VJC 2023 JC1 Promos 9649

VICTORIA JUNIOR COLLEGE JC 1 PROMOTIONAL EXAMINATION 2023 H2 Further Mathematics 9649/01 Paper 1 3 hours Additional Materials: Answer Paper Graph Paper List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name and CT group on 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. Give non-exact numerical answers correct to 3 significant figur es, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calculator are not allowed in a question, you are required to present the mathematical steps us ing mathematical notations and not calculator commands. 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. This document consists of 6 printed pages. [Turn over

2 1 Do not use a calculator in answering this question. Let w be a non-real cube root of unity. (a) Prove that 2 1.ww  [ 2 ] (b) Hence, or otherwise, show that 22(1 2 3 )(1 2 3 ) 3.ww w w    [ 3 ] 2 The set of points  ,x y which are equidistant from the line x d and the point (0,2 ),d where d is a positive constant, is denoted by P. Using this geometrical definition, show that 2(2 ) (2 ) .yd d dx  [2] Determine the volume of the solid formed when the finite region bounded by P and the y-axis is rotated completely about the x-axis. Leave your answer in exact form in terms of  and d. [5] 3 (a) Show that 22 2 21 23 kk kk  for 0.k [ 2 ] (b) The sequence 123,,,uu u  is such that 1 2 3u  and 1 2 21nn nuu n   . Prove by induction that 2 21 n n nu n    for all positive integers n. [5] 4 Let f( ) , kx xa where k is a positive integer and a > 0. The number 1/ka is a root of the equation f( ) 0.x  (a) Show that for any initial estimate 11 0xx  of 1/ ,ka the Newton-Raphson method gives the iteration formula 1 1 1 (1 ) .nn k n axk x k x      [2] The iteration formula established in part (a) is used to approximate 3 25. (b) Without using a calculator, explain why 1 3x  is reasonable. [1] (c) Determine the value of 3 25, giving your answer correct to 3 decimal places. You need not verify the correctness of your answer. [2] (d) By sketching suitable graphs, illustrate clearly how the iterates obtained in part (c) converge to the value of 3 25. [3] 5 The sequence  nv is given by 1 = 2,v 2 = 0v and 21 422nn nvvv   for 1.n The sequence  nw satisfies a second order homogeneous recurrence relation and ,nnvwk where k is a constant. (a) Find the value of k and obtain an expression for nv as a function of n. [6] (b) Hence show that 42nv  can be expressed as  1f ( ) ,kn  where the two possible forms of f( )n are to be determined. [2]

3 6 (a) The diagram shows the curve with equation f( )yx for 50. 4x  Some points on the curve are given. The stationary points are 3,42   and 1,. 2  Use Simpson’s rule with 2 strips to estimate 5 4 2 3 4 d.yx   [3] (b) The polar curve C has equation f( ) ,r  50, 4  where the function f is as defined in part (a). For examples, when  are 0 and ,4  the values of r are f( 0 ) 1 and 3f 42   respectively. (i) Sketch C, indicating clearly all key features. [3] (ii) Using your answer to part (a), estimate the area of the inner loop of C. [2] 7 Point F lies on the x-axis at (2c, 0) with c > 0. O is the origin and a variable point G moves on the x-y plane in such a way that the total distance 2,OG GF a where .ac It is given that OG r and .FOG  (a) State what type of conic section the locus of G is. [1] (b) Show that ,1c o s pr q   giving the constants p and q in terms of a and c. [4] (c) State what property of the conic section is represented by the value of q. [1] (d) Write down in terms of , an integral that represents the arc length of the curve traced out by G as  varies from 0 to . State, in terms of  and a, an estimate for the value of this integral for extremely small values of c, giving a reason for your answer. [3] [Turn over O x y

4 8 A Mathematics club consists of 12 members, of which 7 are men and 5 are women. They need to arrange themselves in 2 rows for a photoshoot. The first row has 5 seats while the second row has 7 seats. Find the number of ways in which the members can be arranged if (a) there is no restriction, [1] (b) all women are to sit next to each other, [3] (c) 2 particular women must not sit next to each other. [3] After the photoshoot, the members decide to play br idge. Find the total number of ways in which they can be divided into 3 groups of 4 players each. [2] 9 A tetrahedron has a plane base ABC. The coordinates of the vertices are (6,0,0), (0, 4,0),AB  (0,0,1)C and (6,0,9)D (see diagram). (a) Find a cartesian equation of plane ABC. [ 3 ] (b) Determine the coordinates of the point of intersection of OD with plane ABC. [3] (c) Calculate exactly the shortest distance from D to plane ABC. [3] The volume of a tetrahedron is 1 base area height.3 (d) Find the volume of the tetrahedron ABCD. [ 3 ] x y z O A(6,0,0) D(6,0,9) C(0,0,1) B(0,4,0)

5 x y S 10 The diagram below shows the cross-section of a car headlight whose inner reflective surface is modelled by the parabolic curve 2 ,2 , ,xa t y a t at a    where a is a positive constant greater than 1. 2(, 2 )Pa t a t is a point on the curve with parameter t. Q is the point  ,0 .a TS is the tangent to the curve at P, and PR is the line through P and parallel to the x-axis. The angle that PS and QP make with the positive x-direction are and  respectively. (a) By considering gradients of TS and QP, show that 1tan t  and 2. [5] (b) Deduce that angle TPQ is equal to . [ 2 ] (c) Describe the significance of this result in relation to the design of the car headlight. [2] The reflective surface is formed by rotating the parabolic curve completely about the x-axis. (d) Assuming negligible thicknes s, calculate in terms of a and , the area of the reflective surface. [4] [Turn over  P R  T Q O

6 11 Drones are used during hot and dry seasons to monitor the temperatures of forested areas. Two drones 1D and 2D are being flown over an area of rainfo rest to detect hotspots. Taking the observation centre as the origin and the surrounding ground as the x-y plane, the positions of 1D and 2D are