| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Standard product of two binomials |
| Difficulty | Standard +0.3 Part (a) is routine binomial expansion with small positive integer power requiring calculation of three coefficients. Part (b) requires multiplying two binomial expansions and collecting terms for x^10, which is a standard technique but involves more careful bookkeeping across multiple terms. Overall slightly easier than average due to straightforward application of binomial theorem with no novel insight required. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(p = -10; \quad q = 40; \quad r = -80\) | B1, B1, B1 |
| (b) | \((2 + x)^7 = ......... + mx^5 + nx^6 + x^7\) | M1 |
| (b) | \(m = 84, \quad n = 14\) | A1 |
| (b) | Coefficients of \(x^{10}\) terms in expansion of \((1 - 2x)^7(2 + x)^7\) are \(-32m + 80n + r\) | m1 |
| (b) | Coeff. of \(x^{10} = (-32)(84) + (80)(14) + r = -2688 + 1120 + r = -1568 + r\) | A1F |
| (b) | Coeff. of \(x^{10} = -1648\) | A1 |
| Total | 8 |
(a) | $p = -10; \quad q = 40; \quad r = -80$ | B1, B1, B1 | 3 marks | Accept correct embedded values for p, q and r within the expansion
(b) | $(2 + x)^7 = ......... + mx^5 + nx^6 + x^7$ | M1 | Attempting to find **at least two of** $x^5$ term, $x^6$ term, $x^7$ term in the expansion of $(2 + x)^7$
(b) | $m = 84, \quad n = 14$ | A1 | Either correct. (M1 must be scored). PI by later correct work
(b) | Coefficients of $x^{10}$ terms in expansion of $(1 - 2x)^7(2 + x)^7$ are $-32m + 80n + r$ | m1 | Identifying at least two of the three products $-32m, 80n, r$ that give $x^{10}$ terms
(b) | Coeff. of $x^{10} = (-32)(84) + (80)(14) + r = -2688 + 1120 + r = -1568 + r$ | A1F | **Only** if c's value of r in (a). If not shown in any of these forms, can be implied by final answer which matches correct evaluation of $(-1568+\text{c's } r)$
(b) | Coeff. of $x^{10} = -1648$ | A1 | 5 marks | $-1648$ or left as ' $-1648 x^{10}$ '. Ignore other powers of x terms
| **Total** | **8** |
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\begin{enumerate}[label=(\alph*)]
\item The expression $( 1 - 2 x ) ^ { 5 }$ can be written in the form
$$1 + p x + q x ^ { 2 } + r x ^ { 3 } + 80 x ^ { 4 } - 32 x ^ { 5 }$$
By using the binomial expansion, or otherwise, find the values of the coefficients $p , q$ and $r$.
\item Find the value of the coefficient of $x ^ { 10 }$ in the expansion of $( 1 - 2 x ) ^ { 5 } ( 2 + x ) ^ { 7 }$.\\[0pt]
[5 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2016 Q7 [8]}}