. log (e e ) + C
C
) C
1
tan (7 4 x) ✁ C
4
22.
24.
x2
4
1 3
( x 1) 3
4
1
log 2sin x 3cos x C
2
21.
1
tan (2 x 3) x C
2
23.
1
(sin
2
25.
1
C
(1 tan x)
1
x) 2 C
3
27.
26. 2sin x C
1
(sin 2 x) 2
3
C
28. 2 1+ sin x C
29.
1
(log sin x)2 C 30. – log 1+cos x C
2
1
31. 1+ cos x C
32.
x 1
log cos x sin x C
2 2
33.
x 1
log cos x sin x C
2 2
36.
1
( x log x)3 C
3
34. 2 tan x
1
cos (tan
4
37.
35.
C
1
1
(1 log x)3 C
3
x4 ) C
38. D
39. B
✐✂✄☎✆✝✞✟ ✠✡☛
1.
x 1
sin (4 x 10) C
2 8
3.
1 1
1
1
sin12 x x sin 8 x
sin 4 x
4 12
8
4
2.
C
1
1
cos 7 x
cos x C
14
2
♠ ✁✂✄✁☎✁
4.
1
1
cos (2 x 1)
cos3 (2 x 1) C
2
6
6.
1 1
1
1
cos 6 x
cos 4 x
cos 2 x
4 6
4
2
7.
1 1
1
sin 4 x
sin12 x
2 4
12
9.
x tan
11.
x
C
2
10.
5.
C
C
8. 2 tan
12. x – sin x + C
13. 2 (sinx + x cos✆) + C
14.
1 3
1
sec 2 x
sec 2 x C
6
2
17. sec x – cosec x + C
16.
1 2
tan x C
2
x2
C
2
x
2
23. A
22.
20. log cos x sin x
1
cos ( x a )
log
sin (a b)
cos ( x b)
1. tan 1 x3 + C
5.
C
C
24. B
✐✝✞✟✠✡☛☞
3. log
1
C
cos x +sin x
1 3
tan x tan x x C
3
18. tan x + C
15.
21.
x
x C
2
3x 1
1
sin 2 x
sin 4 x C
8 4
32
3x 1
1
sin 4 x
sin 8 x C
8 8
64
19. log tan x
1
1
cos6 x
cos 4 x C
6
4
1
2 x
3
2 2
tan
x
1
2
C
4x 5
2 x2 C
✌✍✎
2.
1
log 2 x
2
4.
1 –1 5 x
sin
C
5
3
6.
1
1 x3
log
6
1 x3
1 4x2
C
C
607
608
✥❢✳❦r
x 2 1 log x
7.
x2 1
2
9. log tan x + tan x + 4
11.
1
tan
6
3x 1
2
1
13. log x –
3
2
15. log x –
a +b
2
C
–1
12. sin
3x 2
–1
14. sin
C
( x a )( x b)
2
19. 6 x – 9x + 20 34 log x
20.
– 4x – x 2
1
3x 1
2
9
2
x 2
2
x 2 2 x +3 log x 1
21.
22.
4 sin
1
x 3
4
2x 3
41
C
x 2 9 x 20
C
C
x2 2 x 3 C
4 x +10 7 log x 2
24. B
x2
4 x 10
C
25. B
✐✂✄☎✆✝✞✟ ✠✡☛
1. log
( x 2) 2
x 1
3. log x 1
C
C
C
1
2
6
x 1
log x 2 2 x 5 ✁
log
✁C
2
x 1✁ 6
6
2
23. 5 x
C
x2 2 x 2
C
x 2 1 2log x
17.
5
11
log 3 x 2 2 x 1
tan
6
3 2
a6
C
16. 2 2x 2 + x 3 C
18.
x6
10. log x 1
C
C
x2
1
log x3
3
8.
2.
5log x 2 4log x 3 C
1
x 3
log
C
6
x 3
x2 1 C
♠ ✁✂✄✁☎✁
3
log x 3 C
2
x
log x
5. 4log x +2 2log x 1 C
6.
2
1
1
1
log x 1
log ( x 2 1)
tan 1 x C
7.
2
4
2
4.
8.
10.
11.
1
log x 1
2
2log x 2
2
x 1
log
9
x 2
1
C
3( x 1)
9.
3
log 1 2 x C
4
1
x 1
log
2
x 1
4
x 1
C
5
1
12
log x 1
log x 1
log 2 x 3 C
2
10
5
5
5
5
log x 1
log x 2
log x 2 C
3
2
6
x2
2
1
3
log x 1
log x 1 C
2
2
1
13. – log x 1 + log (1 + x2) + tan–1x + C
2
12.
5
14. 3log x – 2
x
16.
xn
1
log n
n
x 1
18.
x+
2
tan
3
1
2
C
15.
C
x
3
17. log
3tan
1
x4 ✝ 1
log
✞C
20.
4
x4
22. B
1
x
C 19.
2
1. – x cos x + sin x + C
2
3. e (x – 2x + 2) + C
2 – sin x
1– sinx
1
tan
2
✆
1
x C
C
x2 1
1
log 2
2
x 3
C
ex – 1 ✠
☛✞C
ex ✌
☞
✟
21. log ✡
23. A
✐✍✎✏✑✒✓✔
x
1
x 1
log
4
x 1
2.
4.
✕✖✗
1
x
cos3 x
sin 3x C
3
9
x2
x2
log x
C
2
4
609
610
✥❢ ✳❦
r
5.
x2
log 2 x
2
7.
1
(2 x 2 1) sin 1 x
4
2
9. (2 x 1)
x2
4
C
cos –1 x
4
6.
x 1 x2
4
x
1 x2
4
sin –1 x x 2 1 x2 sin 1 x
11.
–
13.
x tan 1 x
15.
1–x 2 cos –1 x
x3
3
12. x tan x + log cos x + C
1
log (1 x 2 ) C
2
x3
x log x
9
14.
x C
19.
x
18. e tan
C
20.
e2 x
(2sin x cos x) C
5
23. A
1.
1
x 4 x2
2
3.
(x +2) 2
x 4 x 6 log x 2
2
4.
(x +2) 2
3
log x 2
x 4x 1
2
2
5.
5
sin
2
x 2
5
1
C
x
C
2
ex
( x 1) 2
C
24. B
✐✁✂✄ ☎✆✝
2sin
x2
x2
log x
2
4
22. 2x tan–1x – log (1 + x2) + C
21.
1
x2
(log x) 2
2
16. ex sin x + C
ex
C
17.
1+ x
ex
x
x 1
tan 1 x C
2 2
2x C
C
x
x2
tan 1 x
2
C
2
10.
C 8.
x3
x3
log x
C
3
9
x
C
2
2.
✞
✟ ✞
1
1
sin 1 2 x
x 1 4 x2
4
2
x2 4 x 6
x2 4 x 1
x 2
1 4 x x2 C
2
C
C
C
♠ ✁✂✄✁☎✁
6.
(x +2) 2
x 4x 5
2
9
log x 2
2
7.
(2x 3)
13
1 3x x 2
sin
4
8
8.
2x +3 2
x 3x
4
x2 4 x 5
2x 3
1
x 2
3
x 9
log x
6
2
10. A
x 2 3x
x2 9
9.
11. D
1 2 2
(b a )
2
2.
4.
27
2
5. e
3.
2. log
4.
1
2
5. 0
7.
1
log 2
2
8. log
4
3.
2 1
2
1
log 6
5
14.
15.
1
(e – 1)
2
16. 5 –
64
3
6. e4 (e – 1)
1
3
log
2
2
1
log 2
2
☞✌✎
3
2
11.
13.
19
3
15 e8
6.
2
1
e
✐✆✝✞✟✠✡☛
✏
☞✌✍
35
2
1.
10.
C
C
✐✆✝✞✟✠✡☛
1. 2
C
13
9
3
log x
8
2
C
9.
3
12.
3
tan
5
1
5
5
5
3
9log
log
2
4
2
✏
2
✏
4
611
612
✥❢✳❦r
4
17.
1024 2
20. 1 +
2
4
2 2
✂
✂
18. 0
19. 3log 2
21. D
22. C
3
8
✐✁✄☎✆✝✞✟ ✠✡☛☞
1.
1
log 2
2
2.
4.
16 2
( 2 1)
15
5.
7.
✍
64
231
✍
e 2 (e 2 2)
4
– log 2
2
1
21 5 17
log
4
17
6.
4
8.
8
10. B
✌
3.
9. D
✐✁✄☎✆✝✞✟ ✠✡☛☛
1.
✍
2.
4
5. 29
✌
✍
4
6. 9
12. ✎
16 2
15
13. 0
16. – ✎ log 2
17.
8.
8
log 2
✍
3.
4
✌
10.
2
14. 0
a
2
✍
4
1
(n 1) (n 2)
7.
9.
4.
log
1
2
11.
✌
2
15. 0
18. 5
20. C
21. C
✈✏✑✒✑ ✓ ✔✕ ✖✗✖✗✘ ✔✙✚✛✒✗✜✢
1.
x2
1
log
2
1 x2
3.
2 (a x)
–
a
x
C
C
2
2.
3(a b)
4.
1
– 1+ 4
x
(x
1
4
3
a) 2
C
(x
3
b) 2
C
♠ ✁✂✄✁☎✁
5.
2 x
1
3x 3
1
6
6x
1
log x 1
2
6.
6log (1
1
log ( x 2 9)
4
7. sin a log sin ( x a )
sin x
2
–1
9. sin
11.
