JPJC 9758 2022 Promo QP

Name:____________________________________ Class:_____________ JURONG PIONEER JUNIOR COLLEGE JC1 Year End Examination 2022 MATHEMATICS 9758/01 Higher 2 29 Sept 2022 Paper 1 3 hours Additional materials: Answer Paper Cover Page List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name and civics 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. This document consists of 6 printed pages. [Turn over Answer all the questions. Give non-exact numerical answers co rrect 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 ans wers 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. At the end of the examination, fasten all your work securely together. The number of marks is given by [ ] at the end of each question or part question.

2 1 (i) On the same axes, sketch the graphs of 1 2y x  and 42yx , indicating clearly the equations of the asymptotes and the coordinates of the axial intercepts. [2] (ii) Hence, or otherwise, solve the inequality 1 422 xx  . [3] 2 (a) Differentiate the following with respect to x. (i)   3ln 9 3e x , [1] (ii) 42sin x , [2] (iii) 1sec3 sin 2xx  . [3] (b) A curve has equation 32 4 9 10 0y y x x     . Find d d y x in terms of x and y. [2] 3 By using the substitution 31,ut find the exact value of   2 5 33 0 d 1 t t t   without using a calculator. [5] 4 The curve C has equation 2 4 1 x kxy x   , where k is a constant. (i) Find the range of values of k if C has two distinct stationary points. [3] For the rest of this question, use 4k  . (ii) Sketch the graph of C, indicating clearly the equations of any asymptotes and the coordinates of any turning points and axial-intercepts. [3] (iii) Using your sketch in (ii), find the range of values of m, where m is a positive real number for which the equation 2 4 ( 3)( 1) 0x kx mx x      has no real roots. [2] 5 The diagram shows a V-shaped tank with dimensions = 4 mL , = 0.6 mW and = 0.7 mH . The tank is initially empty. Water is pumped into the tank at a rate of 0.0025 m3/s. At any instant from the start of water flowing into the tank, the water in the tank has a depth of y m and a surface width of x m. (i) Find the rate of change of the water depth when = 0.4 my , leaving your answer to 4 decimal places. [4] (ii) Find, to the nearest second, the time taken to completely fill up the tank from the instant when = 0.4 my . [2] L = 4 m y H = 0.7 m W = 0.6 m x

3 6 The diagram shows the curve C and the line l with equations 2 8 16y x x   and 6yx  respectively. (i) Find the intersection points of C and l . [1] (ii) The region R is bounded by C , l , the x-axis and the line x = a, where 56 a . Find the area of region R, leaving your answer in terms of a. [4] 7 Find (a) 10e d5 2e x x x , [2] (b) 2 d 18 x x x , [2] (c) 2 (ln ) dx x x . [5] 8 (a) By expressing the equation of the curve 12 11 21 xy x   in the form 21 ByA x  , where A and B are constants, describe a sequence of three transformations which maps the graph of 1 23y x  onto the graph of 12 11 21 xy x   . [4] (b) The diagram shows the graph of f ( ).yx The curve has a maximum point at (0, 5) and a minimum point at (6, 4) . The equations of the asymptotes of the curve are 2,x 2x and 2y . y x 6 6

4 Sketch the graph of  f 2 4yx    , indicating clearly the equations of the asymptotes and the coordinates of the axial intercepts and turning points . [3] 9 In the isosceles triangle ABC, AC = BC, 20AB cm and 30BAC . A rectangle PQRS is inscribed in ABC with points P and Q on AB, point R on BC and point S on AC as shown in the diagram. Taking PS to be x cm, show that the area of PQRS may be expressed as 220 2 3xx . [2] Hence, as x varies, find the exact values of PS and PQ such that the area of PQRS is a maximum, and find the corresponding area of PQRS. [4] y x (0,−5) O O (6, −4) S R A C Q P x B

5 10 Referred to the origin O , the points A and B have position vectors a and b respectively, where a and b are non -zero and non -parallel vectors. The p oint C lies on OA such that OC : CA = 2 : 1. The point D lies on OB produced such that OD : BD = 4 : 1. (i) Find the position vectors OC and OD , giving your answers in terms of a and b. [2] (ii) The lines BC and AD meet at the point E. Show that E has position vector 42ba . [4] (iii) Show that the area of triangle CDE can be written as k ab , where k is a constant to be found. [4] 11 A curve C has parametric equations 233 , 6 ,x t y t where t is a real parameter. (i) Sketch C. [1] (ii) Find the exact coordinates of the point P on C where the tangent is parallel to the line 42yx . [3] (iii) Show that the equation of the tangent to C at the point Q 4 16,39   is 9 18 8yx . [2] (iv) The tangent at Q cuts the x-axis at the point R. Find the area of triangle PQR. [2] (v) The tangent at Q cuts C again at the point S. Find the coordinates of S. [3] 12 The function f is defined by f 2 2: , , 1 1 xx x x x  . (i) Sketch the graph of f and show that f has an inverse. [2] (ii) Find   1f x and state its domain. [4] (iii) Write down the equation of the line in which the graph of f must be reflected in order to obtain the graph of 1 f  and hence find the exact solution of the equation 1 f ( ) fx   (x). [3] 13 The planes 1p and 2p have equations 4 3 1 1 1 3 3 7 12 r st                                 and 31xy respectively, where s and t are parameters. (i) Find the line of intersection of 1p and 2p . [3]