RI Maclaurin Series C7C Tut Sect A (Soln)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 _________________________________________ Tutorial 7C: Maclaurin Series and Binomial Expansion Page 1 of 3 Tutorial 7C: Maclaurin Series Section A (Basic Questions) 1 EJC Promo 9758/2022/Q2 It is given that (a) Show that 2 2 dd 45 .dd yy yxx  [3] (b) Find the Maclaurin series for y up to the term in 2x . [2] (c) Hence, show that the Maclaurin series for ec o sx x is 211, 4xx up to the term in 2 .x [2] (a) 2ec o sxyx Differentiate wrt x:     22 2d es i n c o s 2 e e s i n 2 ( 1 )d xx xy xx x yx       Differentiate wrt x:   2 22 2 2 2 2 2 dd ec o s s i n2 e 2dd dd d 22 2dd d dd 4 5 (shown)dd xxyy xxxx yy y yyxx x yy yxx         (From (1):  2 des i n 2 d x yxy x ) (b) 0, 1xy , d 2d y x  ,  2 2 d 42 5 ( 1 ) 3d y x   Thus, 22331 2 ... 1 2 ...2! 2yx x x x      (c)   1 2 2 1 22 2 22 22 2 e cos e cos 31 2 ... 2 11 13 3 221 2 2 ...22 2 ! 2 31 11 4 ... 1 ...48 4 xx xx xx xx xx xx x xx                       2ec o s .xyx

Raffles Institution H2 Mathematics 2025 Year 5 _________________________________________________________________________________________ _________________________________________ Tutorial 7C: Maclaurin Series and Binomial Expansion Page 2 of 3 2 TMJC Promo 9758/2022/Q4 It is given that  4f 12 xx x   . (i) Using standard series from the List of Fo rmulae (MF 27), find the series expansion for  f x , up to and including the term in 2 .x Give the coefficients as exact fractions in their simplest form. [4] (ii) Hence, by using 1 6x , find an estimation for 6. Leave your answer as an exact fraction in its simplest form. [2] (i)         1 2 1 2 2 22 2 22 4f 12 41 1 24 11 121 222 1 ... 1 1 2 2 ...24 2 ! 4 2 ! 2 1 2 4 ... 8 4 128 17 5432 ... up to 46 4 xx x x x xx xx xx xxx xx x                                             (ii) Using 1 6x , 2 14 6 17 1 543 121 466 4612 6 25 22616 2 768 3 5 2261 11526             61 1 5 2 52 2 6 1 57606 2261   OR 5 6 2261 6 1152 22616 960   Note: Need not give both answers, just one of the above answers will do,

Raffles Institution H2 Mathematics 2025 Year 5 _________________________________________________________________________________________ _________________________________________ Tutorial 7C: Maclaurin Series and Binomial Expansion Page 3 of 3 3 MI PU2 P2 Promo 9758/2022/Q2 In triangle ABC , angle A is 4    radians and angle B is 1 3 radians. Show that when  is sufficiently small for terms in 3 and higher powers of  to be neglected,  26 12 AC kBC   where k is a constant to be found. [6] 3 Using sine rule,   1 sin sin sin sin sin 3 sin 4 3 2 sin cos cos sin44 3 2 1 cos sin 2 6 cos sin2 AC BC AC B BA B CA AC BC                        Therefore, when  is sufficiently small, 12 12 222 2 2 2 6 122 6 122 6 1 ...22 2 6 122 63 122 AC BC                                where 3 2k  B A C