| Date | November 2009 | Marks available | 6 | Reference code | 09N.2.hl.TZ0.4 |
| Level | HL only | Paper | 2 | Time zone | TZ0 |
| Command term | Find and Hence | Question number | 4 | Adapted from | N/A |
Question
When \({\left( {1 + \frac{x}{2}} \right)^2}\) , \(n \in \mathbb{N}\) , is expanded in ascending powers of \(x\) , the coefficient of \({x^3}\) is \(70\).
(a) Find the value of \(n\) .
(b) Hence, find the coefficient of \({x^2}\) .
Markscheme
(a) coefficient of \({x^3}\) is \(\left( {\begin{array}{*{20}{c}}
n \\
3
\end{array}} \right){\left( {\frac{1}{2}} \right)^3} = 70\) M1(A1)
\(\frac{{n!}}{{3!\left( {n - 3} \right)!}} \times \frac{1}{8} = 70\) (A1)
\( \Rightarrow \frac{{n\left( {n - 1} \right)\left( {n - 2} \right)}}{{48}} = 70\) (M1)
\(n = 16\) A1
(b) \(\left( {\begin{array}{*{20}{c}}
{16} \\
2
\end{array}} \right){\left( {\frac{1}{2}} \right)^2} = 30\) A1
[6 marks]
Examiners report
Most candidates were able to answer this question well.
