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, 2−

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2021 End-of-Year Revision Package 33 6. 2017/Prelim/TJC/P1/Q7 The diagram shows a shot put being projected with a velocity v ms-1 from the point O at an angle  made with the horizontal. The point O is 1.5m above the point A on the ground. The x-y plane is taken to be the plane that contains the trajectory of this projectile motion with x-axis parallel to the horizontal and O being the origin. The equation of the trajectory of this projectile motion is known to be 2 22tan 2 cos gxyx v =− , where g ms-2 is the acceleration due to gravity. The constant g is taken to be 10 and the distance between A and B is denoted by h m. Given that v = 10, show that h satisfies the equation 2 10 sin 2 15cos 2 15 0hh − − − = . [3] As  varies, h varies. Show that stationary value of h occurs when  satisfies the following equation 23tan 2 20sin 2 tan 2 20cos 2 20 0   − − − = . [5] Hence find the stationary value of h. [2] O y x  1.5m A B h m

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2021 End-of-Year Revision Package 34 7. MI PU2 Promo 9758/2019/02/Q1 Town A, Town B and Town C are located in Wakandi Country. The distance between Town A and Town B is 60 km and the distance between Town B and Town C is 20 km. A railroad connects Town A to Town B (see diagram). A manufacturer plans to deliver a certain number of containers of its goods daily from Town A to Town C. To support this plan, the Wakandi government decides to build Interchange Station S and a road connecting this station to Town C (see diagram). Once the road is built, the goods manufacturer can deliver its containers from Town A to Town C by a combination of rail and road via Interchange Station S. The cost to deliver the containers daily by rail is $200 per km and the cost to deliver the containers daily by road is $300 per km. (i) Show that the daily total delivery cost, $T, of the containers from Town A to Town C is given by 2200 300 120 4000,T x x x= + − + where x km is the distance between Town A and Interchange Station S. [2] (ii) Hence use differentiation to find the value of x that gives a stationary value of T, giving your answer correct to 2 decimal places. Show that T is a minimum for this value of x. [4] Town A Town B Town C 20 km Interchange Station S 60 km

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2021 End-of-Year Revision Package 35 8. 2017/Prelim/NYJC/P2/Q3 It is given that ( )ln cos sin ,y ax ax=− where a is a non-zero constant. (i) Show that 22 2 2 dd 0.dd yy axx + + = [3] (ii) By further differentiation of the result in (i), find, in terms of a, the Maclaurin series for y, up to and including the term in 3x . [3] (iii) Hence show that when x is small enough for powers of x higher than 2 to be neglected and 2,a= then 2cos 2 sin 2 1x x kx kx−  + + where k is a constant to be determined. [4] (iv) Using appropriate expansions from the List of Formulae ( MF26), verify the correctness of your answer in (iii). [2] 9. 2016/Promo/PJC/Q4 (i) Given that f ( ) 1 cos 2xx=+ , find f (0) , f '(0) , f ''(0) , (3)f (0) and (4)f (0) . Write down the Maclaurin series for f ( )x up to and including the term in 4x . [5] (ii) Deduce the series for 2cos x up to and including the term in 4x . [2] (iii) Use appropriate expansions from the List of Formulae (MF26) to verify the correctness of your answer in part (ii). [2] 10. TMJC JC1 Promo 9758/2019/Q6 (a) Given that ( ) 2ln 2 e xy=− , where 1 ln 22x , show that 22 2 2 dde 4edd yx yy xx  + =−  . Hence, find the Maclaurin series of y, up to and including the term in 3x . [5] (b) Find the expansion of 12 23 x x − + in ascending powers of x, up to and including the term in 2x . State the range of values of x for which this expansion is valid. [5] 11. 2020/Promo/NYJC/Q10 A curve C has parametric equation 2 3, 4 15, where 1 .yx t t tt = − +− = (i) Show that the curve C cuts the x-axis at the point (8, 0) and the y-axis at (0, 12) and (0, 18). [3] (ii) Sketch C, giving the equation of the line of symmetry. [2] (iii) Find the exact coordinates of the point where C crosses itself. [2] (iv) Show that the equation of the tangent at the point (8, 0) is 23 6 184 0xy+ − = . Hence, find the coordinates of the point at which the tangent cuts the curve again. [4]

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2021 End-of-Year Revision Package 36