Past Exam Questions

(a) Find the first three terms in the expansion of

\(\left(4-\frac{x}{16}\right)^6\)

in ascending powers of \(x\), giving each term in its simplest form.

(b) Hence find the term independent of \(x\) in the expansion of

\(\left(4-\frac{x}{16}\right)^6\left(x-\frac1x\right)^2.\)

(a) Expand \((2-x)^5\), simplifying each coefficient.

(b) Hence solve

\(\frac{e^{(2-x)^5}\times e^{80x}}{e^{10x^4+32}}=e^{-x^5}.\)

Do not use a calculator in this question.

(a) Find the term independent of \(x\) in the binomial expansion of

\(\left(3x-\frac1x\right)^6.\)

(b) In the expansion of

\(\left(1+\frac{x}{2}\right)^n,\)

the coefficient of \(x^4\) is half the coefficient of \(x^6\). Find the value of the positive constant \(n\).

Find the coefficient of \(x^2\) in the expansion of

\(\left(x-\frac{3}{x}\right)\left(x+\frac{2}{x}\right)^5.\)

Given that the coefficient of \(x^2\) in the expansion of

\((1+x)\left(1-\frac{x}{2}\right)^n\)

is \(\dfrac{25}{4}\), find the value of the positive integer \(n\).

The first three terms in the expansion of

\((a+bx)^5(1+x)\)

are

\(32-208x+cx^2.\)

Find the value of each of the integers \(a\), \(b\) and \(c\).

(i) Find the first three terms, in ascending powers of \(x\), in the expansion of \(\left(3-\dfrac{x}{9}\right)^6\).

(ii) Hence find the term independent of \(x\) in the expansion of \(\left(3-\dfrac{x}{9}\right)^6\left(x-\dfrac2x\right)^2\).

(i) The first 3 terms, in ascending powers of \(x\), in the expansion of \((2+bx)^8\) can be written as

\(a+256x+cx^2.\)

Find \(a\), \(b\) and \(c\).

(ii) Using the values found in part (i), find the term independent of \(x\) in the expansion of

\((2+bx)^8\left(2x-\frac3x\right)^2.\)

(a) In the binomial expansion of \(\left(a-\dfrac{x}{2}\right)^6\), the coefficient of \(x^3\) is 120 times the coefficient of \(x^5\). Find the possible values of the constant \(a\).

(b) (i) Expand \((1+2x)^{20}\) in ascending powers of \(x\), as far as the term in \(x^3\). Simplify each term.

(ii) Use your expansion to show that the value of \(0.98^{20}\) is \(0.67\) to 2 decimal places.

(a) (i) Given that \(\left(x^2-\dfrac1{px}\right)^8=x^{16}-4x^{13}+qx^{10}+rx^7+\cdots\), find the value of each of the constants \(p\), \(q\) and \(r\).

(ii) Explain why there is no term independent of \(x\) in the binomial expansion of \(\left(x^2-\dfrac1{px}\right)^8\).

(b) In the binomial expansion of \(\left(1-\dfrac{\sqrt{x}}2\right)^n\), where \(n\) is a positive integer, the coefficient of \(x\) is 30. Form an equation in \(n\) and hence find the value of \(n\).

The first three terms in the expansion of

\(\left(1-\frac{x}{7}\right)^{14}(1-2x)^4\)

can be written as \(1+ax+bx^2\). Find the value of each of the constants \(a\) and \(b\).

(i) Expand \((3+x)^4\), evaluating each coefficient.

In the expansion of

\(\left(x-\frac{p}{x}\right)(3+x)^4\)

the coefficient of \(x\) is zero.

(ii) Find the value of the constant \(p\).

(iii) Hence find the term independent of \(x\).

(iv) Show that the coefficient of \(x^2\) is \(90\).

The first four terms in the expansion of \((1+ax)^5(2+bx)\) are

\(2+32x+210x^2+cx^3,\)

where \(a\), \(b\) and \(c\) are integers. Show that

\(3a^2-16a+21=0\)

and hence find the values of \(a\), \(b\) and \(c\).

The first three terms, in ascending powers of \(x\), in the expansion of \((2+ax)^n\) are \(1024-1280x+bx^2\). Find the values of \(n\), \(a\), and \(b\).

Hence find the term independent of \(x\) in the expansion of \((2+ax)^n\left(x-\dfrac1x\right)^2\).

(i) Find the first \(3\) terms in the expansion of

\(\left(2x-\frac{1}{16x}\right)^8\)

in descending powers of \(x\).

(ii) Hence find the coefficient of \(x^4\) in the expansion of

\(\left(2x-\frac{1}{16x}\right)^8 \left(\frac{1}{x^2}+1\right)^2.\)

(i) The first three terms in the expansion of

\(\left(3-\frac1{9x}\right)^5\)

can be written as \(a+\dfrac bx+\dfrac c{x^2}\). Find the value of each of the constants \(a\), \(b\) and \(c\).

(ii) Use your values of \(a\), \(b\) and \(c\) to find the term independent of \(x\) in the expansion of

\(\left(3-\frac1{9x}\right)^5(2+9x)^2.\)

(i) Find the first \(3\) terms in the expansion of

\(\left(2x-\frac{1}{16x}\right)^8\)

in descending powers of \(x\).

(ii) Hence find the coefficient of \(x^4\) in the expansion of

\(\left(2x-\frac{1}{16x}\right)^8 \left(\frac{1}{x^2}+1\right)^2.\)

The coefficient of \(x^2\) in the expansion of \((2-x)(3+kx)^6\) is equal to \(972\). Find the possible values of the constant \(k\).

The 7th term in the expansion of \((a+bx)^{12}\) in ascending powers of \(x\) is \(924x^6\). It is given that \(a\) and \(b\) are positive constants.

(i) Show that \(b=\dfrac1a\).

The 6th term in the expansion of \((a+bx)^{12}\) in ascending powers of \(x\) is \(198x^5\).

(ii) Find the value of \(a\) and of \(b\).

(a) In the expansion of \((2+px)^5\), the coefficient of \(x^3\) is equal to \(-\dfrac{8}{25}\). Find the value of the constant \(p\).

(b) Find the term independent of \(x\) in the expansion of \(\left(2x^2+\dfrac{1}{4x^2}\right)^8\).