SAJC 2017 H2 MATHS Qns Prelims

SAINT ANDREW’S JUNIOR COLLEGE Preliminary Examination MATHEMATICS Higher 2 9758/01 Thursday 24 August 2017 3 hours Additional materials : Answer paper List of Formulae (MF26) Cover Sheet READ THESE INSTRUCTIONS FIRST Write your name, civics group 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 soft pencil for any diagrams or graphs. Answer all the questions. Total marks : 100 Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of agles in degrees, unless a different level of accuracy is specified in the question. You are expected to use a graphic calculator. Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphic calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. At the end of the examination, fasten all your work securely together. This document consists of 6 printed pages including this page. [Turn over

2      [Turn over   1 The volume of a spherical bubble is increasing at a constant rate of  cm3 per second. Assuming that the initial volume of the bubble is negligible, find the exact rate in terms of  at which the surface area of the bubble is increasing when the volume of the bubble is 20 cm3. [5] [The volume of a sphere, 34 3Vr  and the surface area of a sphere, 24A r where r is the radius of the sphere.] 2 The diagram shows the triangle ABC. It is given that the height AD is h units, 3ABD  and  ACD =  4  x . Show that if x is sufficiently small for x 3 and higher powers of x to be neglected, then  2 + + 3 tan 4 hhBC h p qx rx x    , for constants p, q, r to be determined in exact form. [5] 3 It is given that 2 2 2 2 1 for f( ) (2 )1 for 3 xba x aax xaaa x a a        and that f( 4 ) f( )x ax for all real values of x, where a and b are real constants and 0. ab (i) Sketch the graph of f( )yx for 8.ax a  [3] (ii) Use the substitution cosxa  to find the exact value of 4 3 f( ) d a a x x in terms of a, b and . [5]      A B D C h

3      [Turn over   4 (i) State a sequence of transformations that would transform the curve with equation 2 exy onto the curve with equation f( )yx , where 2 f( ) e axx b , 0a and 1b . [2] (ii) Sketch the curve f( )yx and the curve 1 .f( )y x You should state clearly the equations of any asymptotes, coordinates of turning points and axial intercepts. [5] 5 It is given that  uvw is perpendicular to  uvw , where u, v and w are unit vectors. (i) Show that the angle between v and w is 60 . [4] Referred to the origin O , the points U , V and W have position vectors u, v and w respectively. (ii) Find the exact area of triangle OVW. [2] (iii) Given that u and vw are parallel, find the exact volume of the solid .OUVW [2] [The volume of a pyramid is 1 3 bh , where b is the base area and h is the height of the pyramid.] 6 (a) (i) Find ec o s dx nx x , where n is a positive integer. [4] (ii) Hence, without the use of a calculator, find 2 ec o s dx nx x   in terms of n, when n is odd. [3] (b) The region bounded by the curve 216 xy x  , the y-axis and the line 2 12y  is rotated 2 radians about the x-axis. Find the exact volume of the solid obtained. [5]

4      [Turn over   S 7 (i) Show that for any complex number ie,zr  where 0,r  and ,   11 cot i22 2 z zr    . [3] (ii) Given that i 32ez   is a root of the equation 2 24 0 .zz  State, in similar form, the other root of the equation. [1] (iii) Using parts (i) and (ii), solve the equation 2 2 44 40 .(1 ) 1 ww ww   [4] 8 In a training session, an athl ete runs from a starting point S towards his coach in a straight line as shown in the diagrams below. When he reaches the coach, he runs back to S along the same straight line. A lap is completed when he returns to S. At the beginning of the training session, the coach stands at 1A which is 30 m away from S. After the first lap, the coach moves from 1A to 2A and after the second lap, he moves from 2A to 3A and so on. The distance between iA to 1iA is denoted by 1iiAA  , i  . Figure 1 (i) For training regime 1 (shown in Figure 1) , the coach ensures that the distance 1 3 miiAA   for i  . Find the least number of laps that the athlete must complete so that he covers a total distance of more than 3000 m. [3] Figure 2 (ii) For training regime 2 (shown in Figure 2), af ter the first lap, the coach ensures that the distances 12AA = 2 m, 23 6 mAA  and the distance 12 1 3ii i iAA A A   where i  . Show that the distance the coach is away from S just before the athlete completed r laps is   13 29r  m. Hence find the distance run by the athlete after n complete laps. Also find how far the athlete is from the coach after he has run 8 km. [6] S

5      [Turn over   9 The diagram below shows the curve C with parametric equations 12 s i n ,x   43 c o s ,y  for .   The point P is where 6   . (i) Using a non-calculator method, find the equation of the normal at P. [4] (ii) The normal at the point P cuts C again at point Q, where   . Show that 8sin 2 3cos 1 and hence deduce the coordinates of Q. [3] (iii) Find the area of the region bounded by the curve C, the normal at point P and the vertical line passing through the point Q. [4] 10 A population of 15 foxes has been introduced into a national park. A zoologist believes that the population of foxes, x, at time t years, can be modelled by the Gompertz equation given by: d4 0 lnd x cxtx   , where c is a constant. (i) Using the substitution 40lnu x   , show that the differential equation can be written as d .d u cut  [2] (ii) Hence find u in terms of t and show that e40e ctBx  , where B is a constant. [5] After 3 years, the population of foxes is estimated to be 20. (iii) Find the values of B and c. [3] (iv) Find the population of foxes in the long run. [1] (v) Hence, sketch the graph showing the population of foxes over time. [2] 0  y x PC

6      [Turn over   11 A computer-controlled machine can be progra mmed to make plane cu ts by keying in the equation of the plane of the cut, and drill holes in a straight line through an object by keying in the equation of the drill line. A 10cm  20 cm 30 cm cuboid is to be cut and drilled. The cuboid is positioned relative to the x-, y-and z-axes as shown in Figure 1. First, a plane cut is made to remove the corner at E. The cut goes through the points P, Q and R which are the midpoints of the sides ED, EA and EF respectively. (i) Show that 0 5 15 PQ     and 10 5 0 PR    . [2] (ii) Find the cartesian equation of the plane, p that contains P, Q and R. [2] (iii)