VJC 9758 2025 Prelim P1

2025/VJC/Math Dept [Turn over VICTORIA JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION 2025 H2 MATHEMATICS 9758/01 PAPER 1 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. The total number of marks for this paper is 100. This document consists of 6 printed pages and 2 blank pages.

2 2025/VJC/Math Dept 1 Using an algebraic method, solve the inequality 2 43 14 3 1 x xx − +− . [3] Hence, find the set of values of x that satisfy 2 4ln 3 14(ln ) 3ln 1 x xx − +− . [2] 2 The function f is defined by 1f : , xx x − x , 0,x 1x . (a) Show that ( ) ( ) 21ff xx −= . [3] (b) Find ( ) 3f x in simplified form. [1] (c) Find ( ) 2030f5 . [2] Functions g and h are defined by 1g: xx x − , x , 1x , h : sinx ax− , x , where a is a positive constant. (d) Find the value of a given that the range of hg is ( 1,0− . [2] 3 Referred to the origin O, points A and B have position vectors a and b respectively. Point C lies on OA, such that : 2:1OC CA = . Point D lies on OB, such that ::OD DB = . It is given that the area of triangle ABD is half the area of triangle ABC. (a) Show the area of triangle ABD is given by ( )2   + ab . Hence find the ratio : . [4] (b) The point E has position vector 15 48 +ab . Show that A, E and D are collinear. [3] It is further given that the angle AOB is π 4 and O lies on the perpendicular bisector of the line segment AB. (c) Find the length of projection of a on b , giving your answer in terms of b . Hence find the position vector of the point F, the foot of perpendicular from A to OB. [3]

3 2025/VJC/Math Dept [Turn over 4 (a) A sequence is such that 1up= , where p is a constant and 1 5 81 n n n uu u + = + , for 1n… . (i) Describe how the sequence behaves when 1p= . [2] (ii) Find the value of p for which 6 3125 6253u = . [2] (b) Another sequence 1v , 2v , 3v , … is such that for all 1n… , 21 2n n nv v v k++− + = , where k is a constant. Let 1n n nw v v +=− for 1n… . Explain why the sequence  nw is an arithmetic progression. [2] 5 The function f is given by 22 22 2 2 ( 2) , 0 < 4,f ( ) 2 2 ( 6) , 4 < 8. xxx xx  + − − =  − − −  It is given that ( ) ( )f f 8xx=+ for all real values of x. (a) On the diagram in the Printed Answer Book, sketch the graph of ( )fyx= for 6 7,x−   indicating clearly the coordinates of the end points and the points where the graph cuts the axes. [3] (b) Without integrating, write down the exact area of the region bounded by f( )yx= , the line 4x= , the x-axis and the y-axis. [1] The curve C has equation ( ) ( ) 2 2 2 3 21y xa − − + = , where a is a positive real constant. (c) State the equations of the asymptotes of C in terms of a. [1] (d) Determine the range of values of a if there is at most one intersection between C and the graph of ( )fyx= . [2] 6 It is given that ( ) ( )2 2 1 1 1 4 1 2 1 21 n r nnrr= = − +− + . (a) Show that ( ) 2 2 1 1 n r rr= − is less than 1 4 . [2] (b) Give a reason why the series ( ) 2 2 1 1r rr  = − converges, and write down its value. [2] (c) Find the smallest value of n for which ( ) 2 2 1 1r n rr= − differs from ( ) 2 2 1 1r rr  = − by less than 0.0007. [2] (d) Find ( )( )( )1 1 12r N m r m r m r m=+ − − + − + , where m and N are integers with 0Nm . (There is no need to express your answer as a single algebraic fraction.) [2]

4 2025/VJC/Math Dept 7 Do not use a calculator in answering this question. (a) Find the complex number z which satisfies the equation 4 5i15 * z z =− . [3] (b) The complex number w is such that ( ) 3 iiw− =− . (i) Given that one possible value of w is 2i , find the two other possible values of w. Give your answers in cartesian form iab+ . [4] The points 1W , 2W and 3W on the Argand diagram represent the three roots of the equation ( ) 3 iiw− =− , and the point A represents the complex number ik , where k is a positive real number. (ii) Show that the points 1W , 2W and 3W lie on a circle with centre A for some value of k, stating the value of k. [2] 8 (a) A curve C with equation ( )fyx= undergoes in succession, the following transformations. A: A reflection in the x-axis. B: A stretch parallel to the x-axis with scale factor 1 2 , with the y-axis invariant. The resulting curve has equation 2 by ax x=+ , where a and b are real constants. Given that 11,3 − is a turning point of ( ) 1 fy x= , find the values of a and b and state the equation of C. [5] (b) The diagram below shows the curve of ( )gyx= . The curve has a minimum point at ( )2, 3−− and crosses the x-axis at ( )1,0− and ( )4,0− . The line 2x= is the vertical asymptote and the line 3y= is the horizontal asymptote. (i) Sketch the graph of ( )g'yx= , labelling the coordinates of all relevant point(s) and state the equations of any asymptotes. [2] (ii) Find the area of the region bounded by the graph of ( )g'yx= , the lines 4x=− , 2x=− and the x-axis. [2]

5 2025/VJC/Math Dept [Turn over 9 In the diagram below, t he region R is bounded by the curve C with equation 26 ( 2)xy= − − , the lines 8y= , 2yx=− and the y-axis. The region S is bounded by C and the line 2yx=− . (a) Find the exact area of region R. [5] (b) Find the volume of the solid of revolution formed when region S is rotated through 360 about the x-axis, leaving your answer to 2 decimal places. [3] 10 (a) (i) Express 14 x+ in the form (2 2)A x B−+ , where A and B are constants to be determined. [1] (ii) Hence, find 2 14 d25 x xxx + −+  . [4] (b) Find the exact value of π 3 0 sin 2 dx x x . [3] 11 It is given that d22 ln d y xy xx + − = . (a) Use the substitution ln uy x =  to show that the differential equation can be reduced to ( )d f d u x u= , where the function ( )f u is to be found. [3] (b) Given that y has a minimum value at 3x= , solve the differential equation d22 ln d y xy xx + − = , to find the particular solution for y in terms of x. [5] (c) Sketch the graph of this particular solution. [2]

6 2025/VJC/Math Dept 12 A chemical processing plant uses two types of automated dosing pumps, Pump A and Pump B, to regulate the flow of a catalyst into a reaction chamber. Each pump is removed from the plant and tested over a 10-hour trial period, and the volume of liquid dosed per hour is recorded. The following data were collected. • Pump A: The volume of liquid dosed was 4.5 litres in the first hour, and for each subsequent hour, the volume of liquid dosed decreased by a constant percentage of %r . • Pump B: The volume of liquid dosed was 4.7 litres in the first hour, and for each subsequent hour, the volume of liquid dosed decreased by 0.1 litres. (a) Show that the total volume of liquid dosed by Pump A at the end of the trial period is 450 11 100 k r r  −−  litres, where k is a constant to be