Supplementary Exercise For Test Preparation
1. Given that (1 + ax) n = 1 – 12x + 63x 2 + …, find a and n.
[Ans: a = –1.5, n = 8]
2. Find the coefficient of x22 in the expansion of (1 – 3x)(1 + x3)10.
[Ans: –360]
3. Write down the expression of (1 + 2x)(1 – x)8 as far as the term in x4 and use it to evaluate
(1.4)(0.88).
[Ans: 1 – 6x + 12x2 – 42x4, 0.2128]
4. Find the values of a if the coefficient of x2 in the expansion (1 + ax)4(2 – x)3 is 6.
[Ans: 0, 1]
5. Write down and simplify, in ascending powers of x, the first three terms of the following
expansion
(i) (1 + 2x)7
(ii) (2 – x)7
Hence find the coefficient of x2 in the expansion of (2 + 3x – 2x2)7
[Ans: 1 + 14x + 84x2, 128 – 448x + 672x2, 5152]
8
x

6. Find the first four terms in the binomial expansion, in ascending powers of x, of  2   .
 2
2
[Ans: 256 – 512x + 448x – 224x3]
6
7. Find the coefficient of the term in x
–2 in the
1 

binomial expansion of  x 3  2  .
2x 

[3][3][96]
[Ans:
15
]
16
8. Factorise 2 – 3x + x2. Hence expand ( x 2  3 x  2)6 in ascending powers of x as far as the term in
[Ans: (1 – x)( 2– x); 64  576x  2352x 2 ]
x2 .
[5][97]
9. Write down and simplify the first 3 terms in the expansion, in ascending powers of x, of
6

x 
2
 2   . Hence find the coefficient of x in the expansion of (1 – x)
30


2
4
2
[Ans: 64  6 x  x 2  ..... , 6 ]
5
15
3
6

x 
2  .
 30 
10. In the expansion of (1 + 2x)(a – bx) 12, where a  0, the coefficient of x 8 is zero. Find, in its
a
5
simplest form, the value of the ratio .
[Ans:
]
b
16
11. If (1 + ax + bx 2) 4 = 1 + 8x + 32x 2 + …, find a and b.
[Ans: a = 2, b = 2]
12. In the expansion of (2 + 3x) n, the coefficients of x 3 and x4 are in the ratio 8 : 15. Find the value
of n.
[Ans: 8]
13. Write down the third and fourth terms in the expansion of (a + bx) n. If these terms are equal,
n(n  1)an2b2 x 2 n(n  1)(n  2)an3b3 x 3
show that 3a = (n – 2)bx.
[Ans:
]
,
2
6
Note: Q14 – 15 require knowledge from ‘Enrichment 1’ in lecture notes
14. If x is so small that x 4 and higher powers of x are negligible, show that
1
 1  4 x 2  1  3x 
1
0.96 
3
 1.03 
2

2
3
 Ax 2  Bx 3 , where A and B are constants. Hence evaluate
to five decimal places.
1
[Ans:  1  4 x  2  1  2x  2x 2  4 x 3 ,  1  3 x 

2
3
 1  2x  5 x 2 
40 3
28
x , A  7, B  , 0.00069 ]
3
3
5x
in partial fractions.
(1  2 x )(1  3 x )
Hence, find the first four terms in the series expansion f ( x ) and state the range of x for which
15. Express f ( x ) 
the series converges.
[Ans:
1
1

,(1  2x )1  1  2 x  4 x 2  8 x 3  16 x 4  ...,
1  2x 1  3 x
1
1
(1  3 x )1  1  3 x  9x 2  27 x 3  81 x 4  ... , 5 x  5 x 2  35 x 3  65 x 4  ... ,   x  ]
3
3