EXM.2.AHL.TZ0.1

pestleMathematicsAAHLPaper 2EXM· ahl-1-10-perms-and-combs-binomial-with-negative-and-fractional-indicessource ↗

Write down and simplify the first three terms, in ascending powers of $x$ , in the Extended Binomial expansion of ${\left( {1 - x} \right)^{\frac{1}{3}}}$ .

[3]

By substituting $x = \frac{1}{9}$ find a rational approximation to $\sqrt[3]{9}$ .

[3]

Markscheme / solution

${\left( {1 - x} \right)^{\frac{1}{3}}} = 1 + \frac{1}{3}\left( { - x} \right) + \frac{1}{3}\left( {\frac{{ - 2}}{3}} \right)\frac{{{{\left( { - x} \right)}^2}}}{{2{\text{!}}}} \ldots = 1 - \frac{x}{3} - \frac{{{x^2}}}{9} \ldots$ M1A1A1

[3 marks]

${\left( {\frac{8}{9}} \right)^{\frac{1}{3}}} \simeq 1 - \frac{1}{{27}} - \frac{1}{{729}} \Rightarrow \frac{2}{{\sqrt[3]{9}}} \simeq \frac{{701}}{{729}} \Rightarrow \sqrt[3]{9} \simeq \frac{{1458}}{{701}}$ M1A1A1

[3 marks]

Examiners’ report

[N/A]