2022 J1 RVHS End of Year Revision Package Solutions Ch5-7

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2021 End-of-Year Revision Package 91 5 Differentiation & Applications Basic Skills 1. Prove these trigonometric identities. (a) 1 cos sin 2 sin 1 cos sin x x x x x    (b) tan cotsin cos sec cosec x xx x x x    (c)   2 2 1 cot1 sec 22 cot 1 xx x   (d) tan cotsec 2 cot tan x xx x x   (e) sin sin 2 sin 3 tan 2cos cos 2 cos3 x x x xx x x     (a)       2 2 2 2 1 cos sin sin 1 cos 1 cos sin sin 1 cos 1 2cos cos sin sin 1 cos 2 1 cos sin 1 cos 2 sin x x x x x x x x x x x x x x x x x              (b)    2 2 tan cot sec cosec sin cos cos sin1 1 cos sin sin cos sin cossin cos sin cos sin cos sin cos sin cos sin cos x x x x x x x x x x x x x xx x x x x x x x x x x x              (c)   2 2 2 2 2 2 2 2 2 cot cot 1 cos sin cos 1sin cos cos sin cos 2 1 2 cos 2 1 cos 2 1 2 cos 2 1 1 sec 22 x x x x x x x x x x x x x x               (d) 2 2 2 2 tan cot cot tan sin cos sin cos cos sin sin cos 1 cos 2 sec 2 x x x x x x x x x x x x x x       

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2021 End-of-Year Revision Package 92 (e)    sin sin 2 sin 3 cos cos 2 cos3 2sin 2 cos sin 2 2 cos 2 cos cos 2 sin 2 2cos 1 cos 2 2 cos 1 tan 2 x x x x x x x x x x x x x x x x x          2. Differentiate the following expressions with respect to x: (a) 3 24x x (b) 4(3 1) , 33 x xx   (c) cos(sin x) (d) sin1(1 – x) (e) 3ln 3 x x   (f) e e e x x x +  (a) ଵ ଷ(4𝑥−𝑥ଶ)ିమ య(4−2𝑥) (b) (௫ିଷ)(ସ)(ଷ௫ାଵ)య(ଷ)ି(ଷ௫ାଵ)ర(ଵ) (௫ିଷ)మ =(9𝑥−37)(2𝑥+1)ଷ (𝑥−3)ଶ (c) −sin(sin𝑥)∙cos𝑥 (d) ଵ ඥଵି(ଵି௫)మ(−1)= ିଵ √ଶ௫ି௫మ (e) ୢ ୢ௫ቀଵ ଶቁ(ln(𝑥+3)−ln(𝑥−3)) =1 2൬ 1 𝑥+ 3− 1 𝑥− 3൰ = −3 𝑥ଶ−9 (f) ୢ ୢ௫ቀ ୣమೣ ୣమೣାଵቁ =(eଶ௫+1)(2eଶ௫)−eଶ௫(2eଶ௫) (eଶ௫+ 1)ଶ = 2eଶ௫ (eଶ௫+1)ଶ 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 2 1e cos (3 )x x with respect to x, where 1 1 3 3x   . [3]   2 1 2 2 1 2 2 1 2 d 3e cos (3 ) e 2e cos (3 )d 1 3 3 e 2cos (3 )1 9 x x x x x xx x x x                     (b) Given that  x y a , where 0, 0x y  and a is a positive constant , show that 2 2 d d2 1 0. dd y yx xx       [4]

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2021 End-of-Year Revision Package 93 2 2 2 Differentiating with respect to , 1 1 d 0d2 2 d 0d d d (Note that 0)d d d d Differentiating once more with respect to , d d d2 d dd x y a x y xx y yy x x yy yx yx x x yx yx x y y yx x xx                                    2 2 2 2 2 2 2 d d d d d2 1 0d d d d d d0 or 2 1 0d d d d(reject as 0)d d dHence, 2 1 0dd y x y y yxx x x y y y xx x x y x y yx xx                     Alternate Solution 2 2 2 d 1 (1)d d (2)d 2 x y a y x a ax y a x x y a x x x           2 2 d dLHS 2 1 dd 2 1 1, Substitute (1) & (2)2 1 1 0 (shown) y yx xx a ax x x x a a x x                        

