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       Hence find the series expansion of y in ascending powers of ,x up to and including the term in 3,x simplifying your answer. [5] (b) Using standard series from the List of Formulae, verify that the series expansion obtained in part (a) is correct. [3] 7 Do not use a calculator in answering this question. (a) Given that 1 3ix  is a root of the equation 3 2 18 0,x ax x b    find the values of the real numbers a and b, and the other roots. [5] (b) The complex numbers z and w satisfy the following equations. * 4 6i 2 1 10i w z w z       Find z and w, giving your answers in the form i ,c d where c and d are real numbers. [5]

4 H2 MA 9758/2024 RI Year 5 Promotion Examination [Turn over 8 (a) The diagram shows the curve f ( )y x with a maximum point at  4,3 .C The curve crosses the axes at the points  0, 2A and  3,0 .B The line s 2x and 0y are the asymptotes of the curve. Sketch the graph of f ( ),y x  clearly stating the equations of the asymptotes and the coordinates of the points corresponding to A, B and C where appropriate . [3] (b) The curve 1C has equation 2 8 2 ax bxy x    , where a and b are constants. It is given that 1C has an asymptote y = 3  2x. (i) State the value of a and show that b = 7. [3] (ii) Sketch 1,C clearly stating the equations of any asymptotes, the coordinates of any stationary points and of any points where 1C crosses the axes. [3] (iii) The curve 1C is transformed by a translation of 2 units in the negative x-direction, followed by a stretch with scale factor 1 2 parallel to the y-axis, to form the curve 2.C Find the equation of 2.C [2] y = 0 x y O 2 2x

5 H2 MA 9758/2024 RI Year 5 Promotion Examination [Turn over 9 The planes p and q have equations      2 2 3 3 2r i j k               and 1 1 a b         r. respectively, where a and b are constants and  and  are parameters. The line l passes through the point  5, 4, 0 and is parallel to the vector 2 2 .  i j k The planes p and q meet in the line l. (a) Show that 1a and 1 .2b [2] (b) Find the exact acute angle between the planes p and .q [3] (c) Find the distance from the point  2, 0,3A to the plane .q Hence deduce the shortest distance from A to l. [4] The plane q is reflected about the plane p to obtain the plane .q (d) Find a cartesian equation of the plane .q [3] 10 It is given that   21 2f ( ) 1 3 , for 1 2, , for 214 4,4 x x x x x           and that f ( ) f ( 3)x x  for all real values of x. (a) State the value of f (0). [1] (b) Sketch the graph of f ( )y x for 0 5.x  [3] (c) The function g is given by 4g( ) 2 3x x  for .x  By sketching the graph of g( )y x on the same diagram as in part (b), solve the inequality f ( ) g( ).x x [4] (d) The function h is given by h( ) 3 sin( ) cos( ) 1x x x     for 0 2.x  Explain why the composite function hf exists and find its range in exact fo rm. [4]

6 H2 MA 9758/2024 RI Year 5 Promotion Examination [Turn over 11 A financial institution, Future Investments Inc., has introduced a new investment scheme. The scheme pays a compound interest of 5% per annum at the end of the year, based on the amount in the account at the beginning of each year. John and Sarah are both interested in this scheme. (a) John invests $x at the start of the first year and a further $x on the first day of each subsequent year. He chooses to leave the money in his account for the interest to accumulate. (i) Write down the amount in John’s account, including the interest, at the end of the first year. [1] (ii) Show that John will have a total of $  21 1.05 1nx  in his account at the end of n years. [3] (iii) If John invests $10 000 at the start of every year, find the number of years for the total in his account to first exceed $500 000. Determine if this happens at the start or at the end of that year. [4] (b) Sarah invests $6 000 at the start of the first year. On the first day of each subsequent year, she invests $400 more than the amount invested at the start of the previous year. (i) Explain why the amount in Sarah’s account at the end of n years can