1+e x
2+e x
C
2(2x 1)
C
sin
1
2 x
x
✟
20.
–2 1– x cos
23.
✠
25. e 2
27.
29.
31.
4 2
3
2
x2
1
sin 1 ( x 4 ) C
4
1 ✆1
1
x
tan x ✝ tan ✆1 ✞ C
3
6
2
16.
1
log ( x 4 1) C
4
18.
–2
sin
☞
1
sin ( x
)
sin x
C
x C
1
x
x
x2
C
1 x2
C
24.
30.
32.
1
2log x +1
22.
1
1
– 1 2
3
x
x 1
3
2
8
28. 2sin
6
☛
1
sin 2 x C
2
14.
26.
✡
x
C
3
x3
C
3
✟
21. ex tan x + C
1
x cos 1 x
2
8.
C 12.
[f (ax + b)]n +1
C
17.
a (n +1)
19.
1
10.
1
cos 4 x C
4
15.
C
3
tan
2
x cos a C
cos ( x b)
1
log
sin (a – b)
cos( x a )
13. log
1
x6 )
613
1
( 3 1)
2
1
log 9
40
☛
2
( ☛ ☞ 2)
3log x 2 C
log 1
1
x2
2
3
C
614
33.
✥❢ ✳❦
r
19
2
40.
41. A
43. D
42. B
44. B
✐✁✂✄ ☎✆✝
1.
1 2 1
e
3
e
14
3
✞
✟ ✠
2. 16 4 2
3.
5. 6
6.
32 8 2
3
☛
4. 12
✡
a2
1
2 2
✡
10.
✍
8. (4) 3
✎
✏
✑
✒
✓
1
3
2
☞
✌
7.
3
9
8
9.
12. A
11. 8 3
13. B
✔
✐✁✂✄ ☎✆✝
1.
2
6
✖
9
sin
4
✕
1
✞
✟
2 2
3
✘
2
3
✗
2.
✛
✛
✢
21
2
6. B
3.
4. 4
3
2
✙
✜
✜
✣
5. 8
7. B
✈✤✦ ✧
✦
1. (i)
✚
7
3
★
✩✪
✫✬
✫
✬✭
✩✮
✯✰
✧
✬✱✲
(ii) 624.8
2.
1
6
3.
7
3
6.
8 a2
3 m3
7. 27
4. 9
8.
3
(
2
5. 4
✴
✵
2)
♠ ✁✂✄✁☎✁
9.
ab
( ✆ ✝ 2)
4
10.
9
2
11. 2
9✞
8
18. C
7
2
17. C
13. 7
14.
16. D
15.
✐✑✒✓✔✕✖✗
1.
❞✛✜✢✣
3.
❞✛✜✢✣
5.
4; ❄✛✛✤
12.
☛
9 ✟1 ✠ 1 ✡
1
sin ✌ ✍ ☞
4
✎ 3✏
3 2
19. B
✘✙✚
2.
❞✛✜✢✣
1; ❄✛✛✤ 1
2; ❄✛✛✤ 1
4.
❞✛✜✢✣
2; ❄✛✛✤
❞✛✜✢✣
2; ❄✛✛✤ 1
6.
❞✛✜✢✣
3; ❄✛✛✤ 2
7.
❞✛✜✢✣
3; ❄✛✛✤ 1
8.
❞✛✜✢✣
1; ❄✛✛✤ 1
9.
❞✛✜✢✣
2; ❄✛✛✤ 1
10.
❞✛✜✢✣
2; ❄✛✛✤ 1
✥✢✦✧✛✛✢★✛✤ ✩✪✫✬
11. D
11. D
y✯ = 0
y✯ – y✰– 6y = 0
y✯ – 2y✰ + 2y = 0
xy✰ – 2y = 0
xyy✯ + x(y✰)² – yy✰ = 0
B
2.
4.
6.
8.
10.
12.
x
x C
2
3. y = 1 + Ae–x
y 2 tan
5. y = log (ex + e–x) + C
✘✙✮
xy y✯ + x (y✰)² – y y✰ = 0
y✯ – 4y✰ + 4y = 0
2xyy✰ + x2 = y2
xyy✯ + x(y✰)² – yy✰ = 0
(x² – 9) (y✰)² + x² = 0
C
✐✑✒✓✔✕✖✗
1.
✘✙✭
12. D
✐✑✒✓✔✕✖✗
1.
3.
5.
7.
9.
11.
✥✢✦✧✛✛✢★✛✤ ✩✪✫✬
12. A
✐✑✒✓✔✕✖✗
1
3
✘✙✱
2. y = 2 sin (x + C)
4. tan x tan y
C
–1
6. tan y = x +
x3
3
C
615
616
✥❢✳❦r
8. x – 4 + y – 4 = C
7. y = ecx
9. y = x sin–1x +
11. y
10. tan y = C ( 1 – ex)
1– x 2 + C
1
log ( x 1)2 ( x 2 1)3
4
1
tan –1 x 1
2
1
3
x 2 ✂1 ✁ 1
12. y ✄ log ☎ 2 ✆ – log
2
4
✝ x
✞ 2
14. y = sec x
16. y – x + 2 = log (x2 (y + 2)2)
y 2
a
x
15. 2y – 1 = ex ( sin x – cos x)
17. y2 – x2 = 4
13. cos
1
19. (63t ✟ 27) 3
21. Rs 1648
18. (x + 4)2 = y + 3
20. 6.93%
2log 2
22.
11
log
10
23. A
✐✠✡☛☞✌✍✎ ✏✑✒
✓y
2
1. ( x ✔ y ) ✕ Cx e
x
2.
y✘ 1
2
2
✜ ✙ log ( x ✚ y ) ✚ C
✢ x✣ 2
✗
–1
3. tan ✛
1
5.
2 2
log
7. xy cos
x✤ 2y
x✧ 2y
✦ log
y
=C
x
y
9. cy = log
1
x
11. log ( x2 + y2) + 2 tan–1
12. y + 2x = 3x2 y
y✼
✿ ✽ log ex
❀ x❁
✻
x
✤C
y ✄ x log x
✖ Cx
4. x2 + y2 = Cx
6.
y + x2 + y 2
8.
x ✰1 ✬ cos ✮
10.
★
✪
✵
✲
ye
x
y
✷
✄ Cx
2
y✫✩
✪ y✫
✯ ✱ ✭ Csin ✮ ✯
x✴✶
✲ x✴
x✸C
y
✹
= ✺ log 2
2
x
y✘
✜ ✙ log ex
✢x✣
✗
13. cot ✛
y✄
14. cos ✾
15.
16. C
17. D
2x
( x ❂ 0, x ❂ e)
1 ✂ log x
♠ ✁✂✄✁☎✁
✐✆✝✞✟✠✡☛
1
(2sin x – cos x) + C e–2x
5
1. y =
3.
xy ✎
x4
4
✏
C
4. y (sec x + tan x) = sec x + tan x – x + C
9.
11.
2
(1 ✕ log x ) ✕ C
x
1
C
y
cot x
x
x sin x
y log x ✔
✓
x2
(4log x ✒ 1) ✏ Cx ✑2
16
6.
y✎
8.
y = (1+ x 2 )✖1 log sin x ✗ C (1✗ x 2 )✖1
10. (x + y + 1) = C ey
y2 C
✙
3 y
x✘
☞✌✍
2. y = e–2x + Ce–3x
5. y = (tan x – 1) + C e–tanx
7.
12. x = 3y2 + Cy
13. y = cos x – 2 cos2 x
14. y (1 + x2) = tan–1 x –
15. y = 4 sin3 x – 2 sin2 x
17. y = 4 – x – 2 ex
16. x + y + 1 = ex
18. C
✈✛✜✢✜
1. (i)
❞✮✯✰✱
2; ❄✮✮✲ 1
(iii)
❞✮✯✰✱
4; ❄✮✮✲
3.
y✼ ✘
✣
(ii)
–1
x
9. tan y + tan (e ) =
13.
y sin x 2 x 2
✦✧✦✧★
❞✮✯✰ ✱
4
19. D
✤✩✪✫✢✧✬✭
1; ❄✮✮✲ 3
5. (x + yy✽)² = (x – y)2 (1 + ( y✽)2)
6. sin–1y + sin–1x = C
11. log x – y
✤✥
✚
✳✰✴✵✮✮✰✶✮✲ ✷✸✹✺
2 y2 ✻ x2
4 xy
–1
617
✾
10. e
2
x y 1
❃
x
y
✿
(sin x
0)
2
y❀C
12.
y e2
14.
y ❆ log
x
❁
2
2
sec x
8. cos y =
(2 x ❂ C)
2 x ❅1
, x ❇ ❈1
x ❅1
618
✥❢ ✳❦
r
15. 31250
17. C
16. C
18. C
✐
1.