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2021 End-of-Year Revision Package 94 2. 2017/Prelim/VJC/P2/Q1 A curve C is defined by the parametric equations 2, ,1 1t tx yt t   where t takes all real values except 1. Find ddyx, leaving your answer in terms of t. [3]      22 22d d dd d d1 2 1 11 12y y xx t tt t t t tt tt t        (i) Show that the equation of the tangent to C at the point 2,1 1p pp p     is  22 .y p p x p   [2] (i) At point 2,1 1p pp p    , t p Equation of tangent at point 2,1 1p pp p    ,      222 3 22221 1221 1 11212p py p p xp pp p py p p xp p pp py p p xpy p p x p                  (ii) Find the acute angle between the two tangents to C which pass through the point 2,5. [3] (ii) Tangents pass through 2,5   225 2 24 5 05 or 1p p pp pp p       Equations of tangents are 3 1 and 15 25y x y x    Required acute angle between the 2 tangents    1 1tan 15 tan 30.255 rad or 14.6    2,5

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2021 End-of-Year Revision Package 95 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] Differentiating   222x y x y   _____(1) implicitly with respect to x,  d d2 2 2 1d d y yy x yx x         Where tangent is parallel to the x-axis, d 0d y x . 2 2 1 _____(2) x y y x     Sub (2) in (1),          2 2 2 2 2 1 1 4 2 0 4 4 4 2 2 22 x x x x x x x                When 2 2x  , 1 2 2 1 2y      When 2 2x  , 1 2 2 1 2y      The tangents are 1 2y  and 1 2y  . (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]          d d2 2 2 1d d d2 2 2 2 d d2 2 2 2 d 1d d 2 y yy x yx x yx y x y x yx y x y x x yy x x y                    When 0, 0x y  , d d 1 0 y x   . Hence normal to C at the origin is 0y . When 2, 2x y   , d 1 d 2 y x Tangent to C at A2, 2 ,   12 2 2y x     Where the normal and the tangent intersect,  12 2 2 2 x x      Area of triangle OAB  21 2 2 2 units2  2, 2

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2021 End-of-Year Revision Package 96 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 mk h 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  31π 10 100012h   . [4] (ii) Find the rate of change of h, in terms of k, when 120πV. [4] (i) Let h be the height of the smaller cone 50.510tanhh   Let R and H be the radius and height of the bigger cone respectively. 0t .an 52RHH R   Also, we have 10H h        22223331 1π π 5 103 31π 25031π 2503 21π 2503 41π 1000121π 10 100012V R HR HHHHHh                    (ii)   22d 1π 3 10d 12π104Vhhh    

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2021 End-of-Year Revision Package 97    2 2 d d d d 1π 10 d d d d d 4 4 π 1d 0 V V t h h t h t h t h k h h k h          When 120πV ,     3 3 3 3 3 1 π 10 1000 120π12 10 1000 1440 10 2440 10 2440 2440 10 h h h h h               . When 3 2440 10h  ,   3 23 d 4 2440 10 d π 2440 0.0131 m/min (to 3 s.f.) h k t k    5. 2017/Prelim/HCI/P1/Q6 A particle moving along a path at time t, where 30 t   , is defined parametrically by cot 3x t and 2 cosec3 1y t  . (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, 2sin3 1p . [4] 2dcot 3 3cosec 3d xx t t t   d2cosec3 1 6cosec3 cot 3d yy t t t t    d d 2dd d 6cosec3 cot 3 d 3cosec 3 2cot 3 cosec3 2cos3 y t xt y t t x t t t t      At point P, d | 2cos3d t p y px   Equation of tangent at P: 2cosec3 1 2cos3 cot 3y p p x p    When tangent meets y-axi