TMJC H2 Chapter 9 Maclaurin Series Assignment Suggested Solutions 2024

Chapter 9 Maclaurin Series TMJC 2024 Page 1 of 10 H2 Mathematics (9758) Chapter 9 Maclaurin Series Assignment Suggested Solutions 1 2015(9740)/I/6 (i) Write down the first three non -zero terms in the Maclaurin series for  ln 1 2 x , where 1 1 2 2 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 c ax 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 c ax bx , giving your answer as a simplified rational number. [6] (i)       2 3 2 3 ln 1 2 3 82 2 3 2 22 2 x x x xx x x           (ii)       2 2 2 3 ( 1)1 1 2 ( 1) 2 c bx bx bxax ax c cax abcx a c cc b x               By comparing part (i) and (ii), Coefficient of x: 2a Coefficient of 2x : 12 ( 2)abc b a c     Coefficient of 3x : 2( 1) 8 2 3 c c ab  As 1b c , 2 ( 1) 1 8 22 3 c c c        2 1 8( 1) 3c c c      1 8( 1) 3c c      Replace by 2 x x in  ln 1 x in MF26 Replace by x bx and by n c in  1 n x in MF26

Chapter 9 Maclaurin Series TMJC 2024 Page 2 of 10 3 3 8 3 5 c c c    5 3b  Coefficient of 3 4 3 3 8 13 ( 1)( 2) 5 5 5 523! 3! 3 c c cx a b                                104 27 Question only wants coefficient, so don’t give the whole term, i.e. you should not have the ݔସ

Chapter 9 Maclaurin Series TMJC 2024 Page 3 of 10 2 2019(9758)/ACJC Promo/Q7a & 2019(9758)/MI Promo/Q8 (a) Given that  is sufficiently small, show that 2tan 3 4 4 33           . [3] (b) It is given that  f ( ) ln 1 sinx x  . (i) Without the use of a calculator, find f (0), f (0) and f (0). Hence write down the first two non-zero terms in the Maclaurin series for f ( )x . [5] (ii) By substituting 6 x  , show that 2 ln 2 72 6    . [3] (iii) John wants to use the Maclaurin series found in part (i) to estimate the value of f ( )x when x a , where a is a real number close to zero. Suggest one recommendation for John to improve the accuracy of his estimation. [1] 2 ACJC JC1 Promo 9758/2019/Q7a , (a)                2 1 2 2 2 2 tan tan3tan 3 1 tan tan3 3 tan 1 3 tan 3 3 1 23 1 1 ... 2! 3 1 3 3 3 3 ... 3 3 3 3 3 ... 3 4 4 3 ... 1 1 3 3                                                                  Note that 3      is not small, so you cannot do tan 3 3          We need to apply result from MF26. Recall that tan  when  is sufficiently small. Replace by 3 x  in  1 n x