✁
✂
✄
☎✆
✝
✞ ✟
✠
✞
✚
✚
✚
✛
❧✡☛ ☞✌
2. (i)
(vi)
3. (i)
✍✎ ✏
✑ ✒✓
✔
❧
(ii)
✍
✓
✘✙
✎
❧
✕✖
✡ ✗
OP
✓
✘ ✙✎
❧
✏
✎
✡ ✓♦
✔
(iii)
✓
✘✙
✎
✓✏
✜✢
✎✎
✣✌
✤✎ ✖
(iv)
✍
✓
✘✙
✎
✓
✌
✦
✓
✣✔
✤
✧
✔
(v)
✍
✓
✘✙
✎
✎
✍
✓
✘✙
✎
✓
✘✙
✎
(ii)
✍
✓
✘✙
✎
(iii)
✍
✓
✘✙
✎
❧
(iv)
✓
✘✙
✎
❧
(v)
✓
✘✙
✎
✍
✓
✘✙
✎
✛
4. (i)
a
✫
❧
✓✘
✙
✎
✍✎
✩ ✧
✛
(ii)
(iii)
5. (i)
❧
✓✘
✙
✎
b
a
✓
✘✙
✎
b
❧
★
✍
✓
✘✕
★
✪
✡✩
✬
✚
✛
✍✎
✩ ✧
✭
❧
d
c
❧
✕✎
✌
★
✪
✩
✭
✍✎
✩ ✧
❧✡
(ii)
❧✱✲
✧✖
✮
✎
★
✩
✍
❧✱✲
✣
✧
✔
✡
✯
❧
(iii)
✐
✕✎
✌
✌
★
✰
✡
(iv)
✍
❧✱✲
✁
✂
✄
☎✆
★
✪
✡✩
✝
✍
❧✱✲
✞ ✟
✠
✴
✵
1.
2.
3.
3, b
a
✵
❧✡
✎✎✓
✏
✔
✶
❧✡
62, c 1
✵
✎ ✧✎
✖✡
✤✰
❧✡✮
✲ ✎
✍
✌✔
✡
★
✪
✩
✎ ✧✎
✖✡
✤✰
❧✡✮
✲ ✎
✍
✌✔
✡
★
✪
✩
✷
✸
✎✎✓
✏
✔
✶
✷
✸
4. x = 2, y = 3
6.
8.
✻
5. – 7
4 ˆj kˆ
✻
1 ˆ
i
3
★
✪
✩
1 ˆ
j
3
1 ˆ
k
3
✍
✎
✩ ✧
6;
7.
1 ˆ
i
6
1 ˆ
j
6
9.
1 ˆ
i
2
1 ˆ
k
2
7iˆ
✈✹
❥
✺
6j
2 ˆ
k
6
♠ ✁✂✄✁☎✁
40 ˆ
8 ˆ
i
j
30
30
1 2 2
,
,
13.
3 3 3
16. 3iˆ 2 ˆj kˆ
10.
16 ˆ
k
30
1
12.
14
,
2
14
3
,
14
1ˆ 4 ˆ 1 ˆ
i
j
k (ii) 3iˆ 3 kˆ
3 3
3
18. (C)
19. (C)
15. (i)
✐✆✝✞✟✠✡☛ ☞✌✍✎
1.
✏
4
60
114
4.
8.
2. cos –1
6.
5
7
3. 0
16 2 2 2
,
3 7 3 7
✑ 2
7. 6 a
✑ ✑
✑ 2
11a.b – 35 b
✒
✒
a 1, b 1
9.
10. 8
13
✗
12.
❧✓✔✕✖
14.
❞✖✘✙✚ ✛✖✜ ✔✖✘ ✦✧✖✘✣★ ✩✖✤★ ✪★✫✪★ ✬✭✮✯✣✰ ❧✓✔✕✖✖✘✭
b ❞✖✘✙✚
–1
15. cos
✛✖✜ ❧✓✔✕✖ ✢✖✘ ❧❞✣✖ ✢✤✥
10
102
13.
3
2
✩✖✤★
b ❞✖✘
✲
✱
a
✬✜✓✳✴
18. (D)
✐✆✝✞✟✠✡☛ ☞✌✍✵
1. 19 2
27
2
8. ✉✼✽✾; ✿✹❁❂❃
5. 3,
9.
❄✹✽
61
2
❅✹❆❇❀❁❈❉
2.
✶
6.
❀✹
❊✾❉❁❋✹
2.
✒
a
0
✒
✺✻
❊●❍❅✹✹❁✾
✿✹❁
b
3.
P◗
1 1 1
,
,
3 2 2 2
✸
;
0
■✽●❏❑▲
10. 15 2
✈▼◆❖◆
1.
2ˆ 2 ˆ 1 ˆ
i ✷ j✷ k
3
3
3
11. (B)
❘❙
❚❯❚❯❱
❘❲❳❨❖❯❩❬
3ˆ 1 ˆ
i
j
2
2
x2 – x1 , y2 – y1 , z2 z1 ; ( x2 x1 ) 2 ( y2
y1 ) 2 ( z2 z1 ) 2
12. (C)
619
620
3.
✥❢✳❦r
5ˆ 3 3 ˆ
i✁
j
2
2
✆ ✝
✆
4. ✉✂✄☎; a , b ✈✞✟✠ c ❞✞✡ ☛☞✞✌✞✍✎ ❞✄ ✏✄✉✞✡☎ ✌✞✍✎✞✈✞✡☎ ❞✞✡ ☛✉✑☛✒✏ ❞✠✏✡ ✂✍✓ ✔✄☛✎✓✕
1
3
5.
3
10 iˆ
2
6.
10 ˆ
j
2
✝
✖
9. 3 a + 5 b
8. 2 : 3
10.
1
(160iˆ – 5 ˆj – 70kˆ) 13. ✘ = 1
3
17. (D)
18. (C)
12.
3 ˆ
i
22
7.
3 ˆ
j
22
2 ˆ
k
22
1 ˆ
(3i – 6 ˆj ✗ 2kˆ); 11 5
7
16. (B)
19. (B)
✐✙✚✛✜✢✣✤ ✦✦✧✦
1 1
,
2
2
1. 0,
5.
✪2
17
,
✪2
2. ★
3
;
17 17
,
✪2
17
,
1
1
1
,★
,★
3.
3
3
3
✪3
17
,
✪2
17
4
;
42
,
✩9 6 ✩2
5
42
, ,
11 11 11
,
✪1
42
✐✙✚✛✜✢✣✤ ✦✦✧✫
✬
iˆ
✬
2 iˆ
4. r
5. r
2 ˆj
3 kˆ
(3 iˆ
ˆj
4 kˆ
( iˆ
2 ˆj
2 ˆj
2 kˆ) t✭✮✯ ✘ ✱✰ ✲✮✴✵✶✲✰ ✷✸✹✺✮ ✭✻✼
kˆ) ✽✮✻✾ ✰✮✵✿❀✺ ❁❂
x✗2
y✩4
z✗5
❆
❆
3
5
6
✬
7. r ❇ (5 iˆ ❈ 4 ˆj ❉ 6 kˆ) ❉ ❊ (3 iˆ ❉ 7 ˆj ❉ 2 kˆ)
6.
❑
8. ✾❥✹✮✮ ✰✮ ✷✶❋●✮ ✷❍✿✰✾■✮ ❏ r
(5 iˆ
2 ˆj
3 kˆ ) ;
x
y
z
❄
❄
5
❃2 3
9. ✾❥✹✮✮ ✰✮ ✷✶❋●✮ ✷❍✿✰✾■✮ ❏ r ▲ 3iˆ ▼ 2 ˆj ▼ 5kˆ ◆ ❖ (11kˆ)
✾❥✹✮✮ ✰✮ ✰✮✵✿❀✺ ✷❍✿✰✾■✮ ❏
✾❥✹✮✮ ✰✮ ✰✮✵✿❀✺ ✷❍✿✰✾■✮ ❏
x 3
0
y 2
0
z 5
11
x ❃2 y ❅ 1 z ❃ 4
❄
❄
✭✻✼
1
2
❃1
♠ ✁✂✄✁☎✁
10. (i)
✞
19 ✟
✡ ,
21 ☞
(ii)
26 ✎
✑
38 ✑✓
✒9
(ii)
✝
✠
☛
✞
p✛
✌1
= cos ✏✏
70
11
3
16.
17.
19
✌1
= cos ✏✏
✞
✎
✑
5 3 ✑✓
2✖
✘
✙ 3✚
✕
= cos✔1 ✗
3 2
2
8
14.
8
✍
✞
✒
✍
11. (i)
12.
✆1
= cos
15. 2 29
29
✐✜✢✣✤✥✦✧ ★★✩✪
1. (a) 0, 0, 1; 2
2
(c)
✵
14
✫
,
3
14
(b)
,
1
14
;
5
14
1
3
,
1
(d) 0, 1, 0;
3
,
1
3
;
1
3
8
5
3 iˆ ✭ 5 ˆj ✮ 6 kˆ ✬
✲ ✰ 7
✲
70
✳
✴
3. (a) x + y – z = 2
(b) 2x + 3y – 4 z = 1
(c) (s – 2t) x + (3 – t) y + (2s + t) z = 15
2. r ✯ ✱✱
4. (a)
✝
✠
☛
24 36 48 ✟
,
,
✡
29 29 29 ☞
(b)
0,
18 24
,
25 25
1 1 1✟
✶8
✝
✟
, , ✡
, 0✡
(d) ✠ 0,
3 3☞
5
☛3
☛
☞
5. (a) [ r ✷ ( iˆ ✷ 2 kˆ )] ✸ (iˆ ✹ ˆj ✷ kˆ) ✺ 0; x + y – z = 3
(b) [ r ✷ ( iˆ ✹ 4 ˆj ✹ 6 kˆ ) ] ✸ ( iˆ ✷ 2 ˆj ✹ kˆ) ✺ 0; x – 2y + z + 1 = 0
(c)
✝
✠
6. (a) ❢✻✼✽✾ ✿✼❀❁❂❃ ❄❅✼❆ ❢✽❇ ❈❇ ❢✻✼✽❉
✾ ❃❁✼
(b) 2x + 3y – 3z = 5
5
7.
