RI Maclaurin Series C7C Tut (Qn)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 _________________________________________ Tutorial 7C: Maclaurin Series Page 1 of 4 Tutorial 7C: Maclaurin Series Section A (Basic Questions) 1 It is given that 2ec o s .xyx (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, 4x x up to the term in 2 .x [2] [(b) 2312 . . . 2xx  ] 2 It is given that  4f 12 xx x   . (i) Using standard series from the List of Form ulae (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) 217 5432 ... 46 4xx  (ii) 5760 2261or 2261 960 ] 3 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 2k  ]

Raffles Institution H2 Mathematics 2025 Year 5 _________________________________________________________________________________________ _________________________________________ Tutorial 7C: Maclaurin Series Page 2 of 4 Section B (Discussion Questions) 1 (i) Given that f( ) l n ( 1 2s i n )x x , find f (0), f '(0), f ''(0) and f '''(0). Hence write down the first three non-zero terms in the Maclaurin series for f(x). [7] (ii) The first two non-zero terms in the Maclaurin series for f( x) are equal to the first two non-zero terms in the series expansion of es i nax nx . Using appropriate expansions from the List of Formulae, find the constants a and n. Hence find the third non-zero term of the series expansion of es i nax nx for these values of a and n. [5] [(i) 0, 2, –4, 14 ; 23 722 3x xx (ii) a = –1 ; n = 2 ; 31 3 x ] 2 It is given that  2ln 2 e xy  . (a) Show that d 4e 2d yy x  . [2] (b) Hence find the Maclaurin series for y, up to and including the term in 2x . [3] [(b) 224xx  ] 3 It is given that  sin ln 1 2yx  . (i) Show that   2 2 2 dd12 2 12 0dd yyxx k yxx    , where k is a constant to be found. Hence, find the first three non-zero terms of the Maclaurin expansion for y . [6] (ii) Using standard series from the List of Formulae, verify the correctness of your result from part (i) up to and including the term in 3x . [2] Explain why the expansion is not valid when 1 2x . [1] [(i) 4 ; 23 42 2 ... 3xx x ] 4 (a) A curve has equation 11t a nyx  . Show that  22 22 2 dd dd 1 yy xy xx x    . By further differentiation, obt ain the Maclaurin series for y, up to and including the term in 3x . State the equation of the tangent to the curve at 0x . [6] (b) Given that x is a sufficiently small angle, show that 2 12 c o s 3 1 6 xx    . It is also given that 11 0 s i n 1 2 c o sx x  , form a quadratic equation in x and solve it. [4] [(a) 2311 51 28 4 8x xx  ; 11 2yx (b) 23 10 1 3 0; 0.07316 xx x    ]

Raffles Institution H2 Mathematics 2025 Year 5 _________________________________________________________________________________________ _________________________________________ Tutorial 7C: Maclaurin Series Page 3 of 4 5 Let f( ) e s i nxx x . (i) Sketch the graph of y = f(x) for 33 x  . [2] (ii) Find the series expansion of f( x) in ascending powers of x, up to and including the term in 3x . [3] Denote the answer to part (ii) by g( )x . (iii) On the same diagram as in part (i), sketch the graph of y = g(x). Label the two graphs clearly. [1] (iv) Find, for 33 x  , the set of values of x for which the value of g( )x is within 0.5 of the value of f(x). [3] [(ii) 23 1 ...3xx x  (iv)  1.96,1.56 ] 6 (a) Expand 8(1 ) y in ascending powers of y, up to and including the term in 3y . [1] In the expansion of 28(1 ) x kx in ascending powers of x, the coefficient of 3x is zero. Find the value of the constant k. [3] (b) Find the first two terms in the expansion of 1 2(9 ) 12 x x   in ascending powers of x. [4] (c) Find the set of values of x for which the expansion in part (b) is valid. [2] [(a) 23 81 8 28 56 ...yy y y    ; 1k  ; (b) 35 63 ... x (c) 11 22 x ] 7 It is given that f( )yx is a function such that f( 0 ) 1 and 2d 1d y yx  . Find the first three non-zero terms in the series expansion of f( ) 9 x x . [4] [ 21 37 457131 8 2 1 6x x ] 8 (a) In the triangle ABC, AB = 1, BC = 3 and angle ABC  radians. Given that  is a sufficiently small angle, show that 1 222(4 3 )ACa b     , for constants a and b to be determined. [5] (b) Given that x is small, show that 2tan 2 cos 2 2 1 4     x xx x [3] [(a) 32, 4ab ]

Raffles Institution H2 Mathematics 2025 Year 5 _________________________________________________________________________________________ _________________________________________ Tutorial 7C: Maclaurin Series Page 4 of 4 9 In the triangle ABC, AB = 1, angle BAC = radians and angle ABC = 3 4  radians. (i) Show that 1 cos sinAC    . [4] (ii) Given that  is a sufficiently small angle, show that AC 21, ab  for constants a and b to be determined. [4] [(ii) a = 1 , b = 3/2 ] 10 An arc PQ of a circle of radius r subtends an angle    radians at the centre O. The length of chord PQ is L units and the area of the shaded segment PRQ is A square units. (i) Show that  22 21 c o sLr  . [1] (ii) Given that  is small and 2Ak L , show that 222 8 0kk   . [4] Q O P R r