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ADDITIONAL MATHEMATICS SPM 2012 PAPER 1 ( PART TWO ) LAST KOPEK REVISION TRY TO ANSWER THESE QUESTIONS AND CHECK YOUR ANSWERS FROM THE FOLLOWING SCHEME OF ANSWER AND MARKING 14. The variables x and y are related by the equation p y = x . The diagram below shows the straight line graph 3 obtained by plotting log10 y against x. p (a) Express the equation x in its linear form used to obtain 3 the straight line graph as shown in the diagram. (b) Find the value of p. Answer: log10 y (a) log10 y = log10 p – x log10 3 √1 (0,2) (b) log10 p = 2 √ M1 0 x p = 100 √2 15. Given that y = 2x + 1 , x = 3 (x – 3)2 (a) find the value of dy/dx, when x = 4 , (b) the approximate change in y when x increases from 4 to 4.01 Answer: (a) dy/dx = (2x + 1) (2) (x – 3) (1) – (x – 3) (2) 2 √ M1 (x – 3)4 [2(4) + 1] (2) (4 – 3) (1) – (4 – 3)2 (2) = (4 – 3)4 = 16 √ 2 15 (b) Sambungan (b) x = 4 ∂x = 0.01 √ M1 dy/dx = 16 ∂y = dy/dx x ∂x ∂y = 16 (0.01) √ M1 ∂y = 0.16 √2 16. Given y = 5h i + (h – 2) j is a non-zero vector and is parallel to x-axis. Find the value of h. Answer: 5h i + (h – 2) j = x i + 0 j ( parallel to x-axis, vertical component of the vector = 0 ) h–2=0 √ M1 h=2 √2 17. Diagram shows the position of the point A, B and C relative to origin, O. Given B(1, 8), OA = 7i + 2j and OC = ¾ AB. Find in terms of i and j, (a) OC, (b) CB. Answer: y C. .B (a) AB = − OA + OB .A = − (7 i + 2 j) + (i + 8 j) √ M1 x =−6i+6j O OC = −9/2 i + 9/2 j √ 2 (b) CB = − OC + OB = − 9/2 i + 9/2 j + (i + 8 j) √ M1 = − 11/2 i + 7/2 j √ 2 18. Diagram shows a unit circle. T (−h, k) is a point on the rotating ray of an angle θ. Find in terms of h, the value of (a) cos θ. (b) sin 2θ. y (-h, k) Answer: 1 k θ (a) cos θ = −h θ x √1 −p O (b) sin 2θ = 2 sinθ cosθ = 2 (k) (−p) √ M1 = −2pk √ 2 19. Given sin A = 3/5, cos B = 12/13. If both the angle A and B are at the same quarter, find the value of (a) sin (A + B), (b) tan (A − B) 5 Answer: 3 A (a) sin A cos B + cos A sin B 4 = (3/5) (12/13) + (4/5) (5/13) √ M1 13 5 = 56/65 √ 2 B tan A − tan B 12 (b) 1 + tan A tan B = (3/4) – (5/12) √ M1 = 7/16 √ 2 1 + (3 /4) (5 /12) 20. Point A lies on a curve y = 2x4 – x, find the coordinates of point A where the gradient of the normal at point A is -1/7. Answer: dy/dx = mt = 8x3 − 1 √ M1 8x3 – 1 = 7 √ M2 8x3 = 8 x=1 y = 2(1)4 – 1 =1 A = ( 1, 1) √ 3 21. The standard deviation of a set of six numbers is √15. Given that the sum of square for the set of numbers is 144. Find the new mean when a member 10 is added to this set. Answer: N=6 x2 =9 σ2 = 15 x =3 ∑ x2 = 144 ∑x = x (N) 144 = 3 (6) = 18 15 = −x2 √ M1 6 ∑xnew = 18 + 10 √ M2 x new = 28/7 = 4 3 22. A set of positive integers consists of 6, 7, k, 1, 8, 3, 3. (a) Find the value of k if the mean of the data is 5. (b) State the range of the values of k if the median of the data is k. Answer: (a) 28 + k √ M1 =5 7 k=7 √2 (b) 1, 3, 3, k, 6, 7, 8 3≤k≤6 √1 23. A Proton Wira can accommodate 1 driver and 3 adult. Find the number of different ways the selection can be made from 3 men and 4 women if, (a) there are no restriction, (b) the driver must be a man. Answer: (a) 7C4 = 35 (b) 3C1 x 6C3 = 60 THE END Those who have worked hard with the correct method of learning Additional Mathematics surely will receive their rewards

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