[Maximum number: 4]

The coefficient of $x^{3}$ in the expansion of $\left(p+\frac{1}{p} x\right)^{4}$ is 144 .
Find the possible values of the constant p.

[Maximum number: 4]

The coefficient of $x^{4}$ in the expansion of $(3+x)^{5}$ is equal to the coefficient of $x^{2}$ in the expansion of $\left(2 x+\frac{a}{x}\right)^{6}$ .

Find the value of the positive constant a.

(a)

Expand $\left(1-\frac{1}{2 x}\right)^{2}$ .

[ 1 ]

(b)

Find the first four terms in the expansion, in ascending powers of x, of $(1+2 x)^{6}$ .

[ 2 ]

(c)

Hence find the coefficient of x in the expansion of $\left(1-\frac{1}{2 x}\right)^{2}(1+2 x)^{6}$ .

[ 2 ]

[Maximum number: 4]

The coefficient of $x^{3}$ in the expansion of $(1+k x)(1-2 x)^{5}$ is 20 .
Find the value of the constant k.

(a)

Find the coefficient of $x^{2}$ in the expansion of $\left(x-\frac{2}{x}\right)^{6}$ .

[ 2 ]

(b)

Find the coefficient of $x^{2}$ in the expansion of $\left(2+3 x^{2}\right)\left(x-\frac{2}{x}\right)^{6}$ .

[ 3 ]

[Maximum number: 4]

The sum of the first nine terms of an arithmetic progression is 117. The sum of the next four terms is 91 .

Find the first term and the common difference of the progression.

(a)

Expand $(1+y)^{6}$ in ascending powers of y as far as the term in $y^{2}$ .

[ 1 ]

(b)

In the expansion of $\left(1+\left(p x-2 x^{2}\right)\right)^{6}$ the coefficient of $x^{2}$ is 48 . Find the value of the positive constant p.

[ 3 ]

[Maximum number: 3]

Find the coefficient of x in the expansion of $\left(\frac{2}{x}-3 x\right)^{5}$ .

[Maximum number: 5]

The term independent of x in the expansion of $\left(2 x+\frac{k}{x}\right)^{6}$ , where k is a constant, is 540.

(a)

Find the value of k.

[ 3 ]

(b)

For this value of k, find the coefficient of $x^{2}$ in the expansion.

[ 2 ]

[Maximum number: 3]

The coefficient of $x^{3}$ in the expansion of $(1-p x)^{5}$ is -2160 . Find the value of the constant p.