2022 J1 RVHS H2 Math EOY Revision Package (QP) Ch5-7

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2021 End-of-Year Revision Package 31 5 Differentiation & Applications Basic Skills 1. Prove these trigonometric identities. (a) 1 cos sin 2 sin 1 cos sin xx x x x − +− (b) tan cotsin cos sec cosec xxxx xx −+ − (c) ( ) 2 2 1 cot1 sec 22 cot 1 xx x+ − (d) tan cotsec2 cot tan xxx xx + − (e) sin sin 2 sin 3 tan 2cos cos 2 cos3 x x x xx x x −+ −+ 2. Differentiate the following expressions with respect to x: (a) 3 24xx− (b) 4(3 1) ,33 x xx + − (c) cos(sin x) (d) sin−1(1 – x) (e) 3ln 3 x x + − (f) e ee x x x + − Tutorial Review Tutorial 5A Questions 1, 4, 7 and 10. Tutorial 5B Question 1 and 2. Tutorial 5C Question 1 and 2. Tutorial 5D Question 2 and 5. Revision Questions 1. 2020/Promo/SAJC/Q3 (a) Differentiate 21e cos (3 )x x− with respect to x, where 11 33 x−   . [3] (b) Given that +=x y a , where 0, 0xy and a is a positive constant , show that 2 2 dd2 1 0. dd yyx xx + − = [4] 2. 2017/Prelim/VJC/P2/Q1 A curve C is defined by the parametric equations 2 ,,11 ttxy tt==++ where t takes all real values except 1− . Find d d y x , leaving your answer in terms of t. [3] (i) Show that the equation of the tangent to C at the point 2 ,11 pp pp   ++ is ( ) 22.y p p x p= + − [2] (ii) Find the acute angle between the two tangents to C which pass through the point ( )2,5 . [3]

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2021 End-of-Year Revision Package 32 3. 2017/Prelim/NYJC/P1/Q2 The curve C has equation ( ) 222x y x y− = + . (i) Find the equations of the tangents to C which are parallel to the x-axis. [4] (ii) The line l is tangent to C at A . If the normal to C at the origin O meets l at the point B, find the area of triangle OAB. [4] 4. NJC JC1 Promo 9758/2019/Q6 A vessel is formed by removing a smaller cone of radius 5 m from a bigger cone whose semi- vertical angle is  , where tan 0.5 = . Water flows out of the vessel at a rate of 3 mkh per minute, where k is a positive constant. At time t minutes, the height of the water surface from the hole is h m (see diagram). (i) Show that the volume of the water V, in 3m , is given by ( ) 31 π 10 100012 h +− . [4] (ii) Find the rate of change of h, in terms of k, when 120πV = . [4] 5. 2017/Prelim/HCI/P1/Q6 A particle moving along a path at time t , where 30 t  , is defined parametrically by cot 3xt= and 2cosec3 1yt=+ . (a) The tangent to the path at the point ( )cot 3 , 2cosec3 1P p p + meets the y-axis at the point Q . Show that the coordinates of Q is ( )0, 2sin 3 1p+ . [4] (b) The distance of the particle from the point (0, 1)R is denoted by s , where 2 2 2 ( 1)s x y= + − . Find the exact rate of change of the particle’s distance from R at time 4t = . [4] ( )2,