RI Post+Prelim+Revision+P1 Qns

RAFFLES INSTITUTION H2 Mathematics 9758 2025 Year 6 Term 4 Post-Prelim Revision Paper 1 (Source: A level questions) Total number of marks: 100 3 hour Please attempt this paper in one sitting under exam conditions within 3 hours before the lesson on 17 October 2025. 1 The graph of 2 1xy ax bx c   , where ,ab and c are non-zero constants, has an asymptote at 1 2x . The graph also has a turning point at 12, 9  . Find the values of ,ab and c . [4] 2 Find the exact equation of the tangent to the curve  2 ln 11 5yx at the point where 2.x [5] 3 (a) Without using a calculator, solve the inequality 43 .2 x xx  [4] (b) Hence, solve the inequality 34 .2 x xx  [2] 4 A gardener designs a flower bed ABCDE in the shape of a rectangle with an equilateral triangle on one of the shorter sides. Side AE is of length a m and side ED is of length b m (see diagram). The total perimeter of the flower bed is 20 m. Find the maximum possible area of the flower bed, showing that it is a maximum value. Give your answer correct to 4 significant figures. [6]

Raffles Institution H2 Mathematics 2025 Year 6 _______________________________________________________________________________________________ _____________________________________ Y6 H2 Math Term 4 Post Prelim Revision: Paper 1 Page 2 of 4 5 Vectors a and b are such that 1a.b . It is also given that  ab + a is perpendicular to  ab + b . (a) Show that 1ab [3] (b) Hence find the angle between the direction of a and direction of b. [3] 6 (b) Given that 0n , show that 2 sin coscos d , xn x n xx nx x c nn  where c is an arbitrary constant. [3] (c) Using the result in part (b) show that, for all positive integers, n , the value of π 0 cos dx nx x can be expressed as 2 k n , where the possible value(s) of k are to be determined [2] (d) Using the result in part (b) find the exact value of π 2 0 cos 2 d .x xx [3] 7 (a) Use double angle formula, show that  4 1cos cos 4 4 cos 2 3 .8   [2] (b) The region R lies in the first quadrant and is bounded by the curve   342 9yx , the x- axis and the lines 1.5x and 3.x R is rotated about the x-axis through 2π radians. Using the substitution 3sinx  , find the exact volume generated. [6] 8 (a) The complex number 2i is denoted by z and the complex number 13 i is denoted by .w Without using a calculator, evaluate .wwz z . Give your answer in the form i,cd where c and d are real numbers. [4] (b) The equation 32 0za zb z c   , where ,ab and c are constants, has roots 3 and w where w is a complex number. (i) State a condition on ,ab and c for the third root to be w*. [1] (ii) G iven that the condition in part (bi) holds, and that 12 iw  , find the values of ,ab and c . [3]

Raffles Institution H2 Mathematics 2025 Year 6 _______________________________________________________________________________________________ _____________________________________ Y6 H2 Math Term 4 Post Prelim Revision: Paper 1 Page 3 of 4 9 It is given that   2cos 1 e . xy (a) Show that 2 4 2 dd 2e ,dd xyy kyxx   where k is a constant to be found. [3] (b) By differentiation of the result in part (a), find the first three non-zero terms of the Maclaurin expansion of   2cos 1 e . x [4] (c) The first two non-zero terms of the Maclaurin expansion of   2cos 1 e x are equal to the first two non-zero terms of the series expansion of 2 1 ab x , where a and b are constants. Using standard series from the List of Formulae (MF26), find the values of a and b. [2] 10 The line 1l , contains the point A with coordinates  3, 1, 2 and is parallel to the vector 2 1 a    where a is a constant. The line 2l has equation 21 532 xy z   . It is given that 1l and 2l cross at the point B. (a) Find the value of a and the coordinates of B. [5] (b) The plane 1π contains the point A and is perpendicular to 2l . (i) Find the shortest distance from B to 1π . [3] (ii) Hence find the acute angle between 1l and 1π . [2] (c) The plane 2π is perpendicular to 1π and contains 1l . Find a cartesian equation of 2π . [3]

Raffles Institution H2 Mathematics 2025 Year 6 _______________________________________________________________________________________________ _____________________________________ Y6 H2 Math Term 4 Post Prelim Revision: Paper 1 Page 4 of 4 11 (a) Using the substitution 41 ,ux show that 2 2 41 2 dd29 xu x uxu      . [3] (b) The region R lies in the first quadrant and is bounded by the curve 41 2 xy x   and the lines x = 6, x = 12 and y = 0.5. (i) Sketch the graph of 41 2 xy x   for 0x , giving the coordinates of any intercepts with the axes and the equati ons of any asymptotes. Shade the region R on your sketch. [3] (ii) Find the exact area of R, giving your answer in the form lnab c . [5] (iii) Find the volume of the solid generated when R is rotated through 2 radians about the x-axis. Give your answer correct to 2 decimal places. [3] 12 The mass of a person depends both on daily rate of energy intake and on daily rate of energy expenditure. In this question, mass is in kg, time is in days, and energy intake and energy expenditure are measured in Calories per day. The rate of change of a person's mass with respect to time is proportional to the difference between energy intake and energy expenditure. Andrew has a mass of M kg and his energy intake is fixed at C Calories per day. For every kg of his mass, he expends 30 Calories per day. (a) Show that d 30 ,d M kC Mt  where t is time and k is a constant. [1] Andrew's initial mass is 110 kg. (b) Find the energy intake such that he maintains his mass at 110 kg. [1] As part of a health plan, Andrew fixes his en ergy intake at 80% of the value found in part (b). (c) By solving the differential equations in part (a), show that Andrew’s mass while he is on the plan satisfies the equation 3088 22e . ktM  [4] Andrew's mass after 75 days on the plan is 100 kg. (d) Find the number of additional days required on the plan for Andrew's mass to fall below 96 kg. [4] (e) (i) Sketch a graph of Andrew's mass while on this plan. Explain why Andrew cannot achieve a mass of 80 kg using this plan. [2] (ii) State the range of possible values of ener gy intake for which Andrew could achieve a mass of 80 kg. [1]