2026 DHS Prelim P2 (Qns)

© DHS 2026 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) 9758/02 Paper 2 18 September 2026 3 hours Additional Materials : Printed Answer Booklet List of Formulae and Results (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 2 Section A: Pure Mathematics [40 marks] 1 (a) Without using a calculator, solve the inequality 2 2 2 35 1.34xxxx −+ >+− [4] (b) Hence solve the inequality 2 2 23 5 1.13 4 xxxx−+ >+− [3] 2 With reference to the origin O, a variable point R has position vector r and the fixed points A and B have position vectors a and b respectively, where a and b are non-zero and non-parallel vectors. (a) Give a geometrical interpretation of .×ab [1] (b) Given that ( )3, ×−=r ba 0 describe geometrically the set of all possible positions of the point R. [2] (c) Given that ,⋅=⋅rb ra describe geometrically the set of all possible positions of the point R. [2] The point C has position vector 3 6 2 − − . (d ) Find the sine of the acute angle between OC   and the x-axis, leaving your answer in the form of 10,k where k is an exact value to be determined. [3] 3 A curve C has equation ( ) 2 6,x y axy++= where a is a positive constant. A line L has equation ,y xk= −+ intersects curve C for all 2k ≥ where .k∈  (a) Show that the x-coordinate of the point(s) of intersection would satisfy the equation 22 (6 ) 0.ax akx k− +− = [1] (b) Find the value of a. [4]

3 DHS 2026 Year 6 H2 Mathematics Preliminary Examination Paper 2 [Turn over 4 A curve C is defined by the parametric equations sin 3 cos , 3 cos , 0 π.x t ty t t= + = ≤≤ (a) Express x in the form of sin( ),Rt α+ where R is an exact positive constant and α is an acute angle to be found. [2] (b) Deduce the exact maximum value of x on C and explain why the minimum value of x on C is not .R− [2] (c) Sketch the graph of C , showing the exact coordinates of the endpoints and the points of intersection with the x and y axes. [3] (d) Find the cartesian equation of C in the form of f ( ).xy= State clearly the restriction on the values of y. [3] 5 At a high-tech urban farm, a recycling tank initially contains 200 litres of pure water. To support crop growth, a nutrient solution is pumped into the tank at a constant rate of 10 litres per minute. The concentration of nitrogen in the incoming solution is 0.075 grams per litre. The well -mixed solution is pumped out of the tank at the same rate of 10 litres per minute. Let x be the mass (in grams) of nitrogen in the tank at time t minutes. (a) Show that the rate of change of the mass of nitrogen is given by the differential equation d 31 .d 4 20x xt = − [2] (b) It is given that the tank initially contains pure water. Solve the differential equation in part (a) to find an expression for x in terms of t. [5] (c) Sketch the graph of x against t. Explain in this context what happens to the nitrogen mass in the long run. [3]

4 DHS 2026 Year 6 H2 Mathematics Preliminary Examination Paper 2 Section B: Probability and Statistics [60 marks] 6 Tom has a box containing 90 ornaments that are indistinguishable aside from their colours and pictorial designs on them. The numbers of each colour and pictorial design are given in the table. White Blue Pink Green Star 12 6 9 13 Diamond 2 3 7 10 Moon 4 5 11 8 Tom chooses two items from his box of ornaments at random. (a) Given that neither of the items chosen are green, find the probability that both the items chosen are either white or has a moon pictorial design (or both). [2] (b) Find the probability that the two items include exactly one blue ornament and exactly one ornament that has a star pictorial design. [3] (c) Tom has a favourite type of ornament out of the 12 possible colour / pictorial design combinations and the chance of him getting both items being his favourite type of ornament is 284005 . Determine Tom’s favourite type of ornament, showing your working clearly. [2] 7 Ten cards each bear a single letter, and together they can be made to spell the word "ASSESSMENT". Cards bearing the same letter are not distinguishable. (a) Four cards are selected from the ten cards, and the order of selection is not relevant. Find the number of possible selections without restriction. [3] (b) All ten cards are arranged in a row. (i) Find the number of arrangements in which the letters “A”, “M”, “N” and “T” are together. [2] (ii) Find the number of arrangements in which no two letters “S” are together. [2] (iii) Hence find the number of arrangements in which no two adjacent letters are the same. [2]

5 DHS 2026 Year 6 H2 Mathematics Preliminary Examination Paper 2 [Turn over 8 Zayne compares the brightness readings given by two different devices. At five different locations with different lighting conditions, he measured the brightness using the two different devices. The table shows the brightness reading from a laboratory light meter is recorded as x units and the brightness reading from a phone light sensor app is recorded as y units. x 289 329 395 420 508 y 220 280 260 360 400 A scatter diagram for the data is shown below. The school wants to use the more convenient phone app reading to estimate what the laboratory light meter reading would be. (a) Zayne decides to investigate whether the model 98 0.95xy= + is a good fit for the data. (i) In the Answer Booklet, draw the line 98 0.95xy= + on the scatter diagram given. [1] (ii) To check the fit of the model, Zayne wants to analyse the residual for each point on the scatter diagram in part (a)(i). For the model f ( ),xy= the residual for a point (,)ab is f ( ).ab− Explain why this definition of residual is appropriate in this context, and why Zayne should consider the sum of the squares of these residuals. [2] (iii) Mark the residuals for each point on the scatter diagram in part (a)(i). [1] (iv) Calculate the sum of the squares of the residuals for the line 98 0.95 .xy= + [1] (b) (i) Use your calculator to find the equation of the least squares regression line of x on y and state the product moment correlation coefficient. [2] (ii) Calculate the sum of the squares of the residuals for this regression line and verify that it is less than the value found in part (a)(iv). [1] (iii) The equation of the regression line is used to estimate the brightness reading from a laboratory light meter when the reading from a phone light sensor app is 300 units. Explain whether you would expect this estimate to be reliable. [1] y x 200 250 300 350 400 450 500 550 200 250 300 350 400 450

6 DHS 2026 Year 6 H2 Mathematics Preliminary Examination Paper 2 9 A cheese company pro