Formula List

Formula List NUMBERS 1. Natural number (𝑵): counting numbers. Example: 1 , 2 , 3 , 4 , … … 2. Integer (ℤ): Positive and negative whole numbers. … … − 3 , −2 , −1 , 0 , 1 , 2 , 3 … … Example: ℤ− ℤ+ Neutral Integer 3. Real number (ℝ): 1 −3.5 , −2 , 0 , 2 Can be placed in number line. Example: , 𝜋 , √2 … … 4. Unreal number: Cannot be placed in the number line 5. Square number: 1, 4, 9, 25, 36, 49, … … 6. Cubic number: 1, 8, 27, 64, 125, 216, 343 … … 1 0 Example: √−2 , , 𝑒𝑡𝑐 7. Prime number (𝑷): ∎ Integer starts from 2 ∎ Divisible by 1 & itself ∎ Has exactly two factors. Note: 1 is not a prime number. Example: 2 , 3 , 5 , 7 , 11 , 13 , 17, 19 … … Factor of 6 8. Factors of 𝟔: 1, 2, 3, 6 9. Multiples of 𝟔: 6, 12 , 24 , 36 , 48 , 60, … … 10. Even numbers: −4, −2, 0 , 2 , 4 , 6 , 8 , … … 11. Odd numbers: −5, −3, −1, 1 , 3 , 5 , 7 , 9, … …. 1 , 2 , 3 , 6 , 12 , 18 … … Multiple of 6 𝑎 12. Rational Number (𝑸): A number that can be converted into fraction of integers (𝑏 ) . Example: 0 0 =1 2.5 = 25 10 −5.1 =− 51 10 √16 4 =1 𝑎 13. Irrational Number (𝑸′ ): A number that cannot be converted into fraction of integers (≠ 𝑏 ) . 14. Product of prime factors: [Example] 2 48 2 24 2 12 2 6 3 ∴ 48 = 2 × 2 × 2 × 2 × 3 = 24 × 3 2 80 2 40 2 20 2 10 5 ∴ 48 = 2 × 2 × 2 × 2 × 5 = 24 × 5 Formula List SEQUENCE ∎ Type – 1 (Consecutive numbers): 𝑻𝟏 +2 𝑻𝟐 𝑻𝟑 4 5 𝑻𝟐 𝑻𝟑 −1 0 3 𝑻𝟏 −3 −2 𝑻𝒏 ….. +2 𝑛+2 𝑻𝒏 ….. −3 𝑛−3 ∎ Type – 2 (Common Difference): 𝑻𝟏 𝑻𝟐 𝑻𝟑 𝑻𝟒 7 9 11 13 2 2 𝑻𝒏 𝑎 + (𝑛 − 1)𝑑 2 = 7 + (𝑛 − 1) × 2 𝑻𝟏 𝑻𝟐 𝑻𝟑 𝑻𝟒 𝑻𝟓 6 10 16 24 34 4 6 2 8 2 10 2 𝑻𝒏 𝑎 + (𝑛 − 1)𝑑1 + (𝑛−1)(𝑛−2) = 6 + (𝑛 − 1) × 4 + 2 𝑑2 (𝑛−1)(𝑛−2) 2 ×2 Formula List GEOMETRY Acute Angle 0 < 𝜃 < 90° Obtuse Angle 90° < 𝜃 < 180° Reflex Angle 180° < 𝜃 < 360° Right Angle 𝜃 = 90° 𝜃 𝜃 𝜃 𝜃 External ∠ of ∆ Triangle Allied Angle Parallelogram Quadrilateral 𝑏 𝑎 𝒂=𝒃 ∠𝒔 centred at a point Vertically opposite angles Adjacent ∠𝒔 on a straight line Adjacent ∠𝒔 on a right angle Corresponding angles Alternate angles Formula List CIRCLE THEOREM 𝐴 𝑦° 𝑥° 𝑥° 2𝑦° 𝑧° .𝑜 .𝑜 2𝑥° 𝐴 𝐵 𝐶 𝐵 𝑦° 𝐴 𝐵 ∠𝐴𝐶𝐵 = 90° ∠𝐴𝑂𝐵 = 2𝑥 𝑥° = 𝑦° = 𝑧° Reflex ∠𝐴𝑂𝐵 = 2𝑦 𝐵 𝐴 𝐵 𝐴 𝑏° 𝑎° 𝑑° 𝑎° 𝑐° 𝐷 𝐷 .𝑜 𝑐° 𝐶 𝑥° 𝐴 𝐶 𝑎° + 𝑐° = 180° 𝐵 𝐶 ∠𝐴𝐶𝑂 = ∠𝐵𝐶𝑂 = 90° 𝑎° = 𝑥° 𝑏° + 𝑑° = 180° 𝐴 𝐴 .𝑂 𝐵 𝑎° 𝑏° 𝐵 𝑋 𝑥° 𝑦° 𝑌 𝑎° = 𝑦° 𝑏° = 𝑥° 𝐶 𝑍 𝐴𝐵 = 𝐵𝐶 Formula List LOCUS CONSTRUCTION 1. Locus of a point 𝑷, equidistant from 𝑨 and 𝑩: 𝑷 𝑩 𝑨 ← Locus ∴ Perpendicular bisector is the locus of a point 𝑃, equidistant from 𝐴 and 𝐵. 2. Locus of a point 𝑷, equidistant from 𝑨𝑩 and 𝑩𝑪: 𝑨 ← Locus 𝑷 𝑩 𝑪 ∴ Angle bisector is the locus of a point 𝑃, equidistant from 𝐴𝐵 and 𝐵𝐶. 3. Locus of a point 𝑷, so that it is 𝟒 cm from 𝑷: 𝑷′ 𝟒 cm ← Locus 𝑷 ∴ Circle is the locus of a point 𝑃, 4 cm from it. Formula List 4. Locus of a point 𝑷, so that it is always 𝟑 cm from a line 𝑨𝑩: 𝑷 𝟑 cm 𝟑 cm 𝟑 cm 𝑩 𝑨 𝟑 cm Locus ∴ Capsule is the locus of a point 𝑃, so that 𝑃 is always 3 cm from 𝐴𝐵. ̂ 𝑩 is always 𝟗𝟎°: 5. Locus of a point 𝑷, so that 𝑨𝑷 𝑷 𝑨 Locus 𝑩 ∴ Circle is the locus of a point 𝑃, so that 𝐴𝑃̂ 𝐵 is always 90°. 