CJC 9758 2023 Prelim P1

9758/01/J2PRELIM/2023 [Turn Over CATHOLIC JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 JC2 Preliminary Examination CANDIDATE NAME CLASS INDEX NUMBER MATHEMATICS 9758/01 Paper 1 29 Aug 2023 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your class, index number and name on the work you hand in. Write in dark blue or black pen. 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. 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 spec ifically 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. Question 1 2 3 4 5 6 7 8 9 10 Total Marks Total 4 6 8 9 10 11 13 15 12 12 100 This document consists of 3 printed pages, including this cover page.

2 9758/01/J2PRELIM/2023 1 The complex numbers z and w satisfy the following equations. 21zw+= 2 4 24iwz− = + Find z and w , giving your answers in the form iab+ where a and b are real numbers. [4] 2 (a) The diagram below shows a sketch of the curve 21 xy x = + for 0x . Rectangles, each of width 1 n , where n + , are drawn under the curve for 01 x . Show that the total area of all the rectangles, A , can be written as 2 1 1 2 1 n r r n nr − = + . [3] (b) Find the exact value of lim n A → . [3] 3 The points P , Q and R have position vectors p , q and r respectively. The points P and Q are fixed and R varies. (a) Given that p is non-zero and ( )−r q ×p = 0 , find a linear relationship between p , q and r . Describe geometrically the set of all possible positions of the point R . [4] (b) Given that x y z  = r , 2 5 3  =− p , 4 1 2 − = q and ( ) 0−r q p = , find a relationship between x , y and z . Describe geometrically the set of all possible positions of the point R . [4] O x y   1 1 2 3 nnn 3 2 1n n n n n n − − − 21 xy x = +

3 9758/01/J2PRELIM/2023 4 It is given that f ( )x is a cubic polynomial with real coefficients. The diagram shows the curve with equation f ( ).yx= (a) What can be said about all the roots of the equation f ( ) 0?x = [2] Suppose 32f ( ) 2 7 16x x x x c= − + + , where c is a real number. (b) Show that 15c=− , if the equation f ( ) 0x = has a root 1 2i.x=− [1] (c) Without using a calculator, determine the other roots of the equation f ( ) 0.x = [3] (d) Hence, find the roots of the equation 32 0.15 16 7 2w w w =− + − + [3] 5 The first four terms of a sequence of numbers are 4− , 2− , 12 and 38. The sum of the first n terms of this sequence is denoted by nS . (a) Explain why nS cannot be a quadratic polynomial in n . [2] It is given that nS is a cubic polynomial. (b) Find nS in terms of n . [3] (c) Show that the nth term of the sequence, nu is 26 16 6nn−+ . [2] (d) Hence find ( ) 2 1 10 m nn n uu − = − in terms of m . [3] y f ( )yx= x O

4 9758/01/J2PRELIM/2023 6 A curve C has equation 2ax bx cy xd ++= − , where a , b , c and d are constants. It is given that two of its asymptotes are 2yx=+ and 1x= . (a) State the value of d , and show that 1ab== . [2] (b) Using differentiation, f ind the range of values of c such that the graph of C contains two stationary points. [4] Use c =14 for the rest of the question. (c) Sketch C , showing clearly the equations of asymptotes and the coordinates of the turning points. [3] (d) State the maximum number of roots to the equation ( ) 22 222 53 ax bx ck x kxd  ++− + − = − , where 0k . Deduce the range of values of k for the maximum number of roots to occur. [2] 7 It is given that ( ) 2 2 , 0 3,f : 3 8 , 3 4, xxx xx  −   −   (a) (i) Sketch the graph of f ( )yx= , labelling the coordinates of any turning points and end - points. Explain why 1f − does not exist. [3] (ii) If the domain of f is restricted to ( 0, k , state the largest value of k such that 1f − exists. Hence, for this value of k , find 1f ( )x− and state the domain of 1f − . [3] (iii) The function g is such that g : e 3 xx + , 0x . Find the function fg , giving your answer in similar form. [3] (b) It is given further that f ( ) f ( 4)xx=+ . (i) Evaluate f (25) and f ( 8)− . [2] (ii) Sketch the graph of 1f1 2yx =− for 8 10x−   . [2]

5 9758/01/J2PRELIM/2023 8 (a) Use the substitution 15 sinx = to show that 2 2 111 15 15 sin22d 5 5 1 xx x Cxx − − = − + + . [5] The diagram below shows a sketch of the curves 1C and 2C . The curve 1C has parametric equations 6cosx = , 2 2 siny = for 02 π . The curve 2C has equation 22 15xy+= . Given that P is a point of intersection between 1C and 2C , (b) determine the exact coordinates of P. [4] The region R is bounded by curves 1C and 2C and the y-axis in the first quadrant. (c) Show that the area of R is given by ( ) 1sin 15 2 πmn − − where m and n are constants to be determined. [6] y x O C1 C2 P R

6 9758/01/J2PRELIM/2023 9 The following diagram shows a plot of land formed by two semicircles joined to a rectangle ABDE . Point C lies on the arc BD , with CBD = , where π0 2 , and point F lies on the arc AE , with FAE = . As part of a training regime, Nigel runs along the perimeter of the shaded portion ABCDEF as shown in the diagram. It is given that AE is fixed at 2r metres, and AB is twice the length of AE . (a) Show that the perimeter, metresP , of ABCDEF is ( )4 2 cos sinPr = + + . [2] (b) Find the exact value of  which maximises P and hence find the exact maximum distance that Nigel can run in one round of ABCDEF , giving your answer in terms of r . [5] (c) Nigel plans to run one round with maximum distance at a constant speed of 6 metres per second within 3 minutes. Find the maximum value of r, giving your answer in the form 2ab+ , where a and b are constants to be determined. [2] (d) To clearly mark out the shape ABCDEF , the management wishes to plant grass within the shape ABCDEF . It costs $0.15 to plant 21 m of grass and the management has a budget of $10000. Using the value of  found in part (b) and the value of r found in part (c), determine, with justifications, if the management is able to afford to cover the entire shape ABCDEF with grass. [3] C  B D E A  F C  B D E A  F

7 9758/01/J2PRELIM/2023 10 A marketing manager of a company wishes to advertise a new product. He has tasked his team to create an engaging video and upload it on InstaFame social platform. He hopes the video would go viral on the internet so that the product will sell well. According to some analysts, a v