, 5, –5
8. y = 3
2
10. r ❖ ▼ 38 iˆ P 68 ˆj P 3 kˆ ◆ ◗ 153
12. cos
1
15
731
✿❁ ❊❃❋❁ ●❃❍❁ ■❍❃❁✼ ❏❑ ✿✼❂ ▲❃ ❉❋✼■ ❄❃❁ ❈❑❆
9. 7x – 5y + 4z – 8 = 0
11. x – z + 2 = 0
621
622
✥❢✳❦r
13. (a) cos
1✁
✄
✆
2✂
5 ☎✝
(c) ✞✟ ✠✡☛☞
(e) 45o
✌✍✎ ☞✌✡✎✞✖ ✓✔✎✕
(b)
✞✟ ✠✡☛☞ ✌✍✎ ✟✎✏✑✞✒ ✓✔✕✎
(d)
✞✟ ✠✡☛☞ ✌✍✎ ☞✌✡✎✞✖ ✓✔✎✕
13
3
(d) 2
3
13
(c) 3
14. (a)
(b)
✈✗✘✙✘ ✚✚ ✛✜ ✢✣✢✣✤ ✛✦✧★✙✣✩✪
3. 90°
6. k ✬
x
1
4.
✫10
7. r
7
8. x + y + z = a + b + c
10.
✲
✵ 0,
✷
17
,
2
✱13 ✴
✶
2 ✸
y
0
✭
5. 0
iˆ ✮ 2 ˆj ✮ 3 kˆ ✮
, 0,
23 ✺
✼
3 ✾
13. 7x – 8y + 3z + 25 = 0
20. r ✭ iˆ ✮ 2 ˆj ✰ 4 kˆ ✮ ✯ (2 iˆ ✮ 3 ˆj ✮ 6 kˆ)
23. B
12. (1, – 2, 7)
7
3
16. x + 2y – 3z – 14 = 0
18. 13
22. D
✐✿❀❁❂❃❄❅ ❆❇❈❆
1. (0, 4) ☛✖ ✠❉❊❋✞✌ Z = 16
2. (4, 0) ☛✖ ●❍■❏✞✌ Z = – 12
20 45
,
19 19
4.
3 1
,
2 2
☛✖ ✠❉❊❋✞✌
☛✖ ●❍■❏✞✌
( iˆ ✮ 2 ˆj ✰ 5 kˆ )
14. p = 1 or
15. y – 3z + 6 = 0
17. 33 x + 45 y + 50 z – 41 = 0
19. r ✭ iˆ ✮ 2 ˆj ✮ 3 kˆ ✮ ✯ (✰ 3 iˆ ✮ 5 ˆj ✮ 4kˆ)
3.
✯
o
9. 9
✹ 17
✻
✽ 3
11.
z
0
Z=
Z=7
235
19
♠ ✁✂✄✁☎✁
5. (4, 3) ✐✆
Z = 18
✝✞✟✠✡☛
6. (6, 0) ✝✈☞✆ (0, 3) ✠✈❞
7. (60, 0) ✐✆
(120, 0) ✝✈☞✆ (60, 30) ✠✈❞
Z = 600;
8. (0, 50) ✝✈☞✆ (20, 40) ✠✈❞
9. Z ✠✈
10.
Z = 6.
✞☛✌✈✍❞ ✎✈✌✏ ✆❞✑ ✈✈ ✑✈✒✓ ✐✆ ✞✔✕✈✡ ✖✗✈✏ ✞✘✒✙✚✝✈❞✒ ✐✆ ✛✜✢✍✡☛
Z = 300;
✛✜✢✍✡☛
(0, 200) ✐✆
623
✝✞✟✠✡☛
✚ ✈❞✒ ✐✆ ✝✞✟✠✡☛
✞☛✌✈✍❞ ✎✈✌✏ ✆❞✑ ✈✈ ✑✈✒✓ ✐✆ ✞✔✕✈✡ ✖✗✈✏ ✞✘✒✙✝
✚ ✈❞✒ ✐✆ ✛✜✢✍✡☛
✞☛✌✈✍❞ ✎✈✌✏ ✆❞✑ ✈✈✑✈✒✓ ✐✆ ✞✔✕✈✡ ✖✗✈✏ ✞✘✒✙✝
Z = 100.
Z = 400
✠✈❞✣✤ ✝✞✟✠✡☛ ☛✈✍ ✍✥✏✒ ✥☞✦
Z ✠✈
✒ ✡ ✩✈❞✪✈ ✍✥✏✒ ✥☞ ✝✡✫
♣✢✧✞✠ ✠✈❞✣✤ ✖✚✖★
✝✞✟✠✡☛ ☛✈✍ ✍✥✏✒ ✥☞✦
✬✭✮✯✰✱✲✳ ✴✵✶✵
1.
2.
1
8
,0 ✝✈☞✆ 2,
2
3
= Rs 160 .
✠✈❞
✞☛✌✈✍❞
✎❞✷✠✈❞✒ ✠✏ ✝✞✟✠✡☛ ✖✒✑✜✈
3. (i)
(ii)
✎✈✌✏
= 30 ✹✠
✆❞✑✈✈
✑✈✒✓
✎❞✷
✖✗✈✏
✞✘✒✙✚✝✈❞ ✒
✐✆
✛✜✢✍✡☛
☛✢✸✜
✒
✐✺✠ ✈✆ ✠✏ ✡✕✈✈ ✻✼ ✝✛✜ ✐✺✠✈✆ ✠✏ ✥☞✦
✽ ✾❞✞✍✖ ✆☞✠✾ ✡✕✈✈ ✻✿ ✞❀✎❞ ✷✾ ✘✸✌❞
✝✞✟✠✡☛ ✌✈✗✈
= Rs 200
4.
✍✾ ✎❞✷ ✡✏✍ ✐☞ ✞✠✾ ✡✕✈✈ ✎✈❞✸✾ ✎❞✷ ✡✏✍ ✐☞✞✠✾✉ ✝✞✟✠✡☛ ✌✈✗✈
5.
❁✼ ✐☞✞✠✾
A ✐✺✠✈✆
✎❞✷ ✐❞✒♣ ✡✕✈✈ ✿✼ ✐☞✞✠✾
B ✐✺✠✈✆
= Rs 73.50.
✠✏ ✐❞✒♣✈❞ ✎❞✷ ✡✕✈✈ ✝✞✟✠✡☛ ✌✈✗✈
= Rs 410
6.
✽ ✝✈✟✈✆ ✌☞✐
✒
✝✈☞✆ ✽ ✠✈❂ ✠✈ ❃❄✠✍✉ ✝✞✟✠✡☛ ✌✈✗✈
7. A ✐✺ ✠✈✆ ✎❞ ✷
= Rs 160.
8. 200 ✓❞✔✠✾✈▼✐
❅
✔☛❆ ✞✡
✞♣❇
✡✕✈✈
B
✐✺✠ ✈✆
✎❞ ✷
✿✼
= Rs 32
✔☛❆✞✡
✎❞✷ ✍☛✢✍❞ ✡✕✈✈ ❈✼ ✐✈❞✾✘
✤❞ ✌ ✍☛✢✍✉
❞ ✝✞✟✠✡☛ ✌✈✗✈
✞♣❇✉
✝✞✟✠✡☛
✌✈✗✈
= Rs 1150000.
9. Z = 4x + 6y ✠✈ ✛✜✢✍✡☛✏✠✆❉✈ ✠✏✞❊✹ ❊✘✞✠ 3x + 6y ❋ 80, 4x + 3y ❋ 100, x ❋ 0 ✝✈☞✆
y ❋ 0, ❊✥✈✧ x ✝✈☞✆ y ❀☛Ø✈✫ ✗✈✈❞●✜ F1 ✝✈☞✆ F2 ✠✏ ✣✠✈✣✤✜✈❞✒ ✠✈❞ ✙Ø✈✈✤✡❞ ✥☞✉✒ ✛✜✢✍✡☛ ☛✢✸✜
= Rs 104
10.
❍✎✤✆✠
11. (D)
F1 ✎❞ ✷ 100 kg ✝✈☞✆ ❍✎✤✆✠ F2 ✎❞✷ 80 kg; ✛✜✢✍✡☛
☛✢✸✜
= Rs 1000
624
✥❢✳❦r
✈ ✁✂✁ ✄☎ ✆✝ ✞✟✞✟✠ ✆✡☛☞✂✟✌✍
1. 40 ✐✎✏✑✒ ✓✔✔✕✖✗ P ♦✕✘ ✙✔✎✚ ✛✜ ✐✎✏✑✒ ✓✔✔✕✖✗ Q ♦✕✘; ✏♦✒✔✏✢✣ A ✑❞ ✙✏✤✑✦✢ ✢✔✧✔✔ = 285
❜✑✔❜★
2. P ✐✩✑✔✚ ♦✕✘ ✪ ✫✔✎✬✕ ✙✔✎✚ Q ✐✩✑✔✚ ♦✕✘ ✭ ✫✔✎✬✕; ✏✢✮✯✔ ✑✔ ✰✗✱✣✦✢ ✢✱✲✗ = Rs 1950
3. ✏✢✮✯✔ ✑✔ ✰✗✱✣✦✢ ✢✱✲✗ Rs 112 ✴✓✔✔✕✖✗ X ✑✔ 2 kg ✦✫✔✔ ✓✔✔✕✖✗ Y ✑✔ 4 kg).