6. Locus of a point 𝑷, so that triangle 𝑨𝑷𝑩 always has an area of 𝟏𝟐 𝐜𝐦𝟐 : Locus 𝑷 𝑷 𝟑 cm 𝟑 cm 𝑨 𝑩 𝟖 cm 𝑷 1 Locus ∴ Parallel lines 𝟑 cm from 𝐴𝐵 are the locus of a point 𝑃, so that triangle 𝐴𝑃𝐵 is always 2 × 8 × 3 = 12 cm2. Formula List SIMILARITY 1. If three angles equal, triangles are similar. 2. If two angles equal, 3rd angle automatically is equal. 3. Opposite sides of equal angles are corresponding sides, 𝑙1 and 𝑙2 . 4. 𝑙1 𝑙2 5. 𝐴1 𝐴2 6. 𝑚 = 𝑚1 2 𝑙 2 𝑙2 𝑉1 𝑙1 3 𝑉1 𝑚 = ( ) , 𝑉 = 𝑚1 𝑉2 𝑙2 2 2 𝐴 3 𝑉 2 = ( 1) 1 1 2 2 𝑤 = 𝑤1 2 7. (𝐴 ) = (𝑉 ) CONGRUENCY POLYGON A regular polygon has all sides equal and all interior angles equal For a 𝒏-sided polygon: Sum of all exterior angles (Regular + Irregular polygon) 360° Sum of all interior angles (Regular + Irregular polygon) (𝑛 − 2) × 180° Each exterior angle (Regular polygon only) Each interior angle (Regular polygon only) Sum of an exterior angle and an interior angle 360° 𝑛 (𝑛 − 2) × 180° 𝑛 (𝑛 − 2) × 180° 360° + = 180° 𝑛 𝑛 Formula List Naming of polygons: 3-sided: Triangle 4-sided: Quadrilateral 5-sided: Pentagon 6-sided: Hexagon 7-sided: Heptagon 8-sided: Octagon 9-sided: Nonagon 10-sided: Decagon TRIGONOMETRY SOH CAH TOA sin 𝜃 = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 𝑂 = 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝐻 𝐴 cos 𝜃 = 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 𝐴 = 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝐻 Opposite For a right-angled triangle, 𝑨𝑩𝑪: 𝜃 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 𝑂 tan 𝜃 = = 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 𝐴 𝐵 𝐶 Adjacent SINE RULE 𝐴 𝑎 𝑏 𝑐 = = sin 𝐴 sin 𝐵 sin 𝐶 𝑐 𝐵 𝑏 𝑎 𝐶 COSINE RULE To find the length of a side 𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴 𝑏 2 = 𝑎2 + 𝑐 2 − 2𝑎𝑐 cos 𝐵 To find an angle when all three sides are given cos 𝐴 = 𝑏 2 + 𝑐 2 − 𝑎2 2𝑏𝑐 cos 𝐵 = 𝑎2 + 𝑐 2 − 𝑏 2 2𝑎𝑐 cos 𝐶 = 𝑎2 + 𝑏 2 − 𝑐 2 2𝑎𝑏 𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 cos 𝐶 Formula List BEARING The bearing of a point 𝐵 from another point 𝐴 is: • an angle measured from the north at 𝐴, • in a clockwise direction. 𝑁 𝐵 𝑁 𝐵 050° 230° Example: The bearing of 𝐵 from 𝐴 is 050°. The bearing of 𝐴 from 𝐵 is 230°. 