2019 J2 TJC H2 Math Prelim (with ans)

TEMASEK JUNIOR COLLEGE, SINGAPORE JC 2 Preliminary Examination 2019 CANDIDATE NAME CG MATHEMATICS 9758/01 Higher 2 Paper 1 30 Aug 2019 Candidates answer on the Question Paper. 3 hours Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your Civics group and name on all the work that you hand in. Write in dark blue or black pen. 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 in the spaces provided in the question paper. Give non-exact numerical answers correct to 3 significant figur es, or 1 decimal place in the case of angles in degrees, unless a differ ent level of accuracy is specified in the question. The use of an approved graphing calculator is expected, where appropriate . 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. The total number of marks for this paper is 100. For Examiner’s Use Question Number Marks Obtained 1 2 3 4 5 6 7 8 9 10 11 Total Marks www.KiasuExamPaper.com 741

2 © TJC 2019 9758/01 1 The diagram below shows a shape which is symmetrical about the x- and y-axes. The shape is made up of four curves, A, B, C and D. The curve A has equation 1xy for 01 xdd and 01 ydd . (i) State the equations and the range of values of x and y for curves B and C. [3] (ii) The curves A and B are scaled by a factor 1 2 parallel to the x-axis and the curves C and D are scaled by a factor 2 parallel to the y-axis. Sketch the resulting shape. [2] 2 The position vectors of A, B, C and D a r e 1 5 D§· ¨¸ ¨¸ ¨¸ ©¹ , 1 2 1 §· ¨¸¨¸ ¨¸ ©¹ , 2 3 1 §· ¨¸ ¨¸ ¨¸ ©¹ and 1 7 E §· ¨¸ ¨¸ ¨¸ ©¹ respectively, where α and β are real numbers. Given that BD is a perpendicular bisector of AC, find the values of α and β. [5] 3 The hyperbola C passes through the point (2, 0) and has oblique asymptotes 2yx and 2yx . ( i) Sketch C, showing the relevant features of the curve. [2] (ii) Write down the equation of C. [1] (iii) By adding a suitable curve to your sketch in part (i), solve the inequality 2 114 x x . [3] 1 1 x y 0 −1 −1 A B C D www.KiasuExamPaper.com 742

3 © TJC 2019 9758/01 4 A curve C has equation ex y xk , x ≠ k, where k is a positive real number. Show algebraically that C has exactly one stationary point, and show that the stationary point lies in the first quadrant. [3] Sketch C for x > k, indicating clearly the equation of the asymptote and the coor dinates of the stationary point. [2] Deduce that 3 2 1 2 e d xk k xxk

³ < 13 221 3e e3 kk§· ¨¸ ©¹ for all positive real values of k. [2] 5 In geometric optics, the paraxial approximation is a small-ang le approximation used in Gaussian optics and ray tracing of light through an optical system such as a lens. In the diagram below, a light ray parallel to the horizontal a xis is reflected at point B on the circular lens centred at point C and has radius cmr . Let radiansBCF T‘ . FM is the perpendicular bisector of CB . (i) Show that cos rCF k T , where k is a real constant to be determined. [1] (ii) Hence find the series expansion for CF if T is sufficiently small for 3T and terms in higher powers of T to be neglected. [2] Suppose that the source of the light ray is now repositioned su ch that radians6BCF ST§·‘ ¨¸ ©¹ . (iii) Find the corresponding series expansion for CF, up to and including the term in 2T . [4] B F C T Part of the circular lens Light ray Horizontal axis M www.KiasuExamPaper.com 743

4 © TJC 2019 9758/01 6 An arithmetic sequence has first term a and common difference d, where a and d are non- zero. The ninth, tenth and thirteenth terms of the arithmetic sequence are the first three terms of a geometric sequence. (i) Show that 15 2ad . [3] (ii) The sum of the first n terms of the arithmetic sequence is denoted by nS . Find the value of 16S . [2] (iii) Given that the kth term of the arithmetic sequence is the fourth term of the geometric sequence, find the value of k. [3] 7 A curve C has parametric equations 22 , e , for 0 .txt yt t t (i) Find the equation of the tangent to C at the point P with coordinates 22,e ppp , where 0p z . Hence, or otherwise, find the exact equation of the tangent L to C which passes through the origin. [5] (ii) (a) Find the cartesian equation of C. [1] (b) Find the exact volume of the solid formed when the region boun ded by C and L is rotated through 2S radians about the x-axis. [5] 8 Do not use a calculator in answering this question. The complex numbers z and w are given by

4 2 1i 1i z

and

2 8 3i w

. (i) Express z and w in polar form cos i sinr TT , where 0r ! and STS d . Give r and T in exact form. [4] (ii) Given that 2,zw and *w are the roots of the equation 32 0xb xc x d where b, c and d are real values, find the equation. [3] (iii) Sketch on an Argand diagram with origin O, the points P, Q and R representing the complex numbers ,zw and zw respectively. [2] (iv) By considering the quadrilateral OPRQ and the argument of zw , deduce that 5tan 2 312 S . [3] www.KiasuExamPaper.com 744

5 © TJC 2019 9758/01 9 (a) Vectors u and v are such that 1 u.v and ()uv uu is perpendicular to ()uv vu . Show that 1uvu . [3] Hence find the angle between u and v. [3] (b) The figure shows a regular hexagon ABCDEF with O at the centre of the hexagon. X is the midpoint of BC. Given that aOA aOA and bOB bOB , find OFOF and OXOX in terms of a and b. [2] Line segments AC and FX intersect at the point Y. Determine the ratio AY : YC. [4] A B C D E F O x X www.KiasuExamPaper.com 745

6 © TJC 2019 9758/01 10 Mr Ng wants to hang a decoration on the vertical wall above his bookshelf. He needs a ladder to climb up. The rectangle ABCD is the side-view of the bookshelf and HK is the side-view of the ladder where 24AB cm and 192BC cm (see Figure 1). The ladder touches the wall at H, the edge of the top of the bookshelf at B and the floor at K. Figure 1 (i) Given that HKD T‘ , show that the length , L cm of the ladder is given by 24 192 cos sinL TT . [1] (ii) Use differentiation to find the exact value of the shortest len gth of the ladder as T varies. [4] [You do not need to verify that this length of the ladder is the shortest.] Take L to be 270 for the rest of this question. The ladder starts to slide such that H moves away from the wall and K moves towards E (see Figure 2). The ladder maintains contact with the bookshelf at B. Figure 2 The horizontal distances from the wall to H and from the wall to K are x cm and y cm respectively. (iii) By expressing yx in terms of T , determine whether the rate of change of y is greater than the rate of change of x. [3] (iv) Given that the rate of change of T is 10.1 rad s when 160 cmCK , find the rate of change of x at this instant. [5] H B A bookshelf D K C H B A bookshelf D K C E www.KiasuExamPaper.com 746

7 © TJC 2019 9758/01 11 The daily food calories, L, taken in by a human body are partly used to fulfill the daily requirements of the body. The daily requirements is proportional t