5. ✐✩✫✔✢ ✮✕✯✔❞ ♦✕✘ ✵✶ ✏✒✑✒ ✦✫✔✔ ✷✔✤✔✚✯✔ ✮✕✯✔❞ ♦✕✘ ✛✭✶ ✏✒✑✒✸ ✙✏✤✑✦✢ ✬✔✓✔ = Rs 136000.
6. A ✷✕ : 10, 50 ✙✔✎✚ 40 ❜✑✔❜★✗✔✹; B ✷✕ : 50, 0 ✙✔✎✚ 0 ❜✑✔❜★✗✔✹ ✺✢✻✔✼ D, E ✙✔✎✚ F ✑✔✕ ✓✔✕✽❞ ✽✔✦❞
❣✎ ✦✫✔✔ ✰✗✱✣✦✢ ✢✱✲✗ = Rs 510
7. A ✷✕ : 500, 3000 ✙✔✎✚ 3500 ✬❞✒✚; B ✷✕: 4000, 0 ✙✔✎✚ 0 ✬❞✒✚ ✦✕✬ ✺✢✻✔✼ D, E ✙✔✎✚ F ✑✔✕
✓✔✕✽❞ ✽✔✦❞ ❣✎ ✦✫✔✔ ✰✗✱✣✦✢ ✢✱✲✗ = Rs 4400
8. P ✐✩✑✔✚ ♦✕✘ ✵✶ ✫✔✎✬✕ ✙✔✎✚ Q ✐✩✑✔✚ ♦✕✘ 100 ✫✔✎✬;✕ ✣✔❜✒✉✔✽
✕ ✣ ✑❞ ✰✗✱✣✦✢ ✢✔✧✔✔ = 470 kg.
9. P ✐✩✑✔✚ ♦✕✘ ✛✵✶ ✫✔✎✬✕ ✙✔✎ ✚ Q ✐✩✑✔✚ ♦✕✘ 50 ✫✔✎✬✕; ✣✔❜✒✉✔✽
✕ ✣ ✑❞ ✙✏✤✑✦✢ ✢✔✧✔✔ = 595 kg.
10. A ✐✩✑✔✚ ✑❞ ✾✶✶ ✿❀✏❁❂✗✔✹ ✙✔✎✚ B ✐✩✑✔✚ ✑❞ ✵✶✶ ✿❀✏❁❂✗✔✹;✕ ✙✏✤✑✦✢ ✬✔✓✔ = Rs 16000
❃❄❅❆❇❈❉❊ ❋●❍❋
1. P E|F
3. (i) 0.32
4.
1
3
2. P A|B
(ii) 0.64
16
25
(iii) 0.98
11
26
5. (i)
4
11
1
2
7. (i) 1
6. (i)
8.
2
, P F|E
3
1
6
11. (i)
(ii)
4
5
3
7
(ii) 0
(ii)
9. 1
1 1
,
2 3
(ii)
(iii)
2
3
(iii)
6
7
10. (a)
1 2
,
2 3
(iii)
1
1
, (b)
3
9
3 1
,
4 4
♠ ✁✂✄✁☎✁
12. (i)
14.
1
2
(ii)
1
15
1
3
5
9
13.
15. 0
625
16. C
17. D
✐✆✝✞✟✠✡☛ ☞✌✍✎
3
25
4. A ✈✏✑✒ B ✓✒✔✓✒
1.
6. E ✈✏✑✒ F ✓✒✔✓✒
7. (i)
p✢
2.
25
102
✔✕✖✗✘✏ ✙✑✗✚
(ii)
8. (i) 0.12
3
8
11. (i) 0.18
12.
7
8
14. (i)
17. D
p
1
5
(ii) 0.58
10. A ✈✏✑✒ B ✓✒✔✓✒
(ii) 0.12
(iii) 0.72
13. (i)
2
1
, (ii)
3
2
✔✕✖✗✘✏ ✛✙✜✗ ✙✑✗✚
✔✕✖✗ ✘✏ ✛✙✜✗ ✙✑✗✚
1
10
9.
44
91
5. A ✈✏✑✒ B ✓✒✔✓✒
3.
(iii) 0.3
(iv) 0.4
✔✕✖✗✘✏ ✛✙✜✗ ✙✑✗✚
(iv) 0.28
20
40
16
, (ii)
, (iii)
81
81
81
15. (i) , (ii)
16. (a)
1
1
1
, (b) , (c)
5
3
2
18. B
✐✆✝✞✟✠✡☛ ☞✌✍✌
1.
1
2
2.
2
3
3.
9
13
4.
12
13
5.
198
1197
6.
4
9
7.
1
52
8.
1
4
9.
2
9
10.
8
11
11.
5
34
12.
11
50
13. A
14. C
626
✥❢ ✳❦
r
✐
1. (ii), (iii)
4. (i)
☛☞ ✌✍
✁
✂
✄
☎✆
✝
✞ ✟
✠
✡
(iv)
2. X = 0, 1, 2;
X
0
1
2
P(X)
1
4
1
2
1
4
X
0
1
2
3
P(X)
1
8
3
8
3
8
1
8
X
0
1
2
3
4
P(X)
1
16
1
4
3
8
1
4
1
16
X
0
1
2
P(X)
4
9
4
9
1
9
X
0
1
P(X)
25
36
11
36
X
0
1
(ii)
(iii)
5. (i)
(ii)
6.
P(X)
7.
256 256
625 625
2
96
625
X
0
1
2
P(X)
9
16
6
16
1
16
8. (i) k
✏
1
10
✓
✒
3
10
4
16
1
625 625
(ii) P(X 3)
(iv) P(0 X 3)
✓
3
3. X = 6, 4, 2, 0
❣ ☞
✎
3
10
(iii) P(X 6)
✑
✒
17
100
♠ ✁✂✄✁☎✁
1
1
(b) P(X ✝ 2) ✆ , P(X ✞ 2) ✆ 1, P(X ✟ 2) ✆
2
2
1
14
10. 1.5
11.
12.
3
3
13. Var(X) = 5.833, S.D = 2.415
9. (a) k ✆
14.
1
6
X
14
15
16
17
18
19
20
21
P(X)
2
15
1
15
2
15
3
15
1
15
2
15
3
15
1
15
❡✠✡☛
= 17.53, Var(X) = 4.78 ✈✠☞✌ S.D(X) = 2.19
15. E(X) = 0.7 ✈✠☞✌ Var (X) = 0.21
16. B
17. D
✐✍✎✏✑✒✓✔ ✕✖✗✘
1. (i)
2.
3
32
(ii)
25
216
4. (i)
29 ✚ ✙ 19 ✚
3. ✛ ✜ ✛ ✜
✢ 20 ✣ ✢ 20 ✣
✙
1
1024
(ii)
5. (i) (0.95)5
(iv) 1 – (0.95)5
6.
9
10
9.
11
243
7.
99
100
7 5
11.
12 6
✤
✩
✭
1✥
✪
2✮
50
(b)
5
12.
45
512
63
64
(iii)
243
1024
(iii) 1 – (0.95)4 × 1.2
20
✦
✫
20C12 ★
1 99
2 100
35 5
18 6
15. A
(iii)
9
(ii) (0.95)4 × 1.2
4
10. (a) 1
14. C
7
64
20
C13 ★ ... ★
20
C 20 ✧✬
49
(c) 1
4
13.
149 99
100 100
22 ✯ 93
1011
49
627
628
✥❢✳❦r
✈ ✁✂✁ ✄☎ ✆✝ ✞✟✞✟✠ ✆✡☛☞✂✟✌✍
1. (i) 1
2. (i)
3.
1
3
10
10
4. 1
C r (0.9) r (0.1)10
r
r 7
✎ 2✏
✑ ✒
✓5✔
6.
510
2 ✖ 69
9.
31 ✎ 2 ✏
✑ ✒
9 ✓ 3✔
6
15. (i) 0.5
(ii)
✎ 2✏
7✑ ✒
✓ 5✔
7.
4
✎ 2✏
1✕ ✑ ✒
✓ 5✔
(iii)
6
625
23328
10. n ✗ 4
13.
8.
11.
14
29
864
3125
(iv)
4
1 2 8
, ,
15 5 15
17. A
1
2
(ii)
20
21
5. (i)
12.
(ii) 0
14.
2
7
91
54
3
16
16
31
19. B
(ii) 0.05
16.