𝐴 𝐴 ELEVATION – DEPRESSION 1. Angle of Elevation → line of sight above horizontal line. 2. Angle of Depression → line of sight below horizontal line. Angle of elevation of 𝑻 from 𝑴 = 𝜽° Angle of depression of 𝑴 from 𝑻 = 𝜽° Formula List ARITHMETIC ∎ Ratio: If the amount 𝐴 is divided between 𝑃, 𝑄 and 𝑅 in the ratio 𝑝: 𝑞: 𝑟 Share of 𝑃 = 𝑃 𝑝+𝑞+𝑟 × 𝐴, Share of 𝑄 = 𝑞 𝑝+𝑞+𝑟 ×𝐴 Share of 𝑅 = 𝑟 𝑝+𝑞+𝑟 ×𝐴 ∎ Percentage: loss 1. Profit = sale price – cost price 4. Percentage loss = cost price × 100 2. Loss = cost price – sale price increase 5. Percentage increase = original value × 100 profit 3. Percentage profit = cost price × 100 𝐴 6. % of 𝐴 compared with 𝐵 = 𝐵 × 100 ∎ Simple Interest: Here, 𝑃𝑅𝑇 𝐼= 100 𝐼 = Interest 𝑃 = Principle ∎ Compound Interest: 𝑇 = Time (Year) 𝐴 = 𝑃 (1 + 𝑅 ) 100 𝑇 𝑅 = Rate of Interest 𝐴 = Final Amount ∎ Speed, Distance, Time: 1. • 𝑑 =𝑠×𝑡 • 𝑠= • 𝑡= 𝑑 𝑡 𝑑 𝑠 2. 𝑑 𝑠 𝑡 𝟏𝐬𝐭 part • 𝑠 = 𝑥 km/h • 𝑡 = 𝑡1 h • 𝑑 = 𝑥𝑡1 𝟐𝐧𝐝 part • 𝑠 = 𝑦 km/h • 𝑡 = 𝑡2 h • 𝑑 = 𝑦𝑡2 Avg. speed = = Total distance Total time 𝑥𝑡1 +𝑦𝑡2 𝑡1 +𝑡2 ∎ Map Scale: 1. Scale of a map = (map distance): (actual distance) 2. Scale of a map is written in the form 1: 𝑛, where 1 is map distance ratio and 𝑛 is actual distance ratio. 𝑑 1 3. 𝑑1 = 𝑛 where 𝑑1 = map distance and 𝑑2 = actual distance. 2 𝐴 1 2 4. 𝐴1 = (𝑛) 2 𝑉 1 3 5. 𝑉1 = (𝑛) 2 where 𝐴1 = map area and 𝐴2 = actual area. where 𝑉1 = model volume and 𝑉2 = actual volume. Formula List SETS 1. Notation: Number of elements in set 𝐴 𝑛(𝐴) 𝐴 = {𝑝, 𝑞, 𝑟, 𝑠} ∴ 𝑛(𝐴) = 4 Intersection ∩ 𝐴 = {𝑓, 𝑔, ℎ, 𝑖}, 𝐵 = {𝑓, ℎ, 𝑗, 𝑘} ∴ 𝐴 ∩ 𝐵 = {𝑓, ℎ} Union ∪ 𝐴 = {𝑓, 𝑔, ℎ, 𝑖}, 𝐵 = {𝑓, ℎ, 𝑗, 𝑘} ∴ 𝐴 ∪ 𝐵 = {𝑓, 𝑔, ℎ, 𝑖, 𝑗, 𝑘} Universal set 𝜀 Complement of set 𝐴 𝐴′ 𝐴 ∪ 𝐴′ = 𝜀 𝜀 = {1, 2, 3, 4}, 𝐴 = {2, 3} ∴ 𝐴′ = {1, 4} Null set/Empty set 𝜙 𝐴 = {𝑥, 𝑦, 𝑧}, 𝐵 = {2, 3, 4} ∴ 𝐴∩𝐵 = 𝜙 ∴ 𝑛(𝐴 ∩ 𝐵) = 0 Is an element of ∈ 𝐴 = {𝑥, 𝑦, 𝑧} 𝑥 ∈ 𝐴, 2 ∉ 𝐴 Is a subset of ⊂ 𝐴 = {𝑥, 𝑦, 𝑧}, 𝐵 = {𝑦, 𝑥}, 𝐶 = {𝑥, 𝑝} ∴ 𝐵 ⊂ 𝐴 and 𝐶 ⊄ 𝐴 𝜀 2. All possible subsets: • • If 𝑨 = {𝒙, 𝒚, 𝒛}, subsets of 𝑨 are: {𝒙}, {𝒚}, {𝒛}, {𝒙, 𝒚}, {𝒙, 𝒛}, {𝒚, 𝒛}, {𝒙, 𝒚, 𝒛}, 𝝓. Both 𝝓 and 𝑨 are subsets of 𝑨. 3. De Morgan’s rules: 𝐴′ ∩ 𝐵′ = (𝐴 ∪ 𝐵)′ 𝐴′ ∩ 𝐵′ ∩ 𝐶 ′ = (𝐴 ∪ 𝐵 ∪ 𝐶)′ 𝐴′ ∪ 𝐵′ = (𝐴 ∩ 𝐵)′ 𝐴′ ∪ 𝐵′ ∪ 𝐶 ′ = (𝐴 ∩ 𝐵 ∩ 𝐶)′ 4. Subset: Isosceles Rightangled Acute Equilateral Scalene Obtuse Formula List 𝜀 = 𝑃𝑜𝑙𝑦𝑔𝑜𝑛𝑠 • • • • • • Rectangle Rhombus Parallelogram Square Kite Quadrilateral Trapezium Rhombus → all sides equal. Rectangle → all angles equal. Square → all angles 90°, all sides equal. Parallelogram → opposite sides parallel. Kite → one line symmetry. Trapezium → one pair of parallel sides. Triangle 𝜀 = 𝑁𝑢𝑚𝑏𝑒𝑟𝑠 Real, ℝ Rational, ℚ Integer, ℤ • Natural, ℕ • 1, 2, 3, … • • … , −1, 0, 1, … ℕ is the set of natural numbers or positive integers {1, 2, 3, 4, … … }. ℤ is the set of integers {… , −2, −1, 0, 1, 2, … }. ℚ is the set of rational numbers. ℝ is the set of real numbers. ½, ¾, 2.5 𝜋, ln 2, √3 5. Description of a Venn diagram: • • • • • • • • • • • • • • 1 → 𝑂𝑛𝑙𝑦 𝐴. 2 → 𝑂𝑛𝑙𝑦 𝐵. 3 → 𝑂𝑛𝑙𝑦 𝐶. 4 → 𝑂𝑛𝑙𝑦 𝐴 𝑎𝑛𝑑 𝐵 𝑜𝑟 𝑗𝑢𝑠𝑡 𝐴 𝑎𝑛𝑑 𝐵 𝑜𝑟 𝐴 𝑎𝑛𝑑 𝐵 𝑏𝑢𝑡 𝑛𝑜𝑡 𝐶. 5 → 𝑂𝑛𝑙𝑦 𝐴 𝑎𝑛𝑑 𝐶 𝑜𝑟 𝑗𝑢𝑠𝑡 𝐴 𝑎𝑛𝑑 𝐶 𝑜𝑟 𝐴 𝑎𝑛𝑑 𝐶 𝑏𝑢𝑡 𝑛𝑜𝑡 𝐵. 6 → 𝑂𝑛𝑙𝑦 𝐵 𝑎𝑛𝑑 𝐶 𝑜𝑟 𝑗𝑢𝑠𝑡 𝐵 𝑎𝑛𝑑 𝐶 𝑜𝑟 𝐵 𝑎𝑛𝑑 𝐶 𝑏𝑢𝑡 𝑛𝑜𝑡 𝐴. 7 → 𝐴𝑙𝑙 𝐴, 𝐵 𝑎𝑛𝑑 𝐶. 8 → 𝑁𝑜𝑡 𝐴, 𝐵 𝑎𝑛𝑑 𝐶. 4, 7 → 𝐵𝑜𝑡ℎ 𝐴 𝑎𝑛𝑑 𝐵 𝑜𝑟 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝐴 𝑎𝑛𝑑 𝐵. 5, 7 → 𝐵𝑜𝑡ℎ 𝐴 𝑎𝑛𝑑 𝐶 𝑜𝑟 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝐴 𝑎𝑛𝑑 𝐶. 6, 7 → 𝐵𝑜𝑡ℎ 𝐵 𝑎𝑛𝑑 𝐶 𝑜𝑟 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝐵 𝑎𝑛𝑑 𝐶. 1, 4, 5, 7 → 𝐴. 2, 4, 6, 7 → 𝐵. 3, 5, 6, 7 → 𝐶. 𝜀 𝐴 4 1 5 8 7 3 𝐶 2 6 𝐵 Formula List MATRIX 1. Multiplication by a scalar 𝑎 ( 𝑐 𝑎 ( 𝑐 𝑎 ( 𝑐 2. Addition of matrix 3. Subtraction of matrix 4. Multiplication of matrix 𝑎 𝑏 𝑘𝑎 𝑘𝑏 𝑘( )=( ) 𝑐 𝑑 𝑘𝑐 𝑘𝑑 𝑤 𝑥 𝑎+𝑤 𝑏+𝑥 𝑏 ) + ( 𝑦 𝑧) = ( ) 𝑐+𝑦 𝑑+𝑧 𝑑 𝑤 𝑥 𝑎−𝑤 𝑏−𝑥 𝑏 ) − ( 𝑦 𝑧) = ( ) 𝑐−𝑦 𝑑−𝑧 𝑑 𝑎𝑤 + 𝑏𝑦 𝑎𝑥 + 𝑏𝑧 𝑏 𝑤 𝑥 )( )=( ) 𝑐𝑤 + 𝑑𝑦 𝑐𝑥 + 𝑑𝑧 𝑑 𝑦 𝑧 5. If two matrices have order (𝑝 × 𝑞) and (𝑟 × 𝑠), they can be multiplied if 𝑞 = 𝑟 and the order of the resultant matrix will be (𝑝 × 𝑠). 𝑤 𝑥 𝑎 𝑏 ( ) = ( 𝑦 𝑧) 𝑐 𝑑 ∴ 𝑎 = 𝑤, 𝑏 = 𝑥, 𝑐 = 𝑦, 𝑑 = 𝑧. 6. Equal matrix 𝑎 𝑏 7. Determinant of ( ) is 𝑎𝑑 − 𝑏𝑐. 𝑐 𝑑 1 𝑎 𝑏 𝑑 −𝑏 8. Inverse of ( ) is ( ). If 𝑎𝑑 − 𝑏𝑐 = 0, the matrix has no inverse. 𝑎𝑑−𝑏𝑐 𝑐 𝑑 −𝑐 𝑎 9. Identity matrix: (1 × 1) (2 × 2) (3 × 3) 1 0 0 𝐈 = (0 1 0) 0 0 1 10. If any matrix is multiplied by identity matrix, the matrix remains same, i.e., 𝟏 × 𝐀 = 𝐀. The product of inverse matrix and original matrix is identity matrix i.e., 𝐀−1 × 𝐀 = 𝐈, 1 0 where 𝐈 = ( ). 0 1 𝐈 = (1) 1 0 𝐈=( ) 0 1 MATRIX TRANSFORMATION Formula List Formula List 5. Identification of transformation and their fully description: Transformation Identification Need to describe fully Enlargement Change of area or change of side-length Centre and scale factor Translation Change of position without rotation Column vector Reflection 180° rotation Corresponding object – image joining lines are parallel Corresponding object – image joining lines intersects at the same point, which is the centre of 180° rotation Other rotation - 6. 𝟏𝟖𝟎° rotation is same as enlargement, 𝑺. 𝑭. = −𝟏. 7. How to find unknown matrix? Equation of reflection line Centre Angle, direction and centre Formula List Case – 2: For known transformation, use the following matrices. Transformation Rotation 90°, anticlockwise, centre (0, 0) Rotation 90°, clockwise, centre (0, 0) Rotation 180°,centre (0, 0) Reflection in 𝑥 −axis Reflection in 𝑦 −axis Reflection in 𝑦 = 𝑥 Reflection in 𝑦 = −𝑥 Enlargement, scale factor 𝑘, centre (0, 0) Matrix 0 −1 ( ) 1 0 0 1 ( ) −1 0 −1 0 ( ) 0 −1 1 0 ( ) 0 −1 −1 0 ( ) 0 1 0 1 ( ) 1 0 0 −1 ( ) −1 0 𝑘 ( 0 0 ) 𝑘 8. 𝑀𝑎𝑡𝑟𝑖𝑥 × 𝑂𝑏𝑗𝑒𝑐𝑡 = 𝐼𝑚𝑎𝑔𝑒 9. 