TJC 2026 Prelim P1 QP

TEMASEK JUNIOR COLLEGE 2026 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME CENTRE NUMBER S INDEX NUMBER MATHEMATICS 9758/01 Paper 1 2 September 2026 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (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 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 6 printed pages and 2 blank pages. [Turn over

2 @TJC 2026 9758/01 1 Mdm Wang runs a stall in a food court selling three types of noodles: wonton noodles, fishball noodles and beef noodles. For each bowl of wonton noodles, fishball noodles and beef noodles, she makes a profit of exactly 60 cents, 90 cents and $1.20 respectively. On a particular day, Mdm Wang made a total profit of $222.00, and she earned $72.00 more in profit from the sale of beef noodles than from wonton noodles. The total number of bowls of noodles sold that day was 4 times the number of bowls of fishball noodles sold that day. (a) Find the number of bowls of wonton noodles, fishball noodles and beef noodles that Mdm Wang sold that day. [3] Due to a system error, all customers who made online payments for wonton and beef noodles on that day were overcharged. Customers were charged $5.50 instead of the correct price of $4.50 for a bowl of wonton noodles, and $7.00 instead of $6.50 for a bowl of beef noodles. It was found that 1 in 4 bowls of wonton noodles and 1 in 5 bowls of beef noodles were paid for online. (b) Find the total amount Mdm Wang had to refund to all affected customers. [1] 2 A curve C has equation lnxy y= , where 0y . (a) Find d d y x in terms of x and y, and hence show that C has no stationary points. [3] (b) Find the equation of the tangent to C at the point where 2ey= . Give your answer in the form y mx c=+ , where m and c are exact values. [3] 3 Solve the inequality 4 xx −−  , where  is a real constant. Give your answer in terms of . [4] 4 Do not use a calculator in answering this question. The complex number z is such that 10z = and ( ) 1arg tan 3z −=− . (a) Find z in the form a + bi where a and b are integers. [2] (b) In an Argand diagram, O is the origin and the point A represents z. The points B and C are such that OABC is a rectangle described in an anti -clockwise sense and OC = 2OA. Find the complex numbers represented by B and C. [4]

3 @TJC 2026 9758/01 5 It is given that f ( )yx= is such that ( ) 2 2 2 dd 2tan secddyy x p xxx =+ , where p is a constant. The Maclaurin series for f ( )x is given by 2311 ... 2y x x qx= + + + + , where q is a constant. (a) State the values of f (0) and f (0) . [1] (b) Find the values of p and q. [5] (c) Using the given series for f ( )x and standard series from the List of Formulae (MF27), find the first three non-zero terms of the series expansion of f ( )x . [2] (d) The given Maclaurin series for f ( )x is used to estimate 0.2 0 f ( ) dxx and 0.4 0.2 f ( ) dxx . Without any further computations, determine which estimate has a smaller percentage error. Give an explanation for your choice. [1] 6 The function f is defined by f : x 2 1 x x+  , , 1 0.xx −    (a) Find 1f ( )x− and state its domain. [3] (b) Sketch, on the same diagram, the graphs of f and f −1, showing clearly the relationship between the two graphs and giving the equations of the asymptotes. [3] The function g is defined by g : x 1e x−  , , 0 ln 3.xx    (c) Explain why f −1g exists. [2] 7 (a) The sum of the first n terms of a series is ( ) 21 134 nn− , where n is a positive integer. (i) Find an expression for nu , the nth term of the series. [2] (ii) Show that this series is an arithmetic series. [2] (b) A convergent geometric series has first term 3 and fourth term 324 ,x− where , 0xx  . (i) Find the range of values that x can take. [3] (ii) Find, in terms of x, the sum of the geometric series after but not including the 10th term. Give your answer in the form ,()m p x x q− where m, p and q are positive integers. [3] [Turn over

4 @TJC 2026 9758/01 8 A plane π1 is parallel to the two vectors 2 1 0    and 0 1 2    , and passes through the point A with coordinates (1, 0, 1). (a) Show that the cartesian equation of 1 is x − 2y + z = 2. [3] (b) The line l contains the point B with coordinates (2, 1, 0) and is parallel to the vector 1 2 k    . Given that the acute angle between l and π1 is  where 1sin 3 = , find the possible values of k. [4] (c) Given now that k = 1, find the coordinates of the point of intersection between l and π1.[3] (d) A plane π2 contains the point C with coordinates (2, 1, −3) and is parallel to π1. Find the exact distance between π1 and π2. [3] 9 The curves 1C and 2C have equations 2 axy x= + and ( ) 2 4 2 8 2 x a x ay x+ + + += + respectively, where a is a constant. (a) Find the set of values of a such that 1C and 2C do not intersect for all real values of x. [3] It is now given that 0 4.a (b) Find, in terms of a, the coordinates of the stationary points of 2C . [4] (c) Sketch 1C and 2C on a single diagram and give the equations of the asymptotes, in terms of a where appropriate. [5] (d) Hence solve the inequality ( ) 2 4 2 8 22 x a x aax xx + + + +++ . [1]

5 @TJC 2026 9758/01 10 A drone is programmed to fly from its launch pad at the point P(1, 0) along a path described by the equation 1, 1y x x= −  , as shown in the figure above. The pilot of the drone is located at the point Q(10, 0). (a) (i) Show that the distance, L, of the drone from the pilot as it travels along its flight path is given by 2 19 99L x x= − + . [1] (ii) Use differentiation to find the exact minimum value of L, proving that it is a minimum. [4] (b) The distance travelled by the drone along its flight path, denoted by F, is given by the integral ( ) 2 1 d1d d x yF x x x=+  , when the drone is at the point (x, y). Given that the drone is flying at a constant speed of 0.8 units per second along its flight path, find the rate at which the shadow of the drone is moving horizontally when it is directly above its pilot. You may assume that the drone casts a shadow directly below it as it flies. [5] Q P x y [Turn over

6 @TJC 2026 9758/01 11 (a) A curve f ( )yx= passes through the point 0, ,3  and has gradient given by d 1 cos .dy yx =+ Using double angle formula, find an exact expression for y in terms of x. [4] (b) An island is inhibited by two species of animals, A and B, and the populations of A and B are x and y respectively at time t. The rate of change of x and y are proportional to x + y and x − y respectively. The populations of A and B are initially the same at N. At the instant when 2,xy= the