ASRJC FM 2023 Prelim P1 (Questions)

FURTHER MATHEMATICS 9649/01 Paper 1 14 September 2023 3 hours Additional Materials: Answer Booklets List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST An answer booklet will be provided with this question paper. You should follow the instructions on the front cover of the booklet. If you need additional answer booklet, ask the invigilator for a continuation booklet. Write your name and class on the cover page and on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a 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 figures, 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 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. This document consists of 7 printed pages and 1 blank page. ANDERSON SERANGOON JUNIOR COLLEGE JC2 Preliminary Examination 2023 Higher 2

2 ASRJC 2023 9649/JC2 FM Preliminary Examination [Turn Over 1 Given that a is a complex number with 3a  and 0 arg( ) , 4a  shade, on an Argand diagram, the region corresponding to the complex number z for which 23zz and 3.za Hence or otherwise, find the complex number a in cartesian form for which the maximum value of z is 2 3. [5] 2 Points F and 'F are the foci of an ellipse and point P is a point on the ellipse. The line L is the tangent to the ellipse at point P. The angles  and  are the acute angles that FP and 'FP make with the tangent respectively. Using the reflective property of ellipse, prove that    2 2' sin  bFP F P where b is the length of the semi-minor axis of the ellipse. [5] 3 (i) Using the substitution 1 nuy  , show that the differential equation   23 d 1 d 1 nyy yx x n x  where 0y can be written as 23 d 1 1 d un ux xx  . [2] (ii) For 3,n find an expression for y in terms of x. [5] 4 The curve C has polar equation given by 1sin cos 22r  , 0  . (i) Find the exact maximum value of r. [2] (ii) Sketch the curve C. [2] (iii) The point in the first quadrant where r is at its maximum is P. Find the area bounded by the curve C from 0  to P, the tangent at point P and the x-axis, leaving your answer in the form 3a b c  where a, b, c are constants to be determined in exact form. [4]

3 ASRJC 2023 9649/JC2 FM Preliminary Examination [Turn Over 5 Explain geometrically why   11 2 2 0 11 d . 4I x x    [1] Obtain an estimate of the value of I using Simpson’s rule with four intervals giving your answer to 4 decimal places. [2] Show geometrically that   1 1 2 22 0 1 2 1 d .2I x x   [3] Applying Simpson’s rule with four intervals to the integral   1 1 2 22 0 1d xx , obtain a second estimate of the value of I, giving your answer to 4 decimal places. [2] Explain why the second method gives a better approximation to the value of I than the first method. [1] 6 A fishing club’s waters are stocked with a particular variety of fish. Each season, breeding increases the population of fish by a proportion b of the population at the start of the season. Similarly, a proportion d of the population at the start of the season dies during the season. Also, a number c of fish is caught and removed from the waters each season. (a) Write down a recurrence relation in the form 1 ,nnu pu q  where nu is the fish population at the start of the nth season, and p and q are constants to be expressed in terms of b, c and d. [2] (b) Solve the recurrence relation in (a) to find a general expression for nu in terms of p, q and 1u . [3] (c) Given that p = 1.08 and q = -160, find the size of the fish population which would remain constant from season to season. [2] (d) The club is anxious not to have to re -stock by buying in fish each year since this is expensive. They propose to achieve this by ensuring that the initial stock is above the equilibrium population. (i) What does the mathematical model which you have developed predict will happen under these circumstances [2] (ii) What is actually likely to happen and why? [1]

4 ASRJC 2023 9649/JC2 FM Preliminary Examination [Turn Over 7 (a) Using the substitution 22,v u a show that '3 4 2 ' ()u u au b du f v dv      where the expressions for  ', ' and fv are to be written down clearly. [2] (b) The parametric equations of a curve 1C are given by   223 , 2 3 where 0.x t y t t t    Find the range of values of t for which the curve 1C lies below the x-axis. [1] Hence show, by using a linear transformation or otherwise, that the length , r , of the curve 1C which is below the x-axis can be expressed as   1 22 1 6 1 1 dr w w w     [3] Given another curve 2C with parametric equations 333 3 , 3x t t y t   Show that the area of the curved surface, s, obtained by rotating completely the arc of the curve 2C from the points 30 to 2tt  about the x-axis is given by 2 .144 rs  [4]

5 ASRJC 2023 9649/JC2 FM Preliminary Examination [Turn Over 8 A pendulum consists of a ball of mass m kg tied to one end of an inextensible string with length, L cm, as shown in Figure 1 below. The string is inclined to the vertical at an angle of  radians t seconds after the initial release of the ball. Initially, the ball is released at 0 radians. The horizontal and vertical displacement of the ball are defined by x cm and y cm respectively. (i) Show that 222 22 d d d cos sin ddd    x LL ttt and 22 2 22 d ddsin cos ddd    y LL ttt . [2] The resultant horizontal force, xF and resultant vertical force, yF acting on the ball are given by  sinT N and  cos T mg N respectively where T is the tensile force in the string and g is the gravitational acceleration and N represents a newton which is the international unit measure for force. Newton’s 2 nd law of motion states that the acceleration of a body is proportional to the resultant force acting on the body and takes place in the direction of the resultant force i.e. mFa where a is the acceleration. (ii) By applying Newton’s 2nd law of motion in the horizontal direction, we get 22 2 ddsin cos sin dd          T m L L tt -------- (1). Obtain another equation by applying Newton’s 2 nd law of motion in the vertical direction and hence show that 2 2 d d   g tL if  is small. [4]