ACJC 2026 JC2 H2 Prelim Paper 1 QP

ANGLO-CHINESE JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 MATHEMATICS 9758/01 Paper 1 25 August 2026 QUESTION PAPER 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST Answer all questions. Write your answers on the Printed Answer Booklet . Follow the instructions on the front cover of the answer booklet. 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 must present the mathematical steps using mathematical notations and not calculator commands. You must show all necessary working clearly. The number of marks is given in brackets [ ] at the end of each question or part question. _________________________________________________________________________________ This document consists of 7 printed pages and 1 blank page. [Turn over

2 ANGLO-CHINESE JUNIOR COLLEGE 2026 H2 MATHEMATICS 9758/01 1 Use an algebraic method to solve the inequalities ( ) 2 59 3625 xx xxxx +   −++ . [4] 2 Using standard series from the List of Formulae (MF27), find the Maclaurin’s series for ln(2 sin ) x+ , where x is sufficiently small for terms 3x and above to be neglected. [3] Hence deduce the equation of the tangent to the curve ln(2 sin )yx=+ at 0x= . [1] 3 A piece of cardboard has the shape of an equilateral triangle, with fixed side length l cm. Six perpendicular cuts, each of length x cm, are made to the cardboard to remove the corners. The remaining cardboard is then folded to form an open box with an equilateral triangular base. (a) Show that the length of the equilateral triangular base is 23lx− . [1] (b) Find the value of x, in terms of l, that gives the maximum volume of the open box. (You need not show that your answer gives a maximum.) [3] 4 A curve C has equation 23 1 ( )xy x y+ = + . (a) Show that d2 d2 y x y x x y −= − . [2] (b) Hence find the exact x-coordinate of the points on C whose tangent is parallel to the x-axis. [3] (c) By finding the gradient of the tangent to C at ( )0,1 , hence find the acute angle between the tangent to C at ( )0,1 and the x-axis. [2] l x 3  l l x

3 ANGLO-CHINESE JUNIOR COLLEGE 2026 H2 MATHEMATICS 9758/01 [Turn over 5 The function f is defined by 2f : , for , 2.2 kxx x x x + −  (a) If the point with coordinates ( )7,4− lies on the curve 1f ( ),yx −= show that 3.k = [1] Use 3k = for the rest of this question. The function g is defined by ( )g : f , for , 0 and 2.x x x x x   −  (b) Sketch the graph of g( )yx= stating the axial intercept and the equations of asymptotes. [3] (c) Find 1g ( )x− , stating its domain. [4] 6 The diagram shows a sketch of the curve f ( )yx= . The shaded region under the curve between x = 0 and x = 1, as shown in the diagram, is A. This region is split into n vertical strips of equal width, h. (a) State the value of h. [1] (b) Hence show, using a sketch, that 1 f ( ) n r h rh =  is less than the area of A. [2] You are now given that 2 2 ef ( ) 1e x xx − −= + . (c) Hence find the exact value of 1 lim f ( ) n n r h rh → =  . [3] 0 1 x y

4 ANGLO-CHINESE JUNIOR COLLEGE 2026 H2 MATHEMATICS 9758/01 7 It is given that ( ) 2 f : 4 4 , 0 , where is a constant.x x x x − − −    (a) Find the greatest value of  for which the function 1f− exist. [1] The function ( )g x is defined by ( ) 2 for 2 0,g 2 for 0 13. xxx xx + −  = − −   (b) Sketch the graph of ( )gyx= , stating the coordinates of the endpoints. [2] (c) Considering the result in part (a), show that the composite function gf exists. [1] (d) Find the range of function gf. [1] (e) Without finding gf and 1g− , find the exact value of x for which ( )gf 3x =− . [3] 8 With reference to the origin O, the points A, B and C are such that vectors OA =a  , OB =b  and 22OC −= a b  . (a) Given that 2, 3==a b and the angle between a and b is 6  , find the area of triangle OBC. [3] The lines OA and BC intersect at point X. (b) Show that 2 3OX =a  . [3] The plane p has equation d=rn  and contains the point A, where n is a unit vector. (c) Find the shortest distance from point X to plane p, in the form of 1 dk , where k is a positive integer to be determined. [3] a b A B C X O

5 ANGLO-CHINESE JUNIOR COLLEGE 2026 H2 MATHEMATICS 9758/01 [Turn over 9 The energy added to a humanoid robot’s backup cell increases by a constant number of units each hour of its operation. The energy added during the 8 th hour is 26 units, and the total cumulative energy stored over the first 12 hours is 258 units. (a) Find the energy added during the 1 st hour and determine the minimum number of full hours the robot needs to operate before its cumulative energy stored exceeds 1000 units. [5] The robot requires compute tokens to run its onboard AI models. The robot’s server wallet starts off with 10 000 tokens. At the beginning of each month starting from the first month, a fixed number of K tokens is added to the wallet. At the end of each month, the total number of tokens in the wallet increase s by 2% due to network yields. Immediately after, the robot consumes exactly 2500 tokens for a monthly software update. Let nu denote the number of tokens in the wallet at the end of nth month after the monthly software update. (b) Write down the expressions for 1u and 2u in terms of K. [2] (c) Hence show that ( ) ( ) ( )1.02 10 000 1.02 1 125 000 1.02 1n n n nuK= + − − −  , where  is a constant to be determined. [3] (d) Find the minimum integer value of K to ensure the robot’s wallet never runs out of tokens in the long run. [2] 10 (a) Do not use a calculator in answering this question. One of the roots of the equation 32 10z az bz+ + − = , where a and b are real numbers, is 1 i 3 22−+ . (i) Express 32 1z az bz+ + − as the product of a linear factor and a quadratic factor with real coefficients and find the values of a and b. [4] (ii) Hence state the other roots of the equation 32 10z az bz+ + − = . [1] (iii) Consider 1 i 3 22z=− + which is a root of the equation 32 10z az bz+ + − = and using the results in part (i), show that 5710 zz+ + = . [2]

6 ANGLO-CHINESE JUNIOR COLLEGE 2026 H2 MATHEMATICS 9758/01 10 (b) Do not use a calculator in answering this question. It is given that 1 1 i 3z =+ and 2 3iz =− . (i) Sketch on an Argand diagram with origin O, showing the points A, B and C representing 1z , 2z and 12zz+ respectively. [2] (ii) Show that OACB is a square. [2] (iii) Hence b y considering 1arg( )z and 12arg( )zz+ , show that tan 2 312  =− . [3] 11 Two scientists are interested in studying the growth of a deer population after its introduction into a national park. They model the number of deer, P, at time t years after the introduction of an initial population of 50 deer. Both P and t are assumed to be continuous variables. Scientist A uses the Gompertz growth model. ( )d 1 0.2lnd P PPt =− . (a) Using the substitution lnyP= , show that ( )d 1 0.2d y yt =− . [1] (b) Solve the differential equation for y in terms of t and show that y can be expressed as 0.25(1 e ) tyK −=