2025 RI Math Prelim P1_solutions with comments

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 Page 1 of 32 H2 Math Year 6 Preliminary Examination Paper 1: Solutions with comments 1 A curve has equation 2 1 cy ax b x    , where a, b and c are real constants. It is given that the curve crosses the x-axis at 2x . The normal to the curve at the point  0, 3 meets the x-axis at 7.5x . Find the values of a, b and c. [4] Solution Comments [4] 2 1 cy ax b x    Since  2, 0 and  0, 3 lies on the curve, 12 0 --- (1)3a b c   3 --- (2)b c    22 d 2 d 1 y cxax x   Gradient of normal at  0, 3 3 0 2 0 7.5 5   . Gradient of the tangent to the curve at (0, 3) is 5 2 and thus 5 --- (3)2a Using GC, 5, 3 and 62a b c   . There are quite many sign/conceptual errors in the attempt to find d d y x. Do note that the gradient of the normal to the curve at the point (0, 3) means that we need to substitute 0x in d d y x, and get 1 a as the gradient of the normal. The two points (0, 3) and (7.5, 0) gives the gradient of the normal as 2 5 .

2025 H2 Math Year 6 Preliminary Examination Paper 1: Solutions with Comments _____________________________________________________________________________________ Page 2 of 32 2 The point P travels along the curve C with equation 1sin , 1 1y 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] Solution Comments [4] 1sin , 1 1y x x x    Differentiate with respect to x: 1 2 d sind 1 y xm xx x     Differentiate with respect to x again:       2 2 22 2 2 2 2 2 2 32 d 11 21 2 1 11 11 1 1 1 2 1 d x x x x xx x x x x m x x x x                 When 1,2x 3 3 12d d 4 d d 1 d 9 1 1 4 4 d m t t m x x            Therefore, required rate is 14 3. Most students did well but many did not simplify the answer to the lowest terms. Answer mark was not awarded to non-exact answers and unsimplified answers. Some students substituted x=p, assuming the parameter x takes the value of p at point P, note that p will be taken as a constant in this case and we shall not differentiate with respect to p.

2025 H2 Math Year 6 Preliminary Examination Paper 1: Solutions with Comments _____________________________________________________________________________________ Page 3 of 32 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 24   a b a