18. C
—✘—
i jd
i kB ;
v Ł; k;
ò ( ✆✝ ☎ ✞)
✁✂✁✄✁
ge v p j
Av
l ke x h
7
✟✝2 ☎ ✠✝ ☎ ✡ ☛✝.
kj
B bl
i d kj p u r
g fd
é✎
ù
2
☞✌ + ✍ = A ê (✏✑ ✔ ✒✑ ✔ ✓)ú ✔ B
ë ✎✑
û
= A(2 ✕✌ + ✖ ) + B
n ku k i { kk e
✌
o Q x . kkd k v kj v p j i n k d h r y u k d j u i j ] g e s i kI r
2 ✕A = ☞
b u l e hd j . kk d k g y
d ju i j]
v kj
Av
g kr k g
A✖ + B = ✍
kj
B o Q e ku
i kI r g k t kr g A b l
i d kj ] l e kd y f u Eu
e i f j o f r r g k t kr k g
A ò (2 ✕✌ ✙ ✖) ✕✌ 2 ✙ ✖✌ ✙ ✗ ✘✌ ✙ B ò ✕✌2 ✙ ✖✌ ✙ ✗ ✘✌
= AI1 + BI 2 ]
✕✌2 + ✖✌ + ✗ = ✚, j f [
i kB ;
I1 =
ò (2✕✌ ✙ ✖)
k, A r c ]
(2 ✕✌ + ✖ )d✌ = d✚ g A
v r %]
I1 =
b l h i d kj ]
I2 =
i Lr d
t kr k g A
t g k¡
2
3
ò
✕✌2 ✙ ✖✌ ✙ ✗ ✘✌
gA
3
(✛✜2 ✤ ✢✜ ✤ ✣ ) 2 ✤ C1
✕✌ 2 ✙ ✖✌ ✙ ✗ ✘✌
o Q i "B 3 2 8 i j 7 -6 -2 e p p k f d ,
x,
l e kd y
l =k d k i ; kx
d j o Q K kr
fd ; k
630
bl
x f. kr
ò(
i d kj ]
mn kg j . k
gy
✁ ✞ ✂ ) ✄✁ 2 ✞ ☎✁ ✞ ✆ ✝✁
ò✁
✟✠
mi j n ’ kk,
1 ✞ ✁ - ✁2 ✝✁ K kr
x b fo f/
v i u kr
d k e ku v r r % K kr
d j f y ; k t kr k g A
d hf t , A
g , ] g e f y [ kr
g
é✌
2 ù
✡ = A ê ☛1 ✎ ✍ - ✍ ☞ ú ✎ B
ë ✌✍
û
= A (1 ✥ 2 ✡) + B
n ku k i { kk e ]
✡ o Q x . kkd
k v kj v p j i n k d k c j kc j d j u i j ] g e
✥ 2A = 1
v kj
A+B = 0
i kI r
g kr k g A
b u l e hd j . kk d k g y
l e kd y
f u Eu e i j ko f r r
ò✁
A = -
d ju i j] ge
1
2
v kj
B✏
1
2
i kI r
d jr
g A bl
i d kj ]
g k t kr k g
1 ✞ ✁ - ✁2 ✝✁ = -
1
1
(1 - 2✑) 1 ✓ ✑ - ✑ 2 ✒✑ ✓ ò 1 ✓ ✑ - ✑2 ✒✑
ò
2
2
1
1
= - I1 ✓ I2
2
2
I1 =
1 + ✡ ✥ ✡2 = ✖ j f [
bl
1 ✕ ✡ - ✡2 ✔✡
ò✗
1
2
2
3
✘✗ ✙ ✗ 2 ✚ C1
3
3
2
2
= ✛1 ✕ ✡ - ✡ ✜ 2 ✕ C1 , t
3
I2 =
ò
1 ✕ ✡ - ✡2 ✔✡
i j f o p kj d hf t , A
2
; g l e kd y
✡-
1
✦✖
2
=
ò
5 æ
1ö
- ç ✢ - ✤ ✣✢
4 è
2ø
j f [ k, A r c ]
✔✡ = ✔✖ g A
2
v r %]
i j f o p kj d hf t , A
, (1 ✥ 2 ✡) ✔✡ = ✔✖ g A
I 1 = ò (1 - 2 ✡) 1 ✕ ✡ - ✡2 ✔✡ =
i d kj ]
v kx ]
k, A r c
ò (1 - 2✡)
(1)
I2 =
ò
æ 5ö
2
ç 2 ✧ - ✖ ✔✖
è
ø
gk
C1 d
kb v p j g A
i jd
=
1
2
5
4
=
1 ✄ 2✆ - 1☎ 5
1
- (✆ - ) 2
2
2
4
2
=
1
(2✆ - 1) 1 ✁ ✆ - ✆ 2
4
2
✁
1 5 -1 2
✂ sin
2 4
5
✁
✁
✁
i kB ;
l ke x h
C2
5 - 1 æ 2 ✆ - 1ö
sin ç
✁ ✝2
è 5 ✞
ø
8
5 -1 æ 2✆ - 1 ö
sin ç
✁ ✝2 ,
è 5 ✞
ø
8
t g k¡ C d kb v p j g A
2
(1)
e saI v kj
1
I 2 o Q e ku
ò
✟
j [ ku i j ] g e i kI r
3
1
1
2
2
1 ✡ ✟ - ✟2 ✠✟ = - (1 ✁ ✆ - ✆ ) 2 ✁ (2 ✆ - 1) 1 ✁ ✆ - ✆
3
8
5
æ 2✆ - 1 ö
sin -1 ç
,
✁
✞ ✁ ✝ t g k¡
è
16
5 ø
C= -
C1 ☛ C2
2
i ’ u ko y h ☞ ✌☞ o Q v r e ] fu Eu fy f[ kr
✍✎ ✌ ✏
✏ ✑ ✏
g kr k g
2
v U;
i ’ u l fEe fy r
(✆ ✁ 1) 2✆ 2
✍✒ ✌
, d
✁
v p j gA
d hft ,
✍✓ ✌
3
(✆ ✁ 3) 3 - 4✆ - ✆ 2
m kj
✍✎ ✌
✍✒ ✌
✍✓ ✌
1 2
(✆
3
3
2
✁ ✆
) -
3
1
(2 ✆ 2 ✁ 3) 2
6
(2 ✆ ✁ 1)
8
✁
✆
2
3
1
- (3 - 4 ✆ - ✆ 2 ) 2
3
✆
2
✁ ✆
2✆ 2 ✁ 3 ✁
✁
✁
1
1
log | ✆ ✁
16
2
3 2
log ✆ ✁
4
✆
2
✆
✁
✁
3
2
2
✁
✁ ✆
| ✁C
C
7 - 1 æ ✆ ✁ 2 ö (✆ ✁ 2) 3 - 4✆ - ✆ 2
sin ç
✁
è 7 ✞
ø
2
2
✁
C
631
632
x f. kr
v Ł; k;
✁✂✄
v fn ’ k f=kd
10
x . ku i Qy
✝ ✝
✝ ✝
✝
✝
e ku y hf t , f d
☎ ,✆ v kj ✞ d kb r hu l f n ’ k g A ☎ v kj (✆ ´ ✞ )
✝ ✝ ✝
✝ ✝
✝
☎ ✟(✆ ´ ✞ ) d ks ☎ ,✆ v kj ✞ d k b l h o Qe e v fn ’ k f=kd x . ku i Qy
✝
✝ ✝
[ ☎ ✆ ✞ ]) } kj k O; Dr f d ; k t kr k g A b l i d kj ] g e i kI r g
✝ ✝ ✝
✝ ✝ ✝
[ ☎ ,✆ , ✞ ] = ☎ ✟(✆ ´ ✞ )
i { k. k
✂
✝ ✝
(✆ ´ ✞ ) , d l f n ’ k
✝ ✝ ✝
☎ ✟(✆ ´ ✞ ) , d v fn ’ k j kf’ k g ] v
D; ksafd
, d
✠✂
g]
Fkkr
: i
l ] v f n ’ k f =kd
g A b l s[
✝ ✝ ✝
☎ ,✆ , ✞ ] ( ;
✝ ✝
☎ ,✆ v
Hkt kv k l
l e kr j
"kVi Qy d
g kr k g ( n f [ k,
v ko Qfr
1 0 -2 8 ) A
f u l n g ] l e kr j
"kVi Qy d
o ky
bl fy ,
cu
l e kr j
kj
✏
✎
p r Hkt
✝
✞
kj
l
i n f ’ kr
✓
✓
✒ × ✔
d k v k; r u
o Q v k/ kj
d k { k=ki Qy
d k v r f o "V d j u o ky
d h f n ’ kk e
✏
✑
d k
v kl Uu
ry
d k ?kVd
d k c u ku s
☞ ☞
✡ ´☛
v ko Qfr
gA
✏
✝
✝
✝
☎ . (✆ ´ ✞ )
✝ ✝ gAv
(✆ ´ ✞ )
✁✂✠✫
i j v f Hky c o Q v u f n ’ k✑ i { ki g h b l d h mp kb g ]
g A v Fkkr ; g
r %] l e kr j "kVi Qy d
d k v k; r u
✝ ✝ ✝
☎ . (✆ ´ ✞ ) ✝ ✝
☞
✝ ✝ | ✆ ´ ✞ | = ✕☞ . (✡ ´ ☛☞ )
(✆ ´ ✞ )
✖✂
; fn
k
✝ ✝ ✝
[ ☎ ,✆ , ✞ ]
x . ku i Qy
e ku r hu l f n ’ kk
t k
d gr
v f n ’ k j kf ’ k g A
T; kf e r h;
✍
✌v
o Q v f n ’ k x . ku i Qy ] v Fkkr ~
