| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2007 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Standard product of two binomials |
| Difficulty | Moderate -0.8 Part (a) is straightforward application of the binomial theorem formula to find three coefficients. Part (b) requires multiplying the result by (1 + x/2) and collecting x³ terms, which is a standard 'hence' extension but still routine. This is easier than average A-level questions as it involves direct formula application with minimal problem-solving. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \((1 + 2x)^8\) | M1 |
| \(= 1 + \binom{8}{1}(2x) + \binom{8}{2}(2x)^2 + \binom{8}{3}(2x)^3 + ...\) | A1A1 | A1 for each of \(a\), \(b\), \(c\) |
| \(= 1 + 16x + 112x^2 + 448x^3 + ....\) | A1 | |
| \(\{a = 16, b = 112, c = 448\}\) | ||
| Total: 4 | ||
| (b) | \(x^3\) terms from expn. of \((1 + \frac{1}{2}x)(1 + 2x)^8\) | M1 |
| are \(cx^3\) and \(\frac{1}{2}x(bx^2)\) | A1 | \(b\), \(c\) or candidate's values for \(b\) and \(c\) from (a) |
| Coefficient of \(x^3\) is \(c + 0.5b = 504\) | A1ft | Ft on candidate's (\(c + 0.5b\)) provided \(b\) and \(c\) are positive integers \(>1\) |
| Total: 7 |
(a) | $(1 + 2x)^8$ | M1 | Any valid method. PI by correct value for $a$, $b$ or $c$ |
| $= 1 + \binom{8}{1}(2x) + \binom{8}{2}(2x)^2 + \binom{8}{3}(2x)^3 + ...$ | A1A1 | A1 for each of $a$, $b$, $c$ |
| $= 1 + 16x + 112x^2 + 448x^3 + ....$ | A1 | |
| $\{a = 16, b = 112, c = 448\}$ | | |
| | **Total: 4** | |
(b) | $x^3$ terms from expn. of $(1 + \frac{1}{2}x)(1 + 2x)^8$ | M1 | Either |
| are $cx^3$ and $\frac{1}{2}x(bx^2)$ | A1 | $b$, $c$ or candidate's values for $b$ and $c$ from (a) |
| | | |
| Coefficient of $x^3$ is $c + 0.5b = 504$ | A1ft | Ft on candidate's ($c + 0.5b$) provided $b$ and $c$ are positive integers $>1$ |
| | **Total: 7** | |
---7
\begin{enumerate}[label=(\alph*)]
\item The first four terms of the binomial expansion of $( 1 + 2 x ) ^ { 8 }$ in ascending powers of $x$ are $1 + a x + b x ^ { 2 } + c x ^ { 3 }$. Find the values of the integers $a , b$ and $c$.
\item Hence find the coefficient of $x ^ { 3 }$ in the expansion of $\left( 1 + \frac { 1 } { 2 } x \right) ( 1 + 2 x ) ^ { 8 }$.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2007 Q7 [7]}}