2017 SNGS Sec 4 AM Prelim P1

[Turn over Name: _________________________ ( ) Class: ____________ PRELIMINARY EXAMINATION GENERAL CERTIFICATE OF EDUCATION ORDINARY LEVEL ADDITIONAL MATHEMATICS 4047/01 Paper 1 Friday 18 August 2017 2 hours Additional Materials: Answer Paper READ THESE INSTRUCTIONS FIRST Write your name, class, and index number on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a pencil for any diagrams or graphs. Do not use paper clips, highlighters, 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 scientific calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. At the end of the examination, staple all your work together with this cover sheet. 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. FOR EXAMINER’S USE Q1 Q6 Q11 Q2 Q7 Q12 80 Q3 Q8 Q13 Q4 Q9 Q5 Q10 This document consists of 6 printed pages.

2 CHIJ SNGS Preliminary Examinations 2017 - Additional Mathematics 4047/01 [Turn over Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02  cbxax , a acbbx 2 42 Binomial Theorem   nrrnnnnn bbar nbanbanaba                  ......21 221 , where n is a positive integer and       ! 1........1 !! ! r rnnn rrn 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 sin211cos2sincos2cos  A AA 2tan1 tan22tan   Formulae for ∆ABC C c B b A a sinsinsin  Abccba cos2222  Abc sin2 1

3 CHIJ SNGS Preliminary Examinations 2017 - Additional Mathematics 4047/01 [Turn over 1 The mass of a radioactive substance reduces by half every 4 hours. It is given that m0 is the mass of the substance at a particular time and that m is the mass of the substance t hours later. Calculate the value of the constant k in the relationship ktemm  0 . [3] 2 Solve the equation    xxx 212 3227363  . [4] 3 The function f is defined, for all values of x, by xexx 2)(f  . Find the range of values of x for which f is a decreasing function. [6] 4 (i) On the same diagram, sketch the curves 3 1 2xy and 3 1 8   xy . [2] (ii) Find the coordinates of the intersection points of the two curves. [3] 5 The equation of a curve is x xy 25 8   . A particle moves along the curve in such a way that the y-coordinate of the particle is decreasing at a constant rate of 2 units per second. Find the possible y-coordinates of the particle at the instant when the x-coordinate of the particle is increasing at 11 71 units per second. [6] 6 A curve has equation xy 42  and a line has equation )1(  xmy . Find the range of values of m for which the line )1(  xmy intersects the curve xy 42  at two distinct points. [5]

4 CHIJ SNGS Preliminary Examinations 2017 - Additional Mathematics 4047/01 [Turn over 7 The diagram shows part of the graph of cb xay    sin . (i) State the amplitude and period. [2] (ii) Find the value of each of the constants a, b and c. [3] 8 (i) Sketch the graph of xy  4 . [2] A line of gradient m passes through the point (0, 1). (ii) In the case where m = –2, find the coordinates of any point of intersection of the line and the graph of xy  4 . [3] (iii) Determine the set of values of m for which the line intersects the graph of xy  4 at one point. [2] 9 (i) Prove that AA AA secsin1 costan  . [3] (ii) Hence, find the exact solutions of xx xx cosec2sin1 2cos2tan  for 20  x . [5] 10 A particle, moving in a straight line, passes through a fixed point O with a velocity of 3 m/s . The acceleration, a m/s2, of the particle, t seconds after passing through O is given by tea 2.06.0 . (i) Find the value of t when the particle is at instantaneous rest. [4] (ii) Find the distance travelled by the particle in the first 5 seconds. [4] y x 4 –2 –4π –2π 0 2π 4π 6π 8π 10π 12π

5 CHIJ SNGS Preliminary Examinations 2017 - Additional Mathematics 4047/01 [Turn over  P Q R S O 4 cm 11 The diagram shows a trapezium PQRS inscribed in a semicircle with centre O. The radius of the semicircle is 4 cm. Angle POQ = angle SOR = θ radians. (i) Show that the area, A cm2, of the trapezium PQRS is given by  2sin8sin16 A . [2] (ii) Given that θ can vary, find the value of θ for which the area of the trapezium is a maximum. [6] 12 In the diagram, AB is a diameter of the circle with centre O. CS and BT are the tangents to the circle at C and B respectively. UCS, ACT and BST are straight lines. Prove that (i) triangle ABC and triangle ATB are similar, [2] (ii) angle SCT = angle CTS, [3] (iii) S is the midpoint of BT. [2] O A C B S T U 4 cm 

6 CHIJ SNGS Preliminary Examinations 2017 - Additional Mathematics 4047/01 [Turn over 13 The diagram shows part of the curve xy 212 5  and the line 22  yx . The line intersects the curve at point A. (i) Find the coordinates of A. [3] (ii) Find the area of the shaded region bounded by the curve xy 212 5  , the lines 4x and 22  yx . [5] End of Paper. A x = 4 x y O