ACJC 2025 JC2 H2 Prelim Paper 1 QP

ANGLO-CHINESE JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 MATHEMATICS 9758/01 Paper 1 26 August 2025 QUESTION PAPER 3 hr 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 are required to 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 2025 H2 MATHEMATICS 9758/01 1 (a) Without using a calculator, solve the inequality 2 22 x xx +− . [4] (b) Hence, solve the inequality 2 22 x xx +− . [2] 2 The graph of 2ax bx cy xd ++= + has a vertical asymptote with equation 1x=− and an oblique asymptote with equation 22yx=+ . It is given that 2c . (a) Write down the value of d and show that 2a= and 4b= . [3] (b) Hence find the range of values of c if the graph has no stationary points. [2] 3 It is given that xyyx= , where 0x , 0y . (a) Show that ( ) 2 1 lnd d 1 ln yxy xy += − . [3] (b) Hence find the coordinates of the point on the curve xyyx= whose tangent is parallel to the y-axis. [2] 4 The region R is bounded by the curve with equation lny x x= , the lines ex= , 2ex= and the x-axis. Find the exact volume of the solid formed when R is rotated about the x-axis by 2π radians. Give your answer in the form ( ) 3 3πe e27 ab + , where a and b are integers to be found. [6] 5 Relative to the origin O, the points A and B have non-zero and non-parallel position vectors a and b respectively. The plane p has equation 0=rn . (a) Given that 0 =  a n b n , show that AB is perpendicular to n. Hence, describe the geometrical relationship between AB and the plane p. [2] (b) Write down the equation of a line parallel to AB that is contained in the plane p. [1] (c) The point F is the foot of perpendicular from a point C with position vector c to the plane p. Find the position vector of point F, giving your answer in terms of c and n. [3]

3 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 MATHEMATICS 9758/01 [Turn over 6 Figure 1 shows an open cylindrical tank. Figure 2 shows the cross-sectional view of the tank. The external radius of the tank is r cm, the external height is h cm and the tank is made with a material of thickness a cm throughout. The internal volume of the tank is fixed at 1000 cm3. (a) Show that the volume of the material needed to make the cylindrical tank is given by ( ) 2 2 2π 1 π rV k r a ra  =  − +  − , where k is a constant to be determined. [3] (b) Find, in terms of a, the value of r that minimises V. (You need not show that your answer gives a minimum.) [3] 7 A freshly brewed cup of tea, initially at a temperature of 85°C, is left out to cool in a room with a room temperature of 25°C. After 20 minutes, the temperature of the tea is 55°C. It is known that the rate of change of the temperature of the tea is proportional to the difference between the temperature of the tea and the room temperature. (a) Given that θ is the temperature of the tea t minutes after the tea is left out to cool, show that 25 60e kt −=+ , where k is a constant to be determined. [5] (b) Sketch the graph of θ against t. [2] a c a a Figure 2 r x h a a a Figure 1

4 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 MATHEMATICS 9758/01 8 The curve with equation ( )fyx= for xb , where ( )f x is a quadratic function, intersects the x-axis at the points ( ),0a and ( ),0b , and the y-axis at the point ( )0,c . The graph of ( )fyx= is shown in Figure 1. The scales on the x- and y-axes are the same. (a) (i) Sketch the graph of ( )fyx= , labelling the points where the graph intersects or touches the axes. [2] (ii) Sketch the graph of ( ) 1 fy x= , labelling the points where the graph intersects or touches the axes, as well as the equations of any asymptotes. [2] (iii) Describe fully a sequence of transformations which transforms the graph of ( )f 2 1yx=+ onto the graph of ( )fyx= . [2] (b) The function ( )gyx= is such that ( ) ( )gf xx= for xk , and ( ) 1gyx −= exists. (i) Find the largest possible value of k in terms of a and b. [1] (ii) Using the value of k found in (b)(i), sketch the graph of ( ) 1gyx −= on Figure 1 in the Answer Booklet, labelling the intersections with the axes. Write down the range of 1g− in terms of a and b. [2] (iii) Explain why the solution to ( ) 1g xx− = satisfies the equation ( ) ( ) 1gg xx− = . [1] x O Figure 1 y

5 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 MATHEMATICS 9758/01 [Turn over 9 The planes 1π and 2π have equations ( )3 2 0x c y z+ + − = and ( ) ( ) ( )3 2 2 3 st+ − + − + −r = i j k i j+ k i k respectively, where c is a constant, and s and t are parameters. The point A ( )1,3, 2− lies in both planes. (a) Show that 1c=− . [1] (b) Show that the vector equation of the line of intersection of 1π and 2π , line l, is given by ( )3 2 4 −−r = i + j k + i j+ k , where  is a parameter. [3] (c) Find the position vectors of the points on the line l which are a distance of 32 from the point B (2, 3,7)− . [4] (d) Find the equation of the plane 3π which is parallel to 2π and contains the point B. Hence show that the distance between the planes 2π and 3π is 20 33 . [3] 10 Do not use a calculator in answering this question. (a) One of the roots of the equation 4 3 22 10 0 pq   − + + + = , where p and q are real, is 2 3i+ . Find the values of p and q and the other roots of the equation. [6] (b) The complex numbers u and v are such that 2 i 2u=− + , and 3v = and argv = , where π0 4 . The points A, B and C represents u, v and uv+ respectively on an Argand diagram. (i) Find the modulus and argument of u. [2] (ii) Sketch the points A, B and C on an Argand diagram. [2] (iii) By finding the angle OAC in terms of θ or otherwise, show that ( ) 2 cosu v a b K + = + + , where a, b and K are constants to be determined. [3]

6 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 MATHEMATICS 9758/01 11 The curve C is defined by the parametric equations 2x at= , ay t= , 0t  , where a is a positive constant. (a) Show that the equation of the tangent to the curve at the point 2, aP ap p   is 3223p y x ap+= . [2] (b) Find the equations of the tangents to C at the points where xa= . Find also the acute angle between these tangents. [3] (c) (i) Show that ( ) ( ) 3 2 32 322qq qp p q pp− + = + − . [1] (ii) The tangent to the curve at P cuts the curve again at 2, aQ aq q   . Find q in terms of p. [3] The following diagram shows the graph of C for 1a= , and the Cartesian equation of the curve C is 2 1xy = . The tangent to C at the point 2 1,p p   cuts the curve again at the point 2 14, 2p p  − . The region S is bounded by the tangent to C at the point 2 1,p p  