A LEVEL (P1)
BINOMIAL
QUESTION'S
QUESTIONS
1
1
TOPIC:4 BINOMIAL
Find the coeff cient of x3 in the expansion of
(i) (1 + 2x)6 ,
[3]
(ii) (1 − 3x)(1 + 2x)6 .
[3]
5
2
3
4
5
6
2 .
Find the coefficien of x in the expansion of 3x − x
[4]
(i) Find the f rst 3 terms in the expansion of (2 − x)6 in ascending powers of x.
[3]
(ii) Find the value of k for which there is no term in x2 in the expansion of (1 + kx)(2 − x)6 .
[2]
The f rst three terms in the expansion of (2 + ax)n , in ascending powers of x, are 32 − 40x + bx2 . Find
[5]
the values of the constants n, a and b.
2 6
Find the coefficient of x2 in the expansion of x + .
x
[3]
(i)
3 Find the first three terms in the expansion of (2 + u)5 in ascending powers of u.
[3]
(ii) Use the substitution u = x + x2 in your answer to part (i) to find the coefficient of x2 in the
5
expansion of 2 + x + x2 .
[2]
7
8
Find the value of the coefficient of x2 in the expansion of x 2 6
+ .
2 x
[3]
(i) Find the first 3 terms in the expansion of (2 + 3x)5 in ascending powers of x.
[3]
(ii) Hence find the value of the constant a for which there is no term in x2 in the expansion of
(1 + ax)(2 + 3x)5 .
[2]
9
(i) Find the first 3 terms in the expansion of (2 − x)6 in ascending powers of x.
[3]
(ii) Given that the coefficient of x2 in the expansion of (1 + 2x + ax2 )(2 − x)6 is 48, find the value of
the constant a.
[3]
5
10
3
(i) Find the first 3 terms in the expansion of 2x − in descending powers of x.
x
(ii) Hence find the coefficient of x in the expansion of 1 +
[3]
5
2
3
2x − .
2
x
x
[2]
QUESTIONS
11
2
TOPIC:4 BINOMIAL
5
(i) Find the first 3 terms in the expansion of (1 + ax) in ascending powers of x.
[2]
(ii) Given that there is no term in x in the expansion of (1 − 2x)(1 + ax)5 , find the value of the
constant a.
[2]
(iii) For this value of a, find the coefficient of x2 in the expansion of (1 − 2x)(1 + ax)5 .
12
13
14
[3]
2 6
(i) Find the first three terms, in descending powers of x, in the expansion of x − .
x
[3]
2 6
(ii) Find the coefficient of x4 in the expansion of (1 + x2 )x − .
x
[2]
In the expansion of (1 + ax)6 , where a is a constant, the coefficient of x is −30. Find the coefficient
of x3 .
[4]
(i) Find the first 3 terms in the expansion, in ascending powers of x, of (1 − 2x2 )8 .
[2]
(ii) Find the coefficient of x4 in the expansion of (2 − x2 )(1 − 2x2 )8 .
[2]
15
Find the term independent of x in the expansion of x −
16
2
Find the coefficient of x in the expansion of x + 2 .
x
1 9
.
x2
[3]
7
17
[3]
(i) Find the terms in x2 and x3 in the expansion of 1 − 32 x .
6
[3]
(ii) Given that there is no term in x3 in the expansion of (k + 2x) 1 − 32 x , find the value of the
constant k.
[2]
6
18
The coefficient of x3 in the expansion of (a + x)5 + (1 − 2x)6 , where a is positive, is 90. Find the value
of a.
[5]
19
Find the term independent of x in the expansion of 2x +
20
1 6
.
x2
[3]
(i) Find the first 3 terms in the expansion of (2 − y)5 in ascending powers of y.
[2]
5
(ii) Use the result in part (i) to find the coefficient of x2 in the expansion of 2 − (2x − x2 ) .
[3]
21
The coefficient of x2 in the expansion of k + 13
[3]
22
1
Find the coefficient of x in the expansion of 2x − 2 .
x
7
6
3
[4]
QUESTIONS
3
TOPIC:4 BINOMIAL
23
The coefficient of x3 in the expansion of (a + x)5 + (2 − x)6 is 90. Find the value of the positive
constant a.
[5]
24
The first three terms in the expansion of (1 − 2x)2 (1 + ax)6 , in ascending powers of x, are 1 − x + bx2 .
Find the values of the constants a and b.
[6]
25
6
(i) Find the first 3 terms in the expansion of (2x − x2 ) in ascending powers of x.
6
(ii) Hence find the coefficient of x8 in the expansion of (2 + x)(2x − x2 ) .
26
In the expansion of x2 −
a 7
, the coefficient of x5 is −280. Find the value of the constant a.
x
[3]
[2]
[3]
27
Find the coefficient of x3 in the expansion of 2 − 12 x .
[3]
28
It is given that f x = 2x − 53 + x, for x ∈ >. Show that f is an increasing function.
[3]
29
Find the coefficient of x2 in the expansion of
@
A
1 6
,
(i) 2x −
2x
@
A
1 6
2
(ii) 1 + x 2x −
.
2x
30
7
[2]
[3]
(i) Find the first three terms in the expansion of 2 + ax5 in ascending powers of x.
[3]
(ii) Given that the coefficient of x2 in the expansion of 1 + 2x 2 + ax5 is 240, find the possible
values of a.
[3]
31
(i) Find the first three terms when 2 + 3x6 is expanded in ascending powers of x.
[3]
(ii) In the expansion of 1 + ax 2 + 3x6 , the coefficient of x2 is zero. Find the value of a.
[2]
32
@
A
1 8
3
Find the term independent of x in the expansion of 4x +
.
2x
[4]
33
x 4
Find the coefficient of x in the expansion of 1 + x −
2 x
34
@
A
2 5
Find the coefficient of x in the expansion of x2 −
.
x
2
2
@
A6
.
[5]
[3]