Answers Chapter 1 SkillCheck 2 3 1 a 25y 2 a 5x þ 10 d 10 2y 3 81, 25, 100, 16, 64 4 a m 2 þ 10m þ 21 c n 2 5n þ 6 e 4 17p 15p 2 g x 2 þ 8m þ 16 i 4k 2 þ 4k þ 1 k t 2 49 2 b 64m b 4y 12 e 10a 15 b d f h j l c 9x c 3 þ 6w f k þ 2k 2 y 2 3y 4 6d 2 þ 11d þ 3 3a 2 þ 17af þ 10f 2 y 2 6y þ 9 a 2 25 9m 2 16 pffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 C 2 B 3 32; 125; 4:9; 52; 288 4 a R b I c R d R e R f R g R h R i I j R k I l I pffiffiffi pffiffiffiffiffi 5 a 1 47 ; p2 ; 2 b 2 79 ; 3 20; 2:6_ 6 a 1.8 b 0.7 c 0.4 d 3.5 e 2.5 f 2.6 g 1.6 h 1.9 – 3 15 –4 –3 –1 –2 4 5 4 11 74% –1 0 1 π 5 5 187% 2 9 2 3 7 1.41 pffiffiffiffiffi pffiffiffiffiffi pffiffiffi 8 Teacher to check, 5 2:24; 10 3:16; 17 4:12 Exercise 1-02 1 a e 2 a f k p 3 a e i m 4 B 6 a 2 0.09 pffiffiffi 5 2 pffiffiffi 3 5 pffiffiffi 12 2 pffiffiffi 11 2 pffiffiffi 6 5 pffiffiffi 3 pffiffi b 5 c 27 d f 28 g 45 h pffiffiffi pffiffiffi pffiffiffi b 2 3 c 2 7 d 5 6 e pffiffiffi pffiffiffi pffiffiffi g 4 3 h 10 2 i 4 6 j pffiffiffi pffiffiffi pffiffiffi l 6 3 m 5 3 n 7 3 o pffiffiffi pffiffiffi pffiffiffi q 9 2 r 7 5 s 5 5 t pffiffiffi pffiffiffi b p 16ffiffi 2 c 48 2 d pffiffiffi f 37 g 6 6 h pffiffiffi pffiffiffi 5 5 j 3 2 k 3 3 l 2 pffiffiffi pffiffiffiffi pffiffiffiffiffi 15 3 n 14 17 o 313 5 B false b false c true d true e true 250 50 pffiffiffi 10 7 pffiffiffi 3 7 pffiffiffi 4 2 pffiffiffi 16 2 pffiffiffiffiffi 10 pffiffiffiffiffi 18 17 pffiffiffiffiffi 40 10 f false Exercise 1-03 1 a e i 2 a d g j 608 pffiffiffi pffiffiffi pffiffiffi 11 3 b 3 2 c 4 6 pffiffiffiffiffi pffiffiffiffiffi 10 0 f g 8 15 pffiffiffiffiffi pffiffiffi pffiffiffi 2 3 j 10 5 k 6 10 pffiffiffiffiffi pffiffiffi pffiffiffi 7 39 b 7 10 7 2 pffiffiffi pffiffiffi pffiffiffiffiffi pffiffiffi 7 15 þ 8 2 e 2 5 3 7 pffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffiffiffi 13 11 3 h 11 7 6 13 pffiffiffi 3 5 b b f j n r v z A pffiffiffi 3 3 pffiffiffi 7 5 pffiffiffi 8 3 pffiffiffi 5 pffiffiffi 41 2 0 pffiffiffi 4 6 b f j n r v b f j n r pffiffiffiffiffi 66 144 pffiffiffi 30 2 pffiffiffi 24 6 pffiffiffi 160 5 pffiffiffiffiffi 60 10 pffiffiffi 6 21 pffiffiffi 5 p2ffiffi 42 pffiffiffi 21 2 c g k o s w pffiffiffi 2 5 pffiffiffiffiffi 10 pffiffiffi 3 2 pffiffiffi 6 3 pffiffiffi 5 6 pffiffiffi pffiffiffi 6 2þ2 3 pffiffiffi 7 pffiffiffiffiffi 8 11 pffiffiffi 9 2 pffiffiffi 30 3 pffiffiffi 29 2 pffiffiffi pffiffiffi 12 3 þ 3 6 d h l p t x Exercise 1-04 Exercise 1-01 – 12 3 a D pffiffiffi 4 a 6 2 pffiffiffi e 5 6 pffiffiffi i 6 2 pffiffiffi m 11 3 pffiffiffi q 5 7 pffiffiffi u 15 3 pffiffiffi pffiffiffi y 3 2 6 5 pffiffiffi d 4 5 pffiffiffi h 6 pffiffiffi l 5 3 pffiffiffi pffiffiffi c 5 29 3 pffiffiffi pffiffiffi f 2 68 3 pffiffiffi i 6 7 1 a e i m q u 2 a e i m q u 3 a 4 C 6 a pffiffiffiffiffi 30 5 140 112 396 pffiffiffi 36 5 pffiffiffi 3pffiffi 27 pffiffiffi 5 3 pffiffiffiffiffi 2 14 4 pffiffiffi 2 10 pffiffiffiffiffi 15 30 36 80 pffiffiffi 216 2 pffiffiffi 252 3 pffiffiffi 8 7 1 pffiffiffi 4 3 1 12 c g k o s w c g k o s d h l p t x d h l p t 2 3 6 b 7 2 pffiffiffi b 4 6 c 6 5 A pffiffiffiffiffi c 30 d 15y 2 45 d 6 pffiffiffiffiffi 10 21 pffiffiffi 60 2 pffiffiffi 90 6 pffiffiffi 96 6 144 pffiffiffi 2 2 8 pffiffiffi 2 6 10 2 e x pffiffiffi f a a pffiffiffi e 14 3 f 2 Mental skills 1 2 a f k p 11 6 40 135 b 40 g 43 l 65 c 7 h 80 m 11 d 24 i 18 n 14 e 23 j 15 o 12 Exercise 1-05 1 a d g 2 C 3 a c e g 4 C 5 a d g 6 a e 7 C 8 a d pffiffiffiffiffi pffiffiffiffiffi 15 þ 10 pffiffiffiffiffi 3 10 5 pffiffiffi 42 8 7 pffiffiffi pffiffiffi b 2 3 6 pffiffiffi e 6þ6 6 pffiffiffi h 5 5 þ 75 pffiffiffi pffiffiffiffiffi pffiffiffi 10 þ 10 6 5 3 2 pffiffiffi pffiffiffi pffiffiffi 28 6 þ 21 þ 8 2 þ 2 3 pffiffiffiffiffi 109 þ 10 77 pffiffiffiffiffi 16 10 þ 54 b d f h pffiffiffi pffiffiffiffiffi c 6 þ 14 pffiffiffiffiffi pffiffiffiffiffi f 55 4 11 pffiffiffi i 24 þ 3 6 pffiffiffi pffiffiffiffiffi pffiffiffi 7 þ 2 7 21 2 3 pffiffiffiffiffi 20 þ 10 pffiffiffi 72 23 6 pffiffiffiffiffi 16 35 pffiffiffiffiffi pffiffiffiffiffi 8 2 15 b 9 þ 2 14 pffiffiffi pffiffiffiffiffi 19 þ 6 10 e 77 þ 30 6 pffiffiffiffiffi pffiffiffiffiffi 38 þ 12 10 h 23 þ 4 15 1 b 22 c 8 1 f 166 g 13 pffiffiffi 88 30 7 pffiffiffi 73 þ 40 3 pffiffiffi b 21 2 10 e 29 pffiffiffi c 9 4 5 pffiffiffi f 179 20 7 d 2 h 43 pffiffiffiffiffi c 5 35 þ 29 pffiffiffi f 92 12 5 9780170194662 Answers Exercise 1-06 1 B 2 a g 3 A 5 a 6 a pffiffi 2 2 pffiffi 3 6 b h pffiffi 2 2 2 pffiffi pffiffi 2 7þ7 2 14 pffiffi 7 7 pffiffi 7 28 c pffiffi 3 3 pffiffi 7 5 15 i 4 D pffiffi 55 5 pffiffiffiffi pffiffi 10þ5 3 5 b b d j c c pffiffi 3 2 2 pffiffiffiffi 10 15 e k pffiffi pffiffi 5 2þ 6 4 pffiffi pffiffi 3 2 2 pffiffi 2 7 7 pffiffi 3 2 d f l pffiffi 2 6 pffiffiffiffi 15 4 pffiffi pffiffi 2 33 2 18 Power plus pffiffi 1 a Yes, because you are multiplying by 1. b 37 2 pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 2 a 52 3 b 2þ 3 c d 51 7þ 3 pffiffiffiffi pffiffi pffiffi pffiffi pffiffi pffiffiffi 7þ 3 104 3 3 a 2þ 3 b 104 2 c d 4 13 pffiffi pffiffi pffiffiffi 4 s ¼ 33D 5 22 6 ð2 þ 2 3Þ mm Chapter 1 revision 1 A 2 a I b R e R f R pffiffiffi pffiffiffi 3 a 6 2 b 7 2 pffiffiffi pffiffiffi f 14pffiffi 7 g 48 2 pffiffiffi 50 5 k 3 l 2 2 pffiffiffi 4 a 13 2 b pffiffiffi pffiffiffi d 32 5 9 7 e pffiffiffiffiffi pffiffiffiffiffi 21 5 a b 2 10 pffiffiffi pffiffiffiffiffi 6 e f 7pffiffi 14 i 53 j 322 pffiffiffi 6 a 9 2 12 b pffiffiffi d 23 8 7 e g 103 h pffiffiffiffi pffiffi 10 3 2 7 a 10 b 2 c c R d g R h pffiffiffiffiffi c 5 11 d pffiffiffi h 15 5 i pffiffiffiffiffi m 4 11 n pffiffiffi pffiffiffi 5 27 5 pffiffiffi pffiffiffi 38 2 24 3 pffiffiffi c 4 3 d pffiffiffi g 4 6 h k 23 l pffiffiffiffiffi pffiffiffi 10 10 5 pffiffiffi 77 þ 10 6 pffiffiffi 70 þ 38 5 pffiffi pffiffi 7 d 43 35 I I pffiffiffi pffiffiffi 8 2 e 15 6 pffiffiffi pffiffiffi 6 28 3 j pffiffiffi 6 o 6 2 pffiffiffi pffiffiffi c 14 2 þ 17 3 pffiffiffiffiffi f 8 11 pffiffiffiffiffi 55 125 7 5 e pffiffiffiffiffi c 7 35 27 f 43 pffiffi 5 6 6 f pffiffi 2 2þ1 3 Chapter 2 SkillCheck 1 a e 2 a 3 a 4 a 5 a d 6 a 7 a 0.04 0.095 $72 $7350 36 52 4 1152 $5962.59 b f b b b b e b b 0.22 0.0675 $116.25 $4034.10 24 26 12 50 $33 433.46 c g c c c c f c c 0.183 d 0.047 0.1525 h 0.2 $4494 $8737.60 60 365 8 years 4 months 0.06 $18 481.63 d $64 937.10 Exercise 2-01 1 a $874 b $938.80 2 Greta earns more per week by $27.48. 9780170194662 c $367.20 3 4 5 10 14 15 a $3461.86 b $6923.73 c $15 053.33 Job 1: $1104.64; Job 2: $1160; Job 2 by $55.36 $1096.10 6 $735.23 7 $761.24 8 A 9 $13 312.50 $1394.40 11 $2115 12 54 13 $63.95 a $427 b $700 c $956.87 d $625.55 a $972.12 b $680.48 c $4568.96 Exercise 2-02 1 2 3 4 7 9 10 11 12 13 a $45 697 b $6398.53 a $114 719 b $30 393.03 a $90 904 b $21 581.48 C 5 $19 924.99 6 $45 456.10 $696.42 8 $623.52 a $452 b $1711.10 c 25.0% a $458 b $1747.65 c 24.0% a $2296 b $456 c $1646.73 a $2297.59 b $456 c $1550.39 Gross weekly income ¼ $816.90; Total deductions ¼ $369.10; Net income ¼ $447.80 Exercise 2-03 1 a $5040 d $96.95 2 a $87.50 d $820 3 A 4 a $11 200 5 a $1440 7 a $6750 9 2 years 12 C 15 a $18.90 b e b e $2953.50 $71.92 $5925.15 $279 c f c f $102.50 $451.20 $391 000 $723.04 b $1569 c $9392.50 d $11 331.25 b $7440 6 4.5% b 18.75% 8 9.75% p.a. 10 26 weeks 11 137 days 13 2.5 years 14 2.6% p.a. b $1063.90 Exercise 2-04 1 a Check with your teacher. Investment after 1st yr ¼ $24 150; Investment after 2nd yr ¼ $25 357.50 b Compound interest ¼ $2357.50 2 a $16 153.36 b $1153.36 3 a $38 459.48 b $4359.48 4 a $5408, $408 b $30 245.29, $2445.29 c $11 113.20, $1513.20 d $41 905.55, $2405.55 e $19 337.39, $937.39 5 a $4791.80 b $1642.38 c $308.93 d $3913.84 e $6834.42 Mental skills 2 2 a e i 4 a e 18 $7.50 $240 10 37.5 b f j b f $126 10.8 $3.30 166 $5.80 c g k c g 39 $27 900 $50 135 d h l d h $30.30 60 $52.50 $22 $22.60 609 Answers 6 a e i 8 a e 500 81 $195 160 $67.50 b f j b f $20 $35 $425 $1.50 $31.25 c g k c g 4.5 16.5 $31.50 7.5 38 d h l d h $6.25 74.5 290 $32.50 170 Exercise 2-05 1 $14 332.50 2 a i $9754.75 ii $3254.75 b i $13 858.59 ii $3858.59 c i $12 634.81 ii $394.81 d i $43 949.46 ii $9349.46 e i $8427.39 ii $427.39 3 D 4 $1 301 018.83 5 B 6 a i $13 488.50 ii $3488.50 b i $52 751.13 ii $17 251.13 c i $9448.23 ii $548.23 d i 53 366.91 ii $11 366.91 e i $19 473.44 ii $2973.44 f i $5177.03 ii $277.03 7 C 8 a $600 b $615 c Tegan by $15. 9 a $10 510.31 b $1969.48 less 10 a i $7554.45 ii $7688.85 iii $7758.33 iv $7805.54 b Monthly, because it earns the most interest. Exercise 2-06 1 a d 2 a d 3 a 4 a 5 a d 6 a 7 a 8 a 9 a $175.50 $1907.64 $1275 $34 641.75 $1379 $3420 $2080 $4880 $32.90 $1073.40 $2599 $262.50 b e b e b b b e b b b b $1579.50 $105.98 $24 225 $577.36 $2316.72 $720 $8320 58.7% $437.42 $273.40 $3576 7.4% c f c f c c c $328.14 $2083.14 $10 416.75 $35 916.75 $217.58 $1500 d 48% $13 200 c $ 108.42 c 34.2% c $677 8 Yes, it will lose approximately 52% after 7 years. 9 a $1800 b 5 years c $798.67 d Yes, in the 30th year. e No Power plus 1 4 years and 61 days 2 $4444.44 3 $12 838.71 4 $63 367.49 5 $2276.87 6 790 000 7 a 18 years. b 18 years. c No. The size of the interest rate and the number of compounding periods determine how quickly the principal takes to double in value. Chapter 2 revision 1 3 4 5 6 7 9 12 $13 045.75 a $797.45 a $1052.51 a $67 725 a $2400 a $5955.08 $15 374.72 a $487.50 d $6296.06 13 a $8851.45 c $78.75 d $621.37 8 $36 282.78 11 $45 815.75 c $1908.56 f $6783.56 c 59% SkillCheck 1 a (6, 1) b (5, 4) e AC ¼ BC ¼ 4.5 f isosceles 2 a x 0 1 2 3 y 3 2 1 0 c 6 g 13 d 6 h 23 y 4 2 d 10.4% –4 Exercise 2-07 610 c $4946.80 Chapter 3 d 36.62% 1 $933.89 2 a $20 429.69 b $29 560.31 3 a i $659.66 ii 60% b i $2459.54 ii 45.2% c i $5073.42 ii 60% d i $778.24 ii 41% e i $14 020.37 ii 51% f i $851.35 ii 37% g i $403.03 ii 46.3% h i $1097.20 ii 68.6% 4 a i 90% ii 73% iii 53% iv 48% b By trial and error, in approx 6.6 years. 5 a i $10 000 ii $8000 iii $4096 b 32.8% 6 a $11 138.51 b $4661.49 7 a $6472.88 b $3441 c 8 years and 9 months. d 23.2% 2 $1349.18 b $1011.40 b $736.76 b $13 557.63 b $392.50 b $955.08 10 $852.91 b $4387.50 e $174.89 b $6138.55 –2 0 –2 2 x 4 –4 b x y 2 4 0 2 1 1 1 5 y 4 2 –4 –2 0 –2 2 4 x –4 9780170194662 Answers c 0 1 1 3 x y 1 1 12 a e 13 a e 2 3 y 4 2 –4 2 x 4 –4 3 a 2 b 8 c 5 e 14 d 4 pffiffiffiffiffi f 2 17 Exercise 3-01 1 B 4 a 13 5 a i 2.2 b i 10.8 c i 7.1 d i 7.6 e i 10.2 f i 5.7 pffiffiffiffiffi 6 a 89 7 k: m ¼ 15; l: m ¼ 12 y 9 a 4 2 C b 2 ii (6, 2.5) ii (3.5, 3) ii (2.5, 0.5) ii (0.5, 7.5) ii (6, 3) ii (5, 0) pffiffiffiffiffiffiffiffi b 194 8 B –2 A(–1, –1) –2 3 A c 73 iii 12 iii 23 iii 1 iii 37 iii 5 iii 1 pffiffiffiffiffi 82 c 27 37 0.40 1 1 a d 2 a 3 a 4 D 7 a b 8 a c g c g neither neither 4 1 b perpendicular e parallel b 2 b 16 5 B mAB ¼ 43, mCD ¼ 43; [ AB || CD mPQ ¼ 34, mCD ¼ 43; [ PQ ’ CD 1 b 3 3 45 174 0.90 0.05 d h d h 68 146 14.30 0 1 a i 1 3 d 0.2 d 25 ii 1 y 4 2 –4 –2 0 –2 4x 2 –4 ii 5 y 10 C(3, 1) 2 c parallel f neither c 13 c 23 6 A Exercise 3-03 b i 2.5 B(1, 3) 2 –4 b f b f Exercise 3-02 1 2 –2 0 –2 72 117 1.73 0.14 4x 5 –4 pffiffiffi pffiffiffi b AB ¼ AC ¼ 2 5, BC ¼ 2 2 d isosceles y 10 a 10 pffiffiffi c AB ¼ AC ¼ 2 5 e 11.8 –6 –4 –2 0 b d e f g 11 a b c L(7, 2) P(–3, 0) square c mKL ¼ 23, mPM ¼ 23 3 3 x m–10 mLM ¼ 2 KP ¼ 2,–5 5 10 the gradients are equal, they are parallel pffiffiffiffiffi –5 M(3, KL ¼ LM ¼ PM ¼ KP ¼ –4) 2 13 28.8 h 52 sq. units –10 P(2, 1), Q(1, 3) PQ ¼ 3.6, AC ¼ 7.2, AC ¼ 2 3 PQ mPQ ¼ 23, mAC ¼ 23; the gradients are equal. 4 6 8x –5 K(1, 6) 5 2 c i 4 ii 4 y 4 2 –4 –2 0 –2 2 4 x –4 d i 1 ii 2 4 y 2 –4 –2 0 –2 2 4x –4 9780170194662 611 Answers y y 4 2 0 –2 0 –4 –2–2 –4 –6 –8 –10 e i 0 ii 0 4 d 2 –4 –2 4x 2 –4 f i ii 3 6 4 e 4 y –4 0 2 4 –2 x 6 8 10 x y 0 –2 2 4 x –4 –4 y 4 f 2 a 4 2 2 –10 –8 –6 –4 –2 –2 2 2 y 4 –4 –2 0 –2 2 –4 –2 0 –2 2 4 x 4 g –2 x 10 8 y y 2 0 –2 6 4 2 –4 4 –4 –4 b 2 2 4 –4 x –2 0 –2 2 4 x –4 –4 h c y 10 8 6 4 2 0 –10 –8 –6 –4 –2–2 –4 612 2 4x y 10 8 6 4 2 –4 –2–2 2 4 6 8 10 x 9780170194662 Answers y 10 i b y x=6 10 5 –4 –2 0 2 4 6 5 8 10 x y=1 –5 –10 –10 0 –5 y j –5 10 –10 5 0 –5 5 10 15 x –5 –10 k 6 4 2 –4 –2 0 2 –2 4 x 1 a d g j 2 a d 3 a m ¼ 3, b ¼ 2 m ¼ 1, b ¼ 9 m ¼ 12, b ¼ 11 m ¼ 2, b ¼ 6 y ¼ 2x þ 1 y ¼ 25 x þ 3 m ¼ 2, b ¼ 1 y 4 –6 2 –4 –10 l d y ¼ 2 h x ¼ 1 b y-axis m ¼ 2, b ¼ 7 m ¼ 34, b ¼ 6 m ¼ 23, b ¼ 6 m ¼ 3, b ¼ 11 y ¼ 34 x þ 2 y ¼ 2x 3 c f i l c f m ¼ 1, b ¼ 4 m ¼ 1, b ¼ 0 m ¼ 13, b ¼ 8 m ¼ 1, b ¼ 72 y ¼ 7x þ 5 y ¼ 3x þ 12 y = 2x + 1 1 0 –2 2 4 x b m ¼ 3, b ¼ 2 0 2 –2 4 y 4 x y = 3x – 2 2 –4 a no b yes c yes d yes e no f no C a x ¼ 4 b x¼1 c y¼5 d y ¼ 3 a y x = 2.5 10 –4 –2 0 –2 y –5 10 x y = –3 x y = 2x 4 5 4 c m ¼ 2, b ¼ 0 2 y=1 0 2 –4 5 –5 c x ¼ 1 g y¼6 10 a x-axis –4 2 –2 –2 y 4 –10 b e h k b e –4 –8 –4 x = –0.5 b x¼4 f x ¼ 1 9 C 7 a y¼2 e y¼3 8 A Exercise 3-04 y 10 8 3 4 5 6 10 x y = –2 5 –4 –2 0 –2 2 4 x –4 –10 9780170194662 613 Answers d m ¼ 12, b ¼ 1 Mental skills 3 y 4 y=x –1 2 –4 –2 2 x 4 2 –2 8 h 30 mins 8 h 15 mins 5 h 10 mins 7 h 40 mins b 5 h 40 mins e 11 h 25 mins h 5 h 45 mins c 3 h 25 mins f 1 h 40 mins i 7 h 55 mins Exercise 3-05 –4 e m ¼ 2, b ¼ 3 y 4 2 –4 2 a d g j –2 –2 2 x 4 1 a c e g i 2 a c e 3 B xyþ2¼0 5x y þ 8 ¼ 0 x 2y 6 ¼ 0 6x y 3 ¼ 0 3x 5y þ 10 ¼ 0 m ¼ 2, b ¼ 6 m ¼ 32, b ¼ 2 m ¼ 2, b ¼ 5 b d f h 3x y 1 ¼ 0 x þ 2y 3 ¼ 0 8x y þ 2 ¼ 0 x 2y 6 ¼ 0 b d f 4 m ¼ 4, b ¼ 5 m ¼ 2, b ¼ 1 m ¼ 43, b ¼ 4 B –4 y = –2x + 3 f m¼ 34, Exercise 3-06 b¼0 y 4 2 –4 –2 2 –2 x 4 1 a 2x y þ 1 ¼ 0 c 4x y 20 ¼ 0 e x þ 5y þ 38 ¼ 0 g 4x þ y þ 1 ¼ 0 i 2x þ y þ 10 ¼ 0 2 a and b b d f h y –4 xþyþ2¼0 2x 3y 4 ¼ 0 3x þ y 4 ¼ 0 3x 4y þ 10 ¼ 0 4x + y – 10 = 0 y = –3x x–y–5=0 a 4 g m ¼ 52, b ¼ 1 x + 3y + 3 = 0 y 4 0 2 –4 –2 –2 –4 x P 2 x – 5y – 13 = 0 x 4 d (3, –2) b y = –5x +1 2 c h m ¼ 35, b ¼ 4 y 4 2 y = 3x – 4 5 –2–2 2 4 6 8 x –4 –6 4 y ¼ 2x 5 a C b B, D c B 6 a y ¼ 4x þ 3, y ¼ 4x 6 614 d C, D e A, B f D b 3x y þ 7 ¼ 0, y ¼ 3x 2 3 a xy4¼0 b 4x 5y þ 18 ¼ 0 c 5x 6y þ 23 ¼ 0 d 8x þ 3y 10 ¼ 0 e 3x þ 2y 6 ¼ 0 f 5x 3y 1 ¼ 0 g 6x þ 11y þ 38 ¼ 0 h xþy3¼0 i 4x 3y 11 ¼ 0 4 k: x þ 2y 7 ¼ 0, l: 3x y þ 7 ¼ 0 5 4x þ y 20 ¼ 0 6 5x 7y þ 42 ¼ 0 7 2x 3y þ 18 ¼ 0 8 3x þ 5y 30 ¼ 0 9 a 2x y þ 1 ¼ 0 b same 9780170194662 Answers y2 y1 x2 x1 y y1 ¼ mðx x1 Þ y2 y1 ðx x1 Þ y y1 ¼ x2 x1 y y1 y2 y1 ¼ ) x x1 x2 x1 b xy4¼0 10 a m¼ 6 c same 7 Exercise 3-07 1 a d 2 a d g y ¼ 2x þ 5 y ¼ x þ 3 y¼ xþ2 y ¼ 12 x þ 4 y ¼ 3x 10 b e b e h y ¼ 34 x þ 3 y ¼ 12 x þ 3 y ¼ 34 x y ¼ 3x 3 y ¼ 25 x þ 2 c f c f i y ¼ 3x þ 6 y ¼ 3x 3 y ¼ 13 x þ 6 y ¼ x 2 y ¼ 2x 3 9 10 Exercise 3-08 1 a d 2 a d 3 a 4 a 5 a d y ¼ 2x þ 4 b y ¼ 3x þ 6 y ¼ 2x 12 e y ¼ 5x 13 y ¼ 2x 2 b y ¼ 15 x 15 y ¼ 3x 3 e y¼xþ6 m¼2 b M(0, 2) c 12 1 y ¼ 3x þ 1 b 3 y ¼ 45 x þ 8 b A(10, 0) y ¼ 54 x 25 e (0, 12.5) 2 8 y ¼ 12 x þ 11 2 y ¼ 12 x 10 y ¼ 13 x þ 43 y ¼ 13 x 31 3 d y ¼ 12 x þ 2 c y ¼ 3x þ 11 c 54 c f c f 11 12 Exercise 3-09 1 a i 5x þ 2y 18 ¼ 0 ii 3x 4y 16 ¼ 0 iii (0, 9) iv (0, 4) v 26 units2 b i x 5y þ 20 ¼ 0 ii x þ 2y þ 6 ¼ 0 iii (0, 4) iv (0, 3) v 35 units2 c i 3x y 46 ¼ 0 ii 7x þ 15y þ 66 ¼ 0 2 iii ð15 13, 0Þ iv ð9 37, 0Þ v 123 17 21 units 2 a 5x 2y 25 ¼ 0 b 5x þ 7y 25 ¼ 0 c w¼5 d t ¼ 10 25 3 a DE ¼ EF ¼ FG ¼ DG ¼ 5 units b For DE and GF, m ¼ 0 For DG and EF, m ¼ 43 c Diagonal DF, m ¼ 12 Diagonal EG, m ¼ 2 Since 12 3 2 ¼ 1 it is true that DF ’ EG. d Midpoint of DF ¼ (0, 0) Midpoint of EG ¼ (0, 0) The diagonals bisect each other because their midpoints are the same. e Opposite sides are equal and parallel, adjacent sides are equal, diagonals bisect each other at right angles. pffiffiffi 4 b 6 5 units c (1, 1) d No, since mPR 3 mQS 6¼ 1 e Rectangle, diagonals are equal and bisect each other but not at right angles. pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 5 b CE ¼ 130 units, DF ¼ 130 units 1 1 1 1 c 2, 2 and 2, 2 9780170194662 13 14 15 16 3 d mCE ¼ 11 3 , mPR ¼ 11 3 [ CE ’ DF because 11 3 3 11 ¼ 1 e Square, diagonals are equal and bisect each other at right angles. pffiffiffiffiffi pffiffiffiffiffi a BC ¼ DE ¼ 61 units, CD ¼ BE ¼ 65 units b mBC ¼ 56, mCD ¼ 47, mDE ¼ 56, mBE ¼ 47 c Midpoint of BD ¼ 1 12, 2 12 ; Midpoint of CE ¼ 1 12, 2 12 d Parallelogram, opposite sides are parallel and equal. pffiffiffiffiffiffiffiffi a AC ¼ BD ¼ 104 units b Midpoint of AC ¼ (1, 2), midpoint of BD ¼ (1, 2) mAC ¼ 5, mBD ¼ 15, [ AC ’ BD c The diagonals are equal and bisect each other at right angles. Midpoint of KM ¼ Midpoint of LN ¼ 2 12, 12 mKM 3 mLN ¼ 1 3( 1) ¼ 1 Teacher to check. a mJK ¼ 13, mLM ¼ 13, mKL ¼ 52, mJM ¼ 52 b Parallelogram because opposite sides are parallel. a X 3, 12 ; Y 1 12, 3 12 b mXY ¼ 23, mCB ¼ 23 [ XY || CB pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi c XY ¼ 12 117, CB ¼ 117 [ CB || 2XY a mWN ¼ 1 and mCT ¼ 1 mWN 3 mCT ¼ 1 [ WN ’ CT MWN (1.5, 0.5) and MCT (1.5, 0.5) [ Diagonals bisect at right angles. b rhombus Trapezium pffiffiffiffiffi ST ¼ WX ¼ 37 units pffiffiffiffiffi TW ¼ SX ¼ 2 37 units XS ’ ST because mXS ¼ 16, mST ¼ 6 [ STWX is a rectangle because opposite sides are equal and angles are right angles. a Midpoint of TU ¼ A(4, 1) Midpoint of UV ¼ B(0, 3) Midpoint of SV ¼ C(5, 1) Midpoint of ST ¼ D(1, 5) b Gradient of AB ¼ 1 ¼ gradient of CD Gradient of AD ¼ 45 ¼ gradient of BC pffiffiffiffiffi AC ¼ 9 units, BD ¼ 65 units [ ABCD is a parallelogram. a mLM ¼ 13, mLN ¼ 13, mMN ¼ 13 b L, M and N are collinear points. Power plus 1 2 5 6 a 23 b y ¼ 23 x 2 c y¼4 k¼5 3 B(2, 1) 4 X(2, 3) a 32 b 3x þ 2y þ 2 ¼ 0 or y ¼ 3x 2 1 D(3, 5) or (7, 9) or (1, 3) 615 Answers Chapter 3 revision Chapter 4 1 a 12.6 b M(1, 4) pffiffiffiffiffi 2 a HJ ¼ JK ¼ KL ¼ HL ¼ 58 3 7 b mHJ ¼ 7, mJK ¼ 3, mKL ¼ 37, mHL ¼ 73 pffiffiffi pffiffiffi c HK ¼ 10 2, JL ¼ 4 2 d rhombus 3 a 72 b 51 c 135 4 a 12 b 2 5 a y c SkillCheck 1 3 1 a 182 cm2 2 a 1.51 m2 d 146 2 −4 −2 0 2 4 x −2 b 1 a 200.81 m2 b 3573.56 mm2 c 206.47 cm2 d 15.83 m2 2 a 35 m2 b 3478 cm2 2 3 a 1009 m b 2160 m2 c 4 m2 d 1895 m2 2 2 2 e 7m f 14 m g 1131 m h 95 m2 2 4 a 41.6 m b 5L 5 The triangular prism tent by 1.2 m2. y 8 x + 2y = 16 4 282 m2 b 298 cm2 c 2720 mm2 2 2 204 m e 1288 mm f 165 m2 cube, 1944 m2 b rectangular prism, 538 m2 triangular prism, 9720 m2 d open trapezoidal prism, 6378 m2 80 m3, $8400 b 171.4 m2 2 1036 cm b 1020 mm2 c 204 m2 390 cm2 e 672 cm2 f 5672 mm2 32.91 m2 b 74.56 m2 Exercise 4-02 y = –5x –1 −4 c 680 cm2 c 8.73 m2 Exercise 4-01 1 a d 2 a c 3 a 4 a d 5 a 4 b 770 cm2 b 5.59 m2 Exercise 4-03 −8 −4 0 4 8 12 16 x −4 −8 c y 4 2 −4 −2 0 3x + 4y – 12 = 0 2 4 x −2 −4 616 6 C 7 D 8 a m ¼ 2, b ¼ 10 b m ¼ 4, b ¼ 3 c m ¼ 38, b ¼ 12 9 a 3x y þ 5 ¼ 0 b 2x 5y 50 ¼ 0 c x 3y 6 ¼ 0 10 a m ¼ 1, b ¼ 2 b m ¼ 14, b ¼ 1 c m ¼ 3, b ¼ 9 11 a 3x þ y 20 ¼ 0 b 2x 3y þ 26 ¼ 0 12 a 3x 5y 20 ¼ 0 b xþyþ3¼0 13 a 2x y þ 3 ¼ 0 b x þ 2y þ 8 ¼ 0 14 a 3x y 6 ¼ 0 b 2x þ y ¼ 0 15 8x þ 3y 95 ¼ 0 pffiffiffiffiffi 16 PN ¼ LM ¼ PN ¼ PL ¼ 34 units 3 5 MPN ¼ 5, MPL ¼ 3 [ PN ’ PL [ LMNP is a square because all sides are equal and it has a right angle. 