2023 NJC P1 Question

* © NJC 2023 [Turn_over 2023 NJC H2 Mathematics Preliminary Examination Paper 1 1 The equations of three planes 1p , 2p and 3p are 4 6 3 2 1 3 5 2 4 x y x z z y y z x           respectively, where  and  are constants. (i) Given that 5  and 9  , find the coordinates of the point that lies on all three planes. [1] (ii) Find the values of  and  such that the line where 1p and 2p meet lies in 3p . [5] 2 In this question, k is a positive constant. (a) Using an algebraic method, solve the inequality 2 22 2x k x kx    . Express your answer in terms of k. [4] (b) Hence, solve exactly the inequality 22 10 5x xx    . [3] 3 It is given that 1 2(1 tan ) .y x   (a) Prove that 2 2 2 2 2 d d(1 ) 2 (1 ) 2.d d y yx x x x x    [3] (b) Obtain an equation relating 3 2 3 2 d d d, and .d d d y y y x x x [1] (c) Find the Maclaurin’s series for y, up to and including the terms in 3.x [2] (d) Deduce the equation of the normal to the curve 1 2(1 tan )y x   at 0x . [1] 4 (a) Explain why 2 2 2 2 0 4 d π a a x x a  , where a is a positive real value. [1] It is given that   2 24 for 0 2 ,f 3 6 for 2 3 , a x x ax x a a x a        and that    f 3 fx a x  for all real values of x. (b) Find  f 8.5 a . [2] (c) Sketch the graph of  fy x for 3 8.5a x a   . [2] (d) Find   599 2 0 f d a x x in the exact form. [3]

2 © NJC 2023 5 (a) A curve C has equation 4 9 5 2 xy x   . Describe a sequence of three transformations which transforms the graph of C onto the graph of 1y x . [3] (b) It is given that   2 2f 1 xx q  , where q is a positive constant. State the shape of  fy x . [1] On separate diagrams, sketch the curves with equations (i)   1 fy x , (ii)  fy x  , stating the equations of any asymptotes and the coordinates of the points where the curve crosses the axes. [4] 6 A sequence 1u , 2u , 3u , … is given by the relation 1 10 n n uu k   , where k is a constant and 1.n It is given that 1 3.u  (a) Write down 2u and 3u in terms of k. [2] It is further given that 1 10 n nu        for any positive integer. (b) Show that 100 309 k   and find  in terms of k. [3] (c) State lim n nu  in terms of k. [1] (d) Given that 9k  , find 1 N n n u   in terms of N. [3]

3 © NJC 2023 [Turn_over 7 (a) The complex numbers, 1z , 2z and 3z are 2 i 2 ,  2 i 2 3 and πi12e 5 respectively. The complex number w is given by   2 3 1* zw z z . (i) Find w in the form ier  , where 0r  and π π    . [4] (ii) In the same Argand diagram, sketch the points representing w, 5w and *w . [3] (b) Another complex number v is such that v c v c   is purely imaginary, where c is a positive real number. Show that v c . [3] 8 A tetrahedron OABC has four triangular faces with a vertex O at the origin. The coordinates of A, B and C are  9, 0, 0 ,  0, 2, 0