𝐑𝐓𝐄(ℎ, 𝑘) means (ℎ, 𝑘) is transformed by 𝐄, followed by 𝐓, followed by 𝐑. Case – 3: Formula List VECTOR 1 2 Column Vector: 𝑥1 • 𝐚⃗ = (𝑦 ) or 𝑥1 𝐢 + 𝑦1 𝐣 1 6 ⃗⃗⃗⃗⃗ 𝐴𝐵 −−−/−−− Step 1: 𝑃𝑇 = −−−/−−− = 𝑘 (constant) ⃗⃗⃗⃗⃗ • |𝐚⃗| = √𝑥1 2 + 𝑦1 2 • Unit Vector of 𝐚⃗: |𝐚⃗| • ∠ between 𝐚⃗ and 𝑥-axis: tan 𝜃 = • 𝑥1 𝑥2 𝑥 +𝑥 𝐚⃗ + 𝐛 = (𝑦 ) + (𝑦 ) = (𝑦1 + 𝑦2 ). • 𝑥1 𝑘𝑥 𝑘𝐚⃗ = 𝑘 (𝑦 ) = ( 1 ). 𝑘𝑦1 1 Step 2: ∴ 𝐴𝐵 ∥ 𝑃𝑇 ⃗ 𝐚 1 Show that 𝑨𝑩 ∥ 𝑷𝑻: Use the following steps: 2 1 𝑦1 𝑥1 7 2 Show 𝑨, 𝑩 and 𝑪 lie on a straight line (collinear). ⃗⃗⃗⃗⃗ 𝐴𝐵 −−−/−−− Step 1: ⃗⃗⃗⃗⃗ = −−−/−−− = 𝑘 (constant) 𝐵𝐶 Step 2: ∴ 𝐴𝐵 ∥ 𝐵𝐶 Step 3: Since 𝐵 is common, therefore 𝐴, 𝐵 and 𝐶 lie on a straight line. 3 • • 𝒙𝟏 𝒙𝟐 If (𝒚 ) = (𝒚 ), 𝟏 8 𝟐 o 𝑥1 = 𝑥2 ------------ (i) o 𝑦1 = 𝑦2 ------------ (ii) 𝒙𝟏 𝒙𝟐 If (𝒚 ) || (𝒚 ), 𝟏 𝟐 𝑥1 𝑥2 o (𝑦 ) = 𝑘 (𝑦 ) 1 2 o 𝑥1 = 𝑘𝑥2 ------------ (i) o 𝑦1 = 𝑘𝑦2 ------------ (ii) o 𝑥 • 5 • • • 2 9 ℎ 𝐴(ℎ, 𝑘) means ⃗⃗⃗⃗⃗ 𝑂𝐴 = ( ). 𝑘 In terms of position vector: ⃗⃗⃗⃗⃗ = 𝑂𝐵 ⃗⃗⃗⃗⃗ − 𝑂𝐴 ⃗⃗⃗⃗⃗ o 𝐴𝐵 o ∴ 𝐴𝐵𝐶𝐷 is a parallelogram. Or prove ⃗⃗⃗⃗⃗ 𝐴𝐷 = ⃗⃗⃗⃗⃗ 𝐵𝐶 ∴ 𝐴𝐵𝐶𝐷 is a parallelogram. ⇒ 𝑥𝐚⃗ + 𝑦𝐛 = 𝜆(𝑝𝐚⃗ + 𝑞𝐛) ∴ 𝑥 = 𝜆𝑝 ------------ (i) ∴ 𝑦 = 𝜆𝑞 ------------ (ii) ⃗⃗⃗⃗⃗⃗ For two geometrical properties of 𝑨𝑩 and ⃗⃗⃗⃗⃗⃗ 𝑷𝑸: Use the following steps: Step 1: Ratio • ⃗⃗⃗⃗⃗ 𝑃𝑄 = ⃗⃗⃗⃗⃗⃗ 𝑂𝑄 − ⃗⃗⃗⃗⃗ 𝑂𝑃 ⃗⃗⃗⃗⃗ = (5). Example: Given 𝐴(3, 2) and 𝐴𝐵 2 ⃗⃗⃗⃗⃗ Find 𝑂𝐵. 5 • ⇒ ⃗⃗⃗⃗⃗ 𝑂𝐵 − ⃗⃗⃗⃗⃗ 𝑂𝐴 = ( ). 2 3 8 5 ⃗⃗⃗⃗⃗ • ∴ 𝑂𝐵 = ( ) + ( ) = ( ). 2 4 2 Show that 𝑨𝑩𝑪𝑫 is a parallelogram: ⃗⃗⃗⃗⃗ = 𝐷𝐶 ⃗⃗⃗⃗⃗ Either prove 𝐴𝐵 𝑦 𝑞 ⃗⃗⃗⃗⃗ = 𝜆 × 𝐴𝐶 ⃗⃗⃗⃗⃗ Or otherwise, 𝐴𝐵 𝑥 Position Vector: • 𝑥 𝑝 Step 2: Use shortcut = to find 𝜆. Shortcut: 𝑦1 = 𝑦2 1 4 Given 𝑨, 𝑩 and 𝑪 are collinear (or parallel), find 𝝀. Step 1: Find ⃗⃗⃗⃗⃗ 𝐴𝐵 = 𝑥𝐚⃗ + 𝑦𝐛 and ⃗⃗⃗⃗⃗ 𝐴𝐶 = 𝑝𝐚⃗ + 𝑞𝐛. ⃗⃗⃗⃗⃗ 𝐴𝐵 ⃗⃗⃗⃗⃗ 𝑃𝑄 = 𝑚 𝑛 (constant) Step 2: Properties • ⃗⃗⃗⃗⃗ 𝐴𝐵 ∥ ⃗⃗⃗⃗⃗ 𝑃𝑄 . • 10 (ii) 𝐴𝐵: 𝑃𝑄 = 𝑚: 𝑛. Ratio example: ⃗⃗⃗⃗⃗ = 𝐩 ⃗ + 2𝐪 ⃗. Given 𝐴𝐵: 𝐵𝐶 = 5: 3 and 𝐴𝐵 Find ⃗⃗⃗⃗⃗ 𝐴𝐶 . ⃗⃗⃗⃗⃗ 𝐴𝐵 5 • ⇒ ⃗⃗⃗⃗⃗ = 5+3 • ⇒ • ⃗⃗⃗⃗⃗ = 𝐩 + ∴ 𝐴𝐶 5 𝐴𝐶 𝐩+2𝐪 𝐴𝐶 = 8 5 8 16 𝐪 5 Formula List 11 Area Ratio: 1 ∆ 𝑨𝑩𝑫 2 × 2 × ℎ = ∆ 𝑩𝑫𝑪 1 × 3 × ℎ 2 1 ∆ 𝑨𝑷𝑸 2 × 3 × 1 × sin 𝜃 = ∆ 𝑨𝑩𝑪 1 × 4 × 2 × sin 𝜃 2 ∆ 𝑷𝑺𝑻 3 2 =( ) ∆ 𝑷𝑸𝑹 5 PROBABILITY 𝑠 = number of 𝐴, 𝑡 = number of 𝐵, 𝑢 = number of 𝐶, 𝑛 = total number. 𝑠 𝑡 𝑢 1. 𝑃(𝐴) = 𝑛, 𝑃(𝐵) = 𝑛, 𝑃(𝐶) = 𝑛 𝑠 𝑛 𝑠 𝑛 2. (i) 𝑃(𝐴𝐴) = × , [If replaced] 𝑠 𝑠 𝑠 3. (i) 𝑃(𝐴𝐴𝐴) = 𝑛 × 𝑛 × 𝑛 [If replaced] 𝑠 𝑡 4. (i) 𝑃(𝐴𝐵) = 𝑛 × 𝑛 × 2 [If replaced] 𝑠 𝑛 (ii) 𝑃(𝐴𝐴) = × 𝑠 𝑠−1 𝑛−1 [If not replaced] 𝑠−1 𝑠−2 (ii) 𝑃(𝐴𝐴𝐴) = 𝑛 × 𝑛−1 × 𝑛−2 [If not replaced] 𝑠 𝑡 (ii) 𝑃(𝐴𝐵) = 𝑛 × 𝑛−1 × 2 [If not replaced] 5. 𝐴, 𝐵 and 𝐶 are arranged in the following 6 ways: 𝐴𝐵𝐶, 𝐴𝐶𝐵, 𝐵𝐴𝐶, 𝐵𝐶𝐴, 𝐶𝐴𝐵, 𝐶𝐵𝐴. 𝑠 𝑡 𝑢 6. (i) 𝑃(𝐴𝐵𝐶) = 𝑛 × 𝑛 × 𝑛 × 6 [If replaced] 𝑠 𝑡 𝑢 (ii) 𝑃(𝐴𝐵𝐶) = 𝑛 × 𝑛−1 × 𝑛−2 × 6 [If not replaced] 7. For 2 draws, 𝑃(diferent color) = 1 − 𝑃(same color). 