1 2 3 4 5 6 7 a 275 m2 a 166.4 m2 a 843 cm2 a 432 cm2 85 854 m2 a 1344 mm2 a 42 m b b b b 564 mm2 3456 mm2 1592 cm2 2150 cm2 c c c c b 180 cm2 b A ¼ 735 m2 c 35 m 87.4 cm2 743.1 cm2 3116 cm2 173 cm2 c 343.4 m2 d 28 m Exercise 4-04 1 2 3 4 5 6 7 9 10 11 a 101 cm2 b 628 cm2 2 a 392.7 mm b 62.8 m2 2 a 90p m b 224p mm2 2 a 2827.43 mm b 380.13 m2 a 432p m2 b 192p m2 2 a 314 m b 628 m2 c 628 m2 8 2 5.1 3 10 km 8 5525 cm2 a 30.16 cm b 4.80 cm c 8 cm a 21.9 mm b 25.2 mm a 6.9 cm b 85.3 cm c c c c c d 2419 cm2 192.4 cm2 450p cm2 366.44 cm2 768p m2 402 m2 d 193.02 cm2 c 30.9 mm c 85 cm Mental skills 4 Exact answers shown 2 a 331 b 157 f 203 g 413 k 276 l 72 37 4 a 28.231 b 14.187 f 5.0237 g 21.75 c 1587 h 734 d 255 i 6723 e 421 j 15 744 c 177.4967 d 416.752 e 2.4156 h 3.69 i 5.81 j 23.9121 9780170194662 Answers Exercise 4-05 1 a 446.96 cm2 2 a 352 cm2 3 a 9721.7 cm2 d 2858.8 cm2 4 a 26.14 m2 6 1028.32 cm2 7 a 857.7 cm2 d 5969.0 cm2 g 282.7 cm2 j 3769.9 cm2 m 6615.4 cm2 b b b e b 49 270 cm2 76 cm2 14 031.4 cm2 2793.5 cm2 19 m2 c 864 cm2 b e h k n 412.3 cm2 250.6 cm2 652.9 cm2 1148.8 cm2 3908.4 cm2 c f i l o 1042.0 cm2 628.3 cm2 501.6 cm2 3017.7 cm2 328 cm2 c f i c 5 c f i l 7.4 m3 216.9 m3 146.3 m3 21.5% 63 L 1989.38 cm3 3084.96 cm3 167.33 cm3 794.12 cm3 c 14 778.1 cm2 f 394.7 cm2 5 2953 cm2 Exercise 4-09 Exercise 4-06 1 a 9.7 m3 d 135.7 m3 g 42.3 m3 2 a 251.3 cm3 3 500 kL 6 a 4825.49 cm3 d 6375.00 cm3 g 536.19 cm3 j 12 900 cm3 7 a 182.83 m3 b e h b 4 b e h k b 94 247.8 m3 5026.5 m3 107.5 m3 320 cm3 13 666 cm3 5026.55 cm3 5301.44 cm3 1884.96 cm3 167.55 cm3 $21.94 per day 1 a 192 cm3 b 200 cm3 d 336 m3 e 1200 cm3 2 a i 12 cm ii 1296 cm3 b i 40 m c i 24 mm ii 1568 mm3 d i 60 mm e i 7.7 m ii 133.1 m3 f i 84 cm 3 a 151 m3 b 314 cm3 d 616 cm3 e 393 cm3 4 a i 6.3 cm ii 59.6 cm3 b i 3.9 m ii 19.9 m3 c i 9.2 cm ii 153.6 cm3 d i 3.5 m ii 2.3 m3 e i 244.6 m ii 296 103.1 m3 f i 71.9 cm ii 129 674.2 cm3 5 a i 14 137 cm3 ii 14 137 mL b i 697 m3 ii 697 kL c i 660 cm3 ii 660 mL d i 3619 m3 ii 3619 kL e i 1072 cm3 ii 1072 mL f i 8579 mm3 ii 9 mL 6 1.1 3 1012 km3 7 a 33.75 m3 8 22.5 m 9 14.1 cm 11 2.8 cm 12 26.9 mm c f ii ii ii c f 80 cm3 80 m3 14 400 m3 28 160 mm3 564 480 cm3 1780 mm3 2545 mm3 b 19 t 10 5.7 m Exercise 4-08 9780170194662 1 2 3 4 8 9 10 11 a 9:1 b 9 : 25 c 81 : 25 a 3:5 b 1 : 10 c 8:5 a 3.5 cm b 18 c 48 54 cm2 5 44.1 cm2 6 7.5 cm The area is quadrupled (3 4). The sides are decreased by a factor of 3. The area has increased by a factor of 6.25. 1 The sides have decreased by a factor of 10 . d d d 7 4:9 4:9 36.75 154 cm Exercise 4-10 b i 4:9 ii 8 : 27 1 a i 9 : 25 ii 27 : 125 c i 16 : 25 ii 64 : 125 d i 4 : 25 ii 8 : 125 2 a 9 : 10 b 729 : 1000 3 a 5:7 b 25 : 49 4 76 800 mm3 5 75.6 mL 6 2531.25 g or 2.531 kg 7 78 L 8 a 2.25 b 3.375 9 There has been a 27 64 decrease in the volume. Power plus Exercise 4-07 1 a 59 m3 2 a 343 cm3 d 6100 cm3 3 a i 31 416 cm3 ii 31.416 L b i 616 cm3 ii 0.616 L c i 264 cm3 ii 0.264 L 4 28.27 kL 5 a 12 balls b 60 balls c 31 416 cm3 d 48% 3 3 6 a 1963 cm b 0.55 cm /s 7 a 250 m3 b 210 kL c $415.80 b 59 kL b 240 cm3 e 6048 cm3 c 1152 cm3 f 2500 cm3 Teacher to check. Chapter 4 revision 1 a d 2 a d 3 a 4 a d 5 a d 6 a 7 a d 8 a d 9 a 1.08 m2 277.6 m2 7389.0 m2 14 294.2 cm2 960 cm2 704 m2 452 m2 3180 cm2 3318 cm2 36 816 m3 322.67 m3 1340.41 cm3 360 498 mm3 60.75 m3 234.375 cm2 b 3150 mm2 e 216 cm2 b 1437.3 m2 e 5871.2 cm2 b 7776 cm2 b 4524 m2 e 681 m2 b 1268 cm2 e 1728 cm2 b 20 160 m3 b 540 cm3 e 10 262.54 mm3 b 145 125 mm3 e 3054 cm3 3 b 6 14 10 a 250 cm c f c f c c f c f c c f c f 5236 cm2 482 mm2 104.3 m2 4427.8 cm2 1370 cm2 2488 m2 5890 m2 395 cm2 3436 cm2 10 016 m3 1568 mm3 904.78 m3 455 cm3 18 096 m3 b 36 : 49 Mixed revision 1 1 2 3 4 5 a 12 124 cm2 143 a $1 001.72 a $425 a 32 b 290 m2 c 1568 mm2 b $701.20 b $860.63 b 23 c $4 708.08 c $1 105 617 Answers 6 a 5629.7 m2 b 135.4 m2 pffiffiffi pffiffiffiffiffi 7 a 13 2 b 4 11 8 a $47 210 b $6890.25 y y = 3x – 2 9 6 c 21 205.8 cm2 pffiffiffi pffiffiffi c 6 66 3 (1, 1) 0 –4 –2 –2 –4 –6 10 12 13 14 16 17 18 21 22 23 24 25 2 4 x y = –2x + 3 13 824 cm2 11 B a 565 m2 b 817 m2 c 804 m2 a $4764.06 b $4782.47 c $4786.73 3x 4y þ 24 ¼ 0 15 gradient 5, y-intercept 3 a $768 b $6912 c $3317.76 d $10 229.76 e $213.12 f $10 997.76 a $19 676.44 b $10 313.56 c 65.6% x þpyffiffiffiffi 2 ¼ 0 19 43 m3 20 5x 2yp pffiffi ffiffi 30 ¼ 0 5þ5 a 1010 b 566 c 4 15 pffiffiffiffiffi Show that all sides have length 34 and two sides are perpendicular with gradients 35 and 53. Teacher to check. a 1408.33 mm3 b 39 810.26 cm3 c 11.49 m3 1000 cm2 pffiffiffi a 218 b 55 14 6 SkillCheck g9 h 10 6e 7 v 5w 5 b r6 f m4 j 3n 4 3 n wv 3 c d 15 g a k 1000w 9 o 1y d k2 h 1 l 25 p y12 2 a 19a b p6 c 53 d 7yx 20 2 3 a 18m þ 66m b 15g þ 40 4 a 4(x þ 6) b 5(4 3a) c q(q þ 1) d 6a(3a 2) e 2(y þ 15) f 6(3w 4) 5 a 3 and 6 b 2 and 4 c 4 and 5 d 8 and 2 Exercise 5-01 1 a e i 2 a 6p 7 30n16t 5 100y 20 l 18m 50 e 1 i 3 a g m 4 a 618 n b 3 k g m3n p2 h 4a 2 5b 7 a 243 1024 c m 30 g 5e 10g 4 2 4 k 3pq2 r c 7 d 9q 6 h 9a 10b 5 l 54u 4v 3w 8 10 d wt 15 f 64k 2y 10 g 15 h 12 125d 9y 15 j 27k 1000 1 b 1 625 h 128 25 n 10 1000 1 1 b 25 32 c 7 i 1 o 1 1 c 20 k 9 d 1 j 64 p 1 1 d 1000 16b 4 81d 4 l 81p 8q12r 16 e 8 f 9 1 k 64 l 1 b 3 1 x3 e p n7 2k 4 c 1 16h 2 d m2 16 81 f k 1 25b 2 p3 q5 l 5 b2 m w3 e 3g 3 5 f 3t 2r d 125 343 6 9 k2 e 2 15 9g 16 i 25t 4d 2 j mh10 8x 36 y6 c r2 4q 12 d 1 12q 7 r e g 32h11 h 8 p 12 h 7 i 64p 10h 20 g 256 a8 k 9p8 25d 6 l 64a6 27c9 8 a 5000x 28y 6 b a 256x 16 o 10f 3 e c d h x 27 f j 1 1331t 3 256 625 f 4 i 1 y 11 t3 c 4 h2 256x 16 a4 Exercise 5-02 1 a 8 g 0.1 pffiffiffiffiffi 10 2 a pffiffiffiffiffi 8r e b 3 h 2 g f q3 b 27 h i b f b f b e 243r 5 i c 64 1 3 6.69 7.62 2 n5 5 a3 8x 3 g ðxyÞ 1 100 j f 64h 6 1 25n 4 d 400 5 1 7 d 25 14.66 0.05 5 d2 3 5 a 2 4y 3 f 0.2 l 5 pffiffiffiffi d 4m pffiffiffiffiffiffiffiffiffiffi h 9 90ab 1 1 c 20 4 1 1 1 2 d 10 e 2 j 9 k 2 pffiffiffi g c pffiffiffiffiffiffi g 5 5j8 b 100 3 e a2 4 a 32 c 25 i 2 pffiffiffiffiffi b 3 12 pffiffiffiffiffi f 6 6h 1 1 3 a 52 1 32 1 h ð36wÞ6 e 1 2 k 1 8000 1 4 1 25 f l c g c g c 5.24 132.96 p3 3 x4 27d 6 d h d h d 3.98 0.30 m4 4 x3 512m 12 g 1 16s 8 h 1 16p 6 1 100x 2 y 4 j 1 3 125t 2 k 343a 6b 15 b 17c 10 5y 16 u 4g 3 4h c l Exercise 5-03 1 a e i m 2 a b 5w 6 f 4x 5y 4 j 64p 3 3 b n8 h 1 35 a b b 2r 6 y5 5 a e 6 a e 7 a Chapter 5 1 a e i m m 8 u3v4 6 a 4 2 g 1 87 1 ab 5 a e i 3 a e i 5n 14 13t 9 pþ3 z 5rþ3t rt f j n 5m8n 40 13c 10 21þ8a 28 b f j 9mþ2 20 2mþ8 15 14mþ49 36 b f j g k o 17t 20 6d11r 33 5e 24 c g k 8xþ10 15 3kþ31 70 1712m 35 c g k 15r 14 13t 36 1 3f 29 12b d 4aþ9h 30 9hþ10a 15 7m2n 14 d 7y5 12 k3 6 337k 6 d h l p h l h l 19y 24 17a 30 2e 5 9p4n 3np 3d2r 48 25þ24w 30 17k 30 9x13 20 23x41 20 1413x 6 Exercise 5-04 1 a g 3m 20 3x 4y b h kw 12 2 3 c i 28 pt d g d j 6 5qy d2 12 e k 3d e 2 3 f l 8 v 12a 2 k2 9780170194662 Answers 2 a g 3 a g 5x 2y 16 27 5x z 50p 2 t 7 b h b h t 6r 15 2 c 3b 1 6h 2 c 4 d i 12 j c 5 4 2s 35 i d j h2 k2 9 b2 25b 3 2 3n 3d 10 e f k 6p l 4t 27 27ac w e k 1 3 3g 20y f 25y 2 l uy Mental skills 5 2 a e i m 176 682 152 288 b f j n 363 707 540 693 c g k o 261 1818 2142 3939 d h l p 405 3564 588 852 Exercise 5-05 1 a d g j 2 C 3 a 4 a d g j 5 a d g j m 6 B 7 a c e g i 4h þ 24 4a þ 20z 6y þ 42y 2 12ab 2 21a 2b b e h k 3r 30 2 þ t 2 12x 2y 2 4xy 6h 2 þ 18h 3 c f i l 7x 63y 20e 2 30e 16rt 2 8r 2t 25x 3 20xy Yes 15m 2 þ 21m 49x 3 10x 4 6x 3 þ 35x 2 þ 8 9y 2 36y þ 35 6(4x þ 5) 10y(3 2y) (a 3)(a þ 6) q(q þ 36) hn(n h) b b e h k b e h k n No 15e 2 9e t 2 þ 7t þ 12 6 þ 3v 2v 2 16m 3 þ 2m 2 9(4 3a) 12d(3d þ 2) (8 þ t)(t 3) 2t(3 5t) 2e(10e þ 11) c c f i l c f i l o Yes 3w 3 15w 12 11h 2h2 w 2 8w þ 3 20xy þ 20x 60y x(x þ 1) 4r(4r 3) (3b þ 5)(b 2) 3y(y þ 2x) 9m(5m 6) 4xy(3x 4) 36mn(m 3n) 16vw(3v þ 4w) p(1 8p 4p 2) 8pg(4p 2 þ g 1) b d f h j 2pr(9p þ 8) 36bc(ab 4) 25gh(3g 2h 5) 3mn(2n þ 1 þ 16m) 3a 2(6a 3 4 þ 5a 2) Exercise 5-06 1 a d g j 2 D 3 a c e g i k 4 a e 5 a c m 2 þ 7m þ 12 a 2 5a 24 15 þ 14k k 2 t2 þ t 2 b e h k w 2 þ 10w þ 25 b 2 þ 7b 18 r 2 18r þ 77 x 2 þ 6x 40 2x 2 þ 11x þ 15 3p 2 þ 7p 10 6f 2 þ 4f 10 6 þ 13h 5h 2 10m 2 þ 23m 12 25y 2 25 8y b w2 18k f þ80f h 2 þ 14h þ 49 x 2 2x þ 1 9780170194662 c f i l y 2 144 u 2 15u þ 56 c 2 9c þ 18 99 2n n 2 9e 2 þ 42e þ 49 49d 2 28d þ 4 12m 2 þ 5m 25 16p 2 40p þ 25 12t 2 4t 1 49a 2 þ 84a 36 c m2 d 14u g 4d 2 þ 12d h 36a 2, 1 b k 2 10k þ 25 d q 2 þ 20q þ 100 b d f h j l e g i k m o q 6 a d g j m 7 a d g 8 a d g 25 10h þ h 2 x 2 2xw þ w2 4m 2 12m þ 9 81a 2 þ 36a þ 4 16 40p þ 25p 2 64a 2 48ay þ 9y 2 t2 2 þ t12 k2 9 b 49 m 2 e 25r 2 16 h 4 81m 2 k t 2 t12 n 9t 2 6td þ d 2 b p 2 þ 4p 4 e 6x2 3y2 7xy h 3x 2 þ 7x þ 2 b y 2 þ 18 e x2 h f 49 þ 14k þ k 2 h a 2 þ 2ag þ g 2 j 25x 2 60x þ 36 l 25 þ 70b þ 49b 2 n 121d 2 44cd þ 4c 2 p 1 þ 2y þ y12 r w92 þ 6 þ w 2 y 2 64 c w 2 121 2 81 k f 9d 2 25 2 16p 49 i 9 64k 2 2 2 81k 16l l 49n 2 64m 2 w2 4 o 1 r12 9 4e 2 1 c 25a 2 16 2 100 36y f h 2 6hg þ 9g 2 2 2 49a 16b i u 2 u12 16k 2 48 c 5xy 6x þ 3y þ 9 8m2 þ 2n2 f 12h þ 18 1 2b 2 Exercise 5-07 1 a c e g i k m o 2 a c e g i k m o q s u w 3 a c (x þ y)(3p þ 2q) (3k þ 4g)(5m þ 2n) (2k 5f)(a þ 4) 4(m þ t)(a þ e) (3m þ p)(n 2) ( f 10)(g h) (2 p)(p c) (a þ y)(x þ 1 k) (d þ 4)(d 4) (p þ 11)(p 11) (5 þ t)(5 t) (2r þ 3d )(2r 3d ) (12 þ 7m)(12 7m) (1 þ 9d)(1 9d) (y þ z)(y z) (b þ 11d)(b 11d) (4 þ 9h)(4 9h) (10 þ 7n)(10 7n) 1 þ 5c 12 5c 2 5h þ 32 5h 32 4(m þ 2p)(m 2p) y(y þ 5)(y 5) b d f h j l n p b d f h j l n p r t v x b d (h þ k)(2w 3u) (x 2a)(4y þ 7a) (d þ y)(c h) 3(k 2b)(y þ 4) (9 þ q)(p 2 3) 3(l þ n)(3k 4m) (l 3)(l 2 þ m 2) (a b þ 3q)(p 2q) (x þ 5)(x 5) (y þ 9)(y 9) (10 þ k)(10 k) (5g þ 2e)(5g 2e) (9y þ 4k)(9y 4k) (m þ 2n)(m 2n) (7 þ 4m)(7 4m) (6c þ 5k)(6c 5k) (5a þ 8m)(5a 8m) (11p þ 12q)(11p 12q) 2t þ 13 2t 13 (1 þ mn)(1 mn) 3(d þ 3)(d 3) 2(3 þ 5g)(3 5g) e k(1 þ 4k)(1 4k) f 2(5q þ 1)(5q 1) g 3(d þ 2v)(d 2v) i 2(ab þ 1)(ab 1) h 5t 3(t þ 5)(t 5) j x 2(y þ w)(y w) k 12(4f þ 3g)(4f 3g) l 5 3d þ 12 3d 12 or 54 ð6d þ 1Þð6d 1Þ m 2(x þ 2a)(x 2a) n 25(2 þ w)(2 w) o 5 12 þ 4e 12 4e or 54 ð1 þ 8eÞð1 8eÞ p 3c þ 2 12 3c 2 12 or 14 ð6c þ 5Þð6c 5Þ 619 Answers c þ 14 c 14 w 3uw 3u c 5þ 4 5 4 5b 4a 5b e 4a 7 þ 2 7 2 4 a b m n 4 þ3 2 m 4 n3 d (k þ 5)(k2 5) f (t þ 3)(t 3)(t 2 þ 9) 2 g (10 þ n)(10 n)(100 þ n ) i (p þ 3q)(3p q) h y(2x þ y) j 2x þ 6y 2x 6y k 4ab Exercise 5-08 1 a 3, 8 b 2 a (x þ 3)(x þ 5) d (e þ 3)(e þ 2) g (n 3)(n þ 1) j (w 9)(w þ 2) m (x þ 4)(x 1) p (a þ 2)(a 1) s (p 6)(p 4) v (m þ 2)(m þ 2) 5, 2 c 3, 5 b (d þ 7)(d þ 2) c e (h þ 2)(h þ 2) f h (r 7)(r þ 2) i k ( f 9)( f þ 3) l n (t þ 8)(t 3) o q (k þ 7)(k 2) r t (n 2)(n 1) u w (p 10)(p 10) x d 3, 4 (m þ 9)(m þ 3) (n þ 1)(n þ 10) (h 4)(h þ 1) (a 6)(a þ 2) (m þ 5)(m 2) (w þ 6)(w 2) (r 3)(r 3) (c 5)(c 5) Exercise 5-09 1 a c e g i k m o 2 a c e g i k 3 a c e g i k m o q 4 a 5 a c e g i k 6 a c e 620 3(m þ 1)(m þ 2) b 2(y þ 2)(y 1) 5(t 10)(t þ 8) d 5e 2(e þ 8)(e 3) x(x 11)(x þ 10) f 4(b 7)(b þ 6) 4(w þ 4)(w 3) h 3a(a 4)(a þ 1) 2(e þ 5)(e þ 4) j (t þ 8)(t 3) (u 7)(u þ 6) l (x 7)(x þ 4) (b þ 4)(b 3) n (k 3)(k 4) (x 5)(x 7) (2d þ 3)(3d þ 5) b (4m þ 3)(2m þ 1) (y þ 5)(2y þ 7) d (d þ 10)(2d þ 7) (w þ 15)(2w þ 1) f (e þ 3)(4e þ 3) (2f þ 3)(4f þ 1) h (d þ 1)(3d þ 2) (b þ 1)(2b þ 7) j ( y þ 1)(5y þ 11) (4g þ 3)(2g þ 5) l (3a þ 7)(2a þ 3) (4k 3)(k 2) b (2w 5)(3w 1) (p 3)(5p 4) d (2g 7)2 (3f 4)(4f 3) f (2h 9)2 (y þ 1)(5y 11) h (4d 5)(d þ 1) (2m þ 3)(m 3) j (2a þ 1)(4a 3) (5u 4)(3u þ 1) l (3c þ 1)(3c 5) (5m þ 7)(m 1) n (3g 4)(2g þ 3) (3p 2)(p þ 2) p (7w 1)(w þ 1) (5y 1)(y þ 3) r (3n 2)(n þ 4) (9w 10)2 b 4(y þ 1)2 c (5h 4)2 3(m þ 4)(3m 2) b 2(2y 5)(y þ 1) 5(3k 2)(2k þ 3) d 4(w 4)(3w þ 1) 4(t þ 2)(3t 1) f (5q þ 3)(5q 2) 2(2m 1)(3m 2) h (3h þ 4)(4h 5) 6(2c þ 3)(2c þ 1) j 3(z þ 1)(2z 5) 2(2d 3)(3d þ 5) l 2(x 3)(3x 2) (w 1)(7w 1) b (h 3)(4h þ 5) (4x 3)(2x þ 1) d (r þ 5)(5r þ 1) (d 7)(2d 1) f (3n þ 1)(2n 3) g (3m 2)(3m þ 4) i (3g þ 2)(5g þ 3) k (x 2)(3x 7) h (5c 3)(c þ 1) j (4q þ 3)(2q 5) l (3d 4)(d þ 4) Exercise 5-10 1 a c e g i k m o q s u 2 a c e g i k m o q s u (m þ 8)2 (d þ 3)(3d 5) (5y þ 8)(5y 8) q(q þ 3 3p) 4(2b þ 5)(3b 2) (b 2 þ 1)(b þ 1) (5d 4)(d þ 1) 2(2 þ v)(2 v) 2(w 6)2 (3r 8t)(5r þ 3t) 9(g þ 2k)(g 2k) e(e 5)(e þ 2) 7(2x þ 1)(2x 1) (c 2)2(c þ 2) (t þ 7)(t 5) (6a 1)(4a þ 1) (a 3)(2ab 3) (5u 1)2 3(4 þ w)(4 w) (k þ 4)2(k 4) mn(m þ 2)(m 2) 4(2c 3)(4c þ 1) b d f h j l n p r t 3(d þ 1)(d 1) (3 þ h)(k 5) 4(5f þ 4)( 5f 4) (g 3)(g þ 1) (5r þ 1)(5r 1) (2x 5)2 (b 1)2(b þ 1) m(n þ 3)(n þ p) (6h þ 1)2 (2d þ 1)2 b d f h j l n p r t 20(2p 3q)(3p 2q) (a b)(a þ b þ 4) (3a 1)(2a þ 5) 2(3p þ 2)2 9(x þ 2)(x 3) 2(a þ 3)2 (k 3)(4k þ 7) 3(1 þ 3s)(1 3s) 5y(y 2 2y þ 3) 2(a þ 2)(a 2) Exercise 5-11 1 a xþy d 1 d kþ5 k5 yþ4 2 sþ2 s3 7mþ10 mðmþ1Þðmþ2Þ k2 k ðkþ1Þðk1Þ g 425r 4ðrþ6Þðr6Þ g j m 2 a f k 3 2ðabÞ 2 3ðx2Þ l e kþ1 kþ4 12c 3c1 2w20 wðwþ3Þðwþ5Þ 5hþ12 4hðhþ1Þ f h d 2 þ3d6 d ðdþ2Þðd2Þ i k 2 þ9k5 ðkþ1Þðk1Þðk4Þ g l f h 3(c 1) i n b r 5ðrþtÞ 1 33 c e w4 k 3q1 j ðqþ1 Þðq1Þ 1 3 a 6m b 1 24 1 2ðtrÞ bc a 5 dþt aþ1 mþn 4aþ5c ac aþ4 2ðpþ2Þ 4b7 ðb1Þðbþ2Þðb3Þ 4d1 ðdþ2Þðdþ1Þ b c h m 1 2 4m m1 ðdþ1Þðd3Þ 6 o c d 6k e i 4 p j n f þ3 4ð f 3Þ o 10 hþ1 3 7 3 f 2 Power plus 1 a x 3 þ 15x 2 þ 55x þ 25 b y 3 6y 2 þ 12y 8 c a 3 þ 3a 2b þ 3ab 2 þ b 3 d 27d 3 þ 270d 2 þ 900d þ 1000 2 a 441 b 2025 c 841 d 3481 e 10 404 f 9604 3 899 4 a 399 b 2499 c 8099 d 6396 5 a 3y 2 þ 12y þ 14 b 3x 2 þ 9 c 34n 2 34 d 4b 2 9780170194662 Answers (n þ 2m) 2 (5x 4y) 2 (c 2 þ 1) 2 x yx y 4þ5 45 (5c 2 10)(5c 2 þ 10) 4a 5b4a 5b 7 þ 2 7 2 b d f h j l (x y) 2 5(a 3b) 2 (t 1)(t þ 1)(t 2 þ 8) (x þ 1)(x 1)(x 2 þ 1) (a þ b þ c)(a þ b c) 4pq Chapter 5 revision 1 a 6v 5w 7 e 4k1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 i a a a a a a a c e a c e g i a c e a c a c e a c e a c e a c a 625b 24y 12 100 000 20 64a 6 d 2y þ 12 yðyþ3Þðy3Þ 3t 20 7r20 10 3m 2 10 2 b 8t 7h 6 3 f 8p27 c 25x 2y 4 1 g 16m 2 j 512t 14u 16 64 b 125t 3 b 3 b 243t 15 b 19g 6 b 4wþ9 10 b 34 k c c c c c c b d f b d f h 56tp 40t 2p 4fg 2 30f 2g 8x 3 þ 7x 4 8ar(2r þ 3) (5x 1)(2 3x) 6p(t 2 þ 2pt 8p 2) 25x 3y 3(2x 3y) (n 2 þ 6)(n 1) b 2 þ 13b þ 30 15t 54 t 2 49y 2 9 n 2 þ 18n þ 81 9y 2 12y þ 4 (r þ t)(3p þ 2q) (b þ 10)(b 10) 5(2x þ 1)(2x 1) (y þ 5)(y þ 5) (n þ 11)(n 3) (m 12)(m þ 7) (3w þ 2)(w þ 1) 5(b þ 4)(b 3) 2(3x 1)(2x 7) 5(q 3)(q þ 3) (t 1)2(t þ 1) 2n þ 3t b b d f b d b d f b d f b d f b d 1 h m42 5c 4d 4 l 4d 2 81 1 8 1 16p8 x 16 5pþ20 12 b 2a 6a 5b d 8 d 25x12 y4 d 4w 3 15h 3 35h 2 93 22n 10y 2 41y þ 21 6(4p 3q) 15xy2(1 2x 2y) 4r 2s 3(8s þ 3r 2) 8p 3q 3(1 6q 3) d 2 þ d 56 20x 2 þ 13x 21 25p 2 80p þ 64 n 2 81 16n 2 121 (2a þ 3c)(2b 3d) (5 4y)(5 þ 4y) 3t(t þ 3)(t 3) (x 20)(x 1) (a 7)(a 4) (p þ 9)(p 6) (2y þ 3)(y 3) (3p 2)(p þ 4) (3n 2)(2n 3) 4(5x 3)(x 2) 6ðmþ2Þ 5 c e 9 f aþ2 aþ1 3ðbþ2Þ bðb3Þ Chapter 6 i 10 i 13 i 7 i6 i 48 i 5 9780170194662 Exercise 6-01 1 a i ii b i ii c i ii d i ii e i ii f i ii g i ii h i ii 2 a symmetrical clustering at 9, no outliers not symmetrical, not skewed clustering in the 30s and 60s, no outliers positively skewed clustering at 1, no outliers negatively skewed clustering at 2324, no outliers positively skewed clustering at 130–150, no outliers symmetrical clustering at 5, no outliers positively skewed clustering at 13, 23 is an outlier symmetrical clustering at 50s and 100s, 136 is an outlier Score 66 67 68 69 70 71 72 73 74 75 76 77 9 8 7 6 5 4 3 2 1 0 Frequency 1 2 1 5 3 5 9 5 4 4 0 1 66 67 68 69 70 71 72 73 74 75 76 77 Score mean SkillCheck 1 a b c d e f 2 a i 31 ii 33.3 iii 62 b 78 c i Median ¼ 30, mean ¼ 28.3, range ¼ 25. ii The outlier has increased the median (by 1), the mean (by 5), and the range (by 37). Frequency 6 a c e g i k ii ii ii ii ii ii 16.5 1.8 11.5 43.6 34.3 2.2 iii iii iii iii iii iii 15 2.5 11 43 34.5 2 iv iv iv iv iv iv 15 3 11 43 24, 35 2 mode, median b no outliers c negatively skewed d The lower the score below par, the fewer the golfers that achieve that score. e clustering at 72 f mode ¼ 72, x ¼ 71.6, median ¼ 72 621 Answers 3 a 45 b 19 hours (stem of 0) c no outliers d positively skewed e Most students spend limited time on their computers, and have other commitments and do activities such as sport. Only a few students spend many hours on the computer during the week. f Mode ¼ 1, x ¼ 14, median ¼ 11 4 a Stem Leaf b c d e 5 a d g 12 3 13 14 2 3 15 0 1 3 3 3 5 5 16 0 0 1 2 2 2 2 3 4 5 5 7 8 9 17 0 0 1 2 3 18 2 Symmetrical 123 is an outlier. Clustering occurs in the 160s. mode ¼ 162, median ¼ 162, x ¼ 160.2 slight positive skew b 13.8 is an outlier c 18.4 19.5 e 19.5 f 10.5 No, the range has been affected by the outlier 13.8. Exercise 6-02 1 a 5, 6.5, 8 b 18, 20, 26.5 c 32, 34.5, 38 2 a range ¼ 7, IQR ¼ 3 b range ¼ 22, IQR ¼ 8.5 c range ¼ 16, IQR ¼ 6 3 a 7.5 b 3 4 a 283 mm b 128 mm 5 a 3 b 2.5 c 17.5 d 19 e 21.5 f 1.5 6 a 34 b 13 c i 68, 72, 72, 75, 77, 78, 79, 80 ii 50% d 75% 7 a i 28 ii 9.5 b The interquartile range, as it is not affected by the score of 35. c 48, 48, 48, 49, 51, 53, 55; 54% Exercise 6-03 1 a 2.66 b 2.63 c 1.19 d 1.33 e 2.01 2 a 7 b i 2.64 ii 2.28 c Decreases the standard deviation. 