2023 VJC Promo (Qn)

2023 VJC H2 Maths Promo Paper Duration: 3 hrs Marks: 100 Attempt all questions. [You may skip Q4 & 6 for now, as they are on Integration. Remaining total marks is 84, Duration is 2 hr 31 mins.] 1 (a) Find an expression for d d y x in terms of x if 12tan 1yx  . [3] (b) It is given that cosxy x for 0x . By taking logarithm first, find an expression for d d y x in terms of x. [3] 2 The diagram shows the curve fy x . The curve passes through the points  1, 0A  ,  0, 2B ,  1, 0C and  3, 3D  . On separate clearly labelled diagrams, sketch the graphs of (a) 1 f2yx , [2] (b) f1y x . [2] 3 Given that a and 1a , use an algebraic method to solve the inequality   2 11 ax a xa x   . [5] Hence find the set of values of x such that   2 1 1 ax a xa x    . [2] 4 (a) Differentiate tane x with respect to x. [1] (b) Hence find tan 4es e cdx x x . [4] 5 The curve C has equation 22 44xy . (a) Sketch C, stating the equations of any asymptotes and the coordinates of any points of intersection with the axes. [3] (b) Describe a pair of transformations which transforms the graph of C onto the graph of  2 224xy  . [2] 0, 2B  1, 0A   1, 0C  3, 3D  1y fyx O y x

6 (a) Find 2 d 54 32 x x xx   . [4] (b) Show that 4 2 d32 xa bx  , where a and b are constants to be determined. [3] (c) Using the substitution sinx u , find the exact value of 3 22 22 2 1 d 1 x x x     . [4] 7 Do not use a calculator in answering this question. (a) One of the roots of the equation 3233 0xp xq x   , where p and q are real, is 12 i . Find the other roots and the values of p and q. [5] (b) The complex numbers w and z, with wz , satisfy the simultaneous equations *3iwz  and *2iwz  . Find w and z. [5] 8 It is given that     2fl n 1 2x x . (a) A polynomial  2p xa xb x c  is used to approximate  f x for 12 .x Given that    1f 1p  ,    1.5 f 1.5p  and    2f 2p  , find the values of the coefficients a, b and c. [3] (b) Using standard series from the List of Fo rmulae (MF26), find the Maclaurin series for  f x in ascending powers of x, up to and including the term in 6x . [2] (c) Find the set of values of x for the expansion in part (b) to be valid. [2] (d) Use 2 1 6x  in your series from part (b) to show that 4ln 31 6 2 m  , where m is an integer to be determined. [2] (e) Using your series from part (b), find the series expansion for 2 2 12ln 12 x x x     , up to and including the term in 4x . Give the coefficients in exact form. [2]

9 Do not use a calculator in answering this question. (a) It is given that 3i   and ππcos isin33  . Find 5 *   in the form  cos isinr   , where 0r and ππ  . [4] (b) The complex number z is given by cos isinz   , where ππ 22  . Show that 2 1* 2cos1 z z   . Hence find the modul