✥
✥
✘ ✣ ✘1✙✗ ✤ ✘ 2 ✗✚ ✤ ✘3 ✛✗ , ✜ ✣ ✜1✗✙ ✤ ✜2 ✚✗ ✤ ✜3 ✛✗
✍ ✍
✌ ´✦ =
★✧ ✧✩ ✪✧
✆1 ✆2 ✆3
✞1 ✞2 ✞3
v kj
✥
✢ ✣ ✢1✙✗ ✤ ✢ 2 ✗✚ ✤ ✢ 3 ✛✗,
g ] r ks
i jd
i kB ;
l ke x h
633
= ✁ 2✂3 ✥ ✁ 3✂2) ☎✄ + ( ✁ 3✂1 ✥ ✁ 1✂3) ✆✝ + (✁ 1✂2 ✥ ✁ 2✂1) ✞✄
r Fkk b l hf y ,
✠ ✠ ✠
✟ .( ✁ ´ ✂ ) = ✟1 (✁2 ✂3 ✥ ✁3 ✂2 ) ✡ ✟2 (✁3✂1 ✥ ✁1✂3 ) ✡ ✟3 (✁1 ✂2 ✥ ✁2 ✂1 )
=
☛☞
; fn
✠ ✠
✟ ,✁ v
✠
✂
kj
✟1
✟2
✟3
✁1
✁2
✁3
✂1
✂2
✂3
d kb r hu l f n ’ k g ] r ks
✠ ✠
✠ ✠ ✠
✠
✠ ✠ ✠
[ ✟ , ✁ , ✂ ] = [ ✁ , ✂ , ✟ ] = [ ✂, ✟ , ✁ ]
( r hu k l f n ’ kk o Q p o Qh;
o Qe p ;
l
v f n ’ k f =kd
x . ku i Qy
o Q e ku e d kb i f j o r u u g h g kr k
g A)
e ku y hf t ,
r c ] o Qo y
fd
✠
✠
✠
✟ ✑ ✟1✍✌ ✡ ✟ 2 ✌✎ ✡ ✟3 ✏✌ , ✁ ✑ ✁1✌✍ ✡ ✁2 ✎✌ ✡ ✁3 ✏✌ r Fkk ✂ ✑ ✂1✍✌ ✡ ✂2 ✌✎3✏✌
n [ kd j g h] g e i kI r
✠ ✠ ✠
[ ✟ ,✁ , ✂ ] =
gA
g kr k g
✒1
✒2
✒3
✓1
✓2
✓3
✔1
✔2
✔3
= ✟ 1 ( ✁ 2✂3 ✥ ✁ 3✂2) + ✟ 2 (✁ 3✂1 ✥ ✁ 1✂3) + ✟ 3 ( ✁ 1✂2 ✥ ✁ 2✂1)
= ✁ 1 (✟ 3✂2 ✥ ✟ 2✂3) + ✁ 2 (✟ 1✂3 ✥ ✟ 3✂1) + ✁ 3 (✟ 2✂1 ✥ ✟ 1✂2)
=
✓1
✓2
✓3
✔1
✔2
✔3
✒1
✒2
✒3
✠ ✠ ✠
= [ ✁ , ✂, ✟ ]
b l h i d kj ] i kB d
v r %]
✠ ✠
b l d h t kp
✠ ✠ ✠
d j l d r
g fd
✠ ✠ ✠
[ ✟ ,✁ , ✕✖ ] = [ ✁ , ✂, ✟ ] = [ ✂, ✟ , ✁ ] g A
✠ ✠
✠ ✠ ✠
[ ✟ , ✁ , ✕✖ ] = [ ✂, ✟ , ✁ ]
gA
634
✁
x f. kr
v f n ’ k f =kd
x . ku i Qy
✆ ✆ ✆
✂ .(✄ ´ ☎ )
e ] M kV
(dot)
v kj
o Qkl
(cross)
d k i j Li j
c ny k t k
l d r k g A f u Ll n g ]
✆ ✆ ✆
✆ ✆ ✆
✆ ✆ ✆
✆ ✆ ✆
✆ ✆ ✆
✆ ✆ ✆
✂ .(✄ ´ ☎ ) = [ ✂ ,✄ , ☎ ] = [ ✄ , ☎ , ✂ ] = [ ☎ , ✂ , ✄ ] = ☎ .( ✂ ´ ✄ ) = ( ✂ ´ ✄ ).☎
✡ ✡ ✡
✡ ✡ ✡
= [ ✞ , ✟ , ✠ ] = ✥ [ ✞ , ✠, ✟ ]. f u Ll n g ]
✝✁
✆ ✆ ✆
✆ ✆ ✆
= [ ✂ ,✄ , ☎ ] = ✂ .(✄ ´ ☎ )
✆ ✆ ✆
= ✂ .(✥ ☎ ´ ✄ )
✆ ✆ ✆
= ✥ (✂ .(☎ ´ ✄ ))
✆ ✆ ✆
= ✥ éë✂ , ☎ , ✄ ùû
✎ ✎ ✎
☛✁
[☞ , ☞ , ✌ ] ✍ 0.
f u Ll n g ]
✆ ✆ ✆
✆ ✆ ✆
[✂ , ✂ , ✄ ] = [ ✂ , ✄ , ✂ ,]
✆ ✆ ✆
= [✄ , ✂ , ✂ ]
✆ ✆ ✆
= ✄ .( ✂ ´ ✂ )
✒ ✒
= ✏ . 0 ✑ 0.
( D;
kf d
✆ ✆ ✆
✂ ´ ✂✓ 0 )
fV I i . kh mi ; Dr 7 e ] f n ; k i f j . kke ] n ku k c j kc j l f n ’ kk o Q f LFkf r ; k o Q f d l h Hkh o Qe e g ku s
i j Hkh l R;
1 0 -7 -1
i es
z;
✔
gA
r hu l fn ’ kk d h l e r y h; r k
✡ ✡
,✟ v
r hu l f n ’ k ✞
kj
mi i f k l o i Fke ] e ku y hf t ,
; fn
✡
✟
v kj
✡
✠
✡
✠
l e r y h;
fd
l e kr j l f n ’ k g ]
g kr g ] ; f n v kj o Qo y ; f n
✡ ✡
✞ , ✟ v kj
✡ ✡
r ks ✟ ´ ✠ =
✡
✠
✗
0
l e r y h;
✡ ✡ ✡
✞ ✕(✟ ´ ✠ ) ✖ 0
g kr k g A
gA
g v kj b l hf y ,
✡ ✡ ✡
✞ ✕(✟ ´ ✠ ) ✖ 0
g kx kA
i jd
l ke x h
635
✁ ✁
✁
✁
✁ ✁
✁
✂ l e kr j u g h g ] r k ´ ✂ l fn ’ k ✄ i j y c g kx k] D; kfd ✄ , v kjS ✂ l e r y h; g AS
✠ ✠ ✠
v r %] ☎ ✞(✆ ´ ✝ ) ✟ 0 g A
✁ ✁
✠ ✠ ✠
✁
f o y ke r %] e ku y hf t , f d ☎ ✞(✆ ´ ✝ ) ✟ 0 g A ; f n ✄ v kj
´ ✂ e l n ku k ’ kU; r j l f n ’ k g ]
✁ ✁
✁ ✁
✁
✁
r k g e f u "d "k f u d ky r g f d ✄ v kj
n
k
y
k
f
c
d
l
f
n
’
k
g
A
i
j
r
´ ✂ n ku k l f n ’ kk
´✂
✁ ✁
✁
✁
v kj ✂ i j y c g A v r %] ✄ ]
v kj ✂ , d l e r y e f LFkr g ku p kf g , ] v Fkkr ; l e r y h;
✁
✁
✁
✁
g A ; f n ✄ = 0 g ] r k ✄ f d Ug h Hkh n k l f n ’ kk] f o ’ k"k : i l
v kj ✂ ] o Q l e r y h; g kx kA ; f n
✠ ✠
✁
✁ ✁
✁
✁
(✆ ´ ✝ ) ✟ 0 g ] r k v kj ✂ l e kr j l f n ’ k g kx r Fkk b l hf y , ✄ , v kj ✂ l e r y h; g kx ]
; fn
✁
i kB ;
D; kf d
v kjS
d kb Hkh n k l f n ’ k l n o , d
t k b u n ku k l f n ’ kk e l
l e r y e g kr
fd l h , d
l e kr j
g ] t k mu l
f u / kf j r g kr k g ] r Fkk d kb l f n ’ k]
g kr k g ] Hkh b l h l e r y
e f LFkr
g kr k g A
fV I i . kh p kj f c n v k d h l e r y h; r k d h p p k] r hu l f n ’ kk d h l e r y h; r k d k i ; kx d j r g , ]
d h t k l d r h g A f u Ll n g ] p kj f c n
l e r y h;
mn kg j . k
gy
A, B, C v
☛
☞✌ ✍☛ .(✎ ´ ✏☛ )
g e i kI r g
mn kg j . k
☞✗
l e r y h;
gA
Kkr d hft , ] ; f n
e r y h; g kr sg ] ; f n l f n ’ k
✡✡✡☛ ✡✡✡☛
AB, AC
v kS
✡✡✡☛
AD
g e i kI r
n ’ kkb ,
g
v r %] i e ;
1
mn kg j . k
☞✩
k e ku K kr
☛
☛
✍ ✕ 2✒✑ ✖ ✓✑ ✖ 3 ✔✑ , ✎ ✕ ✥ ✒✑ ✖ 2 ✓ ✖ ✔
v kjS
☛ ✑ ✑ ✑
✏ ✕ 3✒ ✖ ✓ ✖ 2 ✔
gA
S
2 1 3
☛ ☛ ☛
✍ .(✎ ´ ✏ ) ✕ -1 2 1 ✕ ✥10.
fd
l fn ’ k
1 2
★
✚★ ✦ ✘✛ - 2 ✘✜ ✧ 3✢✘ , ✣ ✦ ✙ 2 ✘✛ ✧ 3 ✜ - 4 ✢✘ v
kj
✤★ ✦ ✘✛ - 3 ✘✜ ✧ 5✢✘
1 -2 3
☛ ☛ ☛
✍ .(✎ ´ ✏ ) ✕ -2 2 -4 ✕ 0.
