CJC P1

9758/01/PRELIMS/2020 [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 1 September 2020 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name and class on all 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. 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. 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 11 Total Marks Total 5 6 7 8 8 8 9 10 12 13 14 100 This document consists of 27 printed pages and 1 blank page.

2 9758/01/PRELIMS/2020 1. A function f is defined by ( ) 23f 35 xx x += + . (i) Show that ( )f x can be written in the form 35 ba x+ + , where a and b are constants to be determined. [1] (ii) Hence describe a sequence of transformations which transforms the graph of 1y x= on to the graph of ( )fyx= . [4] 2. The complex number z has modulus 2 and argument 8 π . It is also given that 1iw= + . (i) Given that n is an integer, find * nz w in terms of n, giving your answer in the form ier θ . [3] (ii) Hence, find the smallest two positive integers n such that * nz w is real and negative. [3] 3. Do not use a calculator in answering this question. The equation 3225 0z z pz q+ + += has a root 1iz= −+ , where and pq are real constants. Find and pq and all the other roots of the equation. [7] 4. The function f is given by ( ) 5 , , 22f 1, , 2 xx x x xxx  −∈<=  ∈≥   (i) Sketch the graph of f( )yx= , stating the equation of any asymptotes and the coordinates of any points where the curve crosses the x- and y-axes. [2] (ii) Explain why f has an inverse. Hence find 1f () x− , giving your answer in a similar form. [4] Let ( ) 3g xx= , , 2xx∈≥ . (iii) Determine whether 1gf− exists. [2]

3 9758/01/PRELIMS/2020 [Turn over 5. [It is given that the surface area and the volume of a sphere of radius r is 24 rπ and 34 3 rπ respectively.] A manufacturer produces barriers which consist of a hemisphere of radius r cm joined to a cylinder of base radius r cm and height h cm (see diagram). The manufacturer decides that the external surface area, excluding the base, of each barrier is 216200cm . Using differentiation, find the exact value of the maximum volume of a barrier. [8] 6. The curves 1C and 2C have equations 2 2 29 4 xy x += − and 22 125 9 yx−= respectively. (i) Sketch 1C and 2C on the same diagram, including the coordinates of the points where the curves cross the axes, the equations of any asymptotes and the coordinates of the points of intersection of the curves. [6] (ii) Hence, solve the inequality 2 2 2 1 2951 94 xx x ++≥ − . [2] 7. (i) It is given that 32 2 23y y yx x++=− , find the Maclaurin series for y up to and including the term in 2x . [6] (ii) Hence find the Maclaurin series for 1 2 y+ up to and including the term in 2x . [3] 8. Curve C has parametric equations 23xt= , 6yt=− . (i) Point P on C has parameter p, where 01 p<< . Find the equations of the tangent and the normal to C at P. [4] (ii) The tangent to C at P meets the x-axis at point T, while the normal to C at P meets the x-axis at point N. Show that the coordinates of the midpoint M of TN is independent of p. [3] (iii) Prove that the acute angle PM makes with the x-axis is always twice the acute angle PN makes with the x-axis. [3] base r h

4 9758/01/PRELIMS/2020 9. (a) (i) Show that, for axa−<< , 222 1 22d sin 2d xx axa axxa − −+ = −   . [3] The region R in the first quadrant is bounded by the x-axis, y-axis, the line 2x= and the curve 22 16 16xy+= . Find (ii) the exact area of R, [3] (iii) the exact volume generated when R is rotated through 360°about the x-axis. [3] (b) The region S is bounded by the curves 2yx= and 22 44xy+= for 0y≥ . Find the volume of the solid generated when the region S is rotated through 180°about the y- axis. [3] 10. Mr Lim is considering an education endowment plan for his child. EduPlan Awesome allows him to contribute $300 into the account on the first day of every month. At the end of each month, the total in the account is increased by 0.3%. (i) Mr Lim contributes $300 on 1 January 2020 and continues to contribute $300 on the first day of each subsequent month. (a) Show that the total amount in the account at the start of the nth month ( where January 2020 was the 1st month, February 2020 was the 2nd month, and so on) is $ ( )1.003 1nA − , where A is an integer to be determined. [3] (b) Hence, find the total amount in the account at the start of 1 January 2021, giving your answer correct to 2 decimal places. [1] EduPlan Blessing allows him to contribute $k into the account on the first day of every month. At the end of each month, the account earns a bonus and is added to the account. This bonus is $0.01k for the first month, and for each subsequent month, the bonus is $0.01k more than the bonus in the previous month. (ii) State the amount of bonus earned for the 20th month, giving your answer in terms of k. [1] (iii) Find the total amount in the account at the start of the nth month, giving your answer in the form ( )0.005nk n B+ , where B is an integer to be determined. [3] Mr Lim wishes to cash out the education endowment plan on 1 January 2040 when he makes his first monthly contribution on 1 January 2020. (iv) Find the value of k such that both EduPlans have equal value when the education endowment plan is cashed out, giving your answer correct to the nearest whole number. [3] (v) Given that Mr Lim has other financial commitments, explain which EduPlan is the better option for him. [2]

5 9758/01/PRELIMS/2020 [Turn over End of Paper 11. A temporary isolation centre is built to manage the increasing number of COVID-19 cases. The roof takes the shape of a triangular prism. Points ( ),,xyz are defined relative to an origin, O , with unit vectors i along OA    , j along OC   , and k along OD   (see diagram). The coordinates of D, E, F, G and I are (0,0,3), (4,0,3), (4,12,4), (0,12,4) and (1,12,7) respectively. The units are measured in metres. (i) Find a cartesian equation of the plane that contains the roof section EFIH. [3] It is given that the roof section DGIH is part of the plane with equation 36 12 36xy z+− = − . (ii) Find a cartesian equation of the line that contains the roof ridge HI. [3] Steel cables are used to hold the isolation centre in place. Cables are laid in straight lines and the widths of cables can be neglected. It is given that cab