8. AND rule: 𝑃(𝐴 and 𝐵) = 𝑃(𝐴) × 𝑃(𝐵). 9. OR rule: 𝑃(𝐴 or 𝐵) = 𝑃(𝐴) + 𝑃(𝐵). 10. 𝑃(𝐴) + 𝑃(𝐵) + 𝑃(𝐶) = 1. 11. The probabilities of disc, ball, bead, sweet etc. may change or not according to replacement. But probabilities of die, coin etc. always remain unchanged. Formula List MENSURATION 1. Triangles 𝐵 𝐵 𝐴 𝐶 𝑏 𝐴 𝐶 𝑏 𝑏 𝑐 1 1 Area of ∆𝐴𝐵𝐶 = × 𝑏𝑐 × sin 𝜃 Area of ∆𝐴𝐵𝐶 = × 𝑏 × ℎ 2 2 2. 𝜃 𝑎 ℎ ℎ SINE RULE 𝐴 𝑎 sin 𝐴 𝑏 𝑐 𝐵 𝐶 𝑎 = 𝑏 sin 𝐵 = 𝑐 sin 𝐶 COSINE RULE For sides: 𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴 For angles: cos 𝐵 = 𝑎 2 +𝑐 2 −𝑏2 2𝑎𝑐 3. Area of Quadrilateral 𝑏 ℎ2 𝜃 ℎ1 𝑏 ℎ 𝑎 Parallelogram Rhombus i) 𝐵𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡 𝑖. 𝑒 (𝑎 × ℎ1 ) 𝑜𝑟 (𝑏 × ℎ2 ) i) 𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡 ii) 𝑎𝑏 sin 𝜃 ii) 𝑎2 sin 𝜃 Kite Trapezium 1 1 𝐴𝑟𝑒𝑎 = (𝑎 + 𝑏) × ℎ 𝐴𝑟𝑒𝑎 = × 𝑑1 × 𝑑2 2 2 1 iii) 𝑑1 𝑑2 2 4. Circle Length Area r 𝑙 = 2𝜋𝑟 Major arc Minor arc r 𝑙= 2𝜋𝑟 2 𝑙= 2𝜋𝑟 4 𝐴 = 𝜋𝑟 2 𝐴= 𝜋𝑟 2 2 𝐴= 𝜋𝑟 2 4 Formula List 5. Sector and Arc 𝑀𝑎𝑗𝑜𝑟 𝑎𝑟𝑐 = 𝑥 360 𝑀𝑎𝑗𝑜𝑟 𝑠𝑒𝑐𝑡𝑜𝑟 = × 2𝜋𝑟 𝑥 360 × 𝜋𝑟 2 𝑀𝑖𝑛𝑜𝑟 𝑎𝑟𝑐 = 𝑥 𝜃 𝜃 360 𝑀𝑖𝑛𝑜𝑟 𝑠𝑒𝑐𝑡𝑜𝑟 = × 2𝜋𝑟 𝜃 360 × 𝜋𝑟 2 6. Area of polygon: Step 1: 𝜃 = 360 𝑛 1 Step 2: 𝑃𝑜𝑙𝑦𝑔𝑜𝑛 𝑎𝑟𝑒𝑎 = [2 × 𝑟 × 𝑟 × sin 𝜃] × 𝑛 𝑻. 𝑺. 𝑨 = 𝑻𝒐𝒕𝒂𝒍 𝒔𝒖𝒓𝒇𝒂𝒄𝒆 𝒂𝒓𝒆𝒂 7. Cylinder (solid) 𝑪. 𝑺. 𝑨 = 𝑪𝒖𝒓𝒗𝒆𝒅 𝒔𝒖𝒓𝒇𝒂𝒄𝒆 𝒂𝒓𝒆𝒂 Cone (solid) Sphere Hemisphere (solid) 𝑟 𝑙 ℎ ℎ 𝑂 𝑟 𝑟 𝑟 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟 2 ℎ 𝑙 = √ℎ2 + 𝑟 2 𝐶. 𝑆. 𝐴 = 2𝜋𝑟ℎ 𝑇. 𝑆. 𝐴 = 2𝜋𝑟ℎ + 2𝜋𝑟 2 4 𝑣𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟 3 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟 2 ℎ 𝐶. 𝑆. 𝐴 = 4𝜋𝑟 2 𝐶. 𝑆. 𝐴 = 2𝜋𝑟 2 𝐶. 𝑆. 𝐴 = 𝜋𝑟𝑙 𝑇. 𝑆. 𝐴 = 𝜋𝑟𝑙 + 𝜋𝑟 2 𝑇. 𝑆. 𝐴 = 4𝜋𝑟 2 𝑇. 𝑆. 𝐴 = 2𝜋𝑟 2 + 𝜋𝑟 2 3 1 Pyramid 2 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟 3 3 3 Cuboid Cube 𝑉 𝐷 𝐵 𝐶 𝑂 𝐴 𝐴 𝐵 (Tetrahedron) 𝑉𝑜𝑙𝑢𝑚𝑒 = 1 3 × 𝐵𝑎𝑠𝑒 𝑎𝑟𝑒𝑎 × ℎ 𝑆. 𝐴 = 𝐵𝑎𝑠𝑒 + 𝑆𝑙𝑎𝑛𝑡 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝑙 × 𝑏 × ℎ 𝑇. 𝑆. 𝐴 (𝐶𝑙𝑜𝑠𝑒𝑑) = 2𝑙𝑏 + 2𝑙ℎ + 2𝑏ℎ 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝑙 3 𝑇. 𝑆. 𝐴(𝑐𝑙𝑜𝑠𝑒𝑑) = 6𝑙 2 Formula List Prism 𝑙 𝑙 Cross Sections: 𝑽𝒐𝒍𝒖𝒎𝒆 𝒐𝒇 𝒑𝒓𝒊𝒔𝒎 = (𝒂𝒓𝒆𝒂 𝒐𝒇 𝒄𝒓𝒐𝒔𝒔 𝒔𝒆𝒄𝒕𝒊𝒐𝒏) × 𝒍𝒆𝒏𝒈𝒕𝒉 8. Volume of Liquid issued (speed in m/s) 𝑖𝑛 1 𝑠𝑒𝑐 = 𝐶. 𝑆 × 𝑠𝑝𝑒𝑒𝑑 𝑖𝑛 1 𝑚𝑖𝑛𝑢𝑡𝑒 = 𝐶. 𝑆 × 𝑠𝑝𝑒𝑒𝑑 × 60 COORDINATE GEOMETRY (i) Length 𝐴𝐵 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 𝑞 𝑥1 +𝑥2 𝑦1 +𝑦2 , ) 2 2 (ii) Midpoint = ( (iii) Gradient = (iv) Co-ordinates of 𝑃 = ( (v) Equation of 𝐴𝐵, • 𝑝 𝑦2 −𝑦1 𝑥2 −𝑥1 𝑝𝑥2 +𝑞𝑥1 𝑝𝑦2 +𝑞𝑦1 , 𝑝+𝑞 ) 𝑝+𝑞 𝐵 (𝑥2 , 𝑦2 ) 𝑃 𝐴 (𝑥1 , 𝑦1 ) 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) ∎ To find gradient: Case-1: Case-2: Case-3: 𝐵 (𝑥2 , 𝑦2 ) 2 • 𝐴 3 𝐴 (𝑥1 , 𝑦1 ) 𝑦 −𝑦 Gradient, 𝑚 = 𝑥2 −𝑥1 2 m= 1 (vi) For parallel lines, Rise Run 2 𝑑𝑦 =3 mat 𝐴 = 𝑑𝑥 (vii) For perpendicular lines, 𝑚1 = 𝑚2 𝑚1 × 𝑚2 = −1 𝑚1 𝑚2 𝑚2 1 ∴ 𝑚2 = − 𝑚 1 𝑚1 Formula List ∎ To find equation: Case-1: Case-2: 𝑦 𝑦 (3, 5) • • 5 𝒚 = 𝒎𝒙 + 𝒄 𝑙1 𝑦=5 For 𝑙1 ⟶ 𝑐 = 1, 𝑚 = 1 𝑥=3 𝑥 = −2 1 2 1 ∴ 𝑦 = 2𝑥 + 1 𝑥 0 For 𝑙2 ⟶ 𝑐 = 1, 𝑚 = −1 −2 3 0 𝑥 ∴ 𝑦 = −𝑥 + 1 𝑙2 ↪ How to know if a point lies on the line or not Case-3: 𝑦 • (4, 8) does not lie on the line 3𝑥 + 7𝑦 = 18 𝑚 • (𝑥1 , 𝑦1 ) 6 as when 𝑥 = 4, 𝑦 = 7 (≠ 8) • (6, 0) lies on the line as when 𝑥 = 6, 𝑦 = 0 𝑥 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) (viii) 𝐶 𝐵 (𝑥1 , 𝑦1 ) Area = • 𝐴 (𝑥2 , 𝑦2 ) = • 1 |𝑥 2 3 𝑦3 1 |( ↘ ) 2 𝐵 𝑥2 𝑦2 𝐴 𝑥1 𝑦1 𝐶 𝑥3 | 𝑦3 − ( ↙ )| • 𝐶 (𝑥1 , 𝑦1 ) 𝑦 (x) 𝑄 𝑎𝑥 + 𝑏𝑦 = 𝑐 0 𝑃 𝑥 For 𝑷 For 𝑸 𝑦=0 𝑎𝑥 + (𝑏 × 0) = 𝑐 𝑐 𝑥=𝑎 𝑥=0 (𝑎 × 0) + 𝑏𝑥 = 𝑐 𝑐 𝑦=𝑏 Formula List (xi) − − − −(i) For intersecting point 𝑃, • Solve equations (i) and (ii) 𝑃 − − − −(ii) (xii) Tangent, 𝑑𝑦 𝑚 𝑇 = 𝑑𝑥 (𝑥1 , 𝑦1 ) Normal, 1 𝑚𝑁 = − 𝑚 𝑇 Equation of tangent: 𝑦 − 𝑦1 = 𝑚 𝑇 (𝑥 − 𝑥1 ) Equation of normal: 𝑦 − 𝑦1 = 𝑚𝑁 (𝑥 − 𝑥1 ) FUNCTION 1. 𝑎 𝑝 𝑎 𝑏 𝑞 𝑐 𝑟 𝑏 𝑐 𝑝 𝑞 𝑏 𝑞 𝑟 One to Many Many to One One to One 𝑝 𝑎 2. Vertical line test: 𝑦 𝑜 𝑦 𝑦 𝑥 𝑥 𝑜 𝑥 𝑜 One to one Many to one One to many Function Function Not Function Inverse Inverse does not exist 𝑁/𝐴 Formula List 3. 𝒇(𝒙) to 𝒇−𝟏 (𝒙) 4. 𝑓(𝑥) = 𝑎 Step – 1: Let 𝑓(𝑥) = 𝑦 Step – 2: 𝑦 → 𝑥, 𝑥 → 𝑦 Step – 3: Make 𝑦 subject Step – 4: 𝑦 = 𝑓 −1 (𝑥) 𝑥 = 𝑓 −1 (𝑥) 5. 𝐷 of 𝑓(𝑥) = 𝑅 of 𝑓 −1 (𝑥) 𝑅 of 𝑓(𝑥) = 𝐷 of 𝑓 −1 (𝑥) 6. 7. 𝑓(𝑥) = 2𝑥 + 2 𝑓𝑔ℎ(𝑥) 𝑔(𝑥) = 4𝑥 = 𝑓𝑔(𝑥 3 ) ℎ(𝑥) = 𝑥 3 = 𝑓(4𝑥 3 ) = 2(4𝑥 3 ) + 2 STATISTICS ∎ Histogram: Q1 50 < 𝑤 ≤ 60 60 < 𝑤 ≤ 75 75 < 𝑤 ≤ 80 𝒇 80 90 60 80 < 𝑤 ≤ 100 80 Weight (in kg) Step – 1: 𝐹 𝑓. 𝑑. = 𝑐. 𝑤. Step – 2: Bar 1: 𝑓. 𝑑. = Bar 2: 𝑓. 𝑑. = Bar 3: 𝑓. 𝑑. = Bar 4: 𝑓. 𝑑. = 80 10 90 15 60 5 80 20 =8 =6 = 12 =5 Formula List ∎ Median: (a) Median of 𝒑, 𝒒, 𝒙, 𝒚, 𝒛 = 𝑥 (b) Median of 𝟎, 𝟎, 𝟐, 𝟑, 𝟓, 𝟓 = (c) = 2.5 𝑴𝒂𝒓𝒌𝒔 0 1 2 3 𝒇 3 10 12 15 𝒄. 𝒇. 3 13 25 40 4 5 8 2 48 50 = 𝑁 𝑴𝒂𝒓𝒌𝒔 0 1 2 3 𝒇 3 10 12 15 𝒄. 𝒇. 3 13 25 40 4 5 8 3 48 51 = 𝑁 𝒙 0 < 𝑥 ≤ 10 10 < 𝑥 ≤ 12 12 < 𝑥 ≤ 20 𝒇 5 8 10 𝒄. 𝒇. 5 13 23 = 𝑁 (d) (e) 2+3 2 𝑁 2 = 50 2 = 25 Median = 25𝑡ℎ 𝑡𝑒𝑟𝑚 + 26𝑡ℎ 𝑡𝑒𝑟𝑚 2 ∴ Median = 𝑁+1 2 = 51+1 2 2+3 2 = 2.5. = 26 ∴ Median = 26𝑡ℎ 𝑡𝑒𝑟𝑚 = 3. 𝑁 2 = 23 2 = 11.5. ∴ Median group = 10 < 𝑥 ≤ 12. ∎ Mean: (a) Mean of 𝒙, 𝒚, 𝒛, 𝒘 = (b) (c) 𝑥+𝑦+𝑧+𝑤 4 𝒙 0 1 𝒇 4 19 𝒇𝒙 0 19 2 3 4 25 29 23 50 87 92 𝒙 0 < 𝑥 ≤ 10 10 < 𝑥 ≤ 12 𝒙 5 11 𝒇 2 5 𝒇𝒙 10 55 12 < 𝑥 ≤ 20 16 4 64 ∑ 𝑓 = 100, ∑ 𝑓𝑥 = 248 ∴ Mean = ∑ 𝑓𝑥 ∑𝑓 = 2.48 ∑ 𝑓 = 11, ∑ 𝑓𝑥 = 129 ∴ Mean = ∑ 𝑓𝑥 ∑𝑓 = 11.73 Formula List ∎ Mode: (a) 𝑎, 𝑏, 𝑐, 𝑐, 𝑑, 𝑑, 𝑒 → Mode = 𝑦 and 𝑑. (b) 𝑎, 𝑏, 𝑐, 𝑐, 𝑑, 𝑑, 𝑒 → Mode = 𝑐 and 𝑑. (c) 𝒙 𝒇 (d) 0 < 𝑥 ≤ 10 10 < 𝑥 ≤ 12 12 < 𝑥 ≤ 20 0 2 1 20 2 25 3 25 15 10 2 4 16 5 3 Modal group → 0 < 𝑥 ≤ 10. ∎ Bar Chart: 1. Aadil observed the number of people in each of 20 cars entering a car park. The results are shown in the table below. Number of people Frequency 1 2 2 7 3 6 (a) Draw a bar chart to represent this information. (b) Write down the mode. (c) Calculate the mean number of people in each car. Answer: (a) (b) Mode = 2. (c) Mean = (1×2)+(2×7)+(3×6)+(4×4)+(5×1) 2+7+6+4+1 = 2.75. 