2017 SNGS Sec 4 AM Prelim P2

[Turn over Name: _________________________ ( ) Class: ____________ PRELIMINARY EXAMINATION GENERAL CERTIFICATE OF EDUCATION ORDINARY LEVEL ADDITIONAL MATHEMATICS 4047/02 Paper 2 Monday 21 August 2017 2 hours 30 minutes Additional Materials: Answer Paper Graph 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 100. FOR EXAMINER’S USE Q1 Q5 Q9 Q2 Q6 Q10 100 Q3 Q7 Q11 Q4 Q8 This document consists of 6 printed pages.

2 CHIJ SNGS Preliminary Examinations 2017 - Additional Mathematics 4047/02 [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 4047/02 [Turn over 1 The curve  xy f is such that 542)('f 2  xxx . (i) Explain why the curve )(f xy has no stationary points. [2] (ii) Given that the curve passes through the point (–1, –6), find an expression for )(f x . [3] 2 The function f is defined by 82)(f 234  kxxxx , where k is a constant. It is given that 0)(f x has a repeated root 2. (i) Find the value of k, [2] (ii) Determine, showing all necessary working, the number of real solution(s) of the equation   0f x . [4] 3 The roots of the equation 042 2  pxx , where p is a constant, are α and β. The roots of the equation 096 2  qxx , where q is a constant, are   2 and   2 . Find the value of p and of q. [6] 4 The diagram shows part of the curve 2)21( 50 xy  . The tangent 2685  xy at the point A on the curve cuts the x-axis at B. The normal at A cuts the x-axis at C. Find the area of triangle ABC. [9] A B C O y x

4 CHIJ SNGS Preliminary Examinations 2017 - Additional Mathematics 4047/02 [Turn over 5 (a) (i) Write down the general term in the binomial expansion of 182 2 4        x x . [1] (ii) Write down the power of x in this general term. [1] (iii) Hence, or otherwise, determine the term independent of x in the binomial expansion of 182 2 4        x x . [2] (b) In the binomial expansion of   n kx2 , where 3n and k is a constant, the coefficients of x and x2 are equal. Express k in terms of n. [4] 6 Do not use a calculator in this question. (i) Express 133 326  in the form 3ba , where a and b are integers. [3] A toy is modelled in the shape of a right pyramid with a square base as shown in the diagram. The vertical height AB of the pyramid is  43 cm. Given that the length of the slant edge AC is 133 326  cm, (ii) find an expression for BC2 in the form 3dc , where c and d are integers, [3] (iii) express the volume of the pyramid in the form  3393 2 k cm3, where k is an integer. [2] A B C

5 CHIJ SNGS Preliminary Examinations 2017 - Additional Mathematics 4047/02 [Turn over 7 (i) Differentiate  12ln 2 x with respect to x. [2] (ii) Express   121 124 2 2   xx xx in partial fractions. [4] (iii) Hence evaluate     2 1 0 2 2 d121 124 xxx xx . [4] 8 The diagram shows a cycling circuit formed from four straight roads OA, AB, BC and CO. OA = 7 km, AB = 2 km, angle OAB = angle BCO = 90°, and angle COA = θ where  900  . A cyclist cycled along the circuit OABCO. (i) Show that cCOBCABOA   cos5sin9 , where c is a constant to be found. [2] (ii) Express COBCABOA  in the form   cR cos . [4] (iii) Find the values of  for which the cyclist cycled a total distance of 19 km. [3] (iv) State the maximum possible value for the total distance of the circuit. [1] 9 The equation of a curve is    xxy 2321 3  . (i) Find the coordinates of the stationary points of the curve. [6] (ii) Determine the nature of these stationary points. [3] (iii) Hence, sketch the curve. [2] O C A B θ 2 km 7 km

6 CHIJ SNGS Preliminary Examinations 2017 - Additional Mathematics 4047/02 [Turn over 10 A circle, C1, passes through the points A (–3, –25) and B (7, –25). The centre and radius of the circle are  ba , and r respectively. The x-axis is a tangent to the circle. (i) Find the value of a. [1] (ii) Show that b = –13. [3] (iii) Find the equation of the circle C1. [1] (iv) AT is a diameter of the circle. Find the equation of the tangent to the circle at T. [5] (v) Given that a second circle, C2, is a reflection of C1 in the y-axis, find the equation of C2. [2] 11 A student records the population, y, of a certain type of bacteria, x hours after the start of her experiment, in the table below. It is believed that an error was made in recording one of the values. The student wants to use the equation xaby  , where a and b are constants, to model the population of this type of bacteria. x 1.5 3 4.5 6 7.5 y 8.37 13.8 22.8 47.5 62.0 (i) Plot ylg against x and hence determine which value of y in the table above is the incorrect recording, and estimate the correct value of y. [5] (ii) Hence, estimate the value of each of the constants a and b. [5] (iii) The population of a second type of bacteria is modelled by the equation xy 2 . Using your values of a and b, calculate the value of x for which xxab 2 . [2] (iv) Explain how another straight line drawn on your diagram can lead to an estimate of the value of x for which the populations of the two types of bacteria are equal. Draw this line and hence verify your value of x found in part (iii). [3] End of Paper.