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 6000 m3 of water on the first day. For each subsequent day, Robot A fills the tank with 50 m3 of water lesser than the previous day. Robot B fills the tank with 9000 m3 of water on the first day. For each subsequent day, Robot B fills the tank with 85% of the volume of water it fills the tank in the previous day. (a) Find the number of days for robot A to fill up the tank. [3] (b) Determine with clear reasoning whether robot B would be able to fill up the tank with water. [1] (c) Find the total amount of water that robot B fills in the tank by the end of the 10 th day. [2] (d) At the start of the 11th day, robot B is reprogrammed. At the end of the 11 th day, it fills the tank with 5% more volume of water it fills on the previous day and continues to do so for each subsequent day. Show that the total volume of water, in m 3, that Robot B fills in the tank after reprogramming can be expressed as 9 189000 0.85 1.05 1n , where n is the number of days starting from the 11th day. Hence, determine with clear reasoning which robot will be faster in fil ling up the tank with the above change. [5]

16 ©RIVER VALLEY HIGH SCHOOL 9758/01/2024 9 The closed curve C, which is symmetrical about the line 0x , has parametric equations cos3 cos , 2cos2 ,xt ty t for π 4 3π 4tdd . (a) Sketch C. [1] (b) Find the exact equation of the tangent of C at the p oint when π 4t . [3] (c) F ind the acute angle between the two tangents of curve C at π 4t and 3π 4t . [2] (d) Show that the area enclosed by the curve C is given by 3π 4 π 4 3sin 5 sin 3 2sin dtt t t³ . Hence find the area enclosed by the curve C correct to 3 decimal places. [4]

18 ©RIVER VALLEY HIGH SCHOOL 9758/01/2024 10 In a robotics competition, toy cars move along straight lines t o complete tasks. Points are defined relative to the origin 0, 0, 0. The x-, y- and z-axes are in the directions east, north and vertically upwards respectively, with units in centimetres. The position vectors of two toy cars A and B, with respect to time t in seconds, are given as 5t Ari k and 64 t Brij k i j k respectively. (a) Show that after two seconds, car B is at the point with coordinates 9, 4,1 and find the distance that car A has travelled in the same duration. [2] (b) Determine whether cars A and B meet. [3] (c) Explain why cars A and B travel on a common plane surface and show that the cartesian equation of the surface is 50xy z . [5] A drone flies above the cars to capture images of the cars during the competition. The shortest distance between the drone and the surface where cars A and B travel is maintained at 50 cm. (d) Find the cartesian equation of the plane containing the flight path of the drone. [2]