VJC 2024 H2 JC2 Math Prelim P2 Questions Student

2024 VJC Prelim Paper 2 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]

2024 VJC Prelim Paper 2 (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]

2024 VJC Prelim Paper 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 θ , where 0r > and ππ θ−<≤ . [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]

2024 VJC Prelim Paper 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]

2024 VJC Prelim Paper 2 3 The line 1l has equation ( )345 2 λ=−−+ −−rij k ij k , where λ is a real parameter. The point A has position vector 2−+i jk . (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]

2024 VJC Prelim Paper 2 (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 l 2 has the equation 37 23 yz−− = , 2x= . Given that l2 intersects p at point S, find the area of the triangle OAS. [4]

2024 VJC Prelim Paper 2 4 The curve C is defined by the parametric equations 11xa t = + and 2 1y at 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]

2024 VJC Prelim Paper 2 (c) Sketch C. [3]

2024 VJC Prelim Paper 2 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]

2024 VJC Prelim Paper 2 6 A random variable X has the probability distribution given in the following table. x 1 4 6 8 ( )P Xx= a b c d Given that ( )E4X = , ( ) 19Var 4X = and ( ) ( )P 4P 4XX<= > , find the values of a, b, c and d. [5]