DHS Prelims 2025 H2 Math P1 Questions

© DHS 2025 This document consists of 7 printed pages and 1 blank page. Name: Index Number: Class: DUNMAN HIGH SCHOOL Preliminary Examination Year 6 MATHEMATICS (Higher 2) 9758/01 Paper 1 16 September 2025 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.

2 DHS 2025 Year 6 H2 Mathematics Preliminary Examination Paper 1 1 It is given that 22d2 d yxy x yx = − where 0, 0.xy>< Using the substitution ,yu x= show that the differential equation can be transformed to 2 2d1 .13 d uu uxx =− Hence find the general solution of y in terms of x. [6] 2 A series is given by 1 2(4 3 ) n r r x = −∑ where x is constant. (a) Explain why this is a geometric series . Determine the range of values of x for the sum to infinity of this series to exist. [3] (b) Using 1,x= and given that find ( )( ) 1 0 2(4 3 ) 1 2 5 , n r r xr r − =  − ++ +∑ leaving your answer in the form of 2( ),n an bn c++ where a, b and c are constants to be determined. [4] 3 (a) Find the first three non-zero terms in the Maclaurin series for e sin( ).x x π+ [3] (b) It is given that the three terms found in part (a) are equal to the first three terms in the series expansion of ( )1 c ax bx+ for small x, where a, b and c are constants. Find the exact values of a, b and c. Use these values to find the coefficient of x 4 in the expansion of ( )1, c ax bx+ giving your answer as a simplified rational number. [5] 4 A sequence of real numbers x 1 , x2 , x3 , … satisfies the recurrence relation 1 52 23 n n n xx x + += + for all 1.n≥ (a) Given that the sequence converges to l, find the possible exact values of l. [3] (b) Describe how the sequence behaves when 1 3.x = [1] (c) Given that 5 3503 ,2158x = find the value of 1.x [3] 2 1 ( 1)(2 1),6 n r nr nn = = ++∑

3 DHS 2025 Year 6 H2 Mathematics Preliminary Examination Paper 1 [Turn over 5 The graphs of f '( )yx= and f( )yx= are shown below. (a) State the nature of all turning point(s) of the graph of f ( ).yx= [2] (b) State the range of values of x where f is decreasing. [2] (c) Sketch the graph of f( )yx= indicating clearly the equations of the asymptotes, coordinates of the turning point(s) and the intersections with the axes. [3] (d) On the copy of the graph of f( )yx= in the Printed Answer Booklet, sketch and label a line 3,y kx k= + where k is a constant. Hence state the range of values of k for which there is no real solution to the equation f( ) 3 .x kx k= + [2] 6 A curve C is defined parametrically by 2tan , sec sin ,xa t ya t t= = ππ ,44 t− << where a is a positive constant. (a) Show that 2 tan sin . 1 tan t t t = + [1] (b) By using part (a) or otherwise , find the cartesian equation of C in the form ( )f,yx= simplifying your answer. [2] (c) Show that ( ) ( )f f. xx−= − Hence sketch C. [2] (d) F ind the exact area of the region bounded by ,C the x-axis and the lines xA=− and ,xA= where 0, Aa<< leaving your answer in terms of A and a. [3] x (1.5, 0) (−2, 0) y y =1 O y = 3 x y O (−3, 0) (−2, 2) y = x + 3 y = − x − 3 (1, 0) (2.5, 0) (1.5, 1)

4 DHS 2025 Year 6 H2 Mathematics Preliminary Examination Paper 1 7 The function f is defined by .1f : fo e, 22 r 1xxx xx ∈+ + ≠− A function g, defined for , 1,x x∈≥ is such that y→∞ as x increases. It is also given that g(1) 0.5.=− (a) Explain why the composite function fg exists and find the corresponding range of fg. [2] (b) Given that 1fg( ) ,2 ln 1e xx x= + + find an expression for g( )x in terms of x. [2] (c) The domain of f is now further restricted to .x k> State the least value of integer k for which the function 1f− exists. [1] For the rest of the question, use the value of k found in part (c). (d) Without finding 1f,− (i) sketch, on the same diagram, the graphs of f, f −1 and 1ff− showing clearly the relationships between the graphs, [2] (ii) find the gradient of the tangent to the graph of 1f ()yx −= at 1e. 4x= + [3] 8 A complex number z varies with t such that ( )2cos i 3sin ,zt t= + where 0 2.t π≤< (a) By taking Re( )xz= and Im( ),yz= sketch on an Argand diagram the curve that shows the positions of the points representing the complex number z . Find the cartesian equation of this curve. [3] Two complex numbers 1z and 2z for two distinct values of t are such that 12zz = and 10 arg( ) . 2z π<< (b) By referring to the Argand diagram in part (a), find the possible values of ( )12arg .zz+ [2] It is given further that 1z and 2z are roots to the quadratic equation 2 0.zz αβ+ += (c) Explain whether it is necessary for α to be real. [2] (d) Given that α is not real and 12 26 2zz = = , find the values for α and β. [4] 9 The point A has coordinates (1, 2, 4)− and the plane 1π has equation 2 2 5.xy z−+ =

5 DHS 2025 Year 6 H2 Mathematics Preliminary Examination Paper 1 [Turn over (a) Find the exact shortest distance between A and 1π. [2] The plane 2π has equation 3 3,x y az+−= where a is a constant. (b) Find the vector equation of line l , which is the line of intersection of 1π and 2π, in terms of a. [4] The plane 3π has equation 2 4 3,bx y z−+= where b is a constant. (c) Show that ( 6)( 4) 0ab− −= if l is parallel to 3π. [2] (d) State the conditions that a and b must follow for the three planes to form a triangular prism, where all the planes are non-parallel and they do not have any point in common. Justify your answer. [4]

6 DHS 2025 Year 6 H2 Mathematics Preliminary Examination Paper 1 10 The diagram above shows a sketch of the graph of f( )yx= for 0.x≥ There are n rectangles each of width h drawn under the curve for 1 3.x≤≤ Each rectangle when rotated through 2π radians about the x-axis, will result in a c ylindrical disc. The total volume of the n cylindrical discs 1,V can be used to estimate the volume V , which is the actual volume generated when the region bounded by the curve, the lines 1 and 3xx= = , and the x-axis is rotated through 2π radians about the x-axis. (a) Show that 1V , the total volume of the n cylindrical discs, is given by [ ] 1 2 1 0 π f (1 ) . n r V h rh − = = +∑ State the value of h in terms of n. [3] (b) From the above diagram, it can be observed that 1 .VV< Write down 2,V a similar expression as 1V where 2 .VV> [1] It is now given that 2 f( ) 1e x xx = + for 0.x≥ (c) Find the value of 1lim . n V →∞ [2] (d) Find the area of the region bounded by f( )yx= and another curve with equation 22( 1) ( 3) 9,xy−+− = for 3.y≤ [4] x y O 1 3 …

7 DHS 2025 Year 6 H2 Mathematics Preliminary Examination Paper 1 [Turn over h x 15 15 11 [The surface area and volume of a sphere are given by 24πr and 34 π3 r respectively.] An aquarist who has a fixed budget of $k wants