3 a C b A 4 a 1.99 b 13.43 5 a x ¼ 165.89, s ¼ 8.37 b i less than 157 or greater than 175 (to the nearest cm) ii between 157 and 174 (to the nearest cm) 6 a x ¼ 11.37, s ¼ 0.43 b i less than 10.9, greater than 11.8 (correct to one decimal place) ii between 10.9 and 11.8 (correct to one decimal place) 7 C 622 Exercise 6-04 1 a Men: x ¼ 71.40, s ¼ 6.77; Women: x ¼ 77.53, s ¼ 6.96 b Yes, the mean of women’s pulse rates is much higher, which may be due to stresses involved in shopping (and looking after children at the same time). The standard deviation for women is slightly higher. 2 a Dominant hand: x ¼ 0.40, s ¼ 0.11; Non-dominant hand: x ¼ 0.52, s ¼ 0.51 b Yes, the mean reaction time and standard deviation of the dominant hand are much lower than the mean and standard deviation of the non-dominant hand. c i 0.61 and 0.75 ii x ¼ 0.37, s ¼ 0.05 iii Removing the outliers has reduced the mean 0.40 to 0.37 and more than halved the standard deviation. d x ¼ 0.40, s ¼ 0.06 e The removal of the outlier from the non-dominant hand had the greater effect on the mean and standard deviation as the outlier of 2.60 was a more extreme score than the outliers for the dominant hand. 3 a Western Tigers: x ¼ 122.92, s ¼ 26.98; Barrington City: x ¼ 120.92, s ¼ 23.62. b The Barrington City team is slightly more consistent as the standard deviation is 23.62 compared with 26.98 for Western Tigers. 4 a Vatha: x ¼ 13.76, s ¼ 0.55; Ana: x ¼ 14.14, s ¼ 0.66 b Vatha is more consistent as the standard deviation for her times is significantly lower than the standard deviation for Ana’s times. 5 B 6 a Maths: i range ¼ 47 ii IQR ¼ 14 iii s ¼ 10.97 Science: i range ¼ 45 ii IQR ¼ 19 iii s ¼ 13.16 b Maths: x ¼ 67.82; Science: x ¼ 61.25 c The students performed better in Maths as the mean was 67.82 compared to 61.25 for Science. The marks for maths were also more consistent as the IQR and standard deviation are both lower than those of Science. 7 a Roosters: i range ¼ 48 ii IQR ¼ 20 iii x ¼ 26.67 iv s ¼ 12.55 Dragons: i range ¼ 32 ii IQR ¼ 8 iii x ¼ 15.88 iv s ¼ 7.36 b The range, IQR and the standard deviation for the Dragons are significantly lower than those of the Roosters, which show that the Dragons are more consistent in the number of points they scored per game. However the mean of the Roosters is significantly greater than the mean of the Dragons, which would indicate that they are a better team as they were able to score many more points per game. Mental skills 6 2 a f k p 160 900 18 12 b g l q 70 140 34 40 c h m r 240 300 46 8 d i n s 900 180 26 14 e j o t 2600 770 18 24 9780170194662 Answers 2 a b c d Exercise 6-05 1 a 1, 4.5, 6.5, 10, 18 b 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Number of orders/h 2 a 26 b 1, 2, 5, 13, 50 c 0 4 8 12 16 20 24 28 32 36 40 44 48 52 Amount of snow (cm) b 5, 21.5, 49.5, 96, 266 3 a 261 c 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 Monthly Rainfall (mm), Penrith 4 a 27.5 h b 26 h 5 a 26 b 21 6 a 6, 10, 19, 23, 29 c 30 h c 14 b 13 d 4h d i 25% c i 14 ii 7 7 a 20, 46, 51, 68, 88 iii 7 20 30 40 b 10, 13, 15, 16, 20 10 11 12 13 c 30, 51, 65.5, 75, 95 30 40 8 a 4, 6, 7, 9, 15 e 50% ii 75% 50 60 iv 21 70 80 90 e i 33 ii 28 Thunderbirds: 50; Swifts: 49 Thunderbirds: 15.5; Swifts: 9.5 The range for both teams is similar but the IQR of the Swifts is less than the IQR of the Thunderbirds, indicating that the Swifts are more consistent in their performance. The position of the Thunderbirds’ boxplot shows that the Thunderbirds scored more points in games than the Swifts and so performed better in the season. 10K: 9; 10N: 10 b 10K: 6.5; 10N: 5.5 10K: 3; 10N: 4 d 10K lower range and IQR. 75% 3 a c e 4 C 5 a Brisbane: 26.9, 9.3, 4.7; Sydney: 23.5, 8.5, 4.9; Melbourne: 21.4, 13, 8.6; Hobart: 18.6, 11.2, 7 b Melbourne – it has the highest range and IQR. c Brisbane, more than half of the mean monthly temperatures are higher than most of the mean monthly temperatures of the other cities. d Sydney’s median temperature is significantly higher than Melbourne’s, so Sydney is the warmer city. e Sydney has the smaller range and IQR of mean monthly temperatures, so it is more consistent. 6 a Male: 0,1, 2, 4, 7; Female: 2, 4, 5, 7, 10 b Males Females 14 50 15 60 16 70 17 80 18 19 90 20 100 0 1 2 3 4 5 6 7 8 9 10 Text messages c Male: 3; Females: 3 d Males: 7; Females: 8 e Both are positively skewed, the interquartile range is the same, and the range of females is one more than that of the males. Females do receive more text messages, as the boxplot shows that 75% of females receive more messages than 75% of males. 7 a Male: 145, 165, 167, 172.5, 189; Female: 150, 162.5, 165.5, 173.5, 186 Females 4 9 10 11 12 13 14 15 Marks b Dot plot is positively skewed. The length of the boxplot from the median to the highest score is greater than the length from the median to the lowest score. c 15 d 4, 6, 7, 9, 12 4 5 5 6 6 7 7 8 8 9 10 11 12 13 14 15 Marks e i The boxplots are the same up to Q3. ii The whisker from Q3 is reduced without the outlier. Exercise 6-06 1 a i Year 10: 3.5; Year 8: 8 iii Year 10: 1; Year 8: 2 b i 25% ii 75% 9780170194662 ii Year 10: 7.5; Year 8: 8.5 c i 10 ii 0 Males 140 150 160 170 180 190 Height of students b Male: Range 44 IQR 7.5 Female: Range 36 IQR 11 c Male students have a greater range (44 compared to 36), but a smaller interquartile range (7.5 compared to 12). 8 a Low: 64, 73.5, 80, 86, 92; High: 49, 58, 68, 75, 96 High Low 40 50 60 70 80 90 100 Pulse rate b Low: 28, 12.5 High: 47, 17 c The range and interquartile range of the High Frequency group are both greater than that of the Low Frequency group. d The high frequency group. 623 Answers 9 a Sydney: 17.6, 20.4, 23.45, 25.25, 26.1; Brisbane: 21.1, 23.65, 26.9, 28.45, 30.4 Brisbane f Sydney b c 10 a d f g 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Average monthly temperature (ºC) Sydney: 8.5, 4.85 Brisbane: 9.3, 4.8 Sydney average monthly temperatures were slightly more consistent than Brisbane, since its range was 0.9 less than Brisbane’s, while Brisbane’s IQR was only 0.05 less than Sydney’s. Simone b Simone: 12; Amal: 10 c Amal, smaller range. Simone: 10; Amal: 9 e Simone: 4; Amal: 5 Not enough information given to make a valid decision. The interquartile range and range only differ by 1. 25% Exercise 6-07 1 a Boys: $34.58, Girls: $31.78 b Boys: $33.50, Girls: $28 c Boys: Range ¼ 72, IQR ¼ 25, Girls: Range ¼ 69, IQR ¼ 30.5 d i Boys are positively skewed slightly, girls are positively skewed. ii There are no outliers, clustering occurs for the boys in the 20 30s and for the girls in the 10 20s. e Boys generally carry more cash they have a higher mean than the girls and the shape of the data for girls is more positively skewed. 2 a 21 games b i 34 ii 51 c Scorpions: x ¼ 1.6 goals; Vale United: x ¼ 2.4 d Scorpions 5, Vale United 6 e The shape of both teams’ results is positively skewed. Clustering for Scorpions occurs at 1 and 2 and for Vale United it occurs at 2. f Vale United performed better as its mean was 2.4 goals/ game compared to Scorpions 1.6 goals/game. 3 a Sydney: x ¼ 26.2, median ¼ 26.5, mode ¼ 28 Perth: x ¼ 34.3, median ¼ 35, mode ¼ 38 b Sydney: Range ¼ 9, IQR ¼ 3 Perth: Range ¼ 16, IQR ¼ 8 c The temperatures for Sydney and Perth are both negatively skewed, there are no outliers. Sydney’s temperatures are clustered in the high 20s, while Perth’s are clustered at 34 38. d Sydney’s temperatures are lower than Perth’s, as evidenced by the significantly lower mean, median and mode. The range and interquartile range for Perth are greater than the range and interquartile range for Sydney, indicating greater spread. 4 a 30 b Quiz 1: x ¼ 5.6, mode ¼ 6; Quiz 2: x ¼ 6.3, mode ¼ 7 c Quiz 1: 6; Quiz 2: 7 d Quiz 1: i Range ¼ 7 Quiz 2: i Range ¼ 8 ii IQR ¼ 2 ii IQR ¼ 2 e Quiz 1: Results are symmetrical with clustering at 56, no outliers. 624 5 a b c e f 6 a b c d 7 a b c d e 8 a b c d e f 9 a b c d Quiz 2: Results show negative skewness with clustering at 5 and 78, no outliers. Scores for Quiz 2 are just better than Quiz 1, as the mean of Quiz 2 is higher than the mean of Quiz 1. The spread for both quizzes are similar as there is only a difference of 1 between the ranges and the IQRs are equal. 39 i mode ¼ 2 ii median ¼ 2 iii range ¼ 6 iv IQR ¼ 1.5 positively skewed, no outliers d 50% i By the highest columns. ii By the short length of the box when compared to the whole length of the boxplot. i The shape of the distribution, the frequency for each household size and the mode. The mean can also be calculated from the histogram. ii The shape of the distribution, the median and the quartiles Q1 and Q2. i 5 ii 16 i mode ¼ 22 ii range ¼ 18 iii IQR ¼ 24 16 ¼ 8 Negatively skewed. i The tail of the dot plot goes to the left. ii The length of the boxplot from the lowest score to the median is longer than from the median to the highest score. i dot plot ii boxplot iii dot plot iv boxplot Sunbeam Valley: range ¼ 24, median ¼ 71, IQR ¼ 75 67 ¼ 8 Bentley’s Beach: range ¼ 30, median ¼ 73, IQR ¼ 82 67 ¼ 15 Sunbeam Valley: negatively skewed (slight) Bentley’s Beach: positively skewed Sunbeam Valley’s speeds are clustered in the 70s. 25% Bentley’s Beach higher median, positively skewed. 25% of drivers drive faster than all drivers in Sunbeam Valley. This may be due to more main roads with higher speed limits. 36 Lamissa: mode 7, median ¼ 7 Anneka: mode ¼ 7, median ¼ 6 Lamissa: range ¼ 8, IQR ¼ 8 6 ¼ 2 Anneka: range ¼ 9, IQR ¼ 7 4 ¼ 3 Lamissa’s distribution of scores is negatively skewed with clustering at 7. Anneka’s distribution is negatively skewed with clustering at 6 and 7. i 30.5% ii 55.6% Lamissa is the better archer. Her median score is higher than Anneka’s, 30.5% of scores are less than 6 compared to Anneka’s 55.6%. Also, from the boxplot, 50% of Lamissa’s scores are equal to or better than 75% of Anneka’s. The range (47) is too large. Women: 31 Men: 37 Women: Range ¼ 38, IQR ¼ 40 24 ¼ 16 Men: Range ¼ 47, IQR ¼ 46 25 ¼ 21 Distribution for women is positively skewed with clustering in the 20s. Distribution for men is symmetrical with clustering in the 30s. 9780170194662 Answers Exercise 6-08 5 a 700 Points scored against, A e Men have the greater spread in the number of sit-ups completed, as the range and IQR are both greater than those for women. 10 a i 56 ii 38 b i 10 Blue, 10 Yellow ii 10 Green iii 10 Red c i 10 Green ii 10 Yellow iii 10 Red, 10 Blue d 10 Blue. It shares the highest median with 10 Red but its lowest score is still higher than 25% of 10 Red’s scores. 600 500 400 300 300 1 a 25 200 24 200 21 20 19 18 17 16 Stride length, L (cm) 140 150 160 170 180 190 Height, H (cm) b linear c As the heights of students increase, their handspans tend to increase. 2 a weak negative relationship b no relationship c strong positive relationship 3 Weak positive. 4 a Stride length depends on a person’s height; the taller the person, is, the longer their legs are. b 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 140 150 160 170 180 190 Height, H (cm) c linear d Students’ stride length increases with height. e strong positive relationship f Near 72.5 73 cm 9780170194662 b Yes 6 a 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 300 400 500 600 Points scored for, F 700 c Weak negative relationship Computer, C (hrs) 22 5 10 15 20 25 Homework, H (hrs) b no relationship 7 a 190 0 180 Height, H (cm) Hand span, S (cm) 23 170 160 150 140 130 10 11 12 13 14 15 16 17 18 Age, A (years) b Age, because as a young person ages, he usually grows in height. c weak positive relationship 625 Answers 5 a Exercise 6-09 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 1 a Spring stretch, S (cm) Height, H (cm) 190 180 170 160 150 140 19 20 21 22 23 24 25 26 27 28 Length of radius, r (cm) b H ¼ 5r þ 48.5 2 a c 173.5 cm d 184 cm 5 10 15 20 25 30 35 40 45 50 55 60 Mass, M (g) 14 13 b 22.4 cm c 56 g d Yes, because a spring has an elastic limit, which is the point at which a spring will not return to its original length as a result of the mass attached to the spring being too heavy. 6 a 10.1 Shoe size, S 12 11 10 9 8 7 10.0 b S ¼ 0.393H 59 d 12 3 a Time (seconds) 170 172 174 176 178 180 182 184 186 188 Height, H (cm) c 8.5 e 13.5 Temperature, T (°C) 20 10 0 –10 1000 2000 3000 4000 5000 6000 7000 8000 900010000 9.7 9.5 –20 1960 1970 1980 1990 2000 2010 2020 Year –30 –40 b T ¼ 0.0068h þ 16 c 5.8C 4 a 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80 90100 Maths results, M b 80 c 97 Science result, S 9.8 9.6 Height, h (m) b 9.60 c There is a limit to how fast a person can run. –50 626 9.9 d 3800 m Exercise 6-10 1 a i 25 ii 42 iii 15 b December, more customers due to summer and Christmas holiday season. c June, fewer customers due to winter, busy end-of-financial year season. d Number of people employed peaks in December, then falls, only to increase in March, April (the Easter holiday period). It then falls again to a low in June, July and then slowly the number of people employed rises to a peak in December. From 2010 to 2012, the number of people employed is showing a slow increase. 9780170194662 Answers 2 a then fell at a rapid rate between 1980 1990 and continued to fall by about 200/5 years until 2010. c Improved safety in cars with seat belts being compulsory, then drink driving laws introduced. 23 22 21 4 a i 20 24 Populations (millions) 19 18 23 Temperature (°C) 17 16 15 14 22 21 13 20 12 11 10 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 Year 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 ii Year 24 b 2005 – 2010 c Australia’s population increased at 1.1–1.2 million every 5 years up to 1975. The population growth then slowed down for 5 years. From 1980, the population grew at a steady rate of just over a million people every 5 years but from in 2005–2010, the rate increased to 1.9 million for the 5-year period. d i 2627 million ii 3234 million, teacher to check 3 a 1400 Temperature (°C) 23 22 21 20 1300 1200 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 1100 Year 1000 b i 900 Fatalities 800 700 600 500 400 300 200 100 1970 1980 1990 2000 2010 Year b From 1960, road fatalities rose at a steady rate, reaching a peak of approximately 1300 in 1970 1980. Road fatalities 1950 9780170194662 1960 Starting at 22.3 in 1990, the temperature has seen a series of increases of less than 1, followed by falls of less than 1. The increase from 1990 to 2000 is 0.4. ii Starting at 23.1 in 2001, the temperature falls to 22.7 and then increases to a high of 23.4 for two years before falling to 22.1. From 2009, the temperature rose, then remained steady before a slight increase to 22.7 in 2012. c The temperature from 1990 – 2000 continually increased and decreased by less than 1. The temperature in 2001 – 2012 started at 23.1, rose to a high of 23.4 in 2004, before falling for three years. This was followed by a slight increase. The range of annual temperatures for both periods is 1.3, but the minimum and maximum temperatures for 2001 – 2012 are 0.6 higher than for 1990 – 2000. 627 Answers 5 a 560 Annual emissions Mt CO2-e 550 540 530 520 510 500 20022003200420052006 200720082009201020112012 Year b Carbon emissions increased by 55 Mt. c Carbon emissions stabilised. d More environmentally-friendly policies and practices in Australia. e i, ii Teacher to check and discuss. 6 a Approximately 4 million. b 18 million c 300 000 persons per year d 26.5 million 7 a Gradual increase in passenger movements with peaks in October and troughs in February. b i 3.9–4.0 million ii 4.25 million iii 4.2 million iv 4.5 million c 15% Exercise 6-11 1 a Just surveying 300 people between 9 a.m. and 11 p.m. in shopping centres only targets a narrow group of people in certain areas. b The sample needs to be more random and over a large area, not just in shopping centres. A telephone survey should produce more accurate feedback. 2 The report does not say what conditions are needed for the hot water system to work effectively. The temperature in Queensland is much warmer than in NSW and Victoria. Consequently, with the cooler climate in NSW and Victoria, especially in winter, the heat pump system may not provide the savings that people in Queensland obtain. 3 a i The price of petrol has shown little increase from December to February. ii The price of petrol has shown marked rises and falls over the period from December to February. b Both graphs could be improved by starting the vertical scale at 0 cents/litre. 4 a That there is a marked difference between the fuel consumption of the different cars. b i 0.2 L/100 km ii 1 L/100 km iii 0.2 L/100 km c Begin the scale on the vertical axis with 0 and use a scale of 1 cm ¼ 0.5 L/100 km instead of 1 cm ¼ 0.2 L/100 km. 628 5 Yes, as there is no option for a customer to rate the product as unsatisfactory or poor. 6 a An example of a biased question could be: Which of these colours do you prefer red, black, silver, blue? b Apart from surveying people, they need to look at sales figures of all cars. This will give information about the most popular car colour. 