2023 NJC P2 Question

* © NJC 2023 [Turn_over 2023 NJC H2 Mathematics Preliminary Examination Paper 2 Section A: Pure Mathematics [40 marks] 1 Relative to the origin, the fixed point A has position vector a. The line l has equation sr d , where s is a parameter and d is a unit vector not parallel to a. (a) Interpret geometrically the vector equation ,t t  r a d  . [2] (b) Find, in terms of a and d, the position vector of the foot of perpendicular from A to l. [1] (c) A variable point P with position vector p lies on the plane containing A and l. Describe the locus of P if   p d a d . [2] 2 Figure 1 shows the net of an open square-based pyramid cut from a square of cardboard of fixed side length d cm. The net consists of four congruent isosceles triangles, each with base x cm and perpendicular height 1 2 d cm. The net is folded to form a pyramid which has a square base of side length x cm and vertical height h cm, as shown in Figure 2. Use differentiation to find the maximum volume of the pyramid in terms of d. [6] You do not need to show that this value is a maximum volume. [The volume of a pyramid is given by 1 base area height3  .] x cm h cm x cm Figure 2 d cm cm Figure 1

2 © NJC 2023 3 (a) Using integration by parts on   2 2 ln dx xx   , show that     2 2 2 2 ln ln2 ln d d x xx x x cx x x      , where c is an arbitrary constant. [3] (b) Region R is bounded by the curves 2 lnxy x and ln xy x . Find the exact volume generated by rotating R about the x-axis through 2π radians. [5] 4 In this question, a and b are real numbers such that 0a  and 0b . The function f is defined as f : ax x b , x b . (a) Sketch the graph of  fy x and explain why 2f does not exist. [3] (b) The function g is defined as 2 g : bx x b  , 0x and x b . (i) Show that 2g exists and has the same rule as 1g . [5] (ii) Hence find   2024g 2 b in terms of b. [2] 5 A complex number w is given by 2 i k , where k is a real constant and 0k  . (a) Find 3w in the cartesian form ix y , where x and y are real numbers in terms of k. [2] (b) Show that w is a root of the equation       23 2 3 4 2 4 0z k z k z k k       . [3] (c) Hence find the other two roots of the equation in part (b) in terms of k. [4] (d) Find the exact value of k such that the points representing the 3 roots of the equation in part (b) are the vertices of an equilateral triangle in the Argand diagram. [2]

3 © NJC 2023 [Turn_over Section B: Statistics [60 marks] 6 The random variable X is normally distributed with mean  and variance 2 . (a) Find P( 0.05 )X    . [2] (b) The mean of a random sample of n independent observations of X is denoted by X . Find the least value of n such that the probability that X differs from  by at least 0.05 is less than 0.4. [3] (c) In another random sample of m independent observations of X, th