3.50: Binomial independent term and surd difference

0606/23/M/J/24 — Question 4 · 8 marks

Do not use a calculator in part (b).

(a)

Find and simplify the term independent of

x

in the expansion of

\left(x^{2}-\dfrac{1}{2x^{3}}\right)^{10}

[2]

(b)(i)

Use the binomial theorem to show that

\bigl(\sqrt{2}+1\bigr)^{4}-\bigl(\sqrt{2}-1\bigr)^{4}=k\sqrt{2}

, where

k

is an integer to be found.

[4]

(b)(ii)

Hence write

\dfrac{(\sqrt{2}+1)^{4}-(\sqrt{2}-1)^{4}}{(\sqrt{2}+1)^{4}+(\sqrt{2}-1)^{4}}

in the form

a+b\sqrt{2}

[2]