Raffles Institution 2022 Y4 RP Math Supp WS

Page 1 of 4 RAFFLES INSTITUTION MATHEMATICS DEPARTMENT 2022 YEAR 4 RP MATHEMATICS TOPIC 3B: DIFFERENTIATION (MATHS 2) SUPPLEMENTARY WORKSHEET Name: Class: Sec 4 ( ) Date: 1 2020/Y4RP/M2/T1/Q1 Differentiate ( ) 5ln cos 3 x with respect to x, leaving your answer as a single trigonometric term. [2] [Ans: 15 tan 3x− ] 2 2020/Y4RP/M2/T1/Q3 Given that tan 5y x= , show that 2 2 d d y x can be written in the form d d yky x , where k is a constant to be determined. [4] [Ans: k = 10] 3 2019/Y4RP/M2/T1/Q1 Differentiate the following with respect to x, simplifying your answer: (i) 2 sin 3x x , [2] (ii) 23eln cos3 x x       . [2] [Ans: (i) 3 3 cos3 2sin 3x x x x − (ii) 6 3tan 3x x+ ] 4 2019/Y4RP/M2/T1/Q4 (i) By writing sec x as ( ) 1 cos x − , show that ( )d sec sec tand x x xx = . [1] (ii) Hence find the following in terms of sec x , (a) ( )d sec tand x xx ’ [2] (b) ( )d ln sec tand x xx +  . [2] [Ans: (ii)(a) ( ) 2sec 2sec 1x x − (b) sec x ] 5 2018/Y4RP/CT/Q1 Differentiate each of the following with respect to x, leaving your answer in the simplest form. (b) 5tan 5 2x( ) . [2] [Ans: (b) ( ) ( ) 4 250 tan 2 sec 2x x ] 6 2018/Y4RP/CT/Q6 It is given that 1 siny x=− . Downloaded by chocomint (mohamadamiyaz@gmail.com) lOMoARcPSD|19327920

Page 2 of 4 (i) Find an expression for d d y x , leaving your answer in the simplest form. . [2] (ii) Show that 2 2 d d y kyx = , where k is a constant to be determined. [4] [Ans: (i) cos 2 1 sin x x − − (ii) 1 1 sin4 x−− ] 7 2017/Y4RP/CT/Q1 A curve has the equation y = xtan(1- 3x). Find the gradient of the curve at the point where 1x= . [3] [Ans: − 15.1] 8 2017/Y4RP/CT/Q3 Given that y = cos2 2x , find dy dx and d 2 y dx2 , simplifying your answers. Hence show that d2 y dx2 æ èç ö ø÷ 2 +16 dy dx æ èç ö ø÷ 2 - 128y = k cos4x, where k is a constant to be determined. [5] [Ans: 2 2 d d 2sin 4 , 8cos 4d d y y x xx x=− =− ] 9 2016/Y4RP/CT/Q1 Differentiate each of the following with respect to x, expressing your answers as a single fraction. (ii) ( ) 4 2 3 1 tan 5 x x + . [3] [Ans: (ii) ( ) ( ) 3 2 3 2 3 1 6 tan 5 5 3 1 sec 5 tan 5 x x x x x + − + ] 10 2016/Y4RP/CT/Q4 It is given that y = 1+ cosx . (i) Find dy dx , leaving your answer in its simplest form. [1] (ii) Show that d2 y dx2 = - 1 4 y . [4] [Ans: (i) sin 2 1 cos x x − + ] 11 2015/Y4RP/CT/Q2 Given that ( )ln 1 siny x=+ , (a) write down an expression for dy dx in terms of x, [1] (b) show that d2 y dx2 = - secx dy dx æ èç ö ø÷ . [3] [Ans: (a) cos 1 sd i d n y x x x= + ] Downloaded by chocomint (mohamadamiyaz@gmail.com) lOMoARcPSD|19327920