7 –8 Teacher to check. Exercise 6-12 Teacher to check the investigations. Power plus 1 a 1 and 1 b There is no relationship between the variables. c i 1 ii 0.2 iii 0.8 2 b, d, f 3 a x ¼ 13.35, median ¼ 14, mode ¼ 14 b Range ¼ 10, IQR ¼ 15 12.5 ¼ 2.5 c The mean, median, and mode will increase by 4, the range and the interquartile range remain unchanged. Chapter 6 revision 1 a b c 2 a 3 a b c 4 a b 5 a b c i negatively skewed ii clustering at 16 and 17, 10 is an outlier i positively skewed ii clustering at 40s and 50s, no outliers i symmetrical ii clustering at 4, no outliers 6.5 b 6 c 2.5 d 12.5 e 2 x ¼ 0.40, s ¼ 0.08 Range ¼ 0.33, IQR ¼ 0.08 The interquartile range is the better measure as the standard deviation is affected by the outlier 0.62. Girls: x ¼ 67.73, s ¼ 16.08 Boys: x ¼ 61.67, s ¼ 12.35 The girls performed better than the boys as their mean mark was about 6 more than the mean mark of the boys. However the boys’ marks were less spread out than the girls. Range ¼ 7, IQR ¼ 3 1 ¼ 2 0, 1, 2, 3, 7 0 1 2 3 4 5 Goals scored per game 6 7 6 a Before: 50, 64, 69, 76, 80; After: 82, 89, 95.5, 126, 146 After Before 50 60 70 80 90 100 110 120 130 140 150 b i Range ¼ 30, IQR ¼ 12 ii Range ¼ 64, IQR ¼ 37 c The pulse rates for after exercise are significantly higher. In fact, all the rates for after exercise are above all the rates for before exercise. The median pulse for after exercise is 95.5 compared to the median pulse of 69 for before exercise. The range and interquartile range are also greater for the after exercise pulse rate. 9780170194662 Answers i Both ii Stem-and-leaf plot The range (126 70 ¼ 56) is too large. i median ¼ 92 ii IQR ¼ 99.5 84 ¼ 15.5 50% Weeks in storage this determines how many oranges stay good. Number of good oranges, N b 60 50 Mixed revision 2 40 30 20 10 c For the last 10 years, the mean maximum temperatures at Blacktown, after staying at 30.6, have ranged from a low of 27.4 to a high of 31.7, finishing at a temperature of 30.0 in 2013. This shows there has been little change in temperature for the month of January over the last 10 years. 11 a That the product is healthy. b There is no data given on the actual fat content in the product. This should also be stated in terms of daily percentage requirement of fat or in mg of fat. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Weeks in storage, W c linear d The longer the oranges remain in storage, the fewer good oranges there are in the box. e strong negative correlation 9 a 1000 1 a b 2 a 3 a 4 a 5 a b i negatively skewed ii no outliers, clusters at 7 and 8 i positively skewed ii outlier 98, clustering in 40s and 50s b ¼ 7 b x ¼ 10 81n 4m 8 b 4a 4b 3 c 19 6a20 5xþ36 b 24 c 7r6 15 10 Range ¼ 33; Interquartile range ¼ 7 Interquartile range, because it is not affected by the outlier of 112. pffiffiffi 6 a x ¼ 6 3 b n ¼ 0 or 64 c u ¼ 15 or 4 7 a 9d b 2 c 8pv 2 2 8 a 1, 3.5, 5, 6, 10 b 1 3.5 5 6 10 90 9 10 11 12 13 Weight, W (kg) 80 70 60 50 40 14 15 16 30 130 140 150 160 170 180 190 Height, H (cm) b W ¼ 0.714 H 51.4 d 65 kg 10 a Year b c 70 kg e i 192 cm ii 135 cm Temperature (ºC) 40 30 20 17 18 1 2 3 4 5 6 7 8 9 10 11 a weak negative b strong negative c weak positive 1 1 a 4 b 16 c 12 d 1000 y ¼ 5.8 a 53gh 45gh 2 b 7y 4 þ 2y 3 a Girls: mean ¼ 62.7, standard deviation ¼ 16.1 Boys: mean ¼ 66.9, standard deviation ¼ 12.2 b The boys performed better as their mean is higher. a x 2 þ 14x þ 49 b 25m 2 20m þ 4 c 9n 2 100 20, 21, 22 a i stem-and-leaf plot ii stem-and-leaf plot b i median ¼ 42 ii interquartile range ¼ 16 a (y 16)(y 2) b (n þ 8)(n 6) c (a 9)(a þ 8) a The independent variable is W, the weeks in storage. The number of weeks in storage is set first after which time the number of good apples is counted. b Number of good apples N 7 a b c d 8 a 60 50 40 30 20 10 0 10 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Year 9780170194662 2 4 6 8 10 12 14 Weeks in storage W c the number of good apples decreases the longer the apples are in storage. d there is a strong negative relationship between the variables W and N 629 Answers 19 a (5n þ 2)(n þ 1) b (2a þ 3)(a 5) 20 n ¼ 3t 2 21 a x –2 –1 0 1 2 3 4 x b –2 0 2 4 6 c x –10 –8 –6 –4 –2 0 –10 –8 –6 –4 –2 0 d 2 x 22 a n 5 d4 d1 Height, H (cm) 23 a 24 a c (3x 5)(x þ 2) b k < 7 c x 1 12 b 16 c 3ðyþ1Þ yðy3Þ 190 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 Exercise 7-02 1 a m ¼ 12 b x ¼ 20 c y ¼ 15 d k ¼ 13 e y ¼ 1 f w ¼ 4 g x ¼ 2 h t ¼ 4 i a ¼ 4 j k ¼ 6 k w ¼ 10 l d ¼ 12 m k ¼ 1 n w ¼ 5 o y ¼ 1 p m ¼ 6 q x ¼ 12 r p ¼ 3 s k¼2 t y ¼ 10 pffiffiffi 2 a m ¼ 2 b a ¼ 9 c m ¼ 2 7 pffiffi p ffiffi ffi p ffiffiffiffiffi d k ¼ 33 e k ¼ 4 6 f k ¼ 3 10 3 a m 1.94 b x 7.58 c y 0.35 d w 7.07 e a 9.24 f y 6.20 4 a x ¼ 2, 1 b y ¼ 4, 1 c y ¼ 4, 12 d x ¼ 4, 3 e x ¼ 3, 1 f x ¼ 8, 5 5 a x ¼ 6, 5 b x¼4 c x ¼ 11, 6 d d ¼ 0, 2 e q ¼ 5, 2 f n ¼ 0, 4 g k ¼ 0, 7 h y ¼ 0, 5 i v ¼ 0, 12 j m ¼ 0, 3 k a ¼ 20, 4 l n ¼ 0, 10 m u ¼ 4, 2 n x ¼ 7, 6 o p ¼ 4, 5 6 You cannot take the square root of a negative number. 7 a, c, f: cannot find square root of negative number. Exercise 7-03 1 a e i 2 a e i 3 a 0 10 20 30 40 50 60 70 80 90 100110120 x¼1 b m¼5 pffiffiffiffiffi y ¼ 9 f n ¼ 3 20 p ffiffiffiffiffiffiffiffiffi 3 m ¼ 15 j m ¼ 4 w ¼ 2.5 b m ¼ 2.5 x¼3 f x ¼ 5.5 a ¼ 5.5 j a ¼ 0.4 yes b when c is positive c g k c g k c a ¼ 11 d u ¼ 2 p ffiffiffiffiffi p ffiffiffiffiffiffiffiffiffi 3 3 11 h¼q 48 ffiffiffiffiffiffi h k ¼ pffiffiffiffiffiffiffiffiffi 3 3 81 x¼ l x ¼ 40 4 m¼6 d t ¼ 3.2 x ¼ 2.4 h x ¼ 0.8 x ¼ 2.3 l t ¼ 4.4 when c is negative d no Weight, W(kg) b H ¼1.2W þ 70: other answers possible. c 48 kg d 170 cm 25 a 1 b 0 c 3 26 x 3.453 Exercise 7-04 Chapter 7 SkillCheck 1 a a ¼ 12 2 a (k þ 4)(k þ 1) d (u þ 13)(u 5) b x¼6 b (y 8)(y 2) e (w 7)(w 3) c x¼8 c (m 8)(m þ 7) f (x 6)(x þ 4) Exercise 7-01 1 a e i m 2 a e i m 3 a 4 a e i 630 y ¼ 15 n ¼ 35 m ¼ 18 n ¼ 14 k ¼ 1 78 y¼3 y ¼ 56 6 a ¼ 11 C 7 x ¼ 15 x ¼ 92 9 w ¼ 11 1 3 6 9 11 13 16 18 100 cm 3 25 cm child: $21, adult: $48 Anand: 3, Sunjay: 27 Vatha: 22, Chris: 14 x ¼ 35 6 14 x ¼ 15.5 25, 50, 105 8 teachers, 120 students 2 18 mm, 36 mm, 36 mm 4 61, 62, 63 5 94, 96, 98 7 26 8 4 10 213, 214, 215, 216 12 117 15 Scott: 11, Mother: 34 17 72 L when full Mental skills 7 b f j n b f j n b b f j a¼9 y ¼ 7 x ¼ 29 n ¼ 35 w ¼ 1 13 a ¼ 8 35 w ¼ 10 y ¼ 60 A p ¼ 109 7 y ¼ 53 5 a ¼ 75 14 c g k o c g k o m¼7 x ¼ 31 x ¼ 24 d ¼ 3 34 x ¼ 1 13 p ¼ 9 23 w ¼ 50 a ¼ 1 11 13 d k ¼ 57 h y ¼ 46 l m ¼ 10 c x ¼ 133 7 g a ¼ 41 d x ¼ 76 7 h a ¼ 107 14 d x¼3 h y¼3 l w ¼ 9 35 2 a e i 4 a e i 3.5 0.8 0.24 66.3 6.63 6630 b f j b f j 2.4 0.027 0.012 6630 663 66.3 c g k c g k 0.12 0.2 1.8 6.63 0.663 0.663 d h l d h l 0.36 8.8 0.028 0.663 663 0.0663 Exercise 7-05 1 a C ¼ 15.1 m b r ¼ 31.8 cm 4 a 36 km/h b 86.4 km/h 6 a 27C b 0C 2 w ¼ 17 m 3 30.6 m/s c 180 km/h 5 43 c 100C d 39C 9780170194662 Answers 7 a 21.0 9 a 436 km 10 a $97.50 b 105.8 kg b 7h b 620 km 8 4.9 m h w 10 11 h ¼ 13.2 cm j a3 Exercise 7-06 k a 1 1 a y ¼ 5x 2 c y ¼ P8 k b y ¼ km p d y ¼ 5m 3 l w < 3 y ¼ 4d5 8 e y ¼ KD M f g y ¼ 2cþk aqffiffiffiffiffiffiffi i y ¼ w5 x 20m9 h y ¼ 20m 3 3 or y ¼ 3 2 j y ¼ kx d k y ¼ 5nd l y ¼ cT 2 k n or y ¼ 5 n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Þ b a ¼ 2ðsut c a ¼ vu 2 a b ¼ c2 a2 t t2 qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffi Aþpr2 Apr 2 d r ¼ 3 3V e R ¼ f l ¼ p pr 4p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S sx g n ¼ Sþ360 or þ 2 h r ¼ i b ¼ x2 þ 4ac 180 180 xs j x ¼ 5y 4 mn k A ¼ 52m þY Þ m a ¼ bðX Y X a l p ¼ aS –1 0 1 2 –4 –3 –7 –6 –5 –4 –3 –2 –1 –5 –4 –3 –2 –1 0 1 2 3 6 7 –2 3 4 5 b 0 1 2 2 3 5 4 d 4 5 8 9 2 3 4 5 6 7 8 f –3 –2 –1 0 1 2 3 4 g –5 –4 –3 –2 –1 1 0 2 3 4 5 h 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 2 a x<4 b x2 c x > 6 d x1 3 B 4 a x1 b x<4 c x>6 d x 1 e x > 6 f x<2 g x 4 h x 25 i x<0 b y4 c m 2 0 1 2 0 1 2 f y > 4 g a 12 9780170194662 3 4 3 5 4 6 5 7 6 8 9 10 11 7 8 –7 –6 –5 –4 –3 –2 –1 0 1 d x 100 e x<5 e p < 6 3 4 5 6 7 8 9 10 –2 –1 0 1 2 3 4 5 6 7 –6 –5 –4 –3 –2 –1 0 –7 b f j 1 2 3 –6 –5 –4 –3 –2 –1 0 m6 c y 8 a<4 g m 72 x<6 k y 2 –1 0 1 2 3 4 5 6 4 1 d x5 h m 212 l a 14 5 7 8 –12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 –16 –15 –14 –13 –12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 –5 –4 –3 –2 –1 0 1 2 3 4 5 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 b k 12 f y 2 j p < 4 c t < 2 25 g a>1 k m<4 1 d x 12 h d < 5 12 l x – 27 Exercise 7-09 d log25 5 ¼ 12 c 2 d 4 i 6 j 8 b log4 64 ¼ 3 e 5 f 3 k 6 l 3 c log10 10 000 ¼ 4 1 e log2 16 ¼ 4 f log3 19 ¼ 2 pffiffiffi g log8 4 ¼ 23 h log10 0.01 ¼ 2 i log4 2 ¼ 14 j log16 4 ¼ 12 k log9 27 ¼ 32 l log6 p1ffiffi6 ¼ 12 pffiffiffi 6 3 a 125 ¼ 53 b 10 ¼ 101 c 27 ¼ 3 pffiffiffi 6 3:5 1 d 8 2¼2 e 64 ¼ 2 f 81 ¼ 34 pffiffiffi 1 1 1 3 2 ¼ 86 g 125 ¼ 5 h i 10 ¼ 100 2 pffiffiffi 3 1 1 1 3 2 j 5 5¼5 k 2¼8 l 100 ¼ 100 4 Because a base raised to any power always gives a positive number. Exercise 7-10 Exercise 7-08 1 a x>7 c k > 11 2 1 a 2 b 3 g 3 h 2 2 a log5 25 ¼ 2 10 11 e 1 b y > 8 1 3 c 0 x3 0 5 a x > 3 e w < 1 i w < 11 1 a 1 x1 w > 3 x < 35 f t 4 Exercise 7-07 0 2 a e i 3 C 4 a d m0 uðy1Þ 2a n x ¼ 52a 5 or 1 5 o b ¼ xay –5 –14 –13 –12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 i a5 –102 –101 –100 –99 0 1 2 3 4 5 6 –98 7 8 –8 –7 –6 –5 –4 –3 –2 –1 0 –2 –1 0 1 2 3 4 2 9 1 a f k 2 a d g 3 a e 4 a g 5 a d 7 2 2 logx 30 logx 2 logx 14 1.2042 0.3979 1 b 3 h 5 b 7 e b 3 c 2 d g 4 h 4 i l 2 m 1 n b logx 5 c logx 8 e logx 40 f logx 10 h logx 15 i logx 12 b 2.6021 c 3.6021 f 2.2042 g 0.3979 3 c 3 d 2 e 2 i 1 j 4 3 c 1 pffiffiffi 3 0 f 32 loga x or logað xÞ 1 1 2 e 12 j 2 d 0.301 05 h 0.801 05 1 f 0.5 631 Answers Exercise 7-11 Chapter 8 1 a k¼9 d x ¼ 2.5 g k ¼ 1.5 2 a x ¼ 1.425 d x ¼ 0.943 g x ¼ 7.555 j x ¼ 1.011 3 a x¼2 b e x ¼ 72 f 4 a x¼8 b e x¼3 f ffi i x ¼ p1ffiffiffi j 10 m x¼2 n q x ¼ 16 r 5 11.89 12 years 7 a A ¼ 106 g b e h b e h k m¼7 y ¼ 4.5 n ¼ 1.5 x ¼ 2.227 x ¼ 0.428 x ¼ 0.107 y ¼ 0.975 x ¼ 53 c x ¼ 54 x ¼ 13 g x ¼ 54 6 1 x ¼ 1000 c x ¼ 25 1 x¼2 g x ¼ 1000 1 x¼8 k x ¼ 128 x ¼ 15 o x ¼ 12 x¼2 s x ¼ 3.915 6 22.43 23 months b t ¼ 20 days c f i c f i l d ¼ 10 a ¼ 3.5 d ¼ 2.75 x ¼ 2.519 x ¼ 0.661 x ¼ 1.121 k ¼ 2.069 d x ¼ 12 h x ¼ 2 1 d x ¼ 64 pffiffiffi h x ¼ 16 2 1 l x ¼ 25 p x ¼ 0.1 t x ¼ 23.04 1 a 1 2 a 4 3 a 625 b 29 b 12 b 3125 1 2 3 4 6 a D ¼ 190T b i a E ¼ 26.2 h a I ¼ 16D 425 A a h c 1 $ 7.50 2 $15 3 $22.50 b c ¼ 7.5 h 7 C 9 A b x ¼ 13 c x ¼ 8 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 c 3 c 85 c 1 d 69 d 3.2 1 d 25 Exercise 8-01 c t ¼ 58 days Power plus 2 1 a x ¼ 1 25 2 a –3 –2 –1 b –3 –2 –1 c SkillCheck 3.8 km ii 8.55 km c 1 h 5 min b $183.40 c 5.5 h b $33.88 c $67.76 5 b ¼ 2.5a c $45 d 11 8 a F ¼ 0.006m 10 a 22.8 kg e 7.5. It is the same. b 15 L/100 km b 84.1 kg Exercise 8-02 1 a T ¼ 920 s 3 a i 15C c T b 10 h 13 min c 92 km/h 2 C ii 1.8C b i 562.5 m ii 200 m 10 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 3 The solution to D ¼ 12 nðn 3Þ ¼ 100 is not a positive integer. 4 p ¼ 4, q ¼ 3 5 a a¼7 b x ¼ 20 c m¼2 d h¼5 8 6 4 Chapter 7 revision 2 1 a w¼6 d m¼3 2 a m ¼ 25 3 3 a y ¼ 2 d m ¼ 1 g h ¼ 9, 1 4 a u ¼ 1.9 5 Jane: 16, Grace: 13 7 a a ¼ yb x 8 a x0 b e b b e h b 6 b 35 4 c f c c f i c y¼ s¼4 y ¼ 57 p ¼ 10 w ¼ 5 u ¼ 7, 11 m ¼ 2.9 120 m mP 2 ¼ a a ¼ 3 x ¼ 1 23 m ¼ 32 x ¼ 3 x ¼ 7, 1 k ¼ 0, 5 x ¼ 1.4 c a ¼ 1M Mþ1 –4 –3 –2 –1 0 1 2 3 4 –2 –1 0 1 2 3 4 5 6 b x<3 c x 2 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 d x > 5 –9 9 a d 10 a 11 a 12 a 13 a 632 y 16 x > 3 63 ¼ 216 1 0.9542 x ¼ 1.490 –8 –7 –6 –5 –4 –3 –2 b y 7 12 e a < 16 1 b 24 ¼ 16 b 0 b 2.4771 c 0.5229 b x ¼ 0.943 c x ¼ 0.236 –1 c f c c 0 a > 5 x 3 pffiffiffi 3 72 ¼ 7 7 3 d 0.9771 d x ¼ 1.420 500 4 7 9 10 1000 1500 2000 2500 3000 4533 people 5 B a 8 min b 4 people a F ¼ 112 b 6 beats/sec L 1 a y ¼ 16 b x ¼ 114 h 6 14 15 8 a b ¼ 8a c 25 cm 11 a 2.5 h b b ¼ 100 a b 5 friends Exercise 8-03 1 a c 2 a b c 3 a b 4 a d 5 a e 6 a b i £26 ii £57 b i $A28 ii $A92 i approx. £7 ii approx. $A30 i 2 km ii 20 km iii 34 km i 50 furlongs ii 122 furlongs iii 180 furlongs 60 km d 500 furlongs i ¥16 000 ii ¥63 000 iii ¥78 000 i $A250 ii $A760 iii $A920 18C b 10C c 26C 32F e 14F f 86F 4.9 ha b 2 ha c 10.8 acres d 12.2 acres i 32 000 m2 ii 3.2 ha iii 7.8 acres i P620 ii P2100 iii P3600 $A12 c $A5 d P4900 9780170194662 Answers Mental skills 8 b Kate, in 9 minutes d 680 m c 1.5 min e 110 m f The graph shows the distance they move down the slope and this increases as more time passes. 3 a The person starts the journey fast (the graph is steep), then slows down (the graph becomes less steep) before increasing speed again (the graph becomes steeper). b The person is travelling or returning towards a specific place. Initially the person’s speed is fast, then he slows down and stops (the graph is horizontal). c The person starts the journey at a high speed and then gradually slows down to a stop. 4 a H b D c AB d F e E f C 5 2 a e 3 a e b f b f 3 2, 3, 5, 6 4, 10 none of these c g c g 2, 3, 6 5 9 9 3, 5 3 10 4, 9, 10 1 a Temperature increases to maximum on first day then cools down to the initial temperature. Then second day temperature increases to slightly higher maximum than first day and then returns to similar minimum temperature as for previous day. b This graph could indicate four high tides and four low tides in a reasonably regular pattern. c Amount of petrol decreases at constant rate. Petrol tank is filled, then petrol is used at a constant rate. d Height of person increases slowly at first then more quickly (possibly teenage years/ growth spurt indicated by steeper graph) then slows down as final height reached. 2 a B b A c C d B e A 3 a b 100 Time c 60 40 20 0 Time Distance from shop 80 d h d h Exercise 8-05 120 Distance (metres) 2, 5 2 4, 9 4, 9 Height 1 a The cyclist leaves the starting point, travels at a speed of 20 km/h for 1 hour; stops for 1 hour; then continues for another hour at a speed of 10 km/h. At D, the cyclist stops for half an hour, then cycles back towards the starting point at a speed of 30 km/h for 1 hour. b No, as the gradients of the intervals are all different. c The cyclist is moving back towards the starting point. 2 a Kate 133 m/min, Colleen 114 m/min Water level Exercise 8-04 Time 2 4 a f 5 a 6 a 7 a 8 a b Distance Distance 6 a 4 6 8 10 12 Time (seconds) Time Time d Distance Time 7 a i b i i ii B H E F c h c c c c iii v B A F B d i d d d d vi vii A F D C e ix e C e A e A f B f D Exercise 8-06 y = x2 y 10 y = x2 + 2 y = x2 – 1 5 Time C ii B iii A Is the steepest (has the greatest gradient) and must be the fastest ( Jade). ii Is the least steep (the smallest gradient) and must be the slowest (Cameron). iii The slope of this graph is between the other two (Kiet). c Jade stopped to talk to a friend. (Other answers possible.) d This person speeds up slightly and maintains speed for a while, slowing down gradually to a stop. 8 a C b D c E d F e B f A 9780170194662 b g b b b b 1 a Distance c viii iv A B C E 2 –4 –2 –1 2 4 x –5 –10 y = –2x2 y = –x2 b i y ¼ x 2, y ¼ x 2 þ 2, y ¼ x 2 1 ii y ¼ x 2, y ¼ 2x 2 iii y ¼ x 2, y ¼ x 2, y ¼ 2x 2 633 Answers 2 A 3 a F b I g B h L 4 a y ¼ x 2 d y ¼ x 2 9 5 a Exercise 8-07 c i b e A d K H j E y ¼ x2 y ¼ 12 x 2 e k c f J f C G l D y ¼ x 2 14 y ¼ x2 þ 9 1 a y 6 y = 2x2 + 1 y 10 y = (x – 3)2 10 (0, 9) 8 4 2 (3, 0) 8 –2 6 4 0 2 –2 4 6 x y 10 b y = (x – 2)2 8 2 6 –2 –1 1 b (0, 1) 6 a 2 x 4 c concave up d y¼1 2 y –2 0 –2 2 2 6 4 x –2 –1 y = (x + 1)2 y 10 c 2x 1 –2 8 –4 6 4 –6 2 –8 b (0, 2) 7 A 8 a B b G g A h K 9 a i narrower b i wider c i narrower d i wider 10 a x ¼ 4 11 a h 100 –4 y = –3x2 + 2 c x¼0 c i ii ii ii ii b D d J H j L up up down down x ¼ 11 d y¼2 e k iii iii iii iii E F 3 1 5 12 f I l C (0, 1) 0 –2 –2 (–1, 0) –4 d 4 x 2 y 2 –4 –2 0 –2 (3, 0) 2 4 6 8 10 x –4 –6 –8 80 (0, –9) –10 60 e 40 –4 20 y = –(x – 3)2 y 2 (–1, 0) 0 (0, –1)2 –2 –2 4 x –4 1 b 80 m 12 a x ¼ 9 634 2 3 4 c 35.9 m d 4.0 s b x ¼ 14.02 5t –6 e 3.91 s –8 –10 y = –(x + 1)2 9780170194662 Answers f 5 y b (5, 0) –2 –5 2 4 6 8 10 x c y 30 y 50 45 25 20 40 –10 15 35 –15 10 30 5 25 –20 –10 –5 0 –5 –25 (0, –25) y = –(x – 5)2 –30 g y 60 50 y = 3 (x + 20 5 10 x 15 –10 10 –15 5 –20 4)2 –25 –10 –5 0 –5 y = –x3 5 10 x –10 –30 (0, 48) y = 2x3 –15 40 –20 30 –25 –30 20 –35 10 –40 –45 –10 –8 h –6 y 1 –4 –2 –4 0 –2 2 4 x d (1, 0) 0 –1 –50 2 4 x 25 –2 –3 e y 30 y = x3+3 25 20 20 15 15 10 10 5 –4 –5 –10 –5 0 –5 y = –2(x – 1)2 i y 2 (–6, 0) –10 y 30 0 –2 –5 5x 5 0 –10 –5 –5 5 10 x –10 –10 –15 –15 –20 –20 –25 –25 –30 –30 5 10 x y = –x3 – 4 –4 f –6 –8 2 B 3 a D e B y = – 1 (x + 6)2 4 c A g F d E h C 35 15 30 25 Exercise 8-08 20 5 0 –10 –5 –5 15 5 10 x –15 10 y –25 –30 5 5 5 10 x –10 –20 y = x3 – 2 10 0 –10 –5 –5 –10 1 a y = 2x3–3 40 20 10 b G f H y 50 45 25 (0, –9) –10 g y 30 –15 –20 y= –x3 +2 –25 –30 –35 –40 –10 –5 0 –2 5 10 x –45 –50 –55 –5 –10 9780170194662 635 Answers h i y 50 45 10 40 5 35 –10 –5 0 –5 30 25 e y y = 1 x3 + 4 2 15 f y y = –x5 – 2 y = x4 5 10 x y = (x − 1)4 –10 20 y y = –x5 0 1 10 5 –10 –5 0 –5 5 10 x x 3 a Move up 4 units. c Move left 3 units. e Move right 3 units. –10 –15 x –2 1 15 0 –2 b Move right 5 units. d Move up 4 units. f Move left 2 units. –20 Exercise 8-10 –25 –30 1 a –35 x y –40 –45 3 2 1 23 1 2 0 1 2 2 1 3 2 3 –50 b c –55 y 10 –60 –65 –70 –75 5 y = –3x3 – 2 –80 (1, 2) 2 a A b E f D g H 3 a y ¼ 2x 3 3 c G d C h I i B b y ¼ 4x 3 þ 1 e F c I h D b e G 0 –10 –5 –5 Exercise 8-09 1 a E f A 2 a 5 y = 2x x 10 –10 b H g F y d C i B y= y 3x2 +1 d y ¼ x, y ¼ x 2 a y 10 y = (x − 2) 2 5 4 y = x2 0 c 2 x 0 d y –10 1 y = 3x2 y y = –x3 0 y = –x3 + 2 –2 5 4 y= x x 10 –5 –10 0 2 0 –5 x (1, 4) x b y 10 x 5 y = –2x4 y = –2(x + 2)4 –10 –5 0 (1, –2) 5 2 y=– x x 10 –5 636 –10 9780170194662 Answers c b y 10 y 10 5 5 (1, 3) –10 3 a –5 3 y= x x 10 5 0 –5 –5 –5 –10 –10 t 1 2 3 4 5 6 7 8 9 10 S 1000 500 333 250 200 167 143 125 111 100 b –10 c 5 10 x 2 y=– x –3 (1, –5) y 10 s 5 1000 (2, 2) 800 –10 0 –5 y = x –2 1 5 10 x 600 –5 400 –10 200 d y 10 t 2 4 6 8 10 c The time taken is always positive and it is impossible to travel with zero time. Also, you cannot divide by zero. d Yes, when t ¼ 2 h, s ¼ 500 km/h and when t ¼ 4 h, s ¼ 250 km/h. 4 a k¼3 b y ¼ 3x 5 a y 10 5 –10 0 –5 y=– 3 x+2 10 x (1, –1) 5 –5 5 (1, 3) y = 1x + 2 0 –10 –5 5 –5 –10 9780170194662 10 x –10 6 a c ¼ 1, k ¼ 6 7 a L 10 20 W 80 40 b y ¼ 6x þ 1 30 27 40 20 50 16 60 13 70 11 80 10 90 100 9 8 b WL ¼ 800 or W ¼ 800 L c If the length or width equals zero, the block of land doesn’t exist. 637 Answers d 4 a W y 10 100 y = 2x 90 5 80 70 (0, 1) 60 50 –5 (0, –1) (0, –1) 5 –10 10 x 40 –5 30 20 10 0 –10 0 10 20 30 40 50 60 70 80 90 100 y= – 2x L e As the length increases, the width decreases. The graph flattens out and gets closer to the horizontal axis, but never touches it (an asymptote). f As the length decreases, the width increases. The graph is steeper and gets closer to the vertical axis, but never touches it (an asymptote). 