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 coefficient of x3 in the expansion of .)3(32 2 6 xx     [3] 7. (a) 1T r r r x xr            2 9 5 19 r r xr 39 5 19            When 039  r , 3r The term independent of x 3 5 1 3 9           672.0or 125 84  (b) 6 32       x ...3)2(3 6 3)2(2 6 3)2(1 62 3 3 2 456                                xxx ...27 160 3 806464 32  xxx  2 6 332 xx       )69(...27 160 3 806464 232 xxxxx       Coefficient of 3x              927 16063 80164 3 128 or 3 242 (8 marks)

7 [Turn over 8. (i) Show that  12)1(d d  xxx can be expressed in the form 12   x bax where a and b are integers. [4] (ii) Integrate 12 3 x x with respect to x. [3] (iii) Given that the curve )(f xy passes through the point      8 ,2 5 and is such that 12 3)(f   x xx , find )(f x . [2] 8. (i)  12)1(d d  xxx )2()12(2 1)1()1()12( 2 1 2 1   xxx 12 112   x xx 12 23   x x (ii)   x x x d 12 3    x x xx d 12 212)1( Cxxx    22 1 )12(212)1( 2 1 Cxxx  12212)1( Cxx  12)1( (iii) Cxxx  12)1()(f At      8 ,2 5 , C          12 5212 58 C 78 1C 112)1()(f  xxx (9 marks)

8 9. The figure shows a sector OPQ of a circle, centre O, radius 20 cm. Angle POQ = 2θ radians where 20   . A circle centre R, radius r cm, touches the arc PQ at the point S. The lines OP and OQ are tangents to the circle at the points U and T respectively. (i) Write down, in terms of r, the length of OR. [1] (ii) Hence show that   sin1 sin20 r . [2] (iii) Given that r is increasing at 2 cm s –1, find the rate at which θ is increasing when 6   . [4] 9. (i) cm )20( rOR  (ii) sinOR UR sin20  r r )20(sin rr    sin20)sin1( r   sin1 sin20 r (iii) d dr 2)sin1( cossin20)sin1(cos20     2)sin1( cos20    When 2d d t r , 6   , td d d d d d r t r  6cos20 6sin1 2 2          260.0 rad per second (7 marks)

9 [Turn over

10 10. The points A and B lie on a circle with centre C. The coordinates of A and B are (1, 7) and (– 3, 9) respectively. The line 48  xy passes through the centre of the circle. (i) Find the coordinates of C and the radius of the circle. [5] (ii) Hence find the equation of the circle. [1] Another circle, with centre D(–3, 6), has a radius of 6 units. (iii) Do the two circles intersect? Support your answer with working. [2] 10. (i) Gradient of AB 2 1 13 79   Gradient of perpendicular bisector of AB = 2 Midpoint of AB = (1, 8) Equation of perpendicular bisector of AB, )1(28  xy 102  xy At C, 10248  xx 66 x 1x 12y Centre (1, 12) Radius = 22 34  cm 5 (ii) 25)12()1( 22  yx (iii) Sum of radius = (5+6) cm =11 cm Distance between the 2 centres 22 64  132 units < Sum of radius The 2 circles intersect. (8 marks)