Page 3 of 4 12 2015/Y4RP/CT/Q4 For each of the following functions, find dy dx , expressing your answers as a single fraction (a) y = e2x+1 cos3 4x , [4] [Ans: (a) ( ) 2 1 4 2e cos 4 6sin 4 cos 4 x x x x + + ] 13 2014/Y4RP/CT/Q1 Find dy dx for each of the following functions, expressing your answers as a single fraction. (a) y = xcosec2 x , [3] [Ans: (a) 3 sin 2 cos sin x x x x − ] 14 2013/Y4RP/MYCT/Q2 If x xxy cos cossin += , show that ( ) 0d d12d d 2 2 =−− x yyx y . [4] 15 2013/Y4RP/MYCT/Q3 Differentiate xe x 4sin2 41 2− with respect to x, simplifying your answer. [3] [Ans: 2 1 34 sin 4 (8cos 4 sin 4 )xe x x x− + ] 16 2012/Y4RP/T2/Q1 Differentiate the following with respect to x, simplifying your answers. (b) 3sinx x , [2] [Ans: (b) ( ) 2sin 3 cos sinx x x x + ] 17 2009/Y4RP/T3/Q2 Differentiate the following with respect to x, simplifying your answers. (b) 2 33 cos 2 3x x  +  , [3] [Ans: (b) 26 cos 2 cos 2 3 sin 23 3 3x x x x x        + + − +           ] 18 2008/Y4RP/T3/Q1 Differentiate the following expressions with respect to x, giving your answers in the simplest form possible: (c) 3 tan 2 ,xe x [2] (d) sin 3 .3 x x [2] [Ans: (c) ( ) 3 2 3tan 2 2sec 2 xe x x + (d) 2 3 cos3 sin 3 3 x x x x − ] 19 2007/Y4RP/T3/Q1 Differentiate the following with respect to x, simplifying your answers. (c) 3 tan 34 4x x   +−        [2] Downloaded by chocomint (mohamadamiyaz@gmail.com) lOMoARcPSD|19327920

Page 4 of 4 [Ans: (c) 23 tan 3 3 sec 34 4 4x x x        − + + −            ] 20 2007/Y4RP/T3/Q2 Find ( ) 4 4sin cosd x xdx + , giving your answer as a single trigonometric function. [3] [Ans: sin 4x− ] 21 2007/Y4RP/T3/Q3 Find the value of t, where 0 1 t , for which ( ) 2 2 cos 2 1d tdt = . [3] [Ans: 0.912] 22 2006/Y4RP/T3/Q1 Differentiate each of the following with respect to x, simplifying your answer. (a) ( ) ( ) 2ln tan 1 3x x − , [3] [Ans: (a) ( ) 2tan(1 3 ) tan 1 3 6 ln sec (1 3 ) 2 x x x x x x − − − − ] 23 2005/Y4RP/T3/Q1 Differentiate the following with respect to x, simplifying your answers. (a) 2sin x , [1] [Ans: (a) 22 cosx x ] 24 2004/S4AM/T2/Q1 Differentiate each of the following with respect to x, simplifying your answers as far as possible. (b) ( ) 3 2sec 2 1x + [3] [Ans: (b) ( ) ( ) 2 3 212 tan 2 1 sec 2 1x x x ++ ] 25 2004/S4AM/T2/Q2 Find the value of a and b for which ( ) 2 2 sin sin tan 3tan cos 3tan cos d x a b x x dx x x x x + = + + . [4] [Ans: a = 1 and b = −3] 26 2004/S4AM/T2/Q2 (V2) Given ( ) 3 2sec 2 1y x=+ , find dy dx and simplify your answers as far as possible. Calculate the gradient of the curve at the point where 2x= , giving your answer correct to 2 decimal places. [4] [Ans: ( ) ( ) 2 4 2 12 sin 2 1 cos 2 1 x x x + + , 14.35] 27 2003/S4AM/T2/Q1 Differentiate each of the following with respect to x, simplifying your answers as far as possible. (b) 22 3cos 4 x− , [2] (d) 2 tan 63x x , [2] [Ans: (b) 2 6sin 8 2 3cos 4 x x− (d) 2 26 (tan 6 2 sec 6 )x x x x + ] Downloaded by chocomint (mohamadamiyaz@gmail.com) lOMoARcPSD|19327920 Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org)