8 A b They are the same graph reflected in the x-axis c y ¼ ax 5 y 10 8 6 4 Exercise 8-11 1 a 2 y = 3x + 1 y = 5x y = 3x y y = 2x 10 –10 –8 –6 –2 –4 2 8 –2 6 –4 4 x Same shape, shifted down 2 units. y 6 a y = 2x 4 10 2 (0, 1) –10 y = 3x – 1 –5 8 5 10 x 6 b 1 c For y ¼ ax where a ¼ 2, 3 or 5, as a increases the graph increases more rapidly as x becomes larger in the first quadrant. 2 a –x y x y=4 10 4 2 (0, 1) y=4 –10 b 8 –5 y = 3–x 6 5 10 10 x y 8 4 6 2 4 (0, 1) –10 b i 3 B 638 –5 x y¼4 5 10 2 x ii y ¼ a (0, 1) x –10 –5 5 10 x 9780170194662 Answers y c Exercise 8-12 y = –4x –10 –5 (0, –1) 10 x 5 –2 –4 –6 –8 –10 y d –10 –5 (0, –1) 10 x 5 1 a c e 2 D 3 a c e g i k 4 B 5 a c e 6 a centre (0, 0), r ¼ 2 centre (0, 0), r ¼ 8 centre (0, 0), r ¼ 9 b centre (0, 0), r ¼ 6 d centre (0, 0), r ¼ 10 f centre (0, 0), r ¼ 5 centre (2, 4), r ¼ 7 centre (9, 12), r ¼ 15 pffiffiffiffiffi centre (6, 1), r ¼ 10 pffiffiffi centre (0, 0), r ¼ 6 2 centre (2, 0), r ¼ 8 centre (3, 4), r ¼ 9 b d f h j l (x 1)2 þ (y þ 2)2 ¼ 9 (x þ 3)2 þ (y 2)2 ¼ 100 (x þ 6)2 þ (y 2)2 ¼ 5 b (x 10)2 þ (y þ 11)2 ¼ 4 d x 2 þ (y þ 1)2 ¼ 1 f (x þ 1)2 þ (y 5)2 ¼ 8 y 2 –2 1 –4 –3 –6 –2 y = –2 –x –10 y 10 1 –1 x 2 –1 (–1, –1) –2 –8 e centre (3, 1), r ¼ 1 centre (0, 3), r ¼ 2 centre (5, 8), r ¼ 4 pffiffiffi centre (2, 1), r ¼ 5 2 centre (4, 3), r ¼ 2.5 centre (0, 1), r ¼ 13 –3 b y = 4x + 1 10 y 8 5 6 (0, 4) 4 –10 –5 5 10 x 2 –5 –10 –5 f 10 x 5 y y 5 c y = 4x – 1 10 (1, 0) –5 5 x 8 –5 6 d y 4 –10 2 –10 –5 –8 –6 –4 (–5, –2) 5 10 x 2x –2 –2 –4 –2 –6 7 y¼4 x 9780170194662 639 Answers 7 a e 8 a d g 1, 1 b 9, 3 c 16, 4 d 4, 2 49 7 5 f 94, 32 g 14, 12 h 25 4, 2 4, 2 (3, 1), r ¼ 5 b (4, 2), r ¼ 7 c (2, 5), r ¼ 6 (10, 6), r ¼ 1 e (2, 4), r ¼ 5 f (6, 3), r ¼ 4 pffiffiffi (3, 10), r ¼ 9 h (4, 1), r ¼ 2 3 y d x2 + y2 = 49 5 –5 Exercise 8-13 1 a g 2 a f 3 a P P G F b h b g L L J C c i c h E P H E d j d i y 10 y= L E D B x2 e k e j C P A I (7, 0) x 5 f L l C –5 e y = 1 x2 2 y 10 –3 8 5 6 (2, 1) –10 –5 5 10 x 4 2 –5 b y 10 (2, 2) y = 5x –10 –5 f 8 5 10 x y 10 6 5 4 4 (2, 0) 2 –10 (0, 1) –10 –5 c –5 y 4 –10 y = –2x + 4 g y = 4 – x2 y 12 x2 + y2 = 144 2 10 (2, 0) –4 –2 x –5 10 x 5 10 5 2 4 x 5 –2 –4 –12 –10 –5 5 10 12 x –6 –5 –8 –10 –10 –12 640 9780170194662 Answers 4 a 1 5 a H e Q 6 a D e H 7 a i b i c i d i e i f i 8 a b b f b f exponential exponential exponential hyperbola hyperbola hyperbola 3 C C E B c c g c g ii ii ii ii ii ii 6 P E C F 1 2 2 none 23 none iii iii iii iii iii iii d 1 d H h H d A h G y¼0 y¼1 y ¼ 3 x ¼ 0, y ¼ 0 x ¼ 3, y ¼ 0 x ¼ 0, y ¼ 2 d 60 40 30 20 10 (5, 0) 2 4 x 6 e 8 10 12 y 10 (1, 6) 5 y= –10 y = 2(x – 5)2 50 y 10 5 y 5 –5 y=– 1 x+4 6 x 10 –4 –10 x (–3, –1) –1 4 5 10 x –5 –5 –10 –10 b f y 10 y 10 y = 3x + 2 8 (–5, 7) 8 6 (–5, 5) 6 4 4 2 (0, 3) 2 –10 –10 –5 5 10 –2 y 10 c 5 –3 9780170194662 –2 –1 y = x3 + 3 –8 –6 –4 Power plus 1 y 10 5 2 3 x x 3 1 2 –2 y= 1 +2 x–1 (2, 3) 1 x –10 –5 5 –5 –5 –10 –10 10 x 641 Answers 2 a centre (0, 0) and r ¼ 4 7 y y y = 2 (x + 3)2 20 (0, 18) 15 y = 16 – x2 4 10 5 2 (–3, 0) –8 –2 –4 2 x 4 –6 –4 8 y = x3 + 2 5 2 y 6 –4 2 –2 y = 25 – x2 x 4 –5 4 –10 9 y 2 –2 –4 x 2 y 10 b centre (0, 0) and r ¼ 5 –6 –2 –2 2 4 6 –1 x 1 –2 2 3 5 x 4 (1, –3) –4 –6 c centre (0, 0) and r ¼ 3 –8 y –10 4 y = – 3x4 y = –3(x – 2)4 10 y 10 2 5 –6 –4 –2 2 6 x 4 2 –2 –10 y = – 9 – x2 –4 –1 5 –5 –10 Chapter 8 revision 1 H ¼ 310.5 4 d –5 2 x+1 10 x y= (1, 1) 11 a 2 10C 3 a £46 y 10 y = 4x b $A85 8 6 4 2 (0, 1) t 5 a B 6 a C 642 b C b F c A c A d E e D f B –10 –5 5 10 x 9780170194662 Answers b Chapter 9 y = 4–x y 10 SkillCheck 8 1 a 64 2 a 0.8480 d 64.9839 3 a 4548 4 a 64370 6 4 (0, 1) –5 c 5 y –10 –5 26 0.7760 13.9884 3311 69410 b e h b e b 5 b b b 14.2 cm 5.1 cm 59.0 cm 567 6437 5.7 m 114 m c c f c c 12 0.1539 13.7044 521 2880 Exercise 9-01 2 –10 b b e b b 10 x y = –4x (0, –1) 5 10 x 1 a d g 2 a d 3 a 4 4 7 a 9 a 10 a 64.7 cm 18.5 cm 48.8 cm 3841 5257 73 5120 11.6 m 49º c f i c f 54.5 cm 17.4 cm 17.5 cm 4256 451 6 6.49 m 8 2.6 km 5 4 11.2 m c 28º 43º –2 11 127 m 14 180 m 18 47.7 m –4 –6 1 a 237 f 140 2 a 000 f 125 3 SW 5 a –10 y –10 –5 13 177 m 16 79440 20 970 m 17 480 m Exercise 9-02 –8 d 12 1132 m 15 14290 19 14.5 m y = –4–x (0, –1) 5 b g b g 4 295 c 312 h 090 c 330 h a NNW N 10 x T W E 42° –2 046 d 253 i 180 d 225 i b 337.5 b W P 115 065 270 072 e 210 e 038 j 187 N M E 80° P –4 c P S N S N d P –6 10° W 65° X E W –8 12 a c e 13 a g centre (0, 0), r ¼ 10 centre (0, 0), r ¼ 7 centre (10, 0), r ¼ 15 D b C c B L h G i I 9780170194662 S S –10 b centre (0, 0), r ¼ 6 d centre (5, 6), r ¼ 9 pffiffiffi f centre (7, 10), r ¼ 4 5 d J e E f H j A k K l F 6 7 8 10 12 14 a 22 km a 37 a 12.2 km 45.7 km a 2122 km a 261.08 km E K b 257 b 163 km b 305 9 11 b 330 13 b 167.82 km c a a a 323 18.5 km b NNW 13.509 km b 321 15 km b 26 km 643 Answers 7 a cos 38 b sin 75 c cos 25 d tan 78 e cos 7.3 f sin 64.5 g cos 4025 h tan 9.2 i sin 5925 j tan 1950 k sin 84.5 l tan 40.5 1 8 a p b 1pffiffi c p1ffiffi2 d 12 e p1ffiffi3 2 ffiffi pffiffiffi f 23 g 23 h 3 i 1 j p1ffiffi2 9 a, b Teacher to check. c The tan graph is broken into three sections and repeats itself after 180. It has asymptotes at 90 and 270. Exercise 9-03 1 2 3 5 7 9 10 11 pffiffiffiffiffi a 4 13 cm b 15.6 cm c 23 pffiffiffi pffiffiffiffiffiffiffiffi 800 cm or 20 2 cm b 34.64 cm a c 3516 4 34 a 20.40 cm b 79 a 9.1 cm b 28 6 a 53 m b 114 m 37.5 m 8 a 50 b 868 m a 85 m, 50 m b 99 m apart a \WHF ¼ 52 , \WFH ¼ 38 and \HWF ¼ 90 b Tower is 49.1 m. a 285 b 7 d No f Exercise 9-04 1 a 43 b 16 c 87.45 d 34.8 e 5143 2 a 0 b 1 c 1 5 12 3 cos b ¼ 13 , cos a ¼ 12 13, sin b ¼ 13 9 9 4 sin F ¼ 40 41, sin E ¼ 41, cos F ¼ 41 pffiffi pffiffi 5 2 5 cos Y ¼ 3 , sin Y ¼ 3, sin X ¼ 35 pffiffi pffiffiffiffi pffiffiffiffi 6 cos / ¼ 45, sin / ¼ 411, cos u ¼ 411 pffiffiffi 7 a 8 b 12 c 3 2 d 45 e 30 pffiffiffi 8 35 3 m f 7222 Exercise 9-06 f 30 Mental skills 9 2 a 23 g 56 m 9:7 3 a 17 40 b 34 h 25 n 4:3 b 23 c 57 i 5:9 o 35 c 16 25 d 12 j 5:9 4 p 35 d 14 e 14 f 16 k 9 : 20 l 4 : 5 e 5 24 f 2 25 Exercise 9-05 1 a tan A ¼ 60 b tan Y ¼ 1: 3_ c tan X ¼ 23 91 pffiffiffiffi 9 d tan P ¼ 40 e tan Q ¼ 340 f sin X ¼ 11 61 7 2 g cos X ¼ 25 h sin X ¼ 3 2 a P b P c N d N e P f N g N h P 3 a 0.89 b 0.19 c 0.77 d 0.11 e 0.51 f 0.58 g 0.05 h 0.42 i 0.78 j 0.87 k 0.18 l 0.28 4 a, b Teacher to check. c The graph has a wave shape that repeats itself after 360. Maximum y ¼ 1 at y ¼ 90; Minimum y ¼ 1 and y ¼ 270 d No e Yes, centre of symmetry at (180, 0). f i 0 < y < 180 (1st and 2nd quadrants) ii 180 < y < 360 (3rd and 4th quadrants) 5 a, b Teacher to check. c The graph has a wave shape that repeats itself after 360 Maximum y ¼ 1 at y ¼ 0 and y ¼ 360; Minimum y ¼ 1 and y ¼ 180 d Yes, axis of symmetry y ¼ 180 e No f i 0 < y < 90 and 270 < y < 360 (1st and 4th quadrants) ii 90 < y < 270(3rd and 4th quadrants) g Similarities: Both graphs have the same wave shapes that run between y = 1 and y ¼ 1 and repeat themselves after 360. Differences: The graphs have different x- and y-intercepts 6 a 10 b 70 c 50 d 83 e 65 f 12 644 e centre of symmetry at (180, 0). i 0 < y < 90 and 180 < y < 270 (1st and 3rd quadrants). ii 90 < y < 180 and 270 < y < 360 (2nd and 4th quadrants). 1 a e i 2 a e i 3 a d g 57, 123 7, 173 114 1459 15258 no solution 137 69 45 b f j b f j b e h 143 135 33, 147 13157 11551 163180 136 42, 138 60 c g k c g k c f i 110 100 105 15926 no solution 126520 61, 119 143 45, 135 d h l d h l 130 25, 155 118 17348 126520 154370 Exercise 9-07 1 a 18.4 b 21.1 c 105.0 2 a a ¼ 20.51 b b ¼ 11.91 c c ¼ 12.58 d d ¼ 4.10 e e ¼ 30.85 f f ¼ 3.55 g k ¼ 5.99 cm h w ¼ 29.17 m i p ¼ 8.29 m 3 79 m 4 25 m 5 b 1042 cm 6 a 110 b 131.6 m 7 561 km 8 d 124.7 m 9 b 595 m Exercise 9-08 1 a 27 2 a 44.5 d 67.3 3 a 1497 d 13533 4 a 46 or 134 5 a 75 or 117 b b e b e b b 37 c 54 46.6 c 32.0 18.8 f 31.8 12900 c 1428 12929 f 16213 39 c 55 or 125 d 44 or 136 41 c 84 Exercise 9-09 1 a 5.6 b 13.1 c 35.8 2 a a ¼ 8.30 b c ¼ 54.52 c e ¼ 88.41 d b ¼ 16.33 e d ¼ 19.44 f f ¼ 40.72 3 0.6 m 4 C 5 a Teacher to check. b \ XYN ¼ 180 130 ¼ 50 (cointerior angles on parallel lines) \ XYZ ¼ 50 þ 25 ¼ 75 c 4.4 km 6 47 km 7 a 0 b a2 ¼ b2 þ c2 c With cos 90 ¼ 0, the cosine rule reverts to Pythagoras’ theorem. 9780170194662 Answers Exercise 9-10 1 a 70 2 a 112 3 20.8 b 33 c 109 d 131 b 108 c 121 d 23 4 6440 5 99 e 60 f 83 Exercise 9-11 413.4 m2 b 463.1 cm2 2 132.9 mm e 320.4 cm2 97.4 m2 b 463.6 m2 2 227.6 m e 93.5 m2 225 m b 2770 m2 4 a 130 2 418.9 cm b 173.2 cm2 112 b 37 cm2 1 a d 2 a d 3 a 5 a 6 a c f c f b c c 326.9 mm2 0.1 m2 246.2 m2 152.2 m2 766.07 m2 245.7 cm2 740 cm3 Exercise 9-12 10.2 m b 16.1 mm c 17.1 cm 13.1 m e 3.9 m f 18.2 m 32 b 142 (or 38) c 29 55 e 37 f 125 32 þ 23 ¼ 55 (exterior angle of a triangle) 108.50 m c 89 m 1 b i 15.4 ii 15.4 The results are the same. The sine rule sind90 ¼ sin12:8 56 becomes d ¼ sin12:8 56 (since sin 90 ¼ 1), which is the same result when using the sine ratio. 5 7.5 km 6 486 km 9 10 11 12 13 14 15 16 a 57 a 1172 a 0.4 m a 8154 or 986 a 6.8 m a 96 a 165 cm2 15.5 cm b b b b b b b 1 a 5 2 a 11 3 a b 13 b 1 b y pffiffiffiffiffiffiffiffiffiffiffi 31 60 120 30, 150 pffiffiffi 2 4 6 y=x+1 4 –4 –2 0 2 4 2 x –2 30 135 45, 135 45, 135 a 10.9 m a 64590 13 4 a 320 a 1281 km a 48 b b b b b 4.4 m 48590 140 024 pffiffiffi 48 3 y 4 c –6 –4 –2 y = x– –1 2 d 4 6 2 4 6 x 6 4 2 –6 –4 –2 0 x –2 y x+y=4 270° 2 y e 180° 0 –2 y = cos θ 90° 4x –4 6 a 33 cm b 65 pffiffiffi c p48ffiffi2 ¼ 24 2 0.5 2 –2 c 11.5 cm c 57120 1.0 0 –2 –4 y=3–x 0 y 2 1 2 c f i 5 d 1 d 5 12 6 Chapter 9 revision 8 c 6 c 7 2 b b e h 4 70 272, 15258 136.4 mm 4937 7.6 cm 125 30 m2 SkillCheck Power plus pffiffiffi 1 a 31 2 a 45 d 150 g 60, 120 3 Teacher to check. c c c c c c c Chapter 10 1 a d 2 a d 3 a b 4 a c 1 2 3 5 7 87 6533 14.8 cm 7724 or 10236 112.1 mm 56 286 m2 6 4 360° θ 2 –0.5 –4 –1.0 9780170194662 –2 0 2 4 6x –2 645 Answers f c m ¼ 43, b ¼ 5 y y 6 6 2x – y = 5 4 4 2 2 0 –2 4 y =– – x + 5 3 2 4 6 x 0 –2 –2 2 4 6 x –2 –4 –6 Exercise 10-01 4 a yes b no c no d yes e no 5 a For y ¼ 2x þ 1, when x ¼ 2, y ¼ 2 3 2 þ 1 ¼ 5 [ (2, 5) lies on y ¼ 2x þ 1 For x þ y ¼ 7, when x ¼ 2, y ¼ 5, 2 þ 5 ¼ 7 [ (2, 5) lies on x þ y ¼ 7 b (2, 5) 6 a m ¼ 2, b ¼ 3 y y = –2x + 3 f no 1 a x ¼ 3, y ¼ 1 2 a x ¼ 1, y ¼ 2 b x ¼ 2, y ¼ 1 c x ¼ 1, y ¼ 5 b x ¼ 5, y ¼ 9 c x ¼ 1, y ¼ 2 d x ¼ 12, y ¼ 2 12 e x ¼ 2, y ¼ 9 f x ¼ 5, y ¼ 4 g x ¼ 12, y ¼ 6 12 h x ¼ 3, y ¼ 2 x ¼ 5, y ¼ 1 12 j x ¼ 5, y ¼ 8 i k x ¼ 1 12, y ¼ 2 12 3 a l x ¼ 4, y ¼ 0 y 6 6 y = 1 – 2x 4 4 2x + y = 4 2 –4 –2 2 0 2 –6 –4 –2 –2 b m¼ b The lines are parallel. b ¼ 2 Exercise 10-02 y y = –5 x – 2 2 4 2 –2 –2 b x ¼ 5, w ¼ 4 c g ¼ 2, h ¼ 25 e q ¼ 5, r ¼ 4 f k ¼ 4 35, x ¼ 5 g c ¼ 1 12, e ¼ 1 2 a d ¼ 14, k ¼ 6 h k ¼ 3, y ¼ 2 b a ¼ 1, c ¼ 1 i a ¼ 2, f ¼ 2 c h ¼ 3, y ¼ 4 d e ¼ 3, x ¼ 13 e q ¼ 3, w ¼ 6 12 f c ¼ 2, p ¼ 3 1 a d ¼ 3, k ¼ 2 6 0 2 x –2 –4 5 2, 0 2 4 6 x d n ¼ 3.25, p ¼ 1 g 3 a d g j 23, m¼ y¼4 q ¼ 3, w ¼ 3 g ¼ 1, n ¼ 3 q ¼ 1, w ¼ 4 a ¼ 2, f ¼ 2 5 12 b e h k h a ¼ 1, r ¼ i x ¼ 2, w ¼ 2 m ¼ 9, x ¼ 7 c d ¼ 23, h ¼ 7 h ¼ 0, m ¼ 2 f e ¼ 4, y ¼ 3 a ¼ 12, d ¼ 12 i k ¼ 5, p ¼ 2 c ¼ 64, r ¼ 38 l x ¼ 4, y ¼ 3 –4 646 9780170194662 Answers Exercise 10-03 1 a d 2 a c e g i Chapter 10 revision x ¼ 2, y ¼ 5 x ¼ 2, y ¼ 2 x ¼ 9, y ¼ 21 x ¼ 14, y ¼ 2 x ¼ 2, y ¼ 2 x ¼ 7, y ¼ 3 x ¼ 2 23, y ¼ 1 b e b d f h j x ¼ 35, y ¼ 3 45 c x ¼ 7, y ¼ 2 x ¼ 5, y ¼ 1 f x ¼ 4, y ¼ 2 x ¼ 5, y ¼ 3 x ¼ 3, y ¼ 1 x ¼ 7, y ¼ 4 x ¼ 3, y ¼ 2 12 x ¼ 3 15, y ¼ 4 35 1 a x ¼ 2, y ¼ 2 2 a b x ¼ 4, y ¼ 0 y 6 y = 6 + 2x Exercise 10-04 1 5 8 9 10 11 12 0 135 50 195 100 255 0 0 50 150 100 300 –6 c –2 0 (–4, –2) –2 –4 2 6x x ¼ 4, y ¼ 2 b y 6 y = 3 – –x 2 4 y = 2x – 7 $ 2 R = 3n (4, 1) 300 –2 C = 135 + 1.2n 200 4 –4 R ¼ 3n n R 4 2 285 2 680 3 b 364 4 12 Aaron 36, Sejuti 12 6 16 7 black 36, colour 24 Pie: $3.60, Sausage roll: $2.70 Supreme 32, Vegetarian 13 Strawberries $3.50; Blueberries $4.99 b 20-cent coins: 154, 50-cent coins: 91 a Teacher to check. b C ¼ 135 þ 1.2n n C y=x+2 0 2 4 8 x 6 –2 –4 100 –6 0 10 20 30 40 50 60 70 80 90 100 d n ¼ 75 x ¼ 4, y ¼ 1 c Mental skills 10 2 3 1 4 2 a e i 5:9 m 9:7 3 a 17 40 4 5 1 6 5 7 5 6 b f j 5:9 n 4:3 b 2 3 c 16 25 c g k 9 : 20 o 35 d 14 y 1 2 2 5 d h l 4:5 4 p 35 5 e 24 y = 4 – 3x 4 f 2 25 1 a x ¼ 1 12, w ¼ 4 12, y ¼ 5 12 7 3 b a ¼ 1 13 , c ¼ 4 13 , d ¼ 8 11 13 3 4 p ¼ 11 13 , m ¼ 18 11 13, n ¼ 13 13 Teacher to check. ae bd ¼ 0 and a fraction cannot have denominator 0. x ¼ 3, y ¼ 1 Teacher to check. 1 x ¼ 2, y ¼ 2 ii x ¼ 28, y ¼ 16 iii x ¼ 11 , y ¼ 2 20 33 9780170194662 y=x 2 Power plus c 2 a b c d i 6 (1, 1) –4 –2 0 2 4 x –2 –4 x ¼ 1, y ¼ 1 647 Answers y d 2 a d g j 3 a 4 a d g y = 2x + 3 8 (2, 7) 6 4 y=9–x 2 0 –2 2 4 6 1 a d g j m 2 a d g j m p 3 a d g j 4 8 x ¼ 2, y ¼ 7 y 8 y = 2x + 1 6 (2, 5) 4 2 –2 –2 x+y=7 0 2 4 6 8 x x ¼ 2, y ¼ 5 y 6 f 4 y = –1 – x y = 5 – 2x 2 4 6 x –4 –6 (6, –7) 3 a c e 4 a d 5 a c e x ¼ 6, y ¼ 7 m ¼ 5, y ¼ 9 12 b x ¼ 2, y ¼ 13 a ¼ 1, d ¼ 1 d x ¼ 6, y ¼ 15 x ¼ 5, y ¼ 2 f d ¼ 3, w ¼ 10 x ¼ 2, y ¼ 11 b m ¼ 1, p ¼ 3 c h ¼ 10, t ¼ 4 a ¼ 3, c ¼ 12 e x ¼ 1, y ¼ 1 f p ¼ 12, q ¼ 4 1600 adults, 900 children b 18 DVDs, 12 CDs $38 d 28 cheesecakes, 47 mudcakes 120 boys, 93 girls Chapter 11 SkillCheck 1 a x ¼ 5 d u ¼ 7 or 4 648 m ¼ 7 or 3 k ¼ 0 or 3 w ¼ 0 or 23 x ¼ 13 or 112 c ¼ 13 or 14 y ¼ 2 or 112 t ¼ 215 or 1 y ¼ 34 or 12 c ¼ 1 or 125 g ¼ 212 y ¼ 3 or 4 x ¼ 212 or 3 m ¼ 17 or 1 t ¼ 32 or 5 f ¼ 0 or 12 b e h k n b e h k n q b e h k d ¼ 3 or 7 t ¼ 7 or 0 n ¼ 12 or 3 c ¼ 52 h ¼ 1 or 12 g ¼ 1 or 112 p ¼ 112 or 4 a ¼ 12 or 113 e ¼ 14 or 112 m ¼ 23 or 56 f¼6 t ¼ 212 or 12 p ¼ 4 or 7 d ¼ 73 or 12 w ¼ 16 or 3 c f i l o c f i l o r c f i l y ¼ 5 or 3 p ¼ 0 or 3 a ¼ 12 or 35 f ¼ 12 e ¼ 57 or 1 d ¼ 1 or 23 x ¼ 25 or 112 w ¼ 114 or 3 q ¼ 3 or 123 w ¼ 4 or 113 h ¼34 or 1 u ¼ 18 or 5 e ¼ 1 or 5 h ¼ 5 a ¼ 2 or 13 Exercise 11-02 2 0 –6 –4 –2 –2 2y(7 y) 2(3w 5)(3w þ 5) (m 8)(m þ 7) (x 6)(x þ 4) d y ¼ 10 (2y 5)(3y þ 8) (2y þ 5)(4y þ 7) (4d þ 5)2 Exercise 11-01 x –2 e (4 m)(4 þ m) b (d 11)(d þ 11) c 5p(2p þ 5) e 5(x 8)(x þ 8) f (k þ 4)(k þ 1) h (y 8)(y 2) i (u þ 13)(u 5) k (w 7)(w 3) l y ¼ 2 b y ¼ 10 c y¼5 (3a þ 1)(a þ 3) b (5x þ 2)(x 3) c (3t 1)(5t þ 4) e (5v þ 3)(v 7) f (3h 4)(5h 1) h (4p 3)(3p þ 5) i b m ¼ 2 e k ¼ 0 or 3 c x ¼ 2 or 1 f w¼5 1 a x 2 þ 2x þ 1 ¼ (x þ 1)2 b p 2 6p þ 9 ¼ (p 3)2 2 2 c m 8m þ 16 ¼ (m 4) d k 2 þ 4k þ 4 ¼ (k þ 2)2 2 2 49 7 2 e y þ 7y þ 4 ¼ y þ 2 f w2 3w þ 94 ¼ w 32 2 5 2 g x2 þ x þ 14 ¼ x þ 12 h h2 5h þ 25 4 ¼ h 2 2 7 5 2 i a2 þ 72 a þ 49 j v2 53 v þ 25 16 ¼ a þ 4 36 ¼ v 6 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 2 a d ¼ 3 þ 7, 3 7 b x ¼ 5 þ 5, 5 5 pffiffiffi pffiffiffi pffiffiffiffiffi pffiffiffiffiffi c p ¼ 1 þ 10, 1 10 d y ¼ 1 þ 2, 1 2 pffiffi pffiffi pffiffi pffiffi 3 23 3 e m ¼ 1þ22 5 , 122 5 f t ¼ 2þ3 , 3 3 pffiffiffiffi pffiffiffiffi pffiffiffiffi pffiffiffiffi 2þ 42 2 42 6þ 82 6 82 g c¼ 2 , h w¼ 2 , 2 2 pffiffi pffiffi pffiffiffiffi pffiffiffiffi i n ¼ 2þ3 7 , 23 7 j e ¼ 3þ2 71 , 32 71 p pffiffi ffiffi pffiffiffi pffiffiffiffi k d ¼ 2 þ 5, 2 5 l x ¼ 3þ44 2 , 344 2 pffiffiffi pffiffiffi pffiffiffi 3 a h ¼ 1 6 b r ¼1 2 c m ¼ 3 7 pffiffiffiffiffi pffiffiffiffiffi d w ¼ 6, 10 e a ¼ 5 30 f y ¼ 4 19 pffiffiffi pffiffiffiffiffi g p ¼ 6 41 h h ¼ 2 2 i u ¼ 7, 2 pffiffiffiffi pffiffiffiffi pffiffiffiffi j d ¼ 12 29 k c ¼ 92 73 l e ¼ 52 17 pffiffiffiffi pffiffi pffiffiffiffi m y ¼ 32 41 n b ¼ 12 21 o q ¼ 32 5 pffiffiffiffi pffiffiffiffi p g ¼ 74 73 q x ¼ 3 12 , 1 r u ¼ 23 22 4 a x ¼ 11.20 or 0.80 b m ¼ 0.43 or 16.43 c g ¼ 4.65 or 0.65 d h ¼ 1.27 or 2.77 e w ¼ 1.27 or 0.47 f y ¼ 1.14 or 1.47 g p ¼ 2 or 1.33 h e ¼ 1.13 or 0.88 i n ¼ 1 or 2.5 9780170194662 Answers Exercise 11-03 pffiffiffi 1 a x ¼ 3 7 pffiffiffiffi d k ¼ 32 29 pffiffiffiffi g u ¼ 72 61 pffiffi j c ¼ 13 7 pffiffiffiffi m d ¼ 2 2 14 b m¼ e h k n pffiffiffiffi 5 37 2 c w¼4 pffiffiffi y¼2 5 pffiffiffiffi a ¼ 34 65 pffiffiffiffi e ¼ 58 57 pffiffiffiffi a ¼ 53 31 pffiffiffiffiffi 13 pffiffiffiffi p ¼ 12 21 q ¼ 15 , 1 pffiffiffiffi x ¼ 43 10 t ¼ 2 12 , 1 pffiffiffiffiffiffi n ¼ 5 4 113 pffiffiffiffi x ¼ 38 41 pffiffi g ¼ 15 6 pffiffi p ¼ 23 7 pffiffiffiffi y ¼ 34 89 f i l o p y ¼ 23 , 2 pffiffiffiffiffiffi 2 a y ¼ 910141 pffiffiffi d k ¼ 2 5 pffiffiffiffi g h ¼ 94 17 pffiffiffiffi j u ¼ 2 5 14 q k ¼ 56 , 1 3 a k ¼ 8.89, 0.11 d n ¼ 3.19, 0.31 b c ¼ 1.41, 1.41 c m ¼ 2.65, 2.65 e p ¼ 0.85, 2.35 f w ¼ 0.30, 1.13 r pffiffiffiffi 22 b m ¼ 13 c pffiffiffiffi e m ¼ 3 6 21 pffiffi h w ¼ 13 7 pffiffiffiffi k a ¼ 26 58 f i l g x ¼ 2.39, 0.28 h h ¼ 3.83, 1.83 i x ¼ 1.62, 0.62 j a ¼ 4, 9 m t ¼ 8.09, 3.09 k v ¼ 1.48, 1.48 l c ¼ 2.31, 0.69 n x ¼ 4.27, 7.73 o d ¼ 3.31, 1.81 Exercise 11-04 d m ¼ 1, 2 e k ¼ 1, 3 pffiffiffi pffiffiffi c w ¼ 2, 6 pffiffiffi f w ¼ 2, 3 4 g x ¼ 1, 13 pffiffiffi j p ¼ 3 5, 1 h y ¼ 45 , 1 25 i a ¼ 1 12 , 12 pffiffiffi 1 a y ¼ 2, 2 2 2 a d 3 a d m 1.2, 2.8 y 1.4, 1.3 w ¼ 5 m ¼ 1.4 b m ¼ 4 k g¼ 1 2, 2 l c ¼ 2, b e b e x 0.9, 1.4 w 1.2, 1.9 x ¼ 2, 1.3 y ¼ 1.2 b e h k $1200, $900 $2700, $1800 $600, $1000 $2100, $2800 c f c f 1 5 a 0.9, 1.3 e 0.5 k ¼ 2 v ¼ 2, 1.4 Mental skills 11 2 a d g j $100, $50 $500, $1500 $3000, $600 $800, $3200 c f i l $160, $560 $900, $2100 $550, $440 $2000, $1200 Exercise 11-05 1 3 5 7 9 10 12 m by 8 m 2 40 m by 35 m 42 m by 24 m 4 Length 58 m, width 38 m Length 13 m 6 15 m by 12 m 0.52 s, 3.08 s 8 a 4000 m b 2000 m c 24.5 s a 34 m b 3.9 s c i 0.5 s and 1.9 s ii 3.5 s 8 or 9 11 24, 25 or 24, 25 12 35 or 34 y 6 5 4 3 2 1 0 –4 –3 –2 –1 –1 ii 3 and 1 iii 3 iv x ¼ 3 and 1, same as x-intercepts b i x 3 2 1 0 1 2 y 0 3 2 3 12 25 x y 9780170194662 3 2 1 0 1 0 3 42 y 6 5 y = 2x2+ 7x + 3 4 3 2 1 0 –4 –3 –2 –1 –1 1 2 3 4 x –2 –3 ii 3 and 12 iii 3 iv x ¼ 3 and 12, same as x-intercepts c i x 3 2 1 0 1 y 18 5 4 9 10 2 7 3 0 2 0 3 3 y = 2x2 – 3x – 9 y 6 4 2 –4 –3 –2 –1 0 –2 –4 –6 –8 –10 –12 1 2 3 4 x ii 112 and 3 iii 9 iv x ¼ 112 and 3, same as x-intercepts d i x 3 2 1 0 1 y 15 8 3 0 1 y 8 7 6 5 4 3 2 1 0 1 a i 1 2 3 4 x –2 –3 –2 –1 –1 Exercise 11-06 y = x2+ 4x + 3 y = x2 – 2x 1 2 3 x –2 0 3 1 8 2 15 3 24 ii 0 and 2 iii 0 iv x ¼ 0 and 2, same as x-intercepts 2 a 5 b 3 c 0 649 Answers 3 a y y b 4 a i 4, 10 iv (3, 49) ii 40 v concave up iii x ¼ 3 y 0 x 4 0 −2 x −4 0 c d y −40 y 10 –3 0 0 y e (3, −49) b i 0, 3 iv 112, 214 –5 x –15 10 x ii 0 v concave up y 1 2 22 x 0 y f 3 1 1 (1 2_, −2 _4) 8 −1 − 2_ 0 x −4 3 −2 g y h −2 0 x c i No x-intercepts iv 34, 278 4 ( − 3_4, 2 7_8 ) 0 5 x 0 x 1 13_ x −20 d i 0.7, 6.7 iv (3, 14) ii 5 v concave down y i iii x ¼ 34 y y −5 x ii 4 v concave up 5 −1 0 iii x ¼ 112 iii x ¼ 3 (3, 14) y 5 –0.7 0 0 1 _ −2 2 2 e i 4.2, 1.2 iv ð112, 30Þ 650 x¼3 b x¼0 c x ¼ 212 x ¼ 1 e x ¼ 112 f x¼3 i x¼3 ii (3, 1) b i x¼5 ii (5, 16) i x¼1 ii (1, 9) d i x¼4 ii (4, 25) i x ¼ 12 ii (12, 2434) f i x ¼ 4 ii (4, 80) i x¼4 ii (4, 48) h i x ¼ 14 ii ð14, 114Þ i x ¼ 16 ii ð16, 114Þ y ¼ (x þ 2)2 3, (2, 3) y ¼ ðx þ 52Þ2 1014, 52, 1014 2 y ¼ (x 1) þ6, (1, 6) y ¼ (x 3)2 9, (3, 9) y ¼ (x 1)2 þ 1, (1, 1) y ¼ 2ðx 54Þ2 98, 114, 118 ii 21 v concave down iii x ¼ 112 y Exercise 11-07 1 a d 2 a c e g i 3 a b c d e f x 6.7 x (−11_, 30) 2 21 –4.2 0 f i 0.4, 7.6 iv (4, 13) 1.2 x ii 3 v concave up iii x ¼ 4 y 3 0 0.4 7.6 x (4, −13) 9780170194662 Answers g i No x-intercepts iv (0.7, 1.55) ii 4 v concave up iii x ¼ 0.7 y 3 4 5 6 7 a a a a pffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi m ¼ 52 37 b d ¼ 7 4 137 c k ¼ 1 6 253 x 0.30, 3.30 b h 1.55, 0.80 c w 5.37, 0.37 pffiffiffi m ¼ 2, 2 b y ¼ 2 c x ¼ 1, 2 32 and 34 b 14 m 3 22 m y = 2x2 + 5x –7 y 4 –3 _12 (−0.7, 1.55) h i 0, 4 iv (2, 8) –4 x 0 ii 0 v concave down –8 iii x ¼ 2 (2, 8) y 8 a x 1 –7 b y (–1 1 , 5 1) y 3 3 x –4 0 0 6 4 x y = –8x – 3x2 Axis : x = 1 2 y = x – 2x – 24 x –2 2 3 –24 0 Axis : x = –1 1 3 (1, –25) i i 112, 2 iv 134, 18 ii 6 v concave down y 0 iii x ¼ 134 c y (13_4 , 1_8) 11 2 2 y = 5x2 + 3x – 8 x –1 −6 (3, 9), (2, 4) (4, 26), (2, 14) (2, 18), (13, 11) (7, 39) (1, 2), (2, 1) (0, 3), (3, 0) (1, 7), 125, 5 1 (1, 3), 12, 2 x 1 Axis : x = –0.3 0 –8 (–0.3, –8.45) 9 a (112, 412), (1, 2) c (0, 4), (4, 0) Exercise 11-08 1 a c e g 2 a c e g 3 5 b d f h b d f (0, 0), (4, 32) (1, 5), 115, 715 (2, 22), (5, 20) (2, 0), (1, 3) (0, 1), (1, 0) (0, 5), (5, 0) (3, 5), 212, 6 b (5, 13), (1,1) d (4, 2), (2, 4) Mixed revision 3 1 y x+y=6 6 4 (3, 3) 2 Power plus 1 a Teacher to check. 