2019 J2 ASRJC H2 Math Prelim (with ans)

1 1( a) Show that 12 12 x x   can be written in the form 2 f( ) 14 x x where f(x) is a polynomial to be determined. Hence, find 12 d12 x xx      [3] (b) Show that 2e e 1 dln xxx    = ln 2.[ 3] 2 It is given that 1tan 22 yx , where 22 y  . (i) Show that  2 d12 2 2 d yx x .[ 2] (ii) Use the result from part (i) to find the first two non-zero terms in the Maclaurin series for y, giving the coefficients in exact form. [ 3] (iii) Hence, using standards series fro m the List of Formulae (MF26), find the expansion of 12t a n 2 cos 2 x x  in ascending powers of x, up to and including the term in x3, giving the coefficients in exact form. [ 3] 3 The position vectors of A, B and C referred to a point O are , and ab c respectively. The point N is on AB such that AN:NB = 2:1. (i) If O is the midpoint of CN, prove that + 2 3ab c 0  . [2] (ii) Show that A, O and M are collinear, and find the ratio AO:OM [3] (iii) If the point P is such that NP AM K K , show that the ratio of the area of PNAM : area of PNOM = 12:7 [3] +0DWK3UHOLP$65-& www.KiasuExamPaper.com 45

2 4( a) The diagram shows the graph of f( )yx . The curve crosses the x-axis at 0x  and 3x  . It has a turning point at (4, 5) and asymptotes with equations 2y  and 2x  . Showing clearly the coordinates of turning points, axial intercepts and equations of asymptotes where possible, sketch the graphs of (i) fyx ;[ 2] (ii) f'yx [3] (b) The curve with equation y = f(x) is transformed by a stretch with scale factor 2 parallel to the x–axis, followed by a translation of 2 units in the negative x– direction, followed by a translation of 3 units in the positive y–direction. The equation of the resulting curve is  3ln eyx . Find the equation of the curve y = f(x). [3] y x y = f(x) 0 (4, 5) 3 y = 2 x = 2 www.KiasuExamPaper.com 46

3 5 A curve C has parametric equations 2sin , cos ,xa t ya t where 0 and 0.2ta   (i) Find the cartesian equation of C, stating clearly any restrictions on the values of x and y. [2] (ii) Sketch C, showing clearly the axial intercepts. [1] (iii) The region bounded by C, the line 5 4yx a and the y-axis is rotated through ʌUDGLDQVDERXW the y-axis. Show that the exact volume of the solid obtained is 3ʌka where k is a constant to be determined. [5] 6 A function f is defined by 2 6 for 2,4f( ) 1 for 2 1, 2 for 1. xxx xx xx x            (i) Sketch the graph of f. [3] (ii) Evaluate exactly 4 3 f( ) dxx  .[ 4] 7 Do not use a calculator in answering this question. The roots of the equation z2 + (2 – 2i)z =– 3–2 i a r e 1z and 2z . (i) Find 1z and 2z in cartesian form x + iy, showing clearly your working. [5] (ii) The complex numbers 1z and 2z are also roots of the equation 43 24 14 4 13 0zz zz    . Find the other roots of the equa tion, explaining clearly how th e answers are obtained. [2] (iii) Using your answer in part (i), solve z2 + (2 + 2i)z = 3 + 2i. [2] www.KiasuExamPaper.co