2024 JC2 H2 Math prelim - YIJC

Th is document consists of 23 printed pages and 1 blank page. [Turn Over YISHUN INNOVA JUNIOR COLLEGE JC 2 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME CG MATHEMATICS Paper 1 C andidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) 9758/01 28 AUGUST 2024 3 h ours READ THESE INSTRUCTIONS FIRST Write your CG, index number and name on the work 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. A nswer all the questions. Write your answers in the spaces provided in the Question Paper. 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. T he 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 Examiners’ Use Question Marks 1 2 3 4 5 6 7 8 9 10 11 Presentation Total / 100

2 ©YIJC 9758/01/JC2PE/24 1 (a) Without using a calculator, solve the inequality 6 14 x xx ≥−+ . [4] (b) Hence, solve 6 14 x xx ≥+− . [2] 2 The diagram below shows a sketch of the graph of 2sinyx= for 0 π 2x≤≤ . Rectangles each of width π 2n are drawn under the curve for 0 π 2x≤≤ . (a) Show that A , the total area of all the rectangles, is given by 1 2 1 sin 2 n k ka n π− = ∑ , where a is to be determined. [2] (b) Find the exact value of 1 2 1 lim 2sin n n k ka n π− →∞ = ∑ . [2] (c) Hence, find the value of 0 11 sin dy y− ∫ . [2] 3 Do not use a calculator in answering this question. (a) Given that f( )x is a polynomial of degree 4 with real coefficients, explain whether it is possible for f( ) 0x = to have 3 non-real roots and 1 real root. [1] (b) One of the roots of the equation 4 322 15 63 0x x ax x b− + − += , where a and b are real, is 3 2i− . Find the other roots of the equation and the values of a and b. [6] y x

3 ©YIJC 9758/01/JC2PE/24 [Turn over 4 (a) It is given that ( )d ln lnd yx xy x y yx = +− . Using the substitution w xy= , show that the differential equation can be transformed to ( )d fd w wx = , where the function ( )f w is to be found. [3] (b) Hence, given that 31 e2y = when 2x = , solve the differential equation ( )d ln lnd yx xy x y yx = +− , to find y in terms of x. [5] 5 (a) The graph of f( )yx= is shown below. The graph has a turning point at ( )1, 4A − , and axial intercepts at ( )0, 6B and ( )3, 0C − . The lines 2x = − and 10y = are the asymptotes. On separate diagrams, and showing clearly the coordinates of the turning points and any points of intersection with the axes and the equations of the asymptotes where possible, sketch the graphs of (i) 1 f( )y x= , and [3] (ii) ( )fyx ′= [2] (b) State a sequence of transformations that will transform the ellipse 224 –4 0xy y+= to the unit circle 22 1xy += . [3] x y O

4 ©YIJC 9758/01/JC2PE/24 6 A curve is defined by the parametric equations 2 22 12 , , 1 0. 11 ttxy t tt −= = −≤≤ ++ (a) Using differentiation, find the equation of the tangent to the curve at the point where 1 2t = − . [4] (b) Sketch the curve and the tangent in part (a) on the same diagram, labelling the coordinates of the points of intersection with the axes. [2] (c) Show that the area bounded by the curve, the tangent and the x -axis can be expressed in the form ( ) 2 32 8 d 1 b a tct t − + ⌠ ⌡ , where a, b and c are constants to be determined. Hence evaluate this area. [3] 7 (a) Show that 46 (2 1)(2 3)(2 5) r rr r − ++ + can be expressed in the form 21 23 25 ABC rr r ++++ + , where A, B and C are constants to be determined. [2] (b) Hence, find an expression for 1 46 (2 1)(2 3)(2 5) N r r rr r= − ++ +∑ in terms of N. You do not need to give your answer as a single fraction. [4] (c) Using your answer in part (b), find the exact value of 7 4 10 (2 1)(2 1)(2 3)r r rrr ∞ = − −++∑ . [3] 8 The function f is defined by ( )f: 5 ,xx x − , 3.xx∈≥ (a) Find ( )1f x− and state the domain of 1f − . [3] It is given that ( ) 44 for 8,10g 12 for 8 12.2 xxx x x  +≤ −=  − <≤ (b) Sketch the graph of ( )gyx= . [2] (c) Find the value of x such that ( )1g 3.5 x− = . [2] (d) Explain why the composite function gf exists and find gf (x). [2] (e) Find the range of gf. [1]

5 ©YIJC 9758/01/JC2PE/24 [Turn over 9 (a) Find 23 12 dx xx +∫ . [2] (b) Find sin 3 sin 4 dx xx∫ . Hence, find the exact value of 3 0 sin 3 sin 4 dx xx π ⌠ ⌡ . [5] (c) Find e cos3 dx xx− ∫ . Hence, find the exact value of 0 e cos3 dx xx π − ∫ . [5] 10 Two aeroplanes are observed flying in straight lines, with respect to an airport control tower located at ( )0, 0, 0 . The flight paths of aeroplanes A and B can be modelled by 10 2 45 36 λ −     = +−       r and 81 33 14 µ −   = +−    r respectively, where λ and µ is the time elapsed in minutes since the start of the observation for each aeroplane. The x, y and z-directions are due east, due north and vertically upwards respectively, with all distances in kilometres. (a) The flight paths intersect at point P. Find the coordinates of P and explain why the two aeroplanes will not collide. [4] (b) Find the acute angle between the flight path of aeroplane A and the horizontal ground. [2] (c) Find a cartesian equation of the plane Π which contains both flight paths. [3] (d) The airport building has a slanted wall which is parallel to the flight path of aeroplane B , and the wall is inclined at an angle of 60 with the horizontal ground. The cartesian equation of the wall is given by 1ax by z+ += . Given that 1b > , find the values of a and b. [4] 11 [ It is given that the volume of a sphere of radius r is 34 3 rπ and that the volume of a circular cone with base radius r and height h is 21 3 rhπ .] A hollow crystal sphere with centre O has a fixed radius of R cm and it is made of material with negligible thickness. A golden right circular cone with base radius r cm and height h cm is inscribed in the sphere such that its vertex and the circumference of the circular base are both in contact with the inner surface of the sphere. It is also given that hR> (see diagram). R h r O

6 ©YIJC 9758/01/JC2PE/24 (a) Show that 22hR R r= +− . [1] (b) Show that the maximum possible volume of the cone is 33 cmkRπ , where k is a constant whose exact value is to be found. You do not need to show that this volume is a maximum. [6] It is now assumed that the volume of the inscribed cone is maximum for the rest of this question. The space between the bottom of the sphere and the circular base of the cone is fully filled with fluorescent liquid. Unfortunately, the liquid is leaking at a constant rate of 312 cm s − at the bottom of the sphere. The volume of the liquid, L cm3, at the instant when the depth of the liquid is x cm is given by ( )21 33L x Rxπ= − . It is now given that 10R = . (c) Find the rate of decrease of x at the instant when the depth of the liquid is 4 cm. [3] (d) How long does it take for the liquid to be completely drained? [2]

This document consists of 23 printed pages and 1 blank page. [Turn Over YISHUN INNOVA JUNIOR COLLEGE JC 2 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME CG