TPJC_H2_MATHS_P2

1 H2 2017 Preliminary Exam Paper 2 Question Section A: Pure Matheatics [40 marks]. 1 The cubic equation 3 2 31 212 0, az z z b − + + = where a and b are real numbers, has a complex root 1 3i. z = − (i) Explain why the equation must have a real root. [2] (ii) Find the values of a and b and the real root, showing your working clearly. [5] 2 Relative to the origin O , the points A , B and C have position vectors , and + a a c c respectively. The point X is on AC produced such that : AC CX is 2:3 and the point Y is such that AXYB is a parallelogram. (i) The lines AY and BX intersect at the point N . Show that ( ) 1 . 4 ON = − uuu r 7c a [3] (ii) Given that the area of triangle OAB is 4 square units, find the area of triangle OAN. [4] (iii) Give a geometrical interpretation of AN OA AN × uuu r uuu r uuu r . Use the results from part (ii) to show that 56 7 5 AN OA AN × = − c a uuu r uuu r uuu r . [3] 3 (a) Find the series expansion of ( ) 2 e ln 1 3 x x + , where 1 1 3 3 x − < ≤ , in ascending powers of x , up to and including the term in 3 x . [3] (b) In the triangle PQR as shown in the diagram below, 1 PR = , angle QPR 3 π 4 = radians and angle 2 PRQ θ = radians. (i) Show that 1 cos 2 sin 2 QR θ θ = − . [4] (ii) Given that θ is sufficiently small angle, show that 2 1 QR a b θ θ ≈ + + , for constants a and b to be determined. [4] 4 (a) Find e sin d x x x ∫ . [3] (b) P R Q 1

The diagram shows the curve with eq k uation ( ) 2 for 0 1. 3 2 x y x x x = ≤ < √ − − The region bounded by the curve, the x -axis and the line , 0 1 x k k = < < is denoted by S . It is given that n rectangles of equal width are drawn between 0 x = and . x k = (i) Show that the area of the first rectangle, 2 1 2 2 . 3 2 k A n n nk k = − − [1] (ii) Show that the total area of all the n rectangles is ( ) 2 2 2 2 1 , 3 n r rk n n anrk br k = √ − − ∑ where a and b are constants to be determined. [2] It is now given that ( ) 3 1 k = √ − . (iii) Use integration to find the actual area of region S . Hence state the exact value of ( ) 2 2 2 2 1 3 . r rk n n anrk br k ∞ = √ − − ∑ [6] Section B: Probability and Statistics [60 marks] 5 An unbiased six-sided die is rolled twice. The random variable X represents the higher of the two values if they are different, and their common value if they are the same. The probability distribution of X is given by the formula P( ) (2 1) for 1, 2, 3, 4, 5, 6. X r k r r = = − = (i) Find the exact value of k , giving your answer as a fraction in its simplest form.