RI Maclaurin Series C7C Add Prac (Qn)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 _______________________________________________ Additional Practice Questions for Chapter 7C: Maclaurin Series Page 1 of 6 Additional Practice Questions for Chapter 7C: Maclaurin Series 1 (a) Find the fifth term in the expansion of 6 3 1 2x x   in descending powers of x . [2] (b) Find the term independent of x in the expansion of 18 19 3 x x   . [3] (c) Find the coefficient of 18x in the expansion of 12 2 13x x   . [3] (d) Expand 21 0(2 3 4 )xx , up to the term in 3x . [3] (e) The coefficient of 2x in the expansion of 6(2 )xk is equal to the coefficient of 5x in the expansion of 8(2 ) kx . Find k . [3] (f) The expansion of 1 n kx up to and including the term in 3x are 22 313 6 6 6xk x h x  , where n is a positive integer. Find the values of n , k and h . [4] 2 Express   2 2 25 1 5 21 2 xx xx   in partial fractions. [2] Hence or otherwise, obtain the expansion of   2 2 25 1 5 21 2 xx xx   in ascending powers of x , giving the first five terms in the expansion. [3] Find the exact set of values of x for which the expansion is valid. [2] 3 MI Promo 9758/2020/PU2/P1/Q6 (a) Using standard series from the List of Formulae, expand   3el n 1x ax as far as the term in 3x , where a is a non-zero constant. Given that there is no term in 2 ,x determine the coefficient of 3.x [5] (b) Find the expansion of 2 13 9 x x   in ascending powers of x, up to and including the term in 3x . State the set of values of x for which the expansion is valid. [4]

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ ________________________________________________ Additional Practice Questions for Chapter 7C: Maclaurin Series Page 2 of 6 4 9740/2015/01/Q6 (i) Write down the first three non-zero terms in the Maclaurin series for ln(1 2 ) x , where 11 22 x , simplifying the coefficients. [2] (ii) It is given that the three terms found in part (i) are equal to the first three terms in the series expansion of (1 ) cax bx for small x. Find the exact values of the constants a, b and c and use these values to find the coefficient of 4x in the expansion of (1 ) cax bx , giving your answer as a simplified rational number. [6] 5 Given that xx y   121 1 where 0.5x . Show that, provided x is non-zero, then   1 12 1yx xx   . [1] Hence (i) express y as a series of ascending powers of x up to and including the term in 2x [4] (ii) show that 160000 79407 101102 10   . [2] 6 Expand 1 12 x x   in ascending powers of x up to and including the term in 2x . [4] (i) In an attempt to estimate 2 , a student substituted 1x  in the above expansion. Explain why this does not give a good estimate. [ 1] (ii) By putting 1 9x in the expansion, show that 2 1296 p , where p is a positive integer to be determined. [2] 7 For ,0n the expansion of , )1( nmx  in ascending powers of x, is .....4881 2  xx . (i) Find the constants m and n. [ 5] (ii) Show that the coefficient of 400x is in the form ka )4( where a and k are constants to be determined. [2] 8 RIPromo9740/2015/Q8 (a) (i) Expand   2 21f 2 1 x x x    as a series in ascending powers of x up to and including the term in 2x . [3] (ii) State the equation of the tangent to the curve fyx at the origin.[1] (b) Using the standard series given in the List of Formulae (MF26) or otherwise, show that the first three non-zero terms in the Maclaurin series for 1e x can be expressed as  3e1 . . .px qx  where p and q are constants to be determined. [6]

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ ________________________________________________ Additional Practice Questions for Chapter 7C: Maclaurin Series Page 3 of 6 9 RVHS Promo 9758/2020/Q3 It is given that  1 .tan 1 e xy  (i) Show that 2de. d 1x y x y [1] (ii) By further differentiation of the expression in part (i), find the Maclaurin series for y, up to and including the term in 2x . [4] (iii) Hence, or otherwise, find the Maclaurin series for 21 ex y , up to and including the term in x . [1] 10 EJC Prelim 9758/2020/01/Q11 (a) The angular diameter of an object is the angle the object makes (subtends) as seen by an observer. As shown in the diagram below,  denotes the angular diameter (measured in radians) of a circle whose plane is perpe ndicular to the line between the point of view (point P) and the centre of said circle. D denotes the distance from point P to the centre of the circle and d denotes the diameter of the circle. (i) Show that, if  is sufficiently small, Dd  . [2] The equation in (i) is often used in astronomy to estimate the diameters of stars from the angular diameter, assuming their shapes to be approximately circular. (ii) If the angular diameter of a star is measured to be 0.00873 rad and the distance of the star from the earth is 129.46 10 km, estimate the diameter of the star. [1] D d P 

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ ________________________________________________ Additional Practice Questions for Chapter 7C: Maclaurin Series Page 4 of 6 (b) An astronaut A is at a large distance x km from the surface of the earth. The radius of the earth is assume d to be a constant R km. The furthest point on the earth’s surface that the astronaut can see is a point P such that AP = y km and the angle OAP = , where O is the centre of the earth (see diagram). (i) Show that 1 221 Ryx x  . [3] (ii) It is given that R is small compared to x. Show that, if R x  , 23tan 1.5    . [4] (iii) It is also given that 0.0345 rad and the astronaut A is 180,000 km from the surface of the earth, find  and hence estimate the radius of the earth. Leave your answer to the nearest km. [3] 11 If )2ln(cos xy  , prove that 22 2 dd 4dd yy xx   . Hence, by repeated differentiation of this result, or otherwise, obtain the Maclaurin expansion of y in ascending powers of x up to and including the term in 4x . Using 6x  , show that 22 ln 2 2 36 27   . [10] 12 Given that x y 2sin1 2   , wherever it is defined, show that (s i n ) c o s12 2 0  x y x yxd d . Hence, by further differenti ation of this result, obtain the Maclaurin expansion of y in ascending powers of x , up to the term in 3x . By putting 12x  , find an approximate value of 2 3 , correct to 2 decimal places. [10] R R P y A O x

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ ________________________________________________ Additional Practice Questions for Chapter 7C: Maclaurin Series Page 5 of 6 13 RI Prelim 9758/2020/02/Q4 Given that 1tan 2e, x y    show that  2 d42 . d yx yx [2] (i) By repeated differentiation of the above result, find the Maclaurin series for 1tan 2e x   up to and including the term in 3.x [5] (ii) Hence find the Maclaurin series for  1 2 2 tan e 1 x x     up to and including the term in 2.x [ 3 ] 14 ACJCPrelims9740/2006/01/Q16OR Let  ln 1 tanyx (i) Show that d e2 e 2d yyy x   . [3] (ii) By differentiating this result repeatedly, show that   232 32 dd d e2 e e2 edd d yy yyyy y x xx 