2024 RI H2 Math Promo (Qn)

© RI 2024 [Turn Over 2024 RI Yr 5 H2 Math Promo Exam Duration: 3 hr Instructions: Do all questions. The solution will be released on 15 Oct (Tue) 1 A curve D has equation ln , 0, cy ax b x x x    where ,a b and c are constants. It is given that D has a stationary point at 3 2x and the tangent to D at the point where 1x is 5.y x  Find the values of ,a b and .c [4] 2 With respect to the origin O, the fixed points A and B have position vectors a and b respectively, where a and b are non-zero and non-parallel. (a) The variable point R has position vector  1 ,   r a b where  is a real parameter. Describe geometrically the set of all possible positions of R. [1] It is given that the angle AOB is 90 . (b) Explain why 0.a.b [1] (c) Among the set of all possible points R, the point *R is the closest to the origin O. Find the position vector of *.R Hence state the ratio * *:AR BR in t erms of magnitudes of a and b. [4] 3 Do not use a calculator in answering this question. Solve the inequality   26 2 3 2 1 .2 1 x x xx     [4] Hence solve   26 2 3 2 1 .2 1 x x xx     [3]

2 H2 MA 9758/2024 RI Year 5 Promotion Examination [Turn over 4 A curve C has parametric equations 31x    and 2 31y    , for 0. The point P is a variable point on .C (a) With reference to the origin O , OP forms the diagonal of the rectangle ,OQPR where vertices Q and R lie on the x- and y-axis respectively. Using differentiation, find the value of  which maximises the area of rectangle .OQPR You need to show that your answer gives a maximum. [5] When the area of rectangle OQPR is at its maximum, the rate of change of the x-coordinate of the point P is 1 unit per second. (b) Find d d y x and hence determine the rate of change of the y-coordinate of the point P at this instant. [4] 5 (a) Find   0 ( 2) , n r n r n    giving your answer in terms of n. [3] [You may use the result   23 2 1 1 14 n r r n n    for the rest of this question.] (b) By writing down the first two and the last two terms in the series, find   3 1 2 , n r r   giving your answer in terms of n. [3] (c) Find     3 33 3 3 3 3 31 2 3 4 5 6 2 1 2 , n n         giving your answer in terms of n. [3]

3 H2 MA 9758/2024 RI Year 5 Promotion Examination [Turn over 6 It is given that e 1 sin 3 .y x  (a) Show that 22 2 d d 9 9e .d d yy y x x  