# Binomial Theorem – Practice Questions

1.  In the expansion of ${\left(a–3b\right)}^{n}$ , the sum of 9th and 10th term is zero. Find the value of  $\frac{a}{b}$ in terms of  $n$ .

2.  If the coefficient of 4th, 5th and 6th terms in the expansion of  ${\left(1+x\right)}^{n}$ are in arithmetic sequence, then find the value(s) of $n$ .

3.  If the last term in the expansion of  is $\frac{–{\mathrm{log}}_{2}81}{{2}^{3}{4}}}$ , find the value of  $n$ .

4.  Find the remainder when  ${2}^{1003}$ is divided by 7. (Non Calculator question)

5.  If  ${x}_{n}–{y}_{n}\sqrt{2}={\left(1–\sqrt{2}\right)}^{n}$  , then show that

a.  ${{x}_{n}}^{2}–2{{y}_{n}}^{2}={\left(–1\right)}^{n}$

6. (i) Find the first three terms in the expansion, in ascending powers of $x$ , of ${\left(1–2x\right)}^{5}$ . [2 marks]

(ii) Given that the coefficient of ${x}^{2}$ in the expansion of $\left(1+ax+2{x}^{2}\right){\left(1–2x\right)}^{5}$  is 12, find the value of the constant $a$ . [ 3 marks]

7. a) Use the binomial theorem to expand  ${\left(a+\sqrt{b}\right)}^{4}$
b) Hence, deduce an expression in terms of  $a$ and  $b$ for ${\left(a+\sqrt{b}\right)}^{4}+{\left(a–\sqrt{b}\right)}^{4}$

Q8. Write down and simplify the general term in the binomial expansion of ${\left(2{x}^{2}–\frac{d}{{x}^{3}}\right)}^{7}$ , where  $d$ is a constant.
(b) Given that the coefficient of $\frac{1}{x}$ is −70 000, find the value of $d$ .