RI Post+Prelim+Revision+P2 Qns

RAFFLES INSTITUTION H2 Mathematics 9758 2025 Year 6 Term 4 Post-Prelim Revision Paper 2 (Source: Other JCs’ Prelim Qns) Total number of marks: 100 3 hour Please attempt this paper in one sitting under exam conditions within 3 hours before the lesson on 24 October 2025. Section A: Pure Mathematics [40 Marks] 1 (a) The diagram shows the curve h( ).yx The curve has maximum points at  6, 4 and the origin, and crosses the x-axis at  5, 0 . The lines y = 0, x = 4 and y = x + 3 are the horizontal, vertical and oblique asymptotes to the curve respectively. (i) On the diagram given above, sketch the graph of   22 269 ,xy r  where r is a positive constant. Find the range of values of r for the equation     22 26h 9xx r  to have at least one real root. [3] (ii) On a separate diagram, sketch the graph of  1 .hy x [3] (b) The graph of 10 1yx  undergoes a sequence of transformations which transform its equation into 1.yx Describe and write down the transformations. [3] y = h(x) x = 4 y = x + 3 (5, 0) (6, 4) y = 0 y O

Raffles Institution H2 Mathematics 2025 Year 6 _______________________________________________________________________________________________ ______________________________________ Y6 H2 Math Term 4 Post Prelim Revision: Paper 2 Page 2 of 7 2 Referred to the origin O, the points A, B and X are such that OA a  , OB  b  and 3OX ab  . It is given that 2a , 5b and 73OX   . (a) By considering the scalar product of OX  with itself, or otherwise, find the value of .ab . [3] (b) Find the area of triangle OAX. [5] (c) The variable point V, not necessarily coplanar with A, B and X, has position vector v. Given that ,va vb describe geometrically the set of all possible positions of the point V. [2] 3 (a) It is given that sin , for 0 3 π,2f( ) 4, for 3π 4ππ x x x x x        and that f( ) f( 4 π)xx for all real values of x. (i) Sketch the graph of f( )yx for π 6π.2 x [3] (ii) Find 6π π 2 f( ) d ,x x  leaving your answer in exact form. [2] (b) From the diagram above, the region A is bounded by the curve ln(2 ),yx the line ,yh ,h   the x-axis and the y-axis while the region B is bounded by the curve ln(2 )y x and the lines e 2x and .yh Given that the volumes of the solids generated when A and B are rotated completely about the y-axis are equal, find the exact value of h. [5]

Raffles Institution H2 Mathematics 2025 Year 6 _______________________________________________________________________________________________ ______________________________________ Y6 H2 Math Term 4 Post Prelim Revision: Paper 2 Page 3 of 7 4 [The volume of a right square-based pyramid is 1 base area height.3 ] Jane designs a model in the shape of a right s quare-based pyramid. The square base has sides cm.x Each of the four lateral faces is a triangle with base cmx and perpendicular height cm.l The four lateral faces converge at the top of the pyramid to form an apex directly above the center of the square base. The vertical height of the pyramid is cm.h The model is assumed to be made of material of negligible thickness. (a) Form an equation involving ,a n dxl h . [1] In the design of the model, Jane hopes to fix the total surface area, 2cmA of the model but maximise the volume, 3cmV of the model. (b) Using the result in part (a), show that 2 22 2 4 xAx x h  . [1] (c) Hence show that  22 2 2 36 Ax A x V   . [2] (d) Use differentiation to s how that the maximum V occurs when 2 Ax and find a simplified expression for the maximum V in terms of A. (You need not show that your answer gives a maximum.) [4] (e) Given that V is a maximum, find the angle made by a lateral face and the base of the model, giving your answer to the nearest degree. [3] h cm x cm x cm l cm

Raffles Institution H2 Mathematics 2025 Year 6 _______________________________________________________________________________________________ ______________________________________ Y6 H2 Math Term 4 Post Prelim Revision: Paper 2 Page 4 of 7 Section B: Statistics [60 Marks] 5 In a certain university, there are 110 Co-Curri cular Activities (CCA s) clustered into 4 categories. There are 22 Arts and Culture CCAs, 16 Community Service CCAs, 38 Physical Sports CCAs and 34 Special Interest CCAs. (i) Albert wishes to find out about approaches to training of the Physical Sports CCAs, so he sends a questionnaire to 22 Physical Sports CCAs. Explain whether these 22 Physical Sports CCAs form a sample or a population. [1] (ii) Benedict wishes to investigate the level of student engagement in CCAs, but does not want to obtain the detailed information n ecessary from all 110 CCAs. Explain how he should carry out his investigation, and why he should do the investigation in this way. [2] (iii) Find the number of different possible samp les of 16 CCAs, with 4 CCAs chosen from each category. [2] 6 Tetrahedral dice have four faces. Two fair te trahedral dice, one red and one blue, have faces numbered 0, 1, 2, and 3 respectively. The dice are rolled, and the numbers faced down on the two dice are recorded. The random variable T is defined as the score on the red die multiplied by the score on the blue die. (i) Find the probability distribution of T. [3] (ii) Find  E T and show that  115Var 16T  . Show your workings clearly. [2] (iii) Evaluate  P2 T   , where E T  and  2 Var T  . [3] 7 The events A and B are such that  P Aa and  P B b . A and B are independent events. (a) Find an expression for P A'' B in terms of a and b, and hence prove that A' and B' are independent events. [2] It is given that  P0 . 8 5A' 'B  and P0 . 8B'  . (b) Find  P A 'B . [2] For a third event C, it is given that A and C are mutually exclusive and  P 0.52.A' ' C  (c) Find  P C . [1] (d) Hence find the set of possible values of  P B'CA '' . [3]

Raffles Institution H2 Mathematics 2025 Year 6 _______________________________________________________________________________________________ ______________________________________ Y6 H2 Math Term 4 Post Prelim Revision: Paper 2 Page 5 of 7 8 In this question, you should state the parameters of any distribution you use. A ceramic shop sells handmade ceramic cups. The mouths of the cups are assumed to be circular in shape. The diameters of the outer circumferences of the top rim of the cups, S , are assumed to follow a normal distribution with mean mm and standard deviation  mm. (a) It is given that    P 80.5 P 84.5SS  and that the probability of the diameter of the outer circumference of the top rim of a randomly chosen cup being more than 85 mm is 1.15%. Find the value of , and show that 1.10  , when corrected to 3 significant figures. [3] (b) The shop also makes covers of circular shape that can be fitted over the mouths of the cups. The diameter of any randomly chosen cover, C, in mm, follows a normal distribution with mean 83 mm and standard deviation 1.5 mm. A cover would be considered to be well-fitted over the mouth of a cup if the diameter of the cover is larger than that of the outer circumference of the top rim of the cup by not more than 2 mm. Find the probability that a randomly chosen cover is well-fitted over the mouth of a randomly chosen cup. [3] (c) A cover and a cup are randomly chosen. If the cover is well-fitted over the mouth of the cup, find the probability that the diameter of the cover is la rger than that of the cup by more than 1.5 mm. [3] 9 A store owner rece