2017 SCGS Sec 4 AM Prelim P1 Solutions

Paper 1 1. Find the value of k for which the line kxy  2 and the curve 22  xy do not intersect. [4] 1. xky 2 22  xy 2)2( 2  xxk 0244 22  xxkxk 0)2()14(4 22  kkxx 0)2)(4(4)14( 22  kk 032161816 22  kkk 0318 k 8 31k (4 marks) 2. (i) On the same axes sketch the curves of xy 273  and 33xy . [3] (ii) Find the length of the line segment which joins all the points of intersection of the two curves. [3] 2. (i) (ii) At point of intersection, xx 2727 9  09  xx 0)1( 8 xx 1 ,1 ,0 x 3 ,3 ,0 y Length 22 312  or 22 62  102 units (6 marks)

2 3. The diagram shows part of the graph of baxxy  2 . The curve touches the x-axis at (2, 0) and at (5, 0) and has a maximum point at M(p, q). (i) Find the value of a and of b. [2] (ii) Find the coordinates of M. [2] (iii) Solve the equation 422  xbaxx . Hence, solve the inequality 422  xbaxx . [3] 3. (i) 3)52( a 10)5(2 b (ii) At M, x 2 52 2 3 y 102 332 3 2          4 49      4 49 ,2 3M or (1.5, 12.25) (iii) 421032  xxx 421032  xxx or )42(1032  xxx 01452  xx 062  xx 0)7)(2(  xx 0)3)(2(  xx 7 ,2x 3 ,2x 421032  xxx 73  x (7 marks)

3 [Turn over 4. The diagram shows part of a straight line drawn to represent the equation .qyx px  Calculate the value of p and of q. [4] 4. Gradient 3 1 9 3  Equation of straight line, )6(3 1 2 xxy 63 2 xxy xxy 63  6p 3q (4 marks) (6, 0) (15, 3) O

4 5. (a) Without using a calculator, show that 2 315sin15cos o4o4  . [2] (b) Given that 220  x and 49 232cos x , calculate the exact value of xsin . [2] 5. (a)  15sin15cos 44 )15sin15)(cos15sin15(cos 2222  )15sin15(cos 22   30cos 2 3 (b) 49 232cos x 49 23sin21 2  x 49 72sin2 2 x 49 36sin 2 x 7 6sin x 2 322   x 4 3 4   x Hence, 7 6sin x (4 marks)

5 [Turn over 6. Without the use of a calculator, find the values of the integers p and q for which the solution of the equation 121089624 xx  is .qp [4] 6. 121089624 xx  xx  982 2232  xx 223)12( x x 12 223                   12 12 12 223 224323  12 2p 1q (4 marks)

6 7. (a) Find the term independent of x in the expansion of 9 25 1       x x . [3] (b) Obtain the first four terms in the expansion, in ascending order of x, of .32 6     x [2] Hence, find the c