2 Intersects twice –6 –4 –2 0 pffiffiffiffi 7 37 6 b x ¼ 43, 1, 3 72 cm 3 24 cm 6 x –4 –6 Chapter 11 revision 9780170194662 4 –2 y = 2x – 3 1 a m ¼ 2, 32 d y ¼ 2, 6 g n ¼ 3, 5 pffiffiffiffiffi 2 a y ¼ 2 11 2 b x ¼ 0, 5 e w ¼ 43, 5 h d ¼ 4 12 , 1 12 pffiffiffiffi b p ¼ 34 33 c p ¼ 6, 8 f k ¼ 34, 12 i x ¼ 1 12 , 2 12 pffiffi c w ¼ 15 6 x ¼ 3, y ¼ 3 2 a E17.5 3 a m ¼ 6.8 cm 4 a t ¼ 99:96 S b AU$86 b k ¼ 22.7 m b 9.52 m/s c E98 c d ¼ 3.7 m c 9.70 s 651 Answers 5 a x¼1 7 a 29 9 b (1, 2) c 43 b 45 6 f ¼ 3, ypffiffi¼ 3 8 x ¼ 23 7 y 3 y = 2x3 – 2 4 3 2 4 1 –4 –3 –2 –1 0 –1 1 2 3 4 x –2 –3 5 –4 –5 10 11 12 15 18 2 340 cm a m ¼ 14 or 1 b y ¼ 45 or 12 c w ¼ 52 or 1 d ¼ 32.5 m 13 b 240 children 14 C a ¼ 40 30 16 x ¼2 12, y ¼ 5 12 17 52, 128 y 7 8 (–1, 4) 0 x pffiffi a p1ffiffi2 b 23 c p1ffiffi3 0 x ¼ 3, 2 21 y ¼ 37 56 a 4w 2 5w 1071 ¼ 0 b 17 m 3 63 m (4, 1) and (1, 4) 24 a 301 km b 114 25 C Chapter 12 SkillCheck 1 C 2 a 3 5 b No, P(10c coin) ¼ 12 , P(20c coin) ¼ 13, P(50c coin) ¼ 14 3 a 13 b 13 c 56 4 a 0 b 1 5 0.4 6 B 7 0.15 Exercise 12-01 1 a b c d i 0.425 ii 0.14 iii 0.21 i 0.375 ii 0.125 iii 0.25 Yes Expected frequency ¼ 100. The observed frequency of red or purple is 115, which is more than the expected frequency. 33 9 2 a i 15 ¼ 0.2 ii 19 iii 100 ¼ 0.33 iv 100 ¼ 0.09 50 ¼ 0.38 7 3 1 b i 14 ¼ 0.25 ii 20 ¼ 0.35 iii 10 ¼ 0.3 iv 10 ¼ 0.1 652 v 87 200 ¼ 0.435 vi 59 200 ¼ 0.295 vii ¼ 0.255 9 a 200 27 b i 200 ¼ 0.135 ii ¼ 0.31 iii 80 200 ¼ 0.4 21 200 v 62 200 1 200 iv 1 19 20 22 23 6 c Yes d Expected frequency ¼ 40. The expected frequency compares very favourably with the observed frequency of 42. a 50 b Teacher to check. c i Teacher to check. ii 12 d Teacher to check. a 600 b i 281 ii 322 600 ¼ 0.468 600 ¼ 0.537 227 iii 600 ¼ 0.378 iv 522 600 ¼ 0.87 c i 0.5 ii 0.5 iii 0.33 iv 0.83 d The probabilities are similar. a Teacher to check. 3 7 b i 12 ¼ 0.5 ii 15 ¼ 0.2 iii 10 ¼ 0.3 iv 10 ¼ 0.7 c Teacher to check. 3 3 1 a i 10 ¼ 0.3 ii 25 ¼ 0.12 iii 12 iv 10 ¼ 0.1 25 ¼ 0.48 b i 0.33 ii 0.17 iii 0.33 iv 0.17 c Yes d Expected frequency of not yellow is 33. This is more than the observed frequency of 26. a 16 0.17 b 16 or 17 times c, d, e Teacher to check. a 200 4 27 13 b i 200 ¼ 0.02 ii 200 ¼ 0.135 iii 200 ¼ 0.065 iv 86 200 51 200 ¼ 0.43 ¼ 0.105 ¼ 0.005 c Ferry, light rail (tram) d Teacher to check. Exercise 12-02 1 a i 11 25 2 a 156 b i 11 52 7 c 31 3 a 135 4 a ii 4 5 iii 1 5 iv 6 25 v 19 25 b 3 10 ii 7 52 iii 7 78 iv 22 39 v 19 78 vi 1 26 b 56 135 c 1 5 d 17 52 S 11 b 5 a c 6 a b i 1 45 i 19 26 123 P 12 31 ii 11 54 b i ii 0 1 45 IN 49 iii ii 31 54 1 3 iv iii 7 9 1 9 iv 19 45 32 123 iii 27 123 iv 81 123 JA 15 32 d i 49 123 27 c 81 ii 9780170194662 Answers 7 a 200 79 51 77 81 b i 200 ii 100 iii 200 iv 121 v 100 vi 200 c 29 40 d No, because all the people surveyed indicated a day on which they preferred to shop. 8 a 204 7 23 31 71 b i 204 ii 204 iii 102 iv 102 c i 2 54 First 3 a H T 43 54 ii b H T 4 25 i 53 150 ii iii 28 75 17 32 T c 63% 2 a 128 b i 68 ii 60 c d 12 55 3 a 93 b i 21.5% ii 11.8% iii 3.2% c i 15.7% ii 45% d The percentage composition of women in the opposition is three times that of the percentage composition of women in the government. 4 a 150 b i 0.5 ii 0.04 iii 0.43 iv 0.23 c 22 ¼ 0.293 75 5 a 160 7 11 9 b i 20 ¼ 0.35 ii 160 ¼ 0.069 iii 80 ¼ 0.113 c 35 ¼ 0.43 82 6 a 200 b 55% c d 7 a b i 49.5% ii 45% 65.5% 878 i 679 ii 878 ¼ 0.773 iii 67 439 ¼ 0.153 iv iii 36% 545 878 21 439 b 8 d i c i 7 8 b 2 a First die CG AJ GE JR RC CJ AE GR EC RA 1 3 3 8 ii ii 7 8 ¼ 0.048 H T 5 a CR GC JA EG RJ AC GA JG EJ RE 1 2 3 4 5 6 b 36 c i 16 9780170194662 3 1, 3 2, 3 3, 3 4, 3 5, 3 6, 3 4 1, 4 2, 4 3, 4 4, 4 5, 4 6, 4 5 1, 5 2, 5 3, 5 4, 5 5, 5 6, 5 6 1, 6 2, 6 3, 6 4, 6 5, 6 6, 6 1st die 1 2 3 4 b 24 c i 1 6 1 4 iii 11 36 iv 14 v 1 2 vi 1st die 5 12 HHT HTH T H HTT THH T THT H TTH T TTT 1 8 iv v 1 2 H2 3 H3 4 5 H4 6 1 H6 T1 2 T2 3 T3 4 5 T4 6 T6 H5 T5 3 1, 3 2, 3 3, 3 4, 3 1 4 iii ii 4 1, 4 2, 4 3, 4 4, 4 1 2 iv 5 1, 5 2, 5 3, 5 4, 5 3 8 6 1, 6 2, 6 3, 6 4, 6 v 0 2nd die b i ii T H 2 2 1, 2 2, 2 3, 2 4, 2 6 a 2 1, 2 2, 2 3, 2 4, 2 5, 2 6, 2 HHH 2nd die 1 1, 1 2, 1 3, 1 4, 1 Second die 1 1, 1 2, 1 3, 1 4, 1 5, 1 6, 1 H 1 8 iii ¼ 0.621 CE AR JC EA RG Outcomes e i 75 ii 25 coin die outcomes 1 H1 iv 31.5% 1 6 c 1 8 4 Exercise 12-04 1 a CA AG GJ JE ER Third H Exercise 12-03 1 a 150 Second 3 50 1 9 v 0 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 ii vi 1 36 5 12 3 4 5 6 7 8 9 4 5 6 7 8 9 10 iii vii 1 6 7 12 5 6 7 8 9 10 11 6 7 8 9 10 11 12 iv viii 1 2 5 12 653 Answers 7 a 1st coin 2nd coin 3rd coin 3 a 4th Sample coin space H H T T H T H H HH TH H T HHHH HHHT HHTH b i 4 a HHTT 1 4 T HT TT ii 1 2 iii 2nd course H H T H T H T T H T H H H T T T H T T 1 b i 16 15 iv 16 c i 63 ii 14 1 v 16 ii 375 B CB FB YB HTHH HTHT HTTH 1st course HTTT THHH b i 5 a THHT THTH 1 3 THTT H T TTHH H T TTTH C F Y ii H CH FH YH 4 15 1st draw 2nd draw 3 6 7 4 TTHT TTTT iii 38 5 vi 16 iii 937 3 4 b $800 g 250 c 400 h $300 d 700 i $300 e $300 j 500 6 7 6 3 4 7 7 3 4 6 Exercise 12-05 1 a Girls Boys b i B C Ew W 1 24 2 a i A B C D E ii A B C D E b i c i 654 ii Ca Em B, Ca B, Em C, Ca C, Em Ew, Ca Ew, Em W, Ca W, Em 1 6 M B, M C, M Ew, M W, M iii B AB BB CB DB EB C AC BC CC DC EC D AD BD CD DD ED E AE BE CE DE EE A B AB C AC BC D AD BD CD E AE BE CE DE CB DB EB ii ii DC EC 4 25 3 5 R S B, R B, S C, R C, S Ew, R Ew, S W, R W, S 1st draw 2nd draw R G Y R G Y R R ED iii iii 12 25 2 5 iv 4 5 S CS FS YS T CT FT YT 1 15 3rd Octcomes draw 346 6 347 7 364 4 367 7 374 4 376 6 436 6 437 7 463 3 467 7 473 3 476 6 634 4 637 7 643 3 647 7 673 3 674 4 734 4 736 6 743 3 746 6 763 3 764 4 i 12 b ii 34 iii 12 iv 13 6 a Teacher to check. 64 outcomes, beginning with 333, 334, 336, 337, 343, 344, 346, 347, …, 776, 777. 1 1 b i 16 ii 12 iii 12 iv 16 7 a 1 6 A AA BA CA DA EA BA CA DA EA 1 5 1 10 Be B, Be C, Be Ew, Be W, Be P CP FP YP iii Mental skills 12 2 a $700 f $400 3 4 b i 1 2 ii G R R Y Y R R G 1 12 3rd Octcomes draw RRG G RRY Y RGR R RGY Y RYR R RYG G RRG G RRY Y RGR R RGY Y RYR R RYG G GRR R GRY Y GRR R GRY Y R GYR R GYR R YRR G YRG R YRR G YRG R YGR R YGR iii 1 9780170194662 Answers Second Third Outcomes child child First child 8 a B B G B G G b i 9 a 1 8 BBB G BBG B BGB G B BGG GBB G B GBG GGB G 1 8 ii B F R F R R b i d i 8 a 3 8 R iv 1 8 FFF FRF R F FRR RFF R F RFR RRF R iii RRR 1 b i 18 ii 38 8 1 10 a 36 b 36 11 a i 125 outcomes. Teacher to check. ii 60 outcomes. Teacher to check. 29 18 64 b i 125 ii 125 iii 125 1 3 c i 10 ii 10 iii 0 1 2 9 27 11 36 7 8 ii 10 i i 3 4 5 6 7 8 9 10 49 1 0 1 2 3 4 5 1 6 1 11 4 5 6 7 8 9 10 5 6 7 8 9 10 11 ii 1 6 ii 27 ¼ 29 1 9 9 13 a b d 2 3 4 5 6 7 8 ¼ 13 1 2 3 4 5 6 iv c i 1 40 FFR F 1 2 3 4 5 6 7 1 2 3 4 5 6 GGG iii F F 7 a 8 9 1 18 4 11 iii c e 2 11 1 6 ¼ 14 c 6 1 13 11 2 1 0 1 2 3 4 ii ii 2 8 b 6 7 8 9 10 11 12 12 3 2 1 0 1 2 3 4 3 2 1 0 1 2 1 6 c e 1 6 5 4 3 2 1 0 1 i i 1 2 6 18 6 5 4 3 2 1 0 ¼ 13 ii 0 ii 1 Power plus iv iv 117 125 2 5 1 a 320 7 b i 40 ¼ 0.175 57 iii 320 ¼ 0.178 c 2 a Exercise 12-06 16 45 1 1 d Yes, 13314 ¼ 12 e independent c 12 5 a i 16 ii 12 b independent b 48 ¼ 12 6 a 59 c Dependent, as the first draw changes the contents of the bag. 7 a i 58 ii 47 b i 38 ii 57 c i 58 ii 37 d i 38 ii 27 8 12 d 1st draw 1 a independent b independent c dependent d independent e dependent f independent g dependent 2 Dependent, as the balls are not replaced when drawn. 3 a independent b 12 1 4 a i 3 ii 14 b 1Y, 2Y, 3Y, 4Y, 5Y, 6Y, 1G, 2G,3G, 4G, 5G, 6G, 1B, 2B, 3B, 4B, 5B, 6B, 1R, 2R, 3R, 4R, 5R, 6R, 3 32 ¼ 3 20 ¼ 45 107 ii iv 3 7 R 4 7 B 2 6 4 6 3 6 3 6 1 ii 7 14 7 ¼ ii 30 15 4 1 v 20 ¼ 5 2 i 152 ¼ 15 ii 3 2 PðA and P(B|A) ¼ 7, PðAÞ b i 3 a i iv b c 2 7 20 30 4 14 ¼ 23 ¼ 27 0.094 0.15 2nd Outcomes draw R RR B RB R BR B iii iii BB 4 7 4 30 iv 2 ¼ 15 6 7 Yes BÞ 2 ¼ 157 ¼ 27 15 Chapter 12 revision Exercise 12-07 1 a 3 a 4 a 5 13 1 3 4 11 2 11 9780170194662 3 1 6¼2 7 11 b b b 4 11 2 c 6 1 6 4 11 2 9 d 5 11 1 a i 0.353 ii 0.427 iii 0.087 iv 0.513 2 b i 13 ¼ 0.867 ii ¼ 0.133 15 15 c Different at least one head occurring excludes zero heads occurring, which is the same as three tails occurring. The events are complementary. 655 Answers 2 a 35 2 35 b ii 1st card 6 35 3 c i ii 19 iii 35 35 d They don’t like the types of music mentioned in the survey. 3 a B 3 i 9 20 3 20 ii 200 i 0.305 25 110 ¼ 0.227 i 47 69 ¼ 0.681 1 2 3 4 b 16 c i 12 6 a i 1 1, 1 2, 1 3, 1 4, 1 iii ii 0.11 ii 2 1, 2 2, 2 3, 2 4, 2 ii 2 5 1 4 3 1, 3 2, 3 3, 3 4, 3 iii 1st card iv 2nd card 4 7 2 4 4 7 2 7 4 7 656 iv 3 5 iv 0.425 b c 7 a d 8 a i 23 i 13 independent independent b 1 4 4 274 2 7 427 7 2 472 3 4 734 4 3 743 v 1 4 3rd sample card space 2 222 4 224 7 227 2 242 4 244 7 247 2 272 4 274 7 277 2 4 7 2 4 7 2 4 7 422 424 427 442 444 447 472 474 477 2 4 7 2 4 7 2 4 7 722 724 727 742 744 747 772 774 777 vi 1 2 i 1 3 2 3 ii ii 1 2 ii 1 iii iii 1 9 1 3 iv 29 iv 16 c dependent b dependent e independent 1 2 3 4 5 1 2 3 4 4 1, 4 2, 4 3, 4 4, 4 2 2 3 20 ¼ 0.319 7 16 7 7 iii 0.225 22 69 sample space 247 4 5 9 b c 4 a b c d 5 a 3rd card 7 2 D 3 2nd card 4 2 3 4 5 6 c 2 7 3 4 5 6 7 d i 1 3 4 5 6 7 8 ii 1 3 e 1 4 Chapter 13 SkillCheck 1 a d g j 2 a h ¼ 71 b p ¼ 105 c x ¼ 126 m ¼ 68 e a ¼ 24 f w ¼ 36 w ¼ 30, k ¼ 90 h r ¼ 83 i p ¼ 26, r ¼ 98 p ¼ 52 k y ¼ 42 l d ¼ 54 m ¼ 65 (angles on a straight line), n ¼ 65 (alternate angles), p ¼ 50 (angle sum of n XWY) b Isosceles triangle (m ¼ n ¼ 65) Exercise 13-01 1 2 3 4 5 6 7 8 9 a a a a a a a a a 1800 b 1440 c 6 b 21 c 16 b 157.5 144 b 135 c 30 b 15 c 72 b 30 c 140 b 162 c 24 b 5 c 18 8 b 10 c 15 1260 13 d 3240 d 30 120 d 45 d 20 d 144 d d 9 d 180 e 2340 e 9 150 25 60 168 e 72 e 24 f f 30 12 Exercise 13-02 1 In n ABE and n CBD: \ABE ¼ \CBD (vertically opposite angles) AB ¼ CB (given) EB ¼ DB (given) [ n ABE ” n CBD (SAS) 9780170194662 Answers 2 In n LMP and n NPM: LP ¼ NM (given) LM ¼ NP (given) MP is common. [ n LMP ” n NPM (SSS) 3 In n QTW and n PWT: \QTW ¼ \PWT ¼ 90 (QT ’ WT and PW ’ WT) QW ¼ PT (given) WT is common. [ n QTW ” n PWT (RHS) 4 In n ABY and n CBX: \BAY ¼ \BCX ¼ 90(equal angles of the square ABCD) BA ¼ BC (equal sides of the square ABCD) AY ¼ CX (given) [ n ABY ” n CBX (SAS) 5 In n CDE and n FED: \CDE ¼ \FED (given) \DEC ¼ \EDF (equal angles opposite equal sides of isosceles n DEY) DE is common. [ n CDE ” n FED (AAS) 6 In n XOY and n VOW: \XOY ¼ \VOW (vertically opposite angles) OX ¼ OV (equal radii of small circle) OY ¼ OW (equal radii of large circle) [ n XOY ” n VOW (SAS) 7 In n KLM and n MNK: \MKL ¼ \KMN (alternate angles, NM || KL) \KML ¼ \MKN (alternate angles, KN || LM) KM is common. [ n KLM ” n MNK (AAS) 8 In n CDH and n EFG: \HCD ¼ \GEF (corresponding angles, CH || EG) \HDC ¼ \GFE (corresponding angles, DH || FG) CH ¼ EG (given) [ n CDH ” n EFG (AAS) 9 In n UXY and n WXY: \UXY ¼ \WXY (YX bisects \UXW) UX ¼ WX (given) XY is common. [ n UXY ” n WXY (SAS) 10 In n ABE and n CBD: \AEB ¼ \CDB ¼ 90 (AE ’ BC, CD’AB) \B is common. BA ¼ BC (given) [ n ABE ” n CBD (AAS) 11 a In n HEF and n GFE: \HEF ¼ \GFE (given) EH ¼ FG (given) EF is common. [ n HEF ” n GFE (SAS) b [ \EHF ¼ \FGE (matching angles of congruent triangles) 12 a In n AOB and n COD: AB ¼ CD (given) OA ¼ OC (equal radii) 9780170194662 b 13 a b 14 a b c 15 a b 16 a b OB ¼ OD (equal radii) [ n AOB ” n COD (SSS) [ \AOB ¼ \COD (matching angles of congruent triangles) In n QRX and n QTY: QR ¼ QT (equal sides of isosceles n QRT) RX ¼ TY (given) \R ¼ \T (equal angles of isosceles n QRT) [ n QRX ” n QTY (SAS) [ QX ¼ QY (matching sides of congruent triangles) [ n QXY is isosceles (two sides of the triangle are equal) In n TAP and n XCP: \TPA ¼ \XPC (vertically opposite angles) TP ¼ XP (given) AP ¼ CP (given) [ n TAP ” n XCP (SAS) [ TA ¼ XC (matching sides of congruent triangles) [ \A ¼ \C (matching angles of congruent triangles) [ TA || XC (alternate angles are equal) In n OAM and n OBM: OA ¼ OB (equal radii) \OMA ¼ \OMB ¼ 90 (OM ’ AB) OM is common. [ n OAM ” n OBM (RHS) AM ¼ BM (matching sides of congruent triangles) [ OM bisects AB. In n GLH and n KLH: GL ¼ KL (given) GH ¼ KH (given) LH is common. [ n GLH ” n KLH (SSS) \GLH ¼ \KLH (matching angles of congruent triangles) [ LH bisects \GLK. \GHL ¼ \KHL (matching angles of congruent triangles) [ LH bisects \GHK. Exercise 13-03 1 \A ¼ \C and \B ¼ \D Now \A þ \C þ \B þ \D ¼ 360 (angle sum of a quadrilateral) [ 2\A þ 2\B ¼ 360 (\C ¼ \A, \D ¼ \B) [ \A þ \B ¼ 180 [ AD || BC (the pair of co-interior angles have a sum of 180) Also, from \A þ \C þ \B þ \D ¼ 360 (angle sum of a quadrilateral) [ 2\A þ 2\D ¼ 360 (\C ¼ \A, \B ¼\D) [ \A þ \D ¼ 180 [ AB || DC (the pair of co-interior angles have a sum of 180) [ ABCD is a parallelogram (opposite sides are parallel). 2 In n LMP and n NPM: LM ¼ NP (given) PM is common. \LMP ¼ \NPM (alternate angles, LM || NP) [ n LMP ” n NPM (SAS) \LPM ¼ \NMP (matching angles of congruent triangles) 657 Answers 3 4 5 6 7 8 9 10 658 [ LP || NM (alternate angles proved equal) [ LMNP is a parallelogram (opposite sides are parallel). In n DXH and n GXE: \DXH ¼ \GXE (vertically opposite angles) HX ¼ EX (given) DX ¼ GX (given) [ n DXH ” n GXE (SAS) \HDX ¼ \EGX (matching angles of congruent triangles) [ HD || EG (alternate angles proved equal) Similarly, \GHX ¼ \DEX (matching angles of congruent triangles HXG and EXD) [ HG || ED (alternate angles proved equal) [ DEGH is a parallelogram (opposite sides are parallel). (Outline of proof only) nFHC ” nFHE ” nDHE ” nDHC (SAS) [ FC ¼ FE ¼ DE ¼ DC (matching sides of congruent triangles) Also, \CFH ¼ \EDH and \CDH ¼ \EFH (matching angles of congruent triangles) [ CDEF is a rhombus (opposite sides are parallel and all sides are equal). (Outline of proof only) Since WY ¼ XV and the diagonals bisect each other, TW ¼ TV ¼ TY ¼ TX [ n TWV ” n TXY (SAS) and n TVY ” n TWX (SAS) [ \VWT ¼ \XYT and \TVY ¼ \TXW (matching angles of congruent triangles) [ VW || YX and VY || XW (alternate angles proved equal) [ VWXY is a parallelogram. Also, n YVW ” n XYV ” n VWX ” n YXW (AAS) [ \V ¼ \W ¼ \X ¼ \Y (matching angles of congruent triangles) Since the angle sum of VWXY ¼ 360 \V ¼ \W ¼ \X ¼ \Y ¼ 90 [ VWXY is a rectangle. \B þ \C ¼ 180 and \B þ \E ¼ 180 [ BE || CD and BC || ED (pairs of co-interior angles have a sum of 180) [ BCDE is a parallelogram with right angles. [ BCDE is a rectangle. Since the sides are equal, TWME is a rhombus (proved in question 4). Since \M ¼ 90, TWME is a square (a square is a rhombus with a right angle). Since the angles of the quadrilateral are right angles, GHKL is a rectangle (proved in question 6). If GL ¼ GH, GL ¼ GH ¼ LK ¼ KH ( opposite sides of a rectangle are equal) [ GHKL is a square (all sides are equal, all angles are 90). The diagonals bisect each other at right angles, so MNPT is a rhombus (proved in question 4). [ MN ¼ NP ¼ PT ¼ MT Also, nMNT ” nNPT ” nPTM ” nMNP (SSS, since TN ¼ PM) [ \M ¼ \N ¼ \P ¼ \T ¼ 90 (angle sum of a quadrilateral and matching angles of congruent triangles) [ MNPT is a square. a In n ABX and n CDY: BX ¼ DY (given) \B ¼ \D (opposite angles of a parallelogram) AB ¼ CD (opposite sides of a parallelogram) [ n ABX ” n CDY (SAS) b AX ¼ CY (matching sides of congruent triangles) XC ¼ BC BX ¼ AD DY ðBC ¼ AD, opposite sides of a parallelogram and BX ¼ DY , givenÞ 11 12 13 14 15 16 ¼ AY [ AXCY is a parallelogram as pairs of opposite sides are equal. a In n DAE and n CEB: AE ¼ EB (given) \DAE ¼ \CEB (corresponding angles, AD || EC) AD ¼ EC (equal sides of a rhombus) [ n DAE ” n CEB (SAS) b ED ¼ BC (matching sides of congruent triangles from a) AE ¼ DC (equal sides of a rhombus) and AE ¼ EB (given) [ DC ¼ EB [ BCDE is a parallelogram because its opposite sides are equal. a In n APS and n CRQ: AP ¼ CR (given) AS ¼ CQ (given) \A ¼ \C (opposite angles of a parallelogram) [ n APS ” n CRQ (SAS) [ PS ¼ QR (matching sides of congruent triangles) In n PBQ and n RDS: PB ¼ RD (given AP ¼ CR and opposite sides of a parallelogram) \B ¼ \D (opposite angles of a parallelogram) BQ ¼ DS (given CQ ¼ AS and opposite sides of a parallelogram) [ n PBQ ” n RDS (SAS) [ PQ ¼ RS (matching sides of congruent triangles) b PQRS is a parallelogram because pairs of opposite sides are equal). AC and DB are diagonals. OD ¼ OB (equal radii of small circle) OA ¼ OC (equal radii of large circle) [ ABCD is a parallelogram because the diagonals bisect each other. SQ and PR are diagonals. OP ¼ OR (equal radii of small circle) OS ¼ OQ (equal radii of large circle) PR ’ SQ (given) [ PQRS is a rhombus because its diagonals bisect each other at right angles. Since WD ¼ WE ¼ GY ¼ YF (W and G are the midpoints of equal sides DE and GF) and DZ ¼ ZG ¼ EX ¼ XF (Z and X are the midpoints of equal sides DG and EF) [ WZ ¼ WX ¼ YX ¼ ZY (by Pythagoras’ theorem) [ WZYX is a rhombus (a quadrilateral with all sides equal) In n APT and n CRQ: AT ¼ CQ (T and Q are the midpoints of equal sides of a parallelogram) 9780170194662 Answers AP ¼ CR (P and R are the midpoints of equal sides of a parallelogram) \A ¼ \C (opposite angles of a parallelogram) [ n APT ” n CRQ (SAS) [ PT ¼ RQ (matching sides of congruent triangles) Similarly, proving n DRT and n BPQ congruent (SAS), TR ¼ QP (matching sides of congruent triangles) [ PQRT is a parallelogram because its opposite sides are equal. Exercise 13-04 1 a In n ABD and n ACD: AB ¼ AC (given) BD ¼ CD (given) AD is common. [ n ABD ” n ACD (SSS) b \ADB ¼ \ADC (matching angles of congruent triangles) c \ADB þ \ADC ¼ 180 (angles on a straight line) [ \ADB ¼ \ADC ¼ 90 [ AD’BC 2 a In n KMP and n KNP: KM ¼ KN (given) \KPM ¼ \KPN ¼ 90 (KP’MN) KP is common. [ n KMP ” n KNP (RHS) b MP ¼ NP (matching sides of congruent triangles) KP’ MN (given) [ Perpendicular from vertex K to MN bisects MN. 3 a In n ABX and n CDX: \ABX ¼ \CDX (alternate angles, AB || CD) \BAX ¼ \DCX (alternate angles, AB || CD) AB ¼ CD (opposite sides of a rectangle) [ n ABX ” n CDX (AAS) b [ AX ¼ CX and BX ¼ DX (matching sides of congruent triangles) [ X is the midpoint of diagonals AB and CD. [ The diagonals of a rectangle bisect each other. 