DHS Y4 Math 2 Supplementary Worksheet Binomial Theorem

Math 1_Binomial Theorem (Compiled by Lim CH & J Goh) 1 Year 4 Mathematics 1 Binomial Theorem Supplementary Worksheet 1 Name : _________________________________ ( ) Class : _____ Date : __________ 1 (i) Write down and simplify the fourth term, in descending powers of x , in the expansion of n bax x  − . [2] (ii) If this fourth term is independent of x , find the value of n . [2] (iii) With the value of n found in part (ii), calculate the value of ab if the fourth term is equal to −160. [2] [2008 ACS (Barker Road) AMaths P1 (modified)] 2 Given that the coefficient of 4x in the expansion of 8 3 mx x  − is –13608, (i) find the value of the constant m , [4] (ii) hence find the term independent of x in the expansion of 4 21 x + 8 3 mx x  − . [3] [2008 ACS (I) AMaths P2] 3 Write down and simplify the first 3 terms, in ascending powers of x , in the expansion of 5 2 3 x − . Given that the first three terms in the expansion of ( ) 5 2 321      − + +xx px are 22 32qx qx+ − , find the value of p. [6] [2008 Anderson Sec AMaths P1] 4 (i) Write down the first three terms of the expansion, in ascending powers of x , of ( ) nax+1 , where a and n are constants. [2] (ii) Hence find the value of a and of n if ( ) 21 1 20 195 ... n ax x x+ = −+ + , taking up to the term in 2x . [6] [2008 Anglican High AMaths P1] 5 Given that ( ) 231 1 32 448 ... n px x x qx+ = − + ++ , find the value of p, of q and of n. [6] [2008 CHIJ St Nicholas Girls’ AMaths P1]

Math 1_Binomial Theorem (Compiled by Lim CH & J Goh) 2 6 (i) Write, in ascending powers of x, the first three terms in the expansion of .32 4      − x [2] (ii) The coefficients of x and x2 in the expansion of 4 2 32 ) (      − −xx p are in the ratio of 2 : 3. Find the value of p. [3] [ 2008 Balestier Hill Sec AMaths P2] 7 (i) Write down the first three terms in the expansion, in ascending powers of x , of 82 21       + px , where p is a constant. [2] ( ii) The first three terms in the expansion of 82 2 21 ) 2 (       + +pxqx , where p and q are integers, are 4 22 5 2x x+ − . Find the values of p and q , and show that 0< −q p .[6] [ 2008 Cedar Girls’ Sec AMaths P1] 8 Find in ascending powers of x , the first three terms in the expansion of (i) ( ) 5 2 x+ , (ii) ( ) 6 1 ax+ . Given that the coefficient of 2x in the expansion of ( ) 5 2 x+ ( ) 6 1 ax+ is 2960, find the possible value(s) of a . [5] [ 2008 CHIJ Toa Payoh Sec AMaths P1] 9 The coefficient of x in the expansion of n x     + 32 is 3 times the coefficient of 2x . (a) Find the value of n . [3] (b) Hence evaluate the coefficient of 2x in the expansion ( ) n xx      + −32 3 2 . [2] [ 2008 Commonwealth Sec AMaths P1] 10 (i) Find the term independent of x in the expansion of 8 32      − xx . [3] (ii) Expand 2) 3 1 ( ) 1 (x xn − + in ascending powers of x up to the term 2x , where n is a positive integer greater than 1. If the coefficient of 2x is 3, find the value of n . [4] [2008 Crescent Girls’ AMaths P1]