4 4 5 1 Mode = 2 and 3. Formula List ∎ Pie Chart: 1. The proportion of drinks sold at the refreshment tent at a summer fair are represented by the pie chart in figure. Only four categories of drink shown in the pie chart were sold and the number of coffees sold was 135. Coffee (a) Calculate the total number of drinks sold. (b) Calculate the number of orange juice drinks sold. 75° 100° 180 lemonade drinks were sold. Lemonade (c) Calculate the angle for lemonade drink. Tea 45° 140° Orange (d) Calculate the number of teas sold. Answer: (a) 75° ≡ 135 135 ⇒ 360° ≡ × 360 75 ∴ Total drinks = 648. (c) 135 ≡ 75° 75 ⇒ 180 ≡ × 180° 135 ∴ Lemonade ∠ = 100°. (b) 75° ≡ 135 ⇒ 140° ≡ 252 ∴ Orange juice = 252. (d) Tea = 360° − 75° − 100° − 140° ∴ Tea ∠ = 45°. 75° ≡ 135 135 ⇒ 45° ≡ × 45° 75 ∴ Tea = 81. 2. (a) Three times as many students said Blue as said Green. Calculate the angle of the sector which represents the number of students who said Green. (b) There are 8 more students who said Red than Yellow. How many students said Red? Answer: (a) Let Green be 𝑥° ∴ Blue = 3𝑥° ∴ 90° + 110° + 𝑥° + 3𝑥° = 360° ⇒ 𝑥 = 40° ∴ Angle for green = 40°. Red Yellow (b) 20° ≡ 8 students 1° ≡ 8 20 110° ≡ students 8 20 × 110° = 44 students ∴ Red = 44 students. Green 110° Blue Formula List EQUATION GRAPH ∎ Distance-Time Graph and Speed-Time Graph: Constant acceleration At rest Constant speed Constant speed (Returning) Constant deceleration Formula List • • • • • # Distance time: • Gradient = Speed • Vertical difference = Distance • Average speed = ∎ Set of values of 𝒙: Total distance Total time 0 to 𝑇1 = constant acceleration 𝑇1 to 𝑇2 = constant speed 𝑇2 to 𝑇3 = constant deceleration 𝑇3 to 𝑇4 = at rest 𝑇4 to 𝑇5 = returning # Speed time: • Gradient = Acceleration • Area under the graph = Distance • Average speed = Total distance Total time Formula List ∎ Gradient at given point: To find gradient at 𝒙 = 𝟏. 𝟓: Step 1: Draw the tangent line at 𝑥 = 1.5 Step 2: Take two suitable values from the tangent 𝑦 −𝑦 Step 3: 𝑚𝑥=1.5 = 𝑥2 −𝑥1 2 = 1 0.3−(−4) 3−0.75 = 1.91 ∎ Solve Graphically: To solve 𝒙𝟐 − 𝒙 − 𝟏 = 𝟎: Graph Equation = Line ⟹ 2𝑥 2 − 2𝑥 − 2 = 0 [× 2] ⟹ 2𝑥 2 − 2𝑥 − (2𝑥) − 2 + (1) = 0 − (2𝑥) + (1) ⟹ 2𝑥 2 − 4𝑥 − 1 = −2𝑥 + 1 ⟹ 𝑦 = −2𝑥 + 1 𝒙 𝒚 0 1 2 −3 ∴ Solution: 𝑥 = 2, −0.6 ∎ Common Sketching:

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