2017 SCGS AM PE P1

SINGAPORE CHINESE GIRLS’ SCHOOL PRELIMINARY EXAMINATION 2017 SECONDARY FOUR O-LEVEL PROGRAMME ADDITIONAL MATHEMATICS 4047/01 Paper 1 Wednesday 2 August 2017 2 hours Additional Materials: Answer Paper Cover Page READ THESE INSTRUCTIONS FIRST Write your class, index number and name on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all the questions. Write your answers on the separate Answer Paper provided. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an approved electronic scientific calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 80. The Question Paper consists of 6 printed pages. [Turn over

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02  cbxax , a acbbx 2 42 Binomial Theorem   nrrnnnnn bbar nbanbanaba                  221 21 , where n is a positive integer and ! )1()1( )!(! ! r rnnn rnr n r n        2. TRIGONOMETRY Identities 1cossin 22  AA AA 22 tan1sec  AA 22 cot1cosec  BABABA sincoscossin)sin(  BABABA sinsincoscos)cos(  BA BABA tantan1 tantan)tan(   AAA cossin22sin  AAAAA 2222 sin211cos2sincos2 cos  A AA 2tan1 tan22tan   Formulae for ABC C c B b A a sinsinsin  Abccba cos2222  Abc sin2 1

3 [Turn over 1. Find the value of k for which the line kxy  2 and the curve 22  xy do not intersect. [4] 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] 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] 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] (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] 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] 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] 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]

5 [Turn over 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] 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]

6 11. The diagram shows two rods, AB and BC, of length 6 m and 2 m respectively. The rods are fixed at B such that angle ABC = 90o and hinged at the ceiling, at A so that they can rotate in a vertical plane. The rod AB makes an acute angle  with the ceiling. (i) Obtain an expression, in terms of  , for h, where h m is the vertical distance from the ceiling to C. [2] (ii) Express h in terms of )(sin R where 0R and oo 900  . [4] (iii) Find the value of  for which C is 1.6 m below the ceiling. [2] 12. The velocity, v ms1, of a particle P, travelling in a straight line, at time t seconds after leaving a fixed point O, is given by 35192 2  tt eev , where 0t . Find (i) the value of t when the particle is instantaneously at rest, [3] (ii) the acceleration of the particle when 7lnt . [2] The displacement of the particle P, at time t seconds after leaving a fixed point O, is denoted by s metres. (iii) Find an expression, in terms of t, for s. [3] Hence, (iv) find the distance travelled by the particle in the first 1.5 seconds. [3] End of Paper 1  A B C h m n= 6 m 2 m