4 a In n DEG and n FGE: \DEG ¼ \FGE (alternate angles, DE || FG) \DGE ¼ \FEG (alternate angles, DG || FE) GE is common. [ n DEG ” n FGE (AAS) b In n DGF and n FED: \DFG ¼ \FDE (alternate angles, FG || DE) \FDG ¼ \DFE (alternate angles, DG || FE) DF is common. [ n DGF ” n FED (AAS) c \GDE ¼ \EFG (matching angles of congruent triangles DEG and FGE) \DGF ¼ \FED (matching angles of congruent triangles DFG and FED) [ Opposite angles of a parallelogram are equal. 5 a In n BED and n BCD BE ¼ BC (given) DE ¼ DC (given) 9780170194662 b 6 a b 7 a b c 8 a b 9 a b BD is common. [ n BED ” n BCD (SSS) \EBD ¼ \CBD (matching angles of congruent triangles) \EDB ¼ \CDB (matching angles of congruent triangles) [ Diagonal BD bisects \EBC and\EDC. In n LXM and n NXP: \MLX ¼ \PNX (alternate angles, LM || NP) \LMX ¼ \NPX (alternate angles, LM || NP) LM ¼ NP (given) [ n LXM ” n NXP (AAS) LX ¼ NX and MX ¼ PX (matching sides of congruent triangles) [ Diagonals of a parallelogram bisect each other. In n UAW and n XAW: UW ¼ XW (equal sides of a rhombus) AW is common. \UWA ¼ \XWA (diagonals of a rhombus bisect the angles) [ n UAW ” n XAW (SAS) In n UAW and n UAY UW ¼ UY (given) AU is common. \WUA ¼ \YUA (diagonals of a rhombus bisect the angles) [ n UAW ” n UAY (SAS) UA ¼ XA (matching sides of congruent triangles UAW and XAW) WA ¼ YA (matching sides of congruent triangles UAW and UAY) [ Diagonals bisect each other. \UAW ¼ \XAW (matching angles of congruent triangles UAW and XAW). But \UAW þ \XAW ¼ 180 [ \UAW ¼ \XAW ¼ 90 [ WA’UX [ Diagonals WY and UX are perpendicular. [ Diagonals bisect each other at right angles. In n DXF and n EXF: \D ¼ \E (given) \DXF ¼ \EXF ¼ 90 (FX ’ DE) FX is common. [ n DXF ” n EXF (AAS) FD ¼ FE (matching sides of congruent triangles) Also, FD is opposite \E and FE is opposite \D. [ Sides opposite the equal angles in a triangle are equal. Join X to B. In n XBW and n XBY: XW ¼ XY (equal sides of equilateral n XYW) WB ¼ YB (B is the midpoint of WY) XB is common. [ n XBW ” n XBY (SSS) Join Y to A. In n YAX and n YAW: YX ¼ YW (equal sides of equilateral n XYW) XA ¼ WA (A is the midpoint of XW) YA is common. [ n YAX ” n YAW (SSS) 659 Answers c \W ¼ \Y (matching angles of congruent triangles XBW and XBY) \X ¼ \W (matching angles of congruent triangles YAX and YAW) [ \W ¼ \Y ¼ \X But \W þ \Y þ \X ¼ 180 [ \W ¼ \Y ¼ \X ¼ 60 Mental skills 13 2 a e i 4 a e i 12:25 p.m. 0610 1100 6:05 p.m. 1245 1545 b f j b f j 1:10 a.m. 0010 2305 6:40 a.m. 0355 0400 c g k c g 10:50 p.m. 9:10 a.m. 12:20 a.m. 12:10 p.m. 10:50 p.m. d h l d h 10:55 p.m. 3:15 a.m. 11:35 a.m. 2:50 a.m. 12:15 p.m. Exercise 13-05 1 \LMK ¼ \LKM ¼ 45 (angle sum of a right-angled isosceles n KML) [ \PMN ¼ 135 (angles on a straight line) [ 2x þ 135 ¼ 180 (angle sum of isosceles n PMN) [ x ¼ 22.5 2 \ABC ¼ 42 (equal angles of isosceles n ABC) [ \BCD ¼ \ABC ¼ 42 (alternate angles, CE || AB) [ \DBC ¼ \BCD ¼ 42 (equal angles of isosceles n BCD) \ EDB ¼ \ ABD ðalternate angles, CE jj ABÞ ¼ \ ABC þ \ DBC ¼ 42 þ 42 [ m ¼ 84 3 \NKL þ 93 ¼ 147 (exterior angle of n NKL equal to sum of interior opposite angles) [ \NKL ¼ 54 [ \NKH ¼ 54 (NK bisects \HKL) [ \HKL ¼ 108 \NHK þ 108 ¼ 147 (exterior angle of n HKL equal to sum of interior opposite angles) [ \NHK ¼ 39 4 \EDC ¼ 180 116 ðco-interior angles, BC jj EDÞ ¼ 64 ) x ¼ 6442 ðdiagonals bisect the angles of a rhombusÞ ¼ 32 5 \AED ¼ \ABC (corresponding angles, ED || BC) \ADE ¼ \ACB (corresponding angles, ED || BC) But \ABC ¼ \ACB (equal angles of isosceles n ABC) [ \AED ¼ \ADE [ n AED is isosceles (two equal angles) 6 Let \XYP ¼ x, \TWP ¼ y [ \PYW ¼ x (YP bisects \XYW) and \PWY ¼ y (WP bisects \TWY) [ 2x þ 2y ¼ 180 (co-interior angles, YX || WT) [ x þ y ¼ 90 But x þ y þ \YPW ¼ 180 (angle sum of n YWP) [ 90 þ \YPW ¼ 180 [ \YPW ¼ 90 660 7 a TZ ¼ TY (given) [ n TZY is isosceles. [ \TZY ¼ \TYZ (angles opposite equal sides) TZ ¼ UX (given) and TZ ¼ WX (equal opposite sides of a parallelogram) [ UX ¼ WX [ n XUW is isosceles. [ \XUW ¼ \XWU (angles opposite equal sides) But \TZY ¼ \XWU (equal opposite angles of a parallelogram) [ \TZY ¼ \TYZ ¼ \XUW ¼ \XWU In n TZY and n XWU: TZ ¼ XU (given) \TZY ¼ \XWU (opposite angles of a parallelogram) \TYZ ¼ \XUW (proven) [ n TZY ” n XWU (AAS) b TY ¼ UX (given) ZY ¼ UW (matching sides of congruent triangles) TW ¼ ZX (opposite sides of a parallelogram) ) TU ¼ TW UW ¼ ZX ZY ¼ XY [ TUXY is a parallelogram (opposite sides equal). 8 a In n MNY and n TMW: MN ¼ MT (equal sides of a square) MY ¼ TW (Y and W are the midpoints of equal sides of a square) \NMY ¼ \MTW ¼ 90 (angles in a square) [ n MNY ” n MTW (SAS). b \MNY ¼ \TMW ¼ x (matching angles of congruent triangles) [ \NMX ¼ 90 x ) \MXN ¼ 180 x ð90 x Þ ¼ 90 [ MW ’ NY 9 \BCD ¼ \BDC (equal angles of isosceles n BCD) ) \ABD ¼ \BCD þ \BDC ðexterior angle of 4BCD equal to sum of interior opposite anglesÞ ¼ 2 \BCD But \ABD ¼ \AED (opposite angles of a parallelogram) [ \AED ¼ 2\BCD 10 a In n ADB and n ACB: AD ¼ AC (equal radii of large circle) BC ¼ BD (equal radii of small circle) AB is common. [ n ADB ” n ACB (SSS) b In n DXB and n CXB: BX is common. BD ¼ BC (equal radii) \DBX ¼ \CBX (matching angles of congruent triangles proved in a) [ n DXB ” n CXB (SAS) DX ¼ CX (matching sides of congruent triangles proved in b) 11 Let \A ¼ \B ¼ x (equal angles of isosceles n ABC) [ \ACB ¼ 180 2x (angle sum of n ABC) \DCE ¼ \ACB ¼ 180 2x (vertically opposite angles) 9780170194662 Answers [ \D ¼ \E ¼ 12[180 (180 2x)] (angle sum of isosceles n DCE) [ \D ¼ \E ¼ x [ \A ¼ \E [ AB || DE (alternate angles are equal) 12 \YUX ¼ \UYX ¼ \UXY ¼ 60 (angles in equilateral n UXY) [ \UXW ¼ 120 (angles on a straight line) \XWU þ \XUW þ 120 ¼ 180 (angle sum of n WXU) [ \XWU ¼ \XUW ¼ 30 (n WXU is isosceles) \WUY ¼ \XUW þ \YUX ¼ 30 þ 60 ¼ 90 13 \WTP ¼ \P and \YTQ ¼ \Q (alternate angles, WY || PQ) \WTP þ \PTQ þ \YTQ ¼ 180 ðangles on a straight lineÞ ) angle sum of 4PQT ¼ \P þ \PTQ þ \Q ¼ \WTP þ \PTQ þ \YTQ ðfrom aboveÞ ¼ 180 14 \BAD þ \DAH þ \BAC þ \CAF ¼ 180 (angles on a straight line) But \BAD ¼ \DAH (AD bisects \HAB) and \BAC ¼ \CAF (AC bisects \FAB) [ 2\BAD þ 2\BAC ¼ 180 [ \BAD þ \BAC ¼ 90 [ \CAD ¼ 90 15 \BAC þ \BCA þ \ABC ¼ 180 (angle sum of n ABC) [ \ABC ¼ 180 (\BAC þ \BCA) \CBD þ \ABC ¼ 180 (angles on a straight line) ) \CBD ¼ 180 \ABC ¼ 180 ½180 ð\BAC þ \BCAÞ ¼ \BAC þ \BCA 16 \ABO ¼ x ¼ \BAO and \CBO ¼ y ¼ \BCO (equal angles of isosceles n ABO and n CBO, equal radii) [ 2x þ 2y ¼ 180 (angle sum of n ABC) [ x þ y ¼ 90 [ \ABO þ \CBO ¼ 90 ¼ \ABC [ \ABC is a right angle. Exercise 13-06 1 a 2 a 3 4 b 2 c 1 2 d 1.5 3 a i \L and \T, \F and \W, \D and \P, \B and \Y. ii LF and TW, FD and WP, BD and YP, LB and TY. iii LFDB ||| TWPY b i \G and \T, \M and \Q, \Y and \S, ii MY and QS, GM and TQ, GY and TS. iii n GYM ||| n TSQ 27 24 18 4 a Yes, (36 12 ¼ 9 ¼ 8 ¼ 6 ¼ 3) b Yes, all equilateral triangles are similar. 24 19 2 c Yes (28 42 ¼ 36 ¼ 28:5 ¼ 3) 9 3 d Yes (15 ¼ 15 25 ¼ 5) 30 6 e Yes, (18 15 ¼ 25 ¼ 5, and the triangle is right-angled) f Yes, all squares are similar. Exercise 13-07 1 a w ¼ 22.4 b m ¼ 10 c p ¼ 20, h ¼ 21 d x ¼ 18 e a ¼ 12.8, w ¼ 7.5 f g ¼ 1119, q ¼ 18 2 3 4 g y ¼ 26 3, b ¼ 9 5 or 9.6 h u ¼ 12 5 or 12.8, t ¼ 6 78 or 6.875 2 h ¼ 1357 3 x ¼ 889 4 w ¼ 16 cm 5 h ¼ 12 m 6 B 7 h ¼ 2.408 m 8 D 9 2.24 m Exercise 13-08 1 a Two pairs of angles are equal (AA). b All three pairs of matching sides are in the same ratio, 9 11 15:5 1 18 ¼ 22 ¼ 31 ¼ 2 (SSS) c Two pairs of matching sides are in the same ratio 6 12 3 8 ¼ 16 ¼ 4 and the included angles are equal (SAS). d Two pairs of angles are equal (AA). e All three pairs of matching sides are in the same ratio 9 9 14:25 3 12 ¼ 12 ¼ 19 ¼ 4 (SSS) f In both right-angled triangles, the pairs of hypotenuses 20:8 4 and second sides are in the same ratio 12 15 ¼ 26 ¼ 5 (RHS). g Two pairs of angles are equal (AA). h All three pairs of matching sides are in the same ratio 18 27:5 20 5 14:4 ¼ 22 ¼ 16 ¼ 4 (SSS). i All three pairs of matching sides are in the same ratio 6 8 10 3 8 ¼ 1023 ¼ 1313 ¼ 4 (SSS). j Two pairs of matching sides are in the same ratio 26 30 10 18:2 ¼ 21 ¼ 7 and the included angles are equal (SAS). 2 a B and C (SAS) b A and C (SSS) c B and D (RHS) 3 a n UWY ||| n HEK (SAS) b n DML ||| n TPA (RHS) c n ABC ||| n QTP (AA) d n GHN ||| n WVS (SSS) Exercise 13-09 1 a In n TCH and n PMB: TC 18 5 PM ¼ 10:8 ¼ 3 b 9780170194662 5 ¼ 25 15 ¼ 3 \C ¼ \M ¼ 90 [ n TCH ||| n PMB (in a right-angled triangle, hypotenuses and two pairs of matching sides are in proportion or RHS) b In n VWG and n LQE: \V ¼ \L ¼ 22 \W ¼ \Q ¼ 123 [ n VWG ||| n LQE (equiangular or AA) TH PB 661 Answers c In n ABC and n TWM: AB 17 2 TW ¼ 25:5 ¼ 3 c i AC 12 2 TM ¼ 18 ¼ 3 BC 16 2 WM ¼ 24 ¼ 3 [ n ABC ||| n TWM (three pairs of matching sides in proportion or SSS) d In n EVH and n DNL: EV 21 7 DN ¼ 12 ¼ 4 7 ¼ 35 20 ¼ 4 \V ¼ \N ¼ 90 [ n EVH ||| n DNL (two pairs of matching sides in proportion and the included angles equal or SAS) 2 a In n ADE and n ABC: AD 1 AB ¼ 2 (D is the midpoint of AB) d VH NL ¼ 12 (E is the midpoint of AC) \A is common. [ n ADE ||| n ABC (two pairs of matching sides in proportion and the included angles equal or SAS) b In n ABF and n FDE: \AFB ¼ \FED (corresponding angles, BF || CE) \FAB ¼ \EFD (corresponding angles, AC || FD) [ n ABF ||| n FDE (equiangular or AA) c In n WXY and n TXW: \WXY ¼ \TXW ¼ 90 (given) \YWX ¼ 90 \WYX (angle sum of n WXY) \XTW ¼ 90 \WYX ðangle sum of 4WTY Þ e AE AC ¼ \YWX [ n WXY ||| n TXW (equiangular or AA) d In n NDL and n NQR: ND 8 1 NQ ¼ 16 ¼ 2 1 ¼ 10 20 ¼ 2 \N is common. [ n NDL ||| n NQR (two pairs of matching sides in proportion and the included angles equal or SAS) e In n XWH and n YXW: HW 18 3 WX ¼ 12 ¼ 2 NL NR 3 ¼ 12 8 ¼2 \HWX ¼ \YXW (alternate angles, HW || YX) [ n XWH ||| n YXW (two pairs of matching sides in proportion and the included angles equal or SAS) f In n NML and n KLP: \NML ¼ \KLP (alternate angles, NM || LK) \N ¼ \K (opposite angles of a parallelogram) [ n NML ||| n KLP (equiangular or AA) 3 a i In n FLN and n FDE: \FLN ¼ \FDE (corresponding angles, LN || DE) \F is common. [ n FLN ||| n FDE (equiangular or AA) ii d ¼ 9 b i In n ACE and n BCD: \EAC ¼ \DBC ¼ 90 (given) \C is common. [ n ACE ||| n BCD (equiangular or AA) 5 ii y ¼ 511 XW YX 662 f 4 a b 5 a b 6 a b In n YRT and n WUT: \YRT ¼ \WUT ¼ 90 (given) \YTR ¼ \WTU (vertically opposite angles) [ n YRT ||| n WUT (equiangular or AA) ii g ¼ 15 i \T þ \PCT ¼ 90 þ 90 ¼ 180 [ TN || CP (co-interior angles are supplementary) In n NMP and n PCB: \NMP ¼ \PCB ¼ 90 (given) \N ¼ \CPB (corresponding angles, TN || CP) [ n NMP ||| n PCB (equiangular or AA) ii w ¼ 7.5 i In n TYN and n YNM: \TYN ¼ \MNY (alternate angles, TY || MN) \TNY ¼ \YMN (given) [ n TYN ||| n YNM (equiangular or AA) ii h ¼ 12 i In n BHU and n XBD: \BUH ¼ \XDB (given) \UBH ¼ \DXB (alternate angles, BU || DX) [ n BHU ||| n XBD (equiangular or AA) ii y ¼ 18 In n MXG and n KXL: MG || LH (opposite sides of a rectangle) \GMX ¼ \LKX (alternate angles, MG || LH) \MGX ¼ \KLX (alternate angles, MG || LH) [ n MXG ||| n KXL (equiangular or AA) x ¼ 16 In n CLW and n LTE: \CWL ¼ \TEL ¼ 90 (given) \L is common. [ n CLW ||| n LTE (equiangular or AA) x¼2 In n PTU and n KPB: \PUT ¼ \KPB (alternate angles, PK || TU) \PTU ¼ \KBP ¼ 90 (given) [ n BHU ||| n XBD (equiangular or AA) PB ¼ 4 Power plus 1 a In n ABC and n CBD: \ACB ¼ 90 (by Pythagoras’ theorem in n ABC) \CDB ¼ 90 (given) [ \ACB ¼ \CDB \B is common. [ n ABC ||| n CBD (equiangular or AA) In n ABC and n ACD: \ACB ¼ 90 (by Pythagoras’ theorem in n ABC) \CDA ¼ 90 (given) [ \ABC ¼ \CDA \A is common. [ n ABC ||| n ACD (equiangular or AA) [ n ABC ||| \CDB ||| n ACD 8 b CD ¼ 413 4.62 9780170194662 Answers 2 G D C T A H B In n DGT and n BHT: \DGT ¼ \BHT (alternate angles, DC || AB) \GDT ¼ \HBT (alternate angles, DC || AB) DT ¼ BT (given) [ n DGT ” n BHT (AAS) [ DG ¼ BH (matching sides of congruent triangles) C 3 Y P X A W B T (Outline of proof only) X and Y are midpoints of BC and AY. Medians AX and BY meet at P. Draw CP to T, so that CP ¼ PT. Prove that n CYP ||| n CAT (SAS) [ YP || AT [ PB || AT Similarly, prove n CXP ||| n CBT (SAS) [ PA || BT [ APBT is a parallelogram (opposite sides are parallel) W is the midpoint of AB (the diagonals of a parallelogram bisect each other). Chapter 13 revision 1 156 2 36 3 a 36 b 15 c 8 d 24 4 B 5 In n WYZ and n XYZ \W ¼ \X (given) \WZY ¼ \XZY ¼ 90 (YZ ’ WX) YZ is common. [ n WYZ ” n XYZ (AAS) 6 a n BYC is isosceles (BC ¼ BY). [ \BCY ¼ \C (angles opposite equal sides) BC ¼ AD (opposite sides of a parallelogram) [ n ADX is isosceles (AD ¼ XD). [ \A ¼ \DXA (angles opposite equal sides) But \A ¼ \BCY (opposite angles of a parallelogram) [ \A ¼ \DXA ¼ \C ¼\BYC In n ADX and n CBY AD ¼ CB (given) \A ¼ \C (opposite angles of a parallelogram) \DXA ¼ \BYC (proven) [ n ADX ” n CBY (AAS) 9780170194662 b AX ¼ CY (matching sides of congruent triangles) AB ¼ CD (opposite sides of a parallelogram) ) XB ¼ AB AX ¼ CD CY ¼ YD DX ¼ YB (given) [ BXDY is a parallelogram (opposite sides equal). 7 a In n PML and n NLM: LP ¼ MN (opposite sides of a rectangle) LM is common. \PLM ¼ \NML ¼ 90 (angles of a rectangle) [ n PML ” n NLM (SAS). b [ PM ¼ NL (matching sides of congruent triangles) c The diagonals of a rectangle are equal. 8 In n NMA and n PQB: NM ¼ PQ (sides of a square) AM ¼ BQ (given) \NMA ¼ \PQB ¼ 90 (angles of a square) [ n NMA ” n PQB (SAS). \NAM ¼ \PBQ (matching angles of congruent triangles) But \NAM ¼ \CAB and \PBQ ¼ \CBA (vertically opposite angles) [ \CAB ¼ \CBA [ n CBA is isosceles (two equal angles). [ AC ¼ BC (sides opposite the equal angles in isosceles n CBA) Also, NA ¼ PB (matching sides of congruent triangles) ) NC ¼ NA þ AC ¼ PB þ BC ¼ PC [ n NPC is isosceles (two sides proved equal). 15 3 9 a Yes, (27 18 ¼ 10 ¼ 2) 22 12 9 4 b Yes, (27:5 ¼ 16 20 ¼ 15 ¼ 11:25 ¼ 5 ¼ 0:8) 10 1137 12 11 a SAS b RHS c AA A X B Y C In n AXY and n ABC: AX 1 AB ¼ 2 (X is the midpoint of AB) AY 1 ¼ AC 2 (Y is the midpoint of AC) \A is common. [ n AXY ||| n ABC (two pairs of matching sides in proportion and the included angles equal or SAS) [ \AXY ¼ \B (matching angles in similar triangles) [ XY || BC (corresponding angles are equal) XY [ BC ¼ AX AB (matching pairs of sides in proportion) XY [ BC ¼ 12 [ XY ¼ 12 3 BC 663 Answers Mixed revision 4 [ \VXY ¼ 2x ¼ \XVY ) \XTY ¼ \VXT þ \XVY ðexterior angle of 4XVT equal to 1 40 2 a 78 b i 25 ii 14 c 35 39 39 78 3 a 80 9 21 7 b i 13 ii 36 iii 42 iv 56 80 80 ¼ 20 80 ¼ 40 80 ¼ 10 1 _ c 6 ¼ 0:16, which is lower than the experimental probability of 17 80 ¼ 0:2125 4 a B 34 10 10 C 10 45 13 P 13 160 5 6 7 8 9 10 11 12 664 38 9 57 b i ii 19 iii 32 iv 109 v 160 80 160 5 c 27 In n ABC and n CDA: \BAC ¼ \DCA (alternate angles, AB || CD) \BCA ¼ \DAC (alternate angles, AD || CB) AC is common. [ n ABC ” n CDA (AAS) 9 a i 29 ii 13 iii 50 ¼ 0.18 75 ¼ 0.39 30 ¼ 0.43 b i 0.33 ii 0.42 iii 0.25 c The probabilities for drawing a black marble are similar. d 350 a 36 b 170 40 8 a 135 b 135 ¼ 27 67 60 72 8 c i 135 ii 135 ¼ 49 iii 135 ¼ 15 In n LMP and n LNP: LM ¼ LN (given) MP ¼ NP (P is the midpoint of MN) LP is common. [ n LMP ” n LNP (SSS) [ \LPM ¼ \LPN (matching angle of congruent triangles) But \LPM þ \LPN ¼ 180 (angles on a line) [ \LPM ¼ \LPN ¼ 90 a In n PRT and n RPQ: PT ¼ RQ (given) RT ¼ PQ (given) PR is common. [ n PRT ” n RPQ (SSS) b \PRT ¼ \RPQ (matching angles of congruent triangles) [ RT || PQ (alternate angles are equal) \RPT ¼ \PRQ (matching angles of congruent triangles) [ PT || RQ (alternate angles are equal) [ PQRT is a parallelogram (opposite sides are parallel) a 12.86 b 6 Let \VXY ¼ 2x [ \VXT ¼ \TXY ¼ x (TX bisects \VXY) Since XY ¼ VY (equal sides of rhombus XYVW), n XYV is isosceles. the interior opposite anglesÞ ¼ x þ 2x ¼ 3x ¼ 3 3 \TXY 13 a Teacher to check. b i 12 ii 14 14 a 2340 b 3240 15 36 16 a SAS 17 10.6 18 a 1 2 1 1 2 2 2 4 3 3 6 4 4 8 5 5 10 6 6 12 b 5 11 c i1 iii 1 2 iv 3 4 v 14 d 8280 c 1080 b SSS 3 3 6 9 12 15 18 4 4 8 12 16 20 24 ii 0 d 5 5 10 15 20 25 30 1 2 6 6 12 18 24 30 36 e 1 19 a In n YXT and n WVT: YX ¼ VW and YX jj VW (opposite sides of a rectangle) \YXT ¼ \WVT (alternate angles, YX jj VW Þ \YTX ¼ \WTV (vertically opposite angles) [ n YXT || n WVT (AAS) b [ YT ¼ TW and XT ¼ TV (matching sides in congruent triangles) [ The diagonals of a rectangle bisect each other. 