1
ld
Dl
g kA
3
gy
kj
o Q v u l kj
; fn l fn’ k
d hf t , A
-3
5
✁ ✁
✁
✄ , v kj ✂ l e r y h; l f n ’ k g A
★
✚★ ✦ ✘✛ ✧ 3 ✘✜ ✧ ✢✘ , ✣ ✦ 2 ✘✛ - ✘✜ - ✢✘ v
kj
✤★ ✦ l✛✘ ✧ 7 ✘✜ ✧ 3 ✢✘
l e r y h; g ] r ks
636
x f. kr
gy
D; kf d
✄ ✄
,✁ v
kj
✄
✂l
e r y h;
1 3 1
2 -1 -1 ✠ 0.
v Fkkr ]
l
7
3
1 (✥ 3 + 7) ✥ 3 (6 + l) + 1 ( 14 + l) = 0
Þ
Þ
l = 0.
mn kgj . k
✡☛
n ’ kkb ,
fd
o Qe ’ k% p kj k f c n q A,
gy
g e t ku r g f d p kj f c n
l e r y h;
g kr
g]
B, C v
kj
Dl
A, B, C v
✑✑✑✟
✑✑✑✟
✑✑✑✟
AD ✞ 4( - ✌☞ ✏
A, B, C v
mn kg j . k
gy
✘✙
g e i kI r
kj
Dl
✏ 5 ☞✍ ✏ ✎☞) ✞ ✥ ✌☞ ✏ 4 ☞✍ ✏ 3 ✎☞
-4 -6 -2
e r y h;
3 ✕ 0.
3
gA
✔ ✔
f l … d hf t ,
fd
✔
é✚✔ ✢ ✛ , ✛ ✢ ✜✔ , ✜✔ ✢ ✚✔ ù ✒ 2 é ✚✔ , ✛ , ✜✔ ù
ë
û
ë
û
g
★ ★
é ✣★ ✧ ✤ , ✤ ✧ ✦★ , ✦★ ✧ ✣★ ù
ë
û
✟ ✟
✟ ✟
4( ✥ ✌☞ ✏ ✍☞ ✏ ✎☞)
✑✑✑✟ ✑✑✑✟
g kA
éAB,AC,ADù ✕ -1 4
ë
û
-8 -1
kjS
g kr g ] ; f n r hu ksl f n’ k AB, AC v kS
✍☞ ✏ ✎☞ ) ✥ (4✌☞ ✏ 5 ✍☞ ✏ ✎☞) ✞ ✥ 8✌☞ - ☞✍ ✏ 3✎☞
✖✖✖✗ ✖✖✖✗ ✖✖✖✗
v r %]
e r y h;
✍☞ ✏ ✎☞) ✞ ✥ 4 ✌☞ - 6 ✍☞ - 2✎☞
AC ✞ (3 ✌☞ ✏ 9 ✍☞ ✏ 4 ✎☞ ) ✥ ( 4✌☞
i d kj ]
Dl
kj
gA
✓✓✓✔ ✓✓✓✔ ✓✓✓✔
AB ✞ ✥ ( ✍☞ ✏ ✎☞ ) ✥ (4 ✌☞ ✏ 5
r Fkk
e r y h;
é AB,AC, ADù ✒ 0
ë
û
v Fkkr ]
v c]
4✌☞ ✏ 5 ☞✍ ✏ ✎☞ , - ( ☞✍ ✏ ✎☞ ), 3✌☞ ✏ 9 ☞✍ ✏ 4 ✎☞ v
f LFkf r l f n ’ kk
o ky
bl
✟✟✟
é☎ , ✆, ✝ ù ✞ 0 ,
ë
û
g ] bl fy ,
✟ ✟
= (☎ ✏ ✆ ). ((✆ ✏ ✝ ) ´ (✝ ✏ ☎ ))
✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟
= (☎ ✏ ✆ ). (✆ ´ ✝ ✏ ✆ ´ ☎ ✏ ✝ ´ ✝ ✏ ✝ ´ ☎ )
✑✑✑✟
AD
i jd
i kB ;
637
l ke x h
☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎
= ( ✄ ✁ ). (✁ ´ ✂ ✄ ✁ ´ ✄ ✂ ´ )
✞ ✞ ✞
( D; kf d ✆ ´ ✆ ✝ 0 g A)
✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞
✝ ✟ .(✠ ´ ✆ ) ✡ ✟ .(✠ ´ ✟ ) ✡ ✟ .(✆ ´ ✟ ) ✡ ✠ .(✠ ´ ✆ ) ✡ ✠ .(✠ ´ ✟ ) ✡ ✠ .( ✆ ´ ✟ )
✒ ✒ ✒
✒ ✒ ✒
✒ ✒ ✒
✒ ✒ ✒
✒ ✒ ✒
✒ ✒ ✒
✏ éë✌ , ✍ , ✎ ùû ✑ éë✌ , ✍ , ✌ ùû ✑ ☛ ✌ , ✎ , ✌ ☞ ✑ éë✍ , ✍ , ✎ ùû ✑ éë✍ , ✍ , ✌ ùû ✑ éë✍ , ✎ , ✌ ùû
☎ ☎ ☎
= 2 éë ,✁ , ✂ ùû
mn kg j . k
gy
✓✔
g e i kI r
( D; ksa? )
✒
fl … fd ft ,
fd
✒
✒
✒ ✒
é✌✒ , ✍ , ✎✒ ✑ ✕ ù ✏ é✌✒ , ✍ , ✎✒ ù ✑ [ ✌✒, ✍ , ✕ ]
ë
û ë
û
g kr k g A
g
☎ ☎ ☎ ☎
é ,✁,✂ ✄ ✖ ù =
ë
û
=
=
☎ ☎ ☎ ☎
.(✁ ´ (✂ ✄ ✖ ))
☎ ☎ ☎ ☎ ☎
.(✁ ´ ✂ ✄ ✁ ´ ✖ )
☎ ☎ ☎ ☎ ☎ ☎
.(✁ ´ ✂ ) ✄ .(✁ ´ ✖ )
✜ ✜ ✜
✜ ✜ ✜
= éë ✗ , ✘ , ✙ ùû ✛ éë ✗ , ✘ , ✚ ùû
i ’ u ko y h 1 0 -5
✔✢
; fn
✭
✭
✤ ✫ ✦✣ ✥ 2 ✣✧ ✬ 3★✣ , ✩ ✫ 2 ✦✣ ✥ 3 ✣✧ ✬ ★✣ v
kS ✪ ✫
3 ✦✣ ✬ ✣✧ ✥ 2 ★✣
g] r k
✜
é ✗✜ ✘ ✜✙ ù K kr
ë
û
d hft , A
( m kj 24)
✮✢
n’ kkb, fd
✓✢
; fn l fn’ k
l f n’
✭
✭
k ✤ ✫ ✦✣ - 2 ✣✧ ✬ 3 ★✣ , ✩ ✫ - 2 ✦✣ ✬ 3 ✣✧ - 4 ★✣ v
✰✯ - ✯✱ ✄ ✲✯ , 3✰✯ ✄ ✯✱ ✄ 2 ✲✯ v
✰✯ ✄ l✯✱ - 3 ✲✯
kj
e ku y hf t ,
(a)
; fn
✭
✪ ✫ ✦✣ - 3 ✣✧ ✬ 5 ★✣
l e r y h;
g]
rk
ld
l e r y h; gA
a
S
k e ku
K kr
( m kj l = 15)
d hft , A
✳✢
kj
fd
✂1 = 1
✽
✽
✵ ✻ ✴✶ ✼ ✷✴ ✼ ✸✴ , ✹ ✻ ✴✶ v
v kj
✂2 = 2
kS
g ] r k ✂ K kr d
3
✽
✺ ✻ ✺1 ✶✴ ✼ ✺ 2 ✷✴ ✼ ✺3✸✴ g A r c ]
❁ ❁
❁
hf t , ] f t l l ✾ , ✿ v kj ❀ l e r y h;
g k t k, A
( m kj ✂ 3 = 2)
638
x f. kr
(b)
; fn
2
= ✥1
l e r y h;
✆✝
n ’ kkb ,
o ky
☞✝
✍✝
fd
v kj
3
=1
g ] r k n ’ kkb ,
fd
1
d k d kb
f LFkf r
l f n ’ kk
p kj k f c n l e r y h;
4✟✞ ☛ 8 ✞✠ ☛ 12 ✡✞, 2✟✞ ☛ 4 ✞✠ ☛ 6 ✡✞,3✟✞ ☛ 5 ✞✠ ☛ 4 ✡✞
ks
A (3, 2, 1), B (4, ✌, 5), C (4, 2, ✥ 2) v
d hf t , A
✓ ✓ ✓ ✓
✎ ✒✏, ✏ ✒ ✑ v
kj
v kj
5✟✞ ☛ 8 ✞✠ ☛ 5 ✡✞
gA
d k e ku K kr
g kxs A
☎ ☎
☎
✁ , ✂ v kj ✄ d
ug h c uk l d r k g A
; f n p kj f c n
; fn
Hkh e ku
kj
D (6, 5, ✥ 1) l
e r y h;
g] r k ✌
( m kj ✌ = 5)
✓ ✓
✑✒✎
l e r y h;
g ] r k n ’ kkb ,
fd
l fn’ k
☎ ☎
☎
✁ , ✂ v kj ✄
l e r y h;
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