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# Form 5 formulae and note

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### Form 5 formulae and note

1. 1. NOTES AND FORMULAE ADDITIONAL MATHEMATICS FORM 51. PROGRESSIONS (iii) (a) Arithmetic Progression b c c Tn = a + (n – 1)d n  a  f ( x )dx  f ( x )dx  b  f ( x)dx a Sn = [2a  ( n  1)d ] 2 (d) Area under a curve n      = [ a  Tn ] AC  AB  BC 2 (b) Geometric Progression (b) A, B and C are collinear if Tn = ar n–1    n AB   BC where  is a constant. Sn  a (1  r )    1 r AB and PQ are parallel if Sum to infinity   b b PQ   AB where  is a constant. a S  1 r A=  a ydx A=  xdy a (c) Subtraction of Two Vectors (c) General Tn = Sn − Sn – 1 T1 = a = S1 (e) Volume of Revolution2. INTEGRATION x n 1       (a)  xn dx  c n 1 AB  OB  OA (ax  b) n 1 (d) Vectors in the Cartesian Plane (b)  ( ax  b) n dx  c (n  1)a (c) Rules of Integration: b b b b V   y 2 dx  V   x 2 dy  (i)  nf ( x)dx  n f ( x)dx a a a a a b 3. VECTORS   (ii)  f ( x)dx   f ( x)dx b a (a) Triangle Law of Vector Addition OA  xi  yj   Magnitude of     OA  OA  x 2  y 2Prepared by Mr. Sim Kwang Yaw 1
2. 2. (g) Double Angle Formulae Unit vector in the direction of OA sin 2A = 2 sin A cos A r xi  yj 2 cos 2A = cos A – sin A 2 r     ˆ 2 = 2cos A – 1  r x2  y 2 2 = 1 – 2sin A 4. TRIGONOMETRIC FUNCTIONS 2 tan A tan 2A = (iii) y = tan x 1  tan 2 A(a) Sign of trigonometric functions in the four 5. PROBABILITY quadrants. (a) Probability of Event A n( A) Acronym: P(A) = “Add Sugar To Coffee” n( S ) (b) Probability of Complementary Event P(A) = 1 – P(A) (c) Probability of Mutually Exclusive Events (iv) y = a sin nx(b) Definition and Relation P(A or B) = P(A  B) = P(A) + P(B) sec x = 1 cosec x = 1 (d) Probability of Independent Events cos x sin x P(A and B) = P(A  B) = P(A) × P(B) 1 sin x cot x = tan x = tan x cos x 6. PROBABILTY DISTRIBUTION (a) Binomial Distribution(c) Supplementary Angles n P(X = r) = Cr p q r n r o sin (90 − x) = cos x a = amplitude o cot (90 – x) = tan x n = number of cycles n = number of trials (e) Basic Identities p = probability of success 2 2(d) Graphs of Trigonometric Function (i) sin x + cos x = 1 q = probability of failure 2 2 (i) y = sin x (ii) 1 + tan x = sec x Mean = np 2 2 (iii) 1 + cot x = cosec x Standard deviation = npq (f) Addition Formulae (i) sin (A  B) (b) Normal Distribution = sin A cos B  cos A sin B X  Z= (ii) cos (A  B)  = cos A cos B  sin A sin B Z = Standard Score (ii) y = cos x (iii) tan (A  B) = tan A  tan B X = Normal Score 1  tan A tan B  = mean  = standard deviationPrepared by Mr. Sim Kwang Yaw 2
3. 3. (b) Condition and Implication: (a) Normal Distribution Graph Condition Implication Returns to O s=0 To the left of O s<0 To the right of O s>0 Maximum/Minimum ds = 0 displacement dt Initial velocity v when t = 0 Uniform velocity a=0 Moves to the left v<0 Moves to the right v>0 Stops/change v=0 direction of motionP(Z < k) = 1 – P(Z > P(Z < -k) = P(Z > k) Maximum/Minimum dv = 0k) velocity dt Initial acceleration a when t = 0 Increasing speed a>0 Decreasing speed a<0 (c) Total Distance Travelled in the PeriodP(Z > -k) = 1 – P(Z < - P(a < Z < b) 0 ≤ t ≤ b Second k) = 1 – P(Z > k) = P(Z > a) – P(Z > b) (i) If the particle does not stop in the period of 0 ≤ t ≤ b seconds Total distance travelled = displacement at t = b second (ii) If the particle stops in t = a second when t = a is in the interval of 0 ≤ t ≤P(-b < Z < -a) = P(a < P(- b < Z < a) b second,Z < b) = P(Z > a) – = 1 – P(z > b) – P(Z > Total distance travelled in b secondP(Z > b) a) = Sa  S0  Sb  Sa7. MOTION ALONG A STRAIGHT LINE(a) Relation Between Displacement, Velocity and Acceleration  vdt  adtPrepared by Mr. Sim Kwang Yaw 3