2024 JC2 H2 Math prelim - RVHS

1 ©RIVER VALLEY HIGH SCHOOL 9758/01/2024 RVHS 2024 H2 Math Prelim P1 1 The curve C has equation 3yx x . It undergoes the transformations in the following order: Translation by 2 units in the negative y-direction, followed by scaling parallel to the x-axis with scale factor 1 2 , followed by reflection about the x-axis. (a) Determine the equation of the resulting curv e. [4] (b) Find the coordinates of the point of intersection between the two curves. [1]

2 ©RIVER VALLEY HIGH SCHOOL 9758/01/2024 2 (a) Solve the inequality 234 021 xx x t by algebraic method. [3] (b) He nce solve the inequality 2 43 021 xx x d . [3]

4 ©RIVER VALLEY HIGH SCHOOL 9758/01/2024 3 It is given that 1 x yx . (a) By considering ln y , find d d y x in terms of x. [4] (b) Find d d w x in terms of x if 2 11 2 xx wx x . [3]

6 ©RIVER VALLEY HIGH SCHOOL 9758/01/2024 4 The origin O and regular octagon OACDEFGB lie in the same plane, where OA a OA a and OB b OB b (see diagram). (a) Explain why BG BG can be expressed as BG s t ab BG s s for real constants s and t. [2] It is given that angle AOB angle 135OBG q . (b) It is known that line BG is perpendicular to line OA. By considering the scalar product BG OA BG OA OA , show that 2ts . [3] (c) By considering a suitable scalar product, or otherwise, deduce the values of s and t. [3] A B C D E F O G a b

8 ©RIVER VALLEY HIGH SCHOOL 9758/01/2024 5 Do not use a calculator in answering this question. (a) The complex number z is given by

11 77 5 3 ππ i cos isinz

. [4] Find |z| and arg(z). (b) (i) The roots of the equation 2 4iw are 1w and 2w . Find 1w and 2w in cartesian form ixy , showing your working. [3] (ii) He nce, or otherwise, find in exact cartesian form the roots 1v and 2v of the equation 2 10 25 i 0vv . [3]

10 ©RIVER VALLEY HIGH SCHOOL 9758/01/2024 6 (a) Show that 1ln 2 sin 2 ln 2 sin ln cosrr TT T . [2] (b) By letting 1 2rT , find 1 1ln cos 2 n r r §§ · ·¨¨ ¸ ¸©© ¹ ¹¦ in terms of n. [3] (c) Hence, show that 1 1ln cos 2r r f §§ · ·¨¨ ¸ ¸©© ¹ ¹¦ converges and state its value. (You may assume that 12s i n 12 n n §· o¨¸©¹ as nof .) [2]

12 ©RIVER VALLEY HIGH SCHOOL 9758/01/2024 7 (a) Given that 3 ec o saxyx , where a is a constant, show that 23d3e s i nd axyya y xx . [2] (b) By further differentiation of this result, find the Maclaurin s eries for y, up to and including the term in 2x . [5] (c ) Given that the first three non- zero terms in the above Maclaurin series are equal to the first three non-zero terms in the series expansion of 2 exb x , where b is a constant, find the values of a and b. [3]

13 ©RIVER VALLEY HIGH SCHOOL 9758/01/2024 8 There are two identical tanks, each of capacity 90 000 m 3. Robots A and B are each programmed to fill up an empty tank with water at the end of each day. Robot A fills the tank with 60