2024 VJC H2 JC2 Math Prelim P2 Qn (modified)

2024 Victoria Junior College Preliminary Examination H2 Mathematics Paper 2 (modified) Section A: Pure Mathematics (40 marks) 1 (a) Show that ( ) 22 222 11 11 211 xx xxx = + + − + − − . [1] (b) Hence use appropriate expansions from the List of Formulae (MF26) to find the first two non- zero terms in the series expansion of 2 2211 x xx+ − − , 0x in ascending powers of x. [3] (c) State the set of values of x for which the series expansion is valid. [1] (d) It is given that the two terms found in part (b) are equal to the first two terms in the series expansion of ( )cos bax . Find the possible value(s) of the constants a and b. [2] 2 Do not use a calculator in answering this question. The complex numbers 1z , 2z and 3z are such that 2πi 3 1 ez =− , 2 3iz =− + and 1 3 2 zz z= . (a) Express each of 1z , 2z and 3z in the form ier  , where 0r and ππ −   . [3] (b) Sketch an Argand diagram showing the points 1P , 2P and 3P where 1P , 2P and 3P represent the complex numbers 1z , 2z and 3z respectively. [2] (c) Find the area of triangle 12OPP . [2] (d) Find the smallest positive integer n for which ( ) * 2 n z is purely imaginary. [2] 3 The line 1l has equation ( )3 4 5 2 = − − + − −r i j k i j k , where  is a real parameter. The point A has position vector 2−+i j k . (a) The plane p contains the line 1l and the point A. Find a cartesian equation of the plane p. [3] (b) Find the position vector of the point 'A , the reflection of the point A in the line 1l . [4] (c) The plane q is such that q is parallel to p and passes through the point with position vector 3−+jk . Find a cartesian equation of q and the exact shortest distance between p and q. [3] (d) The line l2 has the equation 37 23 yz−− = , 2x= . Given that l2 intersects p at point S, find the area of the triangle OAS. [4]

4 The curve C is defined by the parametric equations 11xa t =+ and 2 1y a t t =−  where a is a positive constant and 0t . (a) Show that 3d2 d yt xt  +=−   . [3] (b) Find, in terms of a, the coordinates of the turning point on C, and explain why it is a maximum. [4] (c) Sketch C. [3] Section B: Statistics (60 marks) 5 Two married couples, two single adults and two children formed a team of 8 to take part in a series of games. (a) In the first game, the team sits in a circle. Find the number of arrangements that can be formed if each married couple must be seated together. [2] (b) A group of three people are to be selected from the team for the second game. Find the number of different groups that can be formed if there must not be a married couple in the group. [2] (c) In the third game, each team member selects a unique number from the set  1, 2, , 8 . Find the number of different ways this can be done if the numbers selected by the children are both greater than the numbers selected by the two single adults. [2]