Easy -1.2 This is a straightforward application of the binomial theorem requiring only direct substitution into the formula for three terms with small powers. It's easier than average as it involves no algebraic manipulation, no problem-solving insight, and is a standard textbook exercise with positive integer n=6 and simple coefficients.
Requires correct structure: 'binomial coefficients' (perhaps from Pascal's triangle), increasing powers of one term, decreasing powers of the other term. Allow 'slips'. Acceptable binomial coefficients: \(\binom{6}{1}\) and \(\binom{6}{2}\) or equivalent, or even \(\binom{6}{1}\) and \(\binom{6}{2}\). Decreasing powers of \(x\): can score only the M mark.
(4 marks)
\(64(1 + \ldots)\) even if all terms in bracket correct, scores max B1M1A0A0
| $(2 + x)^6 = 64 + (6 \times 2^5 \times x) + \left(\frac{6 \times 5}{2} \times 2^4 \times x^2\right) + 192x + 240x^2$ | B1, M1, A1, A1 | Requires correct structure: 'binomial coefficients' (perhaps from Pascal's triangle), increasing powers of one term, decreasing powers of the other term. Allow 'slips'. Acceptable binomial coefficients: $\binom{6}{1}$ and $\binom{6}{2}$ or equivalent, or even $\binom{6}{1}$ and $\binom{6}{2}$. Decreasing powers of $x$: can score only the M mark. |
| --- | --- | --- |
| | (4 marks) | $64(1 + \ldots)$ even if all terms in bracket correct, scores max B1M1A0A0 |
Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of $( 2 + x ) ^ { 6 }$, giving each term in its simplest form.\\
\hfill \mbox{\textit{Edexcel C2 2006 Q1 [4]}}