RI 2026 H2 Math Prelim Paper 1 (Qns)

© RI 2026 [Turn over RAFFLES INSTITUTION 2026 YEAR 6 PRELIMINARY EXAMINATION Higher 2 MATHEMATICS Paper 1 9758/01 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. This document consists of 6 printed pages. RAFFLES INSTITUTION Mathematics Department

2 @ RI 2026 9758/01/PRELIM 1 A curve has parametric equations 21x t , 3 2,y at bt ct   where 2t and a, b and c are constants. The curve has stationary points at 1t and 2t. Given that the curve passes through the point (5, 2), find the values of a, b and c. [4] 2 (a) The sequence 1 2 3, , ,u u u is defined by 14u and 112, for 13n nu u n  . (i) Write down the value of 6u. [1] (ii) Given that as , nn u l , find the exact value of l using an algebraic method. [2] (b) Using the result  21( 1) 2 16nrnr n n  , find 0( )( 1)nrn r r , giving your answer in terms of n, in fully factorised form. [3] 3 Water is poured at a rate of 30.5 m per minute into an empty container in the form of an open hemisphere with inside radius 3 m (see diagram). At time t minutes after the start, the radius of the water surface is mr and the volume of the water in the hemispherical container is  219 ,3h h where h m is the depth of the water. When the depth of the water is 1 m, (a) find the rate of change of the depth of water, [3] (b) find the rate of change of the radius of the water surface. [3] 3 h

3 @ RI 2026 9758/01/PRELIM [Turn over 4 With reference to the origin O, three distinct points A, B and C are such that OA a, OB b and OC c. The mid-point of AC is D. (a) Express OD  in terms of a and c. [1] (b) Given that a c , 0b and BD is perpendicular to AC, show that OB is also perpendicular to AC by using a suitable scalar product. [3] (c) Given further that point D lies on line segment OB, 2a , 6b and 60AOC   , find the exact area of quadrilateral OABC. [2] 5 Do not use a calculator in answering this question. It is given that 1 2i is a complex root of the quartic equation 4 3 2 45 0z pz qz rz     , where p, q and r are real numbers. (a) State, with justification, another complex root of the equation in the form i ,a b where a and b are real numbers and 0b . Hence find a quadratic factor corresponding to these two roots. [2] The quartic equation has a repeated negative real root. (b) Find the value of the repeated real root and hence find the values of p, q and r. [4] (c) By means of a suitable substitution, solve 4 3 245 1 0w rw qw pw     . [2] 6 (a) Show that 22 4 3x x  is always positive for all real values of x. Without using a calculator, solve the inequality 2 10 3 .(2 1)( 3) x x xx x     [6] (b) Hence solve the inequality 2 10 3 .(2 1)( 3) x x xx x     [2]

4 @ RI 2026 9758/01/PRELIM 7 Conservationists are investigating the effect of habitat loss on the number of deer on an island. They observe that the population of the deer on the island decreases at a rate of 2% every year. The number of deer on the island is P at a time t years after conservationists begin observations. (a) Write down a differential equation relating P and t. [1] To maintain population stability, the conservationists relocate deer to the island at a constant uniform rate of h deer per year. (b) Write down a differential equation to model the new situation. [1] (c) Without solving the differential equation, determine the value of h if the conservationists would like the number of deer to settle down to 600 after many years. [2] (d) Solve the differential equation to find an expression for P in terms of t and h. Hence, state the population of the deer, in terms of h, in the long term. [5] 8 The curve C has parametric equations sinx a t , sin cosy a t t , for 0 t   and a is a positive constant. (a) Sketch the graph of C, stating clearly the coordinates of the points where the curve C meets the axes. [2] (b) State the equation of the line of symmetry. [1] (c) Find the equation(s) of the tangent to C which is parallel to the x-axis. [3] The tangent and normal to C at the point P where 6t  cut the x-axis at points Q and R respectively. (d) Find the exact area of triangle PQR. [4]

5 @ RI 2026 9758/01/PRELIM [Turn over 9 A curve C has equation e , 0.xy x x  (a) Show that C has exactly one stationary point. [2] (b) Sketch C, giving the exact coordinates of the stationary point. [1] The region R is bounded by C, the y-axis and the line 1.ey (c) Find the exact area of R. [4] (d) An ornament is formed by rotating R through 2 radians about the line 2.y By considering a suitable translation of C, find the volume of the ornament. [3] 10 (a) It is given that 2 20 109f ( ) . 10 x xx x    Describe the transformation which transforms the graph of 9y x x  onto the graph of f ( )y x . [2] (b) Sketch the graph of f ( )y x , labelling clearly the equations of any asymptotes and the coordinates of any axial intercepts and turning points. [3] (c) On a separate