MI 2026 Prelim P1 QP

©Millennia Institute 9758/01/PU3/26 2026 Preliminary Examination Pre-University 3 MATHEMATICS 9758/01 Paper 1 25 August 2026 3 hours Additional Materials: Printed Answer Booklet List of Formulae (MF27) READ THESE INSTRUCTIONS FIRST Answer all the 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 are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 6 printed pages. CANDIDATE NAME ADMIN NUMBER CLASS

2 ©Millennia Institute 9758/01/PU3/26 1 Priya intends to buy 5 jars of strawberry jam, 4 bottles of honey and 6 cans of sardines. She will have to pay $59.80 in total if she buys them at their usual retail prices. However, two supermarkets are each running their own promotions. At Supermarket A, there is a 25% discount for each jar of strawberry jam, and for every 3 bottles of honey purchased, one additional bottle is given free of charge. Priya can also redeem an $8 voucher at Supermarket A which would bring her total bill to $41.05. At Supermarket B, there is a 40% discount for each jar of strawberry jam while 3 bottles of honey are sold for the price of 2 bottles at the usual price. Each can of sardines is also 20 cents cheaper than its usual retail price. Priya notes that shopping at Supermarket B would cost her $44.70 in total. If both supermarkets price all items at their usual retail prices before promotions, find the usual retail price of each jar of strawberry jam, each bottle of honey and each can of sardines. [4] 2 It is given that  1 1 1 11 1nr r r n    . (i) Write out the first three terms and the last term of  1 1 1 n r r r  . Hence, verify the given result for 4n . [2] (ii) Find  5 1 1 N r r r  for integers 5N  . [2] (iii) Explain why the series in part (ii) is convergent and state the value of  5 1 1r r r    . [2] 3 (i) One of the roots of the equation 3 2 3i 6 8i 0z z z    is iz . Find the other roots of the equation. [4] (ii) Hence, find the roots of the equation 3 2 1 3i 6 8i 0w w w    . [2] 4 It is given that 1 2e cosy xy x x    . (a) Show that 22 2 1 1 2 2 d d d de e 2 cos 2d d d dy y y y y y x xx x x x          . [2] (b) Hence, find the Maclaurin series for y up to and including the term in 2x . [4]

3 ©Millennia Institute 9758/01/PU3/26 [Turn over 5 A sequence 1 2 3, , , ...u u u is defined by the recurrence relation as follows 1u k , 1 2 3n nu u   where k and 3k . (a) Describe how the sequence behaves when (i) 3k  , [1] (ii) 5k  . [1] (b) Find an expression for nu in terms of n and k. [4] (c) Explain why nu is an odd integer for all 2n . [1] 6 The curves with equations  1cosy x  and 243y x  , where 0 1 x  intersect at 12x . The shaded region R is bounded by the two curves and the x-axis, as shown in the diagram below. (i) Find exactly, the area of the region R. [5] (ii) Find exactly, the volume generated when R is rotated 2 radians about the y-axis. [4] (iii) Find the volume generated when R is rotated through 2 radians about the x-axis, giving your answer correct to 2 decimal places. [2] O y x R

4 ©Millennia Institute 9758/01/PU3/26 7 (a) Find   22 sin 2 d 1 cos x x x    . [2] (b) Using the substitution u x , find   1 d 1 2 x x x    . [3] (c) (i) Use integration by parts to find  ln 2 dx x x . [2] (ii) Hence, or otherwise, find    2 ln 2 dx x x . [3] 8 The line l has equation 2 1 0 3 , 1 2                    r where  is a parameter. T he point P has coordinates  2, 0, 1 . (i) Find the coordinates of the point N on l where the length of PN is as small as possible. [3] The plane 1 has equation 2 1 1 1 2 0 , 0 1 3                                r where  and  are parameters. (ii) Find the acute angle between l and 1 . [3] It is given that the plane 2 with equation 2 2x y az   does not meet the line l. (iii) Show that 52a   . [1] (iv) Find the perpendicular distance between l and 2 . [3] (v) Find a cartesian equation of the line where both planes 1 and 2 meet. [3]

5 ©Millennia Institute 9758/01/PU3/26 [Turn over 9 The function f is defined by 2f : , , 1 1axx x x x     , where a is a positive constant. (i) Sketch the graph of f ( )y x , stating the equations of any asymptotes and the coordinates of any points where the curve crosses the axes. [3] (ii) Explain why f has an inverse. [1] (iii) Define 1f in a similar form and state its range. [4] Another function g is defined by g : sin 1, where , 0x x x x      . (iv) Find the range of g and use it to explain why the composite function fg exists. [2] (v) Find the range of fg in terms of a and explain why fg( ) 2 1x a  for all 0 x   . [2] 10 A curve C has parametric equations 2 33 , 2 6 , for x t y t t t     . (i) Find ddyx in terms of t. [2] (ii) Find algebraically, the coordinates of all stationary points of C. [2] (iii) Sketch C, indicating the coordinates of any stationary points. [2] The point P on C has parameter p. (iv) Show that the equation of the normal to C at P can be written in the form 2 5 3( 1) 2 5 6px p y p p p     . [3] (v) If the normal to C at P passes through the point (18, 0), find all possible coordinates of the point P. [3]

6 ©Millennia Institute 9758/01/PU3/26 11 A medical researcher is comparing two targeted therapies for a specific type of benign tumour. Therapy A is an established treatment that has been in use for many years and proven to have minimal side effects. The volume of the tumour, V cm3, after t months of treatment under Therapy A is modelled by the differential equation:   2d 10 16dV t V Vt    for 0t . (a) Show that the given differential equation can be written in the form    2 2d d V t V a bt    , where a and b are positive constants to be determined. [2] (b) Given that the initial volume of the tumo ur at the start of the treatment is 4 3cm , solve the differential equation to show that 2 2 3 3 4e 8 2e 1 t tV   . [7] (c) Find, correct to 4 decimal places, the time taken for the tumour to shrink to 33 cm under Therapy A. Explain what happens to the volume of the tumo ur in the long run according to this model. [3] Therapy B is a new experimental therapy which is highly aggressive. When tested on a similar tumour of the same initial volume, it was found that in the long run, Therapy B could shrink the volume of the tumour to a size smaller than that of Therapy A. (d) Suggest one reason why a patient might still choose Therapy A given the context of both therapies. [1]