2026 DHS Prelim P1 (Qns)

© DHS 2026 This document consists of 7 printed pages and 1 blank page. Name: Centre/Index Number: Class: DUNMAN HIGH SCHOOL Preliminary Examination 2026 Year 6 MATHEMATICS (Higher 2) 9758/01 Paper 1 2 September 2026 3 hours Additional Materials: Printed Answer Booklet List of Formulae (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.

2 DHS 2026 Year 6 H2 Mathematics Preliminary Examination Paper 1 1 A curve has equation ( ) 1 fy x= where the function f is a cubic function. The curve has a stationary point at 11,4− and a vertical asymptote at 2x=− . When ( )f x is divided by ( ) 2,x− the remainder is 4. Find the equation of the curve. [4] 2 [The volume of a cone of base radius r and height h is given by 213V rh π= .] Water is leaking at a constant rate of 15 cm3 per second from the bottom of a cup which is shaped like an inverted right truncated cone as shown in the figure above. The radii of the rim and base of the cup are 6 cm and 3 cm respectively. The height of the cup is 8 cm. (a) Show that the volume of water inside the cup, V, is related to the height of the water level, h, through the equation 33π ( 8) 24 π.64Vh= +− [3] (b) Find the rate at which the water level is changing when h is 2 cm. Express your answer in exact form. [3] 3 (a) An increasing sequence is such that 1 1u = and 1 2nnu up+ = + , where p is a constant, for 0n> . Show that 1p>− for all values of n. [2] (b) Another sequence is defined by 1va= , 2vb= , where a and b are constants, and 22 3, for 2.nnvv n −= +> For this sequence, 43 3vv= . (i) Express b in terms of a. [3] (ii) Find an expression in terms of a for 6v . [2] 3 cm 6 cm 8 cm h cm

3 DHS 2026 Year 6 H2 Mathematics Preliminary Examination Paper 1 [Turn over 4 (a) Find 2 ln dx xx⌠⌡ . [3] (b) Hence find the exact value of 4 22 3 12 ln 2+ d 32xx xx xx −+⌠⌡ , giving your answer in the form lnCD+ , where C and D are integers to be found. [4] 5 The function f is defined by f: 10axx xb−+ , for x∈  , xk≠ where a and b are constants. (a) Give an expression for k and explain why this value of x has to be excluded from the domain of f. [2] (b) Find 1f () x− and state its range. [3] (c) Given that 1f () f ()xx −= for all x in the domain of f, determine the possible values of a and of b. [2] The function g is defined by g: 2xx , for x∈  . (d) Find 1gf (1)− in terms of a and b. [1]

4 DHS 2026 Year 6 H2 Mathematics Preliminary Examination Paper 1 6 (a) Describe a sequence of two transformations that would transform the graph of 223(1 ) 1xy−+= onto the graph of 22 1xy+= . [2] (b) The graph of f( )yx= passes through the origin and has an asymptote at 0y= and turning points at ( , )ab and ( , ).cd The diagram shows the graph of f( )yx= . (i) On the diagram in the Printed Answer Booklet, sketch the graph of 1f( )y x= , stating the equations of any asymptotes and the coordinates of any points where 1f( )y x= crosses the axes and of any turning points. [3] (ii) On the diagram in the Printed Answer Booklet, sketch the graph of f '( )yx= , stating the equations of any asymptotes and the coordinates of any points where f '( )yx= crosses the axes where possible. [3] y x O

5 DHS 2026 Year 6 H2 Mathematics Preliminary Examination Paper 1 [Turn over 7 The diagram below shows the layout of a garden plot OYX which is a quadrant of an ellipse with a semi-minor axis 2k m and semi-major axis 4k2 m, where k > 0. The owner decides to put in a rectangular patio OMPN with corner P touching the curved wall and the edges ON and OM against the walls OX and OY respectively. Taking OX and OY as the x-axis and y-axis respectively with O as the origin, let the coordinates of P be (x, y) and A be the area of OMPN. (a) Write down the equation of the ellipse in terms of k. [1] (b) By considering the coordinates of P, find the maximum area of rectangular patio in terms of k as x varies. Show that the area of rectangular patio is a maximum. [6] (c) John claims that the exact ratio of the maximum area of the rectangular patio to the total area of the garden is constant for any positive real values of k . Explain mathematically whether John is correct. [2] [The area of an ellipse with semi-minor axis a and semi-major axis b is given by π.A ab= ] 8 The complex number z has modulus 2 and argument π6− . (a) O n the Argand diagram in the Printed Answer Booklet, mark out the points A , P, Q and R which represent z , *z , *zz+ and *zz− respectively. Show clearly the geometrical relationship between the points A, P, Q and R. [4] (b) State the geometrical shape of OPQR and explain whether A lies on line segment QR. [2] The complex number w satisfies the equation * 2izw−= . (c) On the same Argand diagram in the Printed Answer Booklet, mark out the point B which represents w. Hence, or otherwise, find the value of arg( )w . [3] 4k2 m 2k m N X P(x, y) M O A Y

6 DHS 2026 Year 6 H2 Mathematics Preliminary Examination Paper 1 x y 0.5 O 9 It is given that 2 cosyx= + . (a) Show that 22 2 2 dd22 2 dd yyyy xx + += . [3] (b) By further differentiation of the result in part (a), find the first 3 non- zero terms of the Maclaurin expansion of 2 cosyx= + . [4] (c) Using appropriate standard series from the List of Formulae (MF27), verify that the expansion found in part (b) is correct. [3] 10 The diagram shows the curve C with the equation 3 12yx x= − . The region above the x-axis enclosed by C is denoted by R. (a) Find the exact volume generated when R is rotated through 2π radians about the x-axis. [3] (b) Using the substitution 12vx= − , show that 1 1.5 2 1 3 1 d 3yxk = ⌠⌡ where k is a constant to be determined. [4] (c) Hence find the exact area bounded by the line 3yx= , C and the x-axis, giving your answer in terms of 3 as simply as possible. [3] 11 Points A, B, C and D have position vectors, relative to the origin O , given by 2OA= +−i jk    , 23OB=−+ +i jk   , 24OC = ++ij k   and 2OD p= +ik   , where p is a constant. (a) Find the cartesian equation of the plane ABC . [3] It is given that OD and the plane ABC are parallel. (b) Find the value of p. [2] The line L is perpendicular to the plane ABC, and it passes through the point D. (c) Find the exact coordinates of the point F where the line L meets the plane ABC. [3] (d) Hence find the exact position vector of the reflection of the point D in the plane ABC. [2] R

7 DHS 2026 Year 6 H2 Mathematics Preliminary Examination Paper 1 [Turn over 12 A ball is dropped from a height of H metres above a flat horizontal floor. E