2025 RI Prelim+P1+Qns

© RI 2025 [Turn over RAFFLES INSTITUTION 2025 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 8 printed pages. RAFFLES INSTITUTION Mathematics Department

2 @ RI 2025 9758/01/PRELIM 1 A curve has equation 2 ,1 cy ax b x    where , and a b c are real constants. It is given that the curve crosses the x-axis at 2.x The normal to the curve at the point 0, 3 meets the x-axis at 7.5.x Find the values of , and .a b c [4] 2 The point P travels along the curve C with equation 1sin , 1 1.y x x x    Let the gradient of the curve C at the point P be .m If the x-coordinate of P is increasing at the rate of 9 units per second when 1,2x find the exact value of the rate at which m is changing at this instant. [4] 3 Relative to the origin O, the points A and C have position vectors a and 2b such that 5 2 4p p p  a i j k and 2 2 ,  b i j k where p is a positive constant. It is given that 2 .a b (a) Find the exact value of .p [2] (b) Evaluate 2 2 . a b a b [1] (c) Evaluate a b and hence find the area of triangle .OAC [3] (d) Use a geometrical reason to explain why   1 2 2 .4   a b a b a b [2] 4 (a) Find    2 9 d .2 1 1 x xx x   [4] (b) (i) Differentiate 2 1 1x  with respect to .x [1] (ii) Differentiate 2ln 1x  with respect to .x [1] (iii) Hence find   2 22 ln 1 d .1 x x xx    [4]

3 @ RI 2025 9758/01/PRELIM [Turn over 5 (a) The diagram below shows a sketch of the graph of f .y x The curve passes through the points with coordinates  1, 0 and  3, 0 , and has turning points at  1, 0 and  4, 4 . The asymptotes are 0, 2 and 2.x x y   Sketch on separate diagrams, the graphs of (i) fy x, [3] (ii)  1fyx, [3] showing clearly the main features of the graphs. (b) Describe a sequence of transformations which transform the graph of 21xyx to the graph of 22 5 42x xyx . [3] x y O

4 @ RI 2025 9758/01/PRELIM 6 (a) A geometric series has first term 2 and common ratio 0.8. Find algebraically, the least value of m for the sum of the first 4m terms of the series to be greater than 99% of the sum to infinity. [3] (b) A finite arithmetic progression A has n terms, first term a and common difference .d In another progression ,B the kth term is obtained by adding the kth positive odd integer to the corresponding term of .A That is, the first term of B is obtained by adding 1 to the first term of ,A the second term of B is obtained by adding 3 to the second term of A and so on. It is given that the twelfth term of A is 25, the sum of all the terms of A is 676 and the sum of all the terms of B is twice the sum of all the terms of .A (i) Find the values of , and .n a d [4] (ii) Obtain the sum of the first ten terms of .B [2] 7 The function f is defined by 3f : 1 for , ln 2.e 2xx x x    (a) Find 1f ( )x and state its domain. [3] Another function g is defined by   22 1 2 for 0,g : 2 2 for 0 3. x xx x x          (b) Sketch the graph of gy x and explain why the composite function 1f g exists. [4] (c) Hence find the exact value of k such that  1 5f g ln . 2k  [3]

5 @ RI 2025 9758/01/PRELIM [Turn over 8 Do not use a calculator in answering this question. (a) The point P on the Argand diagram represents the complex number w given by i ,w u v  where u and v are real and positive. (i) Explain algebraically why arg argkw w for any real constant 1.k [1] Points Q and R represent the complex numbers kw and i ,kw where k is a real constant and 1.k (ii) On the same Argand diagram in the Printed Answer Booklet, plot the points Q and R. Show clearly the geometrical relationship between the points ,P Q and .R [2] Im Re O P X

6 @ RI 2025 9758/01/PRELIM (b) In another Argand diagram, the points A, B and C represent the complex numbers z, f z and f f z respectively, where 2f 2 .z z z  (i) Show that f f f 2 3 1 .z z z z z z     [2] It is given that ABC is a right-angled triangle, described in an anticlockwise sense, with a right angle at ,B and ,BC mBA where m is a positive real constant. (ii) By considering   f f f ,f z z z z   or otherwise, show that 2 1 i.z z m   [2] (iii) In the case where 2i,z x  where x is a positive real number, find x and .m Hence obtain the complex number represented by the point .B [5] 9 Decontamination is a water treatment method to remove a certain undesirable chemical from contaminated water. As the liquid agent is continuously added to the contaminated water in a large tank, the undesirable chemical in the water is removed gradually. The rate of decrease of the concentration of the undesirable chemical is proportional to the product of its current concentration and the rate at which the liquid agent flows into the tank. At time t minutes after the start of the treatment process, the concentration of the undesirable chemical is C mg/L and the liquid agent flows into the tank at a rate of 2 1 1 t L/min. It is known that the initial concentration of the undesirable chemical in the tank is 50 mg/L and after 10 minutes, its concentration is measured to be 20 mg/L. (a) By setting up and solving a differential equation relating C and ,t show that 1tan50e ,q tC   giving the value of q correct to 5 decimal places. [7] (b) Determine the value of C when 50.t [1] (c) Sketch the graph of C against t and state what happens to C in the tank for large values of .t [3]

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