2017 SCGS AM PE P2

SINGAPORE CHINESE GIRLS’ SCHOOL PRELIMINARY EXAMINATION 2017 SECONDARY FOUR O-LEVEL PROGRAMME ADDITIONAL MATHEMATICS 4047/02 Paper 2 Friday 4 August 2017 2 hours 30 minutes 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 100. 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. (i) Express 592 97 2   xx x in partial fractions. [3] (ii) Hence find   7 6 2 d 592 97 x xx x [3] 2. Given that 22332)(f 23  xxxx is exactly divisible by ,2 2 cbxx  (i) find the value of b and of c, [4] (ii) show that 022332 23  xxx has only one real root, [3] (iii) find the remainder when )(f x is divided by )32( x . [2] 3. The roots of the quadratic equation 0432 2  xx are  and . (i) Without using a calculator, show that 32 99 22      . [5] (ii) Hence find the quadratic equation whose roots are 2  and 2  . [2] 4. The diagram below shows a quadrilateral BCDF whose vertices lie on the circumference of a circle. The tangent to the circle at the point F meets CB extended at A and CD extended at E. The lines CF and BD intersect at G and BG = GF. (i) State an angle which is equal to angle BFA. [1] (ii) Prove that angle FDE = angle FAB + angle FCB. [2] (iii) Prove that quadrilateral BCDF is a trapezium. [3] A B C D E F G

4 5. Solve the following equations. (i) 32)3( )2( 123  xx [2] (ii) 23)3(7 11   xx [4] (iii) 2log)1(log)5(log 333  xx [4] 6. A curve has the equation xe xy 2 . The point (p, q) is a stationary point on the curve. Determine (i) the exact value of p and of q, [4] (ii) the nature of the stationary point (p, q). [3] Hence (iii) write down the values of x for which y is increasing. [1] 7. The population, P, of a certain species of fish after t years is given by ).21(1750 t keP  When t = 2, the population is 4600. (i) Find the value of k. [3] Any species is considered as an “Endangered Species” if its population falls below 2500. (ii) Determine, with working, whether this species of fish will become an “Endangered Species” after 10 years. [2] (iii) Hence sketch the population-time graph. [2] 8. The depth of water, d metres, at a pier, t hours after low tide, can be modelled by the formula )cos(btacd  , where a, b and c are positive constants. (i) If low tides occur every 12 hours, find the value of b. [1] Given that the depth of water at the pier was 2 metres during low tide and 8 metres during high tide, (ii) find the value of a and of c. [2] The pier will be open when the depth of the water is more than 4 m. (iii) For how long will the pier be open in a 12-hour period after low tide. [3]

5 [Turn over 9. The diagram shows a solid which consists of a cube fixed on top of a cuboid. The c ube has sides x cm. The cuboid has a square base of side 2x cm and a height of y cm. Given that the volume of the solid is 270 cm3, (i) show that the total surface area, A cm2, of the solid is given by xxA 54010 2 . [5] Given that x can vary, (ii) find the value of x for which A has a stationary value and determine whether this value of A is a maximum or a minimum. [5] 10. The diagram shows part of the curve 43 8   x y . The curve intersects the y-axis at P(0, 4). The normal to the curve at P intersects the line x = 4 at the point Q and the line segment QR is parallel to the y-axis. (i) Find the coordinates of Q. [5] (ii) Find the ratio of the area of the shaded region to the area of the trapezium OPQR in the form 1: n. [5] x cm x cm x cm 2x cm 2x cm y cm y x O P(0, 4) Q R x = 4

6 11. (a) (i) Prove that  cos3cos43cos 3  . [3] (ii) Find all the angles between 2  and  which satisfy the equation 2 3sin4 2 3cos cos3cos4 3     . [4] (b) Solve the equation 7sec3cos6  xx for oo 180180  x . [4] 12. Solutions to this question by accurate drawing will not be accepted. The diagram, which is not drawn to scale, shows a trapezium ABCD in which AB is parallel to DC and BD is parallel to the y–axis. The coordinates of A and of B are (0, 2) and (6, 11) respectively. E(h, 5) is the midpoint of BC such that AB = BE and h > 0. (i) Show that the value of h is 15. [2] Find (ii) the coordinates of C, [2] (iii) the equation of DC, [3] (iv) the coordinates of D. [1] Given that the area of triangle 2 158ABE units2, (v) find the perpendicular distance from B to the line segment AE. [2] End of Paper 2 B(6, 11) A(0, 2) C E(h, 5) D O y x