Math 1_Binomial Theorem (Compiled by Lim CH & J Goh) 3 11 (i) Expand ( ) 1021 xx++ in ascending powers of x up to the term in 3x . [3] (ii) Hence, evaluate ( ) 10 1.0101 correct to 3 decimal places. [3] [ 2008 Clementi Woods Sec AMaths P2] 12 (a) Write down the first three terms in the expansion, in ascending powers of x , of 8 2 2 x − . [2] (b) The coefficients of x and 2x in the expansion of ( ) 8 224 2 x ax bx − ++ are 1536 and 512− respectively. Calculate the value of a and of b . [5] [ 2008 Holy Innocents High AMaths P1] 13 Write down and simplify, in ascending powers of x, the first three terms of the expansion of (a) 5 21      + x , (b) ( ) 5 2 3x− . Hence, or otherwise, obtain the first three terms of the expansion of 5 2 23      − −xx and use it to estimate the value of ( ) 5 94 . 2 correct to 3 decimal places. [8] [ 2008 Maris Stella High AMaths P1] 14 In the expansion of 9 2 2  − px x where p is a positive constant, the term independent of x is 5376. (i) Show that p = 4 . [4] (ii) With this value of p , find the coefficient of 9x in the expansion of ( ) 9 29 912  −+ pxx x . [4] [2008 Ngee Ann Sec AMaths P1] 15 (i) Write down the first three terms in the expansion, in descending powers of x , of 6 2ax x  + , where a is a constant. [2] (ii) The first three terms in the expansion, in descending powers of x , of ( ) 6 2 21 abx x x ++  where a and b are integers, are 86 4 25 264 ...bx x x++ + . Find the value of a and of b . [5] [ 2008 Singapore Chinese Girls’ AMaths P2]

Math 1_Binomial Theorem (Compiled by Lim CH & J Goh) 4 16 (a) The coefficient of 3x in the expansion ( ) 10 51 2 xax +−  is 30− . Find the value of a. [4] (b) In the expansion of n xx      − 22 , the fifth term is independent of x . Find the value of n. [3] [ 2008 Tanjong Katong Sec AMaths P2] 17 (a) Expand ( ) 5 2 p− completely. Use this result to writ e down the expansion of 5 22 2 x x −+ , in ascending powers of x , as far as the term in 2x . [4] (b) Find the coefficient of 8x in the expansion of ( ) 10 10 221 x x −+  . [4] [ 2008 Temasek Sec AMaths P2 (modified)] 18 (i) Find, in the simplest form, (a) the full expansion of 5) 3 1 (x− , [2] (b) the coefficient of 3x in the expansion 5) 3 1 )( 2 1 (x x− + . [2] (ii) Find the term independent of x in the expansion of 18 3 19      − xx , expressing your answer in the form ba 3× − where a and b are positive integers. [4] [2008 Unity Sec AMaths P2] 19 (i) Write down the first three terms in the expansion, in ascending powers of x , of ( ) 6 3 ax+ , where a is a constant. [2] (ii) The first three terms in the expansion of ( )( ) 6 33b x ax−+ are 22187 2187 x cx++ . Find the values of the constants a , b and c . [4] [2008 Victoria School AMaths P1] 20 Write down the first three terms in the expansion, in descending powers of x , of 8 3 bax x  − . Hence evaluate the term independent of x of 8 3 15x x  − . [4] [2008 Zhonghua Sec AMaths P1] 21 Write down and simplify the following binomial expansions. (a) ( ) 3 2 x− (b) 4( 2)xy+

Math 1_Binomial Theorem (Compiled by Lim CH & J Goh) 5 22 In the expansion of 10 2 3 2    − xx , find (i) the coefficient of 5 1 x , (ii) the constant term. 23 (i) Determine the coefficients of 5x and 7x in the expansion of 9 2 xx  + . (ii) Given that the coefficient of 7x in the expansion ( ) 9 22 1x pxx  ++ is 1746, find the value of p, where p is a positive real number. 24 Expand ( ) 1223 2 1x x− + in ascending powers of x up to the term in x3. Hence estimate the value of (1.0197)12, leaving your answer correct to 4 significant figures. 25 In the binomial expansion of n x     + 41 in ascending powers of x , the coefficient of the third term is twice that of the fourth term. Calculate the value of n. Hence, evaluate the middle term of this expansion. 26 In the expansion of ( ) 19 2 3+x , the coefficients of the term in kx and the term in 1kx + are equal. Calculate the value of k . 27 It is given that in the expansion of ( ) 621 xxαβ++ in ascending powers of x , the coefficient of x2 is zero and the coefficient of x3 is −440. Determine the value of α and of β . 28 (a) Find the fifth and sixth terms, in ascending powers of x , in the binomial expansion of 9 2 3 11      − x . (b) Hence find the coefficient of 10x in the expansion of ( ) 1 33 1