20 a In n ABW and n CDW: \ABW ¼ \CDW (alternate angles, AB || CD) \BAW ¼ \DCW (alternate angles, AB || CD) [ n ABW ||| n CDW (AA) b CW ¼ 457 21 In n CEO and n DFO: OC ¼ OD (equal radii) \CEO ¼ \DFO ¼ 90 (CE ’ AB and DF ’ AB) \COE ¼ \DOF (vertically opposite angles) [ n CEO ” n DFO (AAS) General revision pffiffiffiffi pffiffiffi pffiffiffiffiffi 1 7 2 2 3 1010 3 98 þ 24 10 4 $15 700 4 5 gradient ¼ 5, y-intercept ¼ 2 6 a 6x(x þ 2) b 25(1 þ 2y)(1 2y) c (a p10)(a d 2(2p þ 1)(p 3) ffiffiffiffi þ 4) 7 x ¼ 72 41 8 x ¼ 50:3, s ¼ 12.2 9 a 360 498 mm3 b 145 125 mm3 10 a 210 b i 16 ii 143 c 31.9% d 29 210 36 o 0 11 y ¼ 142 49 5 12 12 a 56x b x4 c 9y12 d mn7 13 a y ¼ 614 b k ¼ 3 37 c x ¼ 3 9780170194662 Answers 14 16 17 18 d ¼ 19.1 15 C a 1 b 4 a 5 b 1 In n ABC ” n DEF: AB ¼ DE ¼ 10 cm (given) CB ¼ FE ¼ 12 cm (given) \A ¼ \D ¼ 90 (given) [ n ABC ” n DEF (RHS) 19 a y 1 b x< 30 a F ¼ fine, R ¼ rain c 12 c 3 Sat F F R –5 –4 –3 –2 –1 0 1 2 3 4 5 F 712 6 7 8 9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 –1 0 1 3 2 4 5 R 10 R c x < 3 20 2x þ 3y þ 5 ¼ 0 21 x ¼ 2, y ¼ 2 22 x2 þ y2 ¼ 64 23 x-intercepts at 212 and 3, y-intercept at 15, axis of symmetry x ¼ 14, vertex (14, 1518) 1 8 b i 31 19.2 min y 10 5 –5 3 5 x 10 –10 (0.25, –15.125) –20 24 a 7.2 m3 25 $1607.41 pffiffiffiffiffi 26 a i 13, m ¼ 23 pffiffiffiffiffi iii 13, m ¼ 23 b parallelogram 27 a b 78.5 cm3 c 5747.0 cm3 pffiffiffiffiffi ii 2 10, m ¼ 1 3 pffiffiffiffiffi iv 2 10, m ¼ 1 3 324 km A b 40 þ 18 (180 162) ¼ 58 28 a y –10 5 C c 284 km b y y = 3x d 115 y = 3x (1, 3) (1, 3) –5 5 –5 –10 9780170194662 10 x 1 a 13 b 2 a (x 4)(x þ 4) b d 3x(x 3)(x þ 3) e g (x 2)(2x þ 5) h 3 a x ¼ 52 or x ¼ 2 c x ¼ 0 or x ¼ 35 e x ¼ 10 or x ¼ 12 1 a d g j 2 a c e g i 3 a 4 a e N 40° 29 h ¼ 21, p ¼ 20 F FFF R FFR F FRF R FRR F RFF R RFR F RRF R RRR 3 8 iii 7 8 3 c 12 x(x 4)(x þ 4) c 3(x 3)(x þ 3) (x 5)(x þ 3) f (x þ 8)(x 3) x(x 10)(x þ 7) b x ¼ 0 or x ¼ 10 d x ¼ 1 or x ¼ 5 f x ¼ 2 or x ¼ 32 Exercise 14-01 B 162° N 100 km –10 Outcomes SkillCheck –5 –15 ii Mon Chapter 14 –2.5 –10 Sun 1 0 x Yes, not monic b No c Yes, not monic e No f Yes, not monic h No i No k Yes, monic l i 5 ii 9 iii 1 b i 5 ii i 2 ii 11 iii 10 d i 1 ii i 5 ii 7 iii 3 f i 0 ii i 6 ii 1 ii 11 h i 1 ii i 3 ii 13 iii 0 1 b 17 c 7 d 58 1 b 2 c 14 pffiffiffi 47 f 13 5 3 g 56 64 Yes, monic Yes, monic Yes, monic Yes, not monic 6 iii 3 6 iii 0 9 iii 9 54 iii 22 pffiffiffi e 7 25 d 3 h 7 Exercise 14-02 1 a d f g i 2 a c 3 a d 4 a 9x3 þ 8x2 þ 6x 2 b x3 þ 4x þ 2 c 4x2 3x 2 5x4 3x3 5x2 þ 4 e x4 x3 þ x2 5x 2x5 þ 10x4 þ 2x3 4x2 þ 4x þ 22 7x6 þ x5 þ x3 þ x2 þ 3x 2 h 8x4 þ 15x2 þ 4x 7 4x4 8x3 þ 5x2 þ x 2 j 6x3 2x2 6x þ 1 x2 þ 11x 1 b x2 3x 5 x2 þ 3x þ 5 d 2x2 þ 26x 5 2x2 15x þ 9 b x2 þ 4x 15 c 3x2 þ 11x þ 6 2 2 x þ 7x þ 3 e 3x 15x þ 3 f 3x2 7x 15 pffiffiffi x2 7x þ 6 b 24 21 2 c 1, 6 665 Answers Exercise 14-03 4 c 2 2 3 1 a 3x þ 2x 3x 2x b 32x þ 80x 22x c 45x4 13x3 þ 3x2 2x d 18x4 þ 6x3 7x2 11x 6 e 21x7 þ11x6 þ 2x5 3x4 x3 þ 2x2 f 8x5 þ 47x4 þ 41x2 þ 28x 12 2 4x3 þ 25x2 13x 6 3 a 2x4 þ 11x3 þ12x2 66x b 4x3 þ 9x2 þ24x 54 3 2 c 8x 62x þ 99x d y −1 0 1 j 2 a c 3 a c e g x2 þ 7x þ 4 ¼ (x þ 2)(x þ 5) 6 x2 6x þ 2 ¼ (x 3)(x 3) 7 4x2 þ 3x þ 10 ¼ (x 1)(4x þ 7) þ 17 8x2 þ 9x þ 11 ¼ ð2x þ 1Þ 4x þ 212 þ 812 3 2 2 x þ 6x þ 5x 4 ¼ (x 3)(x þ 9x þ 32) þ 92 4x3 þ 2x2 þ x ¼ (x þ 4)(4x2 14x þ 57) 228 2x3 x2 þ 5x þ 3 ¼ (x þ 6)(2x2 13x þ 83) 495 3x3 x2 þ 11 ¼ (x þ 2)(3x2 7x þ 14) 17 x5 x4 þ 8x3 þ 2x2 x 1 ¼ (x þ 1)(x4 2x3 þ 10x2 8x þ 7) 8 x4 x2 10 ¼ (x þ 3)(x3 3x2 þ 8x 24) þ 62 (3x 1), R ¼ 3 b (x þ 7), R ¼ 14 (3x3 þ 14x2 2x þ 21), R ¼ 42 d (4x þ 6), R ¼ 17 (2x 1)(3x þ 2) b (2x 1)(x2 þ x þ 1) (2x 1)(4x þ 7) d (2x 1)(3x2 þ 2x þ 1) (2x 1)(x3 3x2 4x þ 2) f (2x 1)(x3 x þ 3) (2x 1)(3x2 þ 1) h (2x 1)(4 3x x5) 2 3 x –1 0 1 2 3 4 5 x –20 e f y 6 Exercise 14-04 1 a b c d e f g h i y 3 2 −3 −2 −1 g 0 1 0 x 2 h y –3 –2 –1 0 y 1 2 3 4 5 x y x −2 0 –30 i 6 3 1 3 x −12 y 4 –2 2 x 1 0 Exercise 14-05 1 a e 2 a f 5 7 54 174 b f b g c g c h 181 1709 2 0 d h d i 1 85 14 6 179 29 2 115 e 12 2 B 3 A 4 a x-intercepts are 1, 0, 3 and y-intercept is 0. Exercise 14-06 y 1 a B, C b B, C c A d A, B, C e A, B 2 Teacher to check. 3 a x(x þ 2)(x þ 4) b x(x 2)(x þ 1) c (x 1)(x þ 1)(x þ 2) d (x 2)(2x 1)(x þ 4) e (x 1)(x 2)(x 3) f (x 2)(x þ 8)(x 5) g (x 6)(x þ 1)(3x 1) h (x 2)(3x þ 1)(2x 1) i x2(2x 1)(x 2) 4 a x ¼ 4, 12, 3 b x ¼ 4 c x ¼ 2, 52, 3 d x ¼ 5 e x ¼ 4, 3, 2 f x ¼ 7, 0, 2 g x ¼ 3, 2, 3 h x ¼ 2, 1, 4 i x ¼ 4, 1, 5 j x ¼ 12, 1, 3 k x ¼ 14, 23, 1 l x¼3 Exercise 14-07 1 a 0 1 2 3 x y −1 0 3 x 0 b x-intercepts are 1, 1, 3 and y-intercept is 3. y –10 b y –3 –2 –1 –1 1 3 x –3 1 2 x –12 666 9780170194662 Answers c x-intercepts are 6, 0, 1 and y-intercept is 0. e f y y y –2 1 –6 0 x 4 1 –2 g d x-intercepts are 2, 1 and 112 and y-intercept is 6. h y 6 x 1 –12 x 0 y y 6 –2 –3 –2 x 4 1 2 11 x 1 –2 0 3 x –36 2 2 a b y e x-intercept is 1 and y-intercept is 1. y –2 y 3 2x 0 –4 1 x 0 –18 –32 x –1 c d y –1 f x-intercepts are 2 and 3 and y-intercept is 18. x 0 y –2 3 e x 0 –1 –4 18 y 2 f y 3 x y 4 0 –2 1 2 x 0 x –8 Exercise 14-08 1 a b y –4 –1 1 x y 9780170194662 4 x –64 j y y y 1 –3 –2 0 8 –4 x –4 i d y –1 0 2 0 –2 c h y –1 –2 x –2 g y 2 4 x x 0 1 1 2 2 x 0 0.5 2 x –64 –3 667 Answers k l y e y f y y = –3P(x) 2 2x 1 –2 –1 0 y y = P(–x) 4 –1 x –2 –1 x 1 –48 –6 Exercise 14-09 1 a b y (–2, 5) y = P(x) + 2 y = 2 P(x) –3 –1 1 (1, 2) x x 1 y = P(x) y 3 (–2, 6) y x 3 –9 –2 a b y –8 d y x 3 y –3 y = 1 P(x) 2 (–2, 1 1 ) 2 y y = P(–x) 9 1 c y = –P(x) –9 x 1 (1, –3) –2 e c y –7 f y 4 y y = –P(x) –3 1 3 b y y = P(x) – 2 x (1, –2) x (–1, –2) d 4 shift down 2 units shift up 1 unit stretch vertically by a factor of 2 reflect in x-axis and shift up 3 units reflect in x-axis and stretch vertically by a factor of 3 reflect in x-axis and shift up 2 units stretch vertically by a factor of 2 and shift down 5 units reflect in x-axis, stretch vertically by a factor of 3 and shift up 4 units compress vertically by a factor of 12 and shift up 4 units y = –P(x) Chapter 15 –1 1 1 4 a b c d e f g h i y y = 2P(x) x –2 668 x –18 x (1, 1) (–1, 1) –1 1 3 –4 y = P(x) + 1 y y = 2P(x) y (1, –3)(3, –3) x y 3 c d –12 1 (–2, 3) 2 a y = P(x) – 3 x (2, 3) y = P(–x) –1 x –1 y = P(x) – 3 –2 –3 –1 x 1 x SkillCheck 1 a SSS 2 a SSS b SAS b AA c RHS c RHS d AAS d SSS e SAS e SAS f AAS f AA 9780170194662 Answers Exercise 15-01 1 a radius 2 a b chord c circumference d tangent segment O sector b sector drawn from centre of circle and bounded by 2 radii and arc. Segment is bounded by chord and arc 3 d ¼ 2r 4 D 5 a radius b quadrant c tangent d diameter e chord f arc g sector h circumference i segment 6 a diameter b segment c sector d arc Exercise 15-02 1 3 5 8 9 12 14 15 a Proof by SAS 2 a, b Proof by RHS a Proof by SSS 4 c The centre of the circle a, b Proof by SAS 6 5.74 cm 7 60 cm a UC ¼ 4.5 m (the perpendicular from the centre bisects the chord and the chords are equal as they are the same distance from the centre) b DE ¼ 12 m (chords of equal length subtend equal angles at the centre of a circle) c \UVO ¼ 58 (chords of equal length subtend equal angles; angle sum of isosceles n ) d PQ ¼ 30 mm (Pythagoras: the line from the centre is the perpendicular bisector of the chord) e OM ¼ 21 cm (Pythagoras: the line from the centre is the perpendicular bisector of the chord) pffiffiffi f OD ¼ 18 2 (Pythagoras: the line from the centre is the perpendicular bisector of the chord) 52 cm 10 18.4 km 11 MN ¼ 4 cm 77 cm 13 34 cm, 20 cm a AB ¼ 30 cm b area ¼ 840 cm2 a 52 cm b area ¼ 1920 cm2 Exercise 15-03 1 a b c d e f g h i 2 a b c d e f 9 45 > > > 112 > > > > 120 > > > = 232 ðangle at the centre is twice 40 > > the angle on the circumferenceÞ > 74 > > > > > 63 > > ; 104 90 (angle in a semicircle) 9 48 > > > 36 > > > = 30 ðangles at the circumference of 35 > > a circle standing on the same arcÞ > 74 > > > ; 90 9780170194662 3 a reflex \ROT ¼ 216 (angles at a point), \S ¼ 108 (angle at the centre is twice the angle at the circumference) b x ¼ 43 (angles opposite equal sides in an isosceles triangle are equal), y ¼ 86 (exterior angle of triangle equal to sum of two opposite interior angles, or angle at the centre is twice the angle at the circumference) c n ¼ 37 (angles at the circumference standing on the same arc are equal), m ¼ 74 (angle at the centre is twice the angle at the circumference) d p ¼ 37 (angles at the circumference standing on the same arc equal) e w ¼ 50 (angle at the centre is twice the angle at the circumference) f h ¼ 113 (angle at the centre is twice the angle at the circumference, co-interior angles on parallel lines supplementary) 4 a m ¼ 75 (opposite angles of cyclic quadrilateral) b p ¼ 88 (opposite angles of cyclic quadrilateral); q ¼ 121 (opposite angles of cyclic quadrilateral) c x ¼ y ¼ 90 (angles in a semicircle) 5 a n ¼ 106 (exterior angle of cyclic quadrilateral) b w ¼ 60 (angle in an equilateral triangle; exterior angle of cyclic quadrilateral) c x ¼ 84 (exterior angle of cyclic quadrilateral), y ¼9 110 (exterior angle of cyclic quadrilateral) 6 a 23 = ðangle in semicircle, b 9 ; angle sum of a triangle) c 45 d 63 (opposite angles of a cyclic quadrilateral) e 75 ðexterior angle of cyclic quadrilateralÞ f 88 7 a x ¼ 75 (angle at the centre is twice the angle at the circumference) y ¼ 33 (angles at the circumference standing on the same arc) z ¼ 72 (angle sum of a triangle) b x ¼ 108 (angle at the centre is twice the angle at the circumference) y ¼ 126 (opposite angles of a cyclic quadrilateral) z ¼ 252 (angles at a point, or angle at the centre is twice the angle at the circumference) c x ¼ 70 (straight line) y ¼ 110 (exterior angle cyclic quadrilateral) z ¼ 70 (straight line) d x ¼ 96 (angle at the centre is twice the angle at the circumference) y ¼ 42 (angles opposite equal sides of an isosceles n , angle sum of a triangle) z ¼ 264 (angles at a point) e x ¼ 140 (angles opposite equal sides of an isosceles n , angle sum of a triangle) y ¼ 70 (angle at the centre is twice the angle at the circumference) z ¼ 35 (angle sum of an isosceles n , and by subtraction) f x ¼ 62 (angle in a semicircle, angle sum of a triangle) y ¼ 118 (opposite angles of a cyclic quadrilateral) z ¼ 31 (angle sum of an isosceles n ) 8 WXYZ is a cyclic quadrilateral because \W þ \Y ¼ 180 and \X ¼ \Z ¼ 180 i.e., opposite angles are supplementary 669 Answers Exercise 15-04 1 a b c d 2 a b 3 a 4 a \YMP OP ’ AB (angle between a tangent and radius) proof by angle sum of an isosceles triangle proof by angle at centre is twice angle at circumference angle between a tangent and radius proof by angles on a straight line proof by AA (equiangular triangles) a ¼ 56 (the angle between the radius and the tangent is a right angle) b b ¼ 21 (radius is perpendicular to a tangent, and Pythagoras’ theorem) c c ¼ 134 (a tangent is perpendicular to the radius; angle sum of a quadrilateral) d g ¼ 67 (alternate segment theorem) 5 a 15 b 5 c 9 d 7 e 20 f 4 6 x ¼ 7 cm 7 a XP ¼ 10 cm b AB ¼ 24 cm Exercise 15-05 1 \R ¼ \Q \P ¼ \S ðangles at the circumference standing on the same arcÞ [ n PYR ||| n SYQ (equiangular or AA) RY ) PY SY ¼ QY (matching sides in similar triangles) [ PY 3 YQ ¼ RY 3 YS 2 \ADC ¼ \BEC (opposite angles of a parallelogram) \ADC ¼ \CBE (exterior angle of a cyclic quadrilateral) [ \BEC ¼ \CBE [ n CBE is isosceles (two equal angles) 3 Construction: Draw a perpendicular from O to meet DG at P. Since the perpendicular from the centre to a chord bisects the chord: DP ¼ GP and EP ¼ FP. ) DE ¼ DP EP \PQS ¼ x ¼ \SRP (angles at the circumference standing on the same arc) [ \PQS ¼ \PSQ ¼ x [ n SPQ is isosceles (two angles are equal). 7 Join APX and BQX. In n XPQ and n XAB: \X is common \XPQ ¼ \XAB (corresponding angles, PQ || AB) [ n XPQ ||| n XAB (equiangular or AA) PQ ) XP XA ¼ AB (matching sides in similar n s) but XP ¼ 12 XA (radius is half the diameter) 1 XA ) 2XA ¼ PQ AB ) 12 ¼ PQ AB ) PQ ¼ 12 AB 8 Let \PTX ¼ a [ \R ¼ a (alternate segment theorem) Now \QTY ¼ a (vertically opposite angles) [ \S ¼ a (alternate segment theorem) [ \R ¼ \S ¼ a [ PR || SQ (alternate angles are equal). Chapter 16 SkillCheck y 1 a 670 y = x2 – 3 – √3 x 0 0 √3 x –3 y c ¼ GP FP ¼ FG 4 \THJ ¼ \HIJ (alternate segment theorem) \THJ ¼ \HPI (alternate angles, HT || IP) [ \HIJ ¼ \HPI In n HIP and n HJI: \HIJ ¼ \HPI (proved above) \IHJ ¼ \IHP (common angles) [ n HIP ||| n HJI (equiangular) [ \HIP ¼ \HJI (third pair of equal angles in similar triangles) 5 In n UVX and n UWX: \UXV ¼ 90 ¼ \UXW (angle in a semicircle, straight line) UV ¼ UW (given) UX is common. [ n UVX ” n UWX (RHS) [ VX ¼ VW (matching sides in congruent triangles) [ circle bisects base of triangle. 6 Let \QRP ¼ x [ \SRP ¼ x (PR bisects \QRS) [ \PSQ ¼ x ¼ \QRP (angles at the circumference standing on the same arc) y b y = x2 y = x2 + 3x –3 2 a x 0 y (2, 8) y = x3 b y y = x3 + 3 (1, 4) x 0 3 0 c x y y = x3 – 1 0 1 –1 x 9780170194662 Answers y 3 a y b iii y = 1x (1, 1) iv y y= 1 x–1 x = –2 x 0 0 –1 x 1 0 m¼0 1_ 2 –2 y= 1 x+2 No gradient or the gradient is undefined. Exercise 16-02 y 4 a y b y = 2x (–1, 3) (1, 2) 1 y = 3–x 1 x 0 x 0 y c 0 –1 x y = –4–x (1, –4) 5 a x ¼ yþ1 2 pffiffiffiffiffiffiffiffiffiffiffi c x¼ y4 b x ¼ 3y 1 1 a c e g i 2 a 3 a 4 a 5 a 6 a i b 7 a b 8 a 9 a d 10 a c f i l 11 a i 6 ii 2 iii 0 b i 2 ii 2 iii 1 i 24 ii 4 iii 0 d i 17 ii 1 iii 1 pffiffiffi i 3 ii 1 iii 3 f i 125 ii 13 iii 1 i 125 ii 13 iii 1 h i 2 ii 6 iii 1 i 60 ii 12 iii 21 0 b 28 c 9k 9k2 16 b 10 c 6d 20 d 12 e 256 4 b 12 c 2 d x ¼ 212 5 b 0 c 14 d t ¼ 2, 112 i 11 ii 19 iii 10 9 2y ii 2 c 16 i 11 ii 5 iii 6 Teacher to check c m ¼ 3, 1 pffiffiffiffiffiffiffi 3 b 12 c 1 has no value d x ¼ 12 8 b 11 c 40 3k4 2k2 þ 3 e 3k2 þ 2k þ 3 f 4k 3 x 3; 0 y 3 b 4 x 0; 0 y 4 x 0; y 5 d x 1; y 0 e x 1; y 3 all x; y ¼ 3 g all x; y > 0 h all x; y ¼ 3 all x; y > 2 j all x; 1 y 1 k all x; y 4 all x; 0 < y 4 b y y (1, 4) Exercise 16-01 1 a e 2 a e i m 3 a Yes Yes Yes No Yes Yes i b f b f j n y No Yes Yes Yes No No c g c g k o m¼3 9780170194662 Yes No No No Yes Yes ii d h d h l p Yes Yes No Yes Yes Yes i All x c x m ¼ 12 x i All x ii All y d y ii y 1 y (1, 2) (2, 1) 1 0 3 x 0 i All x x 0 0 1 12 x + 2y = 3 1 1 y y = 3x – 2 (1, 1) 0 –2 0 b i, ii and iii c i For all values of m. ii If m is undefined (a vertical line). x 0 –2 x y c y 4 y=4 ii y > 0 0 x x i x 6¼ 0 ii y 6¼ 0 671 Answers e f y d i y f 1(x) ¼ f (x) ¼ 4x ii y 10 5 1 –3 0 5 y = 4x (1, –1) x 0 –10 –15 (1. 4) x –5 10 x 5 –5 (1, –16) –10 i All x ii All y g i All x ii y 16 h y e i f 1 ðxÞ ¼ 2xþ3 2 ii y (1, 7) y = 4x + 3 x 0 y 3 2x + 3 y = _____ 2 2 1 0 –3 –2 –1 –1 4 2x − 3 y = _____ 2 3 0 i All x ii All y i i All x x f i f 1 ðxÞ ¼ 22x 3 ii 2 − 2x y = _____ 3 y ii y > 3 2 1 y 7 −2 −1 0 −1 −2 x 0 ii y 2 Exercise 16-03 1 a i f ðxÞ ¼ xþ5 2 ii –4 –3 –2 –1 0 –1 x y=2x–5 (x) ¼ 3x ii y 4 3 2 1 4 3x 2 110 _ y = 2 3 3 c i f 1(x) ¼ 6 2x ii y=3– x 2 1 2 3 4 x –2 –3 –4 y=x b i f y=x 4 3 2 1 y x+5 y = __ 2 = 0 1 x 2 a Teacher to check. y b y=2–x 1 1 2 3x y = 1 − __ 2 (–3, 2) i All x 1 2 3x –2 –3 y = 3x 1 2 3 4x The graph of f (x) ¼ 2 x is itself symmetrical about the line y ¼ x. 3 b, c and h 4 a y y 6 5 4 3 2 1 y = x2 – 2 0 x –2 0 1 2 3 4 5 6x –6 –5 –4 –3 –2 –1 –1 672 –2 –3 –4 –5 –6 y = 6 – 2x b No 9780170194662 Answers c y¼ pffiffiffiffiffiffiffiffiffiffiffi xþ2 y y= x2 3 Both graphs are increasing. For y ¼ 2x, y-intercept ¼ 1, no x-intercept. For y ¼ log2 x, x-intercept ¼ 1, no y-intercept. y 4 y = 4x – 2, x ≥ 0 y= x+2 0 –2 y = 3x y = log3 x x 1 y = log4 x –2 0 x 1 y=x pffiffiffiffiffiffiffiffiffiffiffi d y¼ xþ2 y y = x2 – 2, x ≤ 0 0 –2 x –2 y= x+2 5 a b No c x 0 or x 0 y 3 y = x2 + 3 a They are all increasing graphs and have a y-intercept of 1. For x > 0, y ¼ 4x is steeper than y ¼ 3x, which is steeper than y ¼ 2x. For x < 0, y ¼ 4x is closer to the x-axis than y ¼ 3x, which is closer to the x-axis than y ¼ 2x. b They are all increasing graphs and have a x-intercept of 1. For x > 1, y ¼ log2 x is steeper than y ¼ log3 x, which is steeper than y ¼ log4 x. For x < 0, y ¼ log4 x is closer to the y-axis than y ¼ log3 x, which is closer to the y-axis than y ¼ log2 x. 5 a 2x b 4x 6 a log4 x b log2 x 7 a 1.3010 b 2.7973 c 3.7345 d 0.9138 e 0.3979 f 0.1192 8 x 0.5 1 2 5 8 10 y 6 a b x 2 or x 2 4 0 x 0.9 1 0 x 1 2 3 4 5 6 7 8 9 10 9 Teacher to check. y b No c x 12 or x 12 2 Exercise 16-05 1 a x 0 –4 –3 –2 –1 –1 c y = f (x) + 1 1 2 3 4 y 4 3 2 1 –4 –3 –2 –10 –1 –2 –3 –4 x d 4 3 2 1 –4 –3 –2 –10 –1 –2 –3 –4 y b –2 –3 –4 Exercise 16-04 1 y ¼ 2x Interchange x and y and make y the subject. [ x ¼ 2y [ log2 x ¼ log2 2y [ log2 x ¼ y log2 2 where log2 2 ¼ 1 [ y ¼ log2 x 2 a Domain: all x; range: y > 0 b Domain: x > 0; range: all y c They have interchanged. y 4 3 2 1 (– 12 , –6 14 ) 9780170194662 0.7 2 1 –2 –1 0 –1 –2 (2, –4) –3 0.3 y y 7 a 0 0.3 x 0 1 2 3 4 x 1 x y 4 3 2 1 1 2 3 4 x –4 –3 –2 –1 0 –1 2 3 4 –2 –3 y = f (x) – 3 –4 y = f (x – 2) 673 Answers e y 3 2 1 4 a y 4 3 2 1 0 –4 –3 –2 –1 –1 1 2 3 4 –3 –2 –1 –1 –2 –3 x –2 y = f (x + 4) –3 –4 y 3 2 1 c 2 a b y y y = f(x) – 3 0 y = f(x) + 2 x 0 –3 d y y = f(x) – 1 0 3 f y y y = f(x + 1) 0 –1 0 (–2, –1) x x 3 y = f(x + 2) –1 3 a y = f(x – 1) 0 x 1 4 3 2 1 1 2 3x –3 –2 –1 y = f(x) + 3 0 1 2 3 y 3 x 1 2 y = f(x + 1) – 3 y b 0 –2 y = f(x) + 2 x x y (1, 1) y = f(x – 1) d y (2, 2) y = f(x – 2) + 1 y 2 0 x y = f(x) – 3 y = f(x) + 2 0 2 x 0 x x y c y = f(x + 2) + 1 (–2, 1) –3 674 b 3 4 x y 0 –3 0 c y 1 2 d –3 –2 –1–10 –2 –3 5 a e 0 –2 –1 –1 –2 –3 1 y y = f(x – 3) x 0 –1 1 2 3x y = f(x – 3) y e c y = f(x) – 2 0 y = f(x + 2) –3 –2 –1 –1 –2 –3 2 x y 4 3 2 1 b 0 x 9780170194662