NJC 9758 2022 Promo QP

11* © NJC 2022 19[Turn_over NATIONAL JUNIOR COLLEGE SENIOR HIGH 1 Promotional Examinations NAME SUBJECT CLASS 1ma2 REGISTRATION NUMBER H2 MATHEMATICS 9758 3 October 2022 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26)

226 © NJC 2022 READ THESE INSTRUCTIONS FIRST This paper constitutes 50% of your overall score for SH1 H2 Mathematics. Write your name, class and registration number in the boxes above. Please write clearly and use capital letters. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use 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. 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. Up to 2 marks may be deducted for improper presentation. The number of marks is given in the brackets [ ] at the end of each question or part question. Question Number Marks Possible Marks Obtained 1 3 2 6 3 6 4(a) 4 4(b) 3 5 8 6 9 7 10 8 12 9 12 10 15 11 12 Presentation Deduction – 1 / –2 TOTAL 100 This document consists of 23 printed pages and 5 blank pages.

33* © NJC 2022 39[Turn_over 1 Ben, Caleb and Dylan went to the supermarket in the morning to buy salmon, tuna and swordfish. Ben paid $246 for 2 kg of salmon, 1 kg of tuna and 3 kg of swordfish. Caleb bought 1 kg of salmon and 1 kg of swordfish. Caleb paid $18 more than Dylan w ho bought 1 kg of tuna. After 12pm, a discount of 15%, 10% and 5% is given for salmon, tuna and swordfish respectively such that the total cost of 1 kg of salmon, 1 kg of tuna and 1 kg of swordfish after discount is $123.30. Find the selling price of 1 kg of salmon, tuna and swordfish respectively before discount. [3] 2 (a) The diagram shows the graph of  gyx with a stationary point of inflexion at  0,3 and a maximum point at  4,5 . The asymptotes of the graph are 2x and 3.y On a separate diagram, sketch the graph of g ( )yx  , labelling the equations of any asymptotes and the coordinates of any points where the curve crosses the axes. [3] O x y

446 © NJC 2022 2 (b) The diagram shows the graphs of  hyx and   1 hyq xp  , where p and q are constants. The curve  hyx cuts the axes at  1,0 and has asymptotes 0x and 2y . The curve   1 hyq xp  has asymptotes 2x and 1y . Find the values of p and q. [3] 3 (i) On the same axes, sketch the graphs of xby x  and 1y x ba , where 01 a and 1b . State the equations of any asymptotes and the coordinates of the points where the curves cross the axes. [3] (ii) Hence solve the inequality 1xb xbxa   . [3] 4 (a) Find 2 1 d .43 x xx    [4] (b) The region bounded by the curve   2 1yx and the line 1x is rotated through 2π radians about the x-axis. Find the volume of the solid obtained, giving your answer correct to 4 decimal places. [3] x y O x y O

55* © NJC 2022 59[Turn_over 5 Referred to the origin O, the points A, B and C have position vectors a, b and c respectively. The points O, A, B and C do not lie on the same plane. In triangle ABC, points Q and R are the midpoints of BC and AC respectively. (i) Find a vector equation of the line BR in terms of a, b and c. [2] (ii) Given that line AQ has equation  2,     r a a b c . Hence find, in terms of a, b and c, the position vector of the point G where the lines BR and AQ meet. [2] (iii) It is given further that the point S has position vector 2 ba and 2π 3AOB . Find, in terms of a and b , the length of projection of OS   onto OA   . [4] 6 (i) Use the substitution tanx m t to find 22 1 dx mx   , where m is a positive constant and π0 2t . [4] (ii) Find 22 dx x mx   . [2] (iii) Hence, find 1 22 tan dxx xmmx      . [3] 7 A curve C has equation 223 2 0x xy y k    where k is a non-zero constant. (i) Show that d3 d y x y x y x   . [2] (ii) Find the range of values of k such that C has no stationary points. [3] (iii) It is given that 1k  . The lines x  and x  are tangent to C, where  . Find  and  in terms of k. Hence show that x  does not meet C again. [5]

666 © NJC 2022 8 The function f is defined by 4 for 1,f ( ) cos for 1 π. xxx xx   (i) Show that f has an inverse. [3] (ii) Find 1f. [3] (iii) Evaluate 2 πf 2   and find 2f. [4] (iv) Find   2022f0 . [2] 9 A curve D has parametric equations  2cos cos 2 sin 2 xk yk     for 0 π , where k is a positive constant. (i) Find, in terms of k, the exact coordinates of the stationary points of D. [3] (ii) Sketch the graph of D, labelling, in terms of k, the points where D meets the x-axis. [2] (iii) Show that the area enclosed by the x-axis and the part of D above the x-axis, is given by   2 1 22 2sin 2 2sin sin 2 dk       , where 1 and 2 should be stated. [3] (iv) Hence find, in terms of k, the exact area enclosed by the x-axis and the part of D above the x-axis. [4]

77* © NJC 2022 79[Turn_over

886 © NJC 2022 10 A player controls an avatar in an exploration game to find treasures. In one of the stages, the avatar scales a cliff. The sloping surfaces of the cliff can be modelled by plane 1p with equation 52xz  and plane 2p with equation 4 30x y nz   , where n is a constant. The top surface of the cliff can be modelled by plane 3p with equation 12 450xz  which contains the point  30, 30, 40T , where a treasure is located. The figure below shows a side view of the cliff. Locations of points  , , x y z are defined relative to a point  0, 0, 0 at the foot of the cliff and it is assumed that the avatar travels from a point to another using the shortest path. The avatar expends 1 unit of stamina for every 2 units travelled. (i) The ground in the game can be modelled by the plane with equation 0z . Given that 2p is steeper than 1p relative to the ground, find the range of values of 2n . [4] It is given further that  30, 30, 40T also lies in 2p . (ii) Show that 3n . [1] (iii) The boundary where 1p and 2p meet can be modelled by the line l. Verify that a vector equation of boundary l is 58 17 ,4 12         r . [2] (iv) The player resumes the game with the avatar on 1p . The avatar moves towards B, a point on boundary l such that BT is minimum. Find the exact position vector of B. Hence, find the total stamina expended by the avatar to travel from B to T. [5] (v) From the point T, the avatar travels along a path parallel to 10 3 9 a a    , where a is a constant. Another treasure is located at  90, 0, 45R on 3p . Determine whether the avatar will eventually reach this treasure. [3] Ground

99* © NJC 2022 99[Turn_over 11 Engineers need to lay pipes to connect two factories A and B that are separated by a canal of uniform width 480 m as shown in the diagram. They plan to lay the pipes under the canal in a line from A