2025 JPJC H2 Math Prelim P1 QP

Name:____________________________________ Class:_____________ JURONG PIONEER JUNIOR COLLEGE JC2 Preliminary Examination 2025 MATHEMATICS 9758/01 Higher 2 3 Sept 2025 Paper 1 3 hours Additional materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST This document consists of 7 printed pages and 1 blank page. [Turn over 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 co rrect 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 by [ ] at the end of each question or part question.

2 1 The curve C has equation 2 1 4y xx= − . (i) Find d d y x . Hence, find the x-coordinate, 1xx= , of the turning point on C and determine its nature. [3] (ii) Using calculus, find the exact area of the region between C, the x-axis and the lines with equations 1x = and 1xx= . [3] 2 (i) Find, in terms of a, the roots of the equation ( ) 2 1 xa xa =− − . [3] (ii) On the same axes, sketch the curves with equations ( ) 2 1y xa = − and y x a=− , where 1a  . Hence solve the inequality ( ) 2 1 xa xa − − . [3] 3 A ladder of length 3.12 m is sliding down a vertical wall such that the foot of the ladder is moving along the floor at a constant rate of 0.2 m/s (see diagram). Find the rate at which the top of the ladder is sliding down the wall when it is 1.2 m above the floor. [4] Top of ladder Foot of ladder wall floor floor ladder

3 4 Functions f and g are defined by   2 2 f : ln( 1) 2, , 3g : 4 3 , . 2 x x x a x x x x − +  + −  (i) It is given that the function 1f− exists. State the smallest value of a. [1] (ii) Find an expression for 1g ( )x− , stating its domain. [3] Using the value of a found in part (i), (iii) determine whether the composite function 11gf−− exists. [1] 5 (a) The curves 1C and 2C have equations 22 194 xy −= and 2 2 2y x k+= respectively, where k is a positive constant. (i) Sketch 1C and 2C on the same diagram, stating the coordinates of any points of intersection with the axes and the equations of any asymptotes. [3] (ii) State the range of values of k for 1C and 2C to intersect. [1] (iii) State the equations of the common lines of symmetry for both 1C and 2C . [1] (b) The function f, with domain the set of all real values, is given by ( ) 2 6 for 0 3,f 3 9 for 3 5, xxx xx − +  = −   and that ( ) ( )f f 5xx=+ . (i) Find ( )f 46 . [1] (ii) Sketch the graph of ( )fyx= for 55 x−   . [2] (iii) Hence, state the roots of ( )f0 x−= for 55 x−   . [1] [Turn over

4 6 (i) Find 22ed xxx −  . [4] (ii) The curve with equation e xyx −= and the line with equation 1 eyx= meet at the origin O and the point P with x-coordinate 1. The region R is bounded by the curve and the line (see diagram). Find the exact volume of the solid formed when R is rotated through 360 about the x-axis. [4] 7 (a) (i) Find 2 d 25 x x x− . [2] (ii) Hence, given that 4 2 d3 25 x x x = − , where 0  , find  algebraically. [3] (b) Using the substitution 4 tanx = , evaluate 24 20 d16 x xx+ exactly. [5] 8 (a) The first three terms of a sequence are given by 1 9u = , 2 27u = and 3 55u = . Given that nu is a quadratic polynomial in n , find nu in terms of n. [3] (b) It is given ( ) 22 3 1 1 4 n r nnr = += and 325rur=+ . (i) Find 1 n r r u =  . [2] (ii) Hence, or otherwise, find ( )( ) 3 2 2 2 5 n r r = ++ . [3] (iii) Explain why the series 1 r r u  =  does not converge. [1] R x y P O

5 9 It is given that f(x) = 2 1 49 x+ . (i) Find f( ) dxx . [2] (ii) Find the binomial expansion for f( x), up to and including the term in x4. Give the coefficients as exact fractions in their simplest form. [2] (iii) Hence, find the Maclaurin series for 1 3tan 2 x− . Give the coefficients as exact fractions in their simplest form. [3] (iv) Use your series from part (iii) to estimate 0.5 1 0 3 tan d ,2 x x−  correct to 3 decimal places. [1] (v) Use your calculator to find 0.5 1 0 3 tan d ,2 x x−  correct to 3 decimal places. [1] (vi) Comparing your answers to parts (iv) and (v), comment on the accuracy of your estimate in (iv) and how it can be improved. [2] 10 (a) Show that 2eln 3y x =  can be written in the form ( )lny a b cx=+ , where a, b and c are integers to be found. Hence, state a sequence of transformations which transform the graph of lnyx= onto the graph of 2eln 3y x =  . [4] (b) The curve f( )yx= passes through the point P with coordinates ( , )ab , where 0b . The tangent to the curve at P has gradient 5. When f( )yx= is transformed onto the curve g( )yx= , P corresponds to the point R on g( )yx= . For each of the following curves, state the coordinates of R and find the gradient of the curve at R. (i) g( ) 2f( 1)xx=− [3] (ii) 1g( ) f ( )x x= [2] [Turn over

6 11 A company posted a video on a social media platform to advertise a new product. The video is uploaded at the start of 1 st April and the number of daily views is recorded at the end of each day. Let nu , where 1n , denotes the number of daily views recorded each day. It is given that 1 (1 )nnu k u+ =+ , where k is a positive constant. (i) Explain why the sequence {}nu is a geometric progression. [1] (ii) Given that the number of daily views recorded at the end of 1 st April is 311 and the total number of daily views recorded from 1st April to 3rd April is 4043. Find the value of k. [3] (iii) Explain why there is no limit to the total number of daily views in the long run. [1] The company also looks at the number of comments being posted on the social media platform. The number of daily comments is recorded at the end of each day. It is given that rv , where 1r , denotes the number of daily comments recorded and it is defined by the following relation: 1 for 1 4, 80 for 5. r r r urv vr− = + (iv) Show that the total number of