| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2013 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Combined expansions then integrate |
| Difficulty | Moderate -0.3 This is a structured multi-part question with clear signposting ('hence') that guides students through each step. Part (a)(i) is routine binomial expansion with n=3, (a)(ii) involves straightforward substitution and simplification where odd powers cancel, and part (b) requires basic integration of polynomial terms. While it spans multiple techniques, each individual step is standard C2 material with no novel insight required, making it slightly easier than average. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n1.08b Integrate x^n: where n != -1 and sums1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| \(\{(2+y)^3\} = 8 + 12y + 6y^2 + y^3\) | M1 | At least 3 terms simplified and correct |
| A1 | All correct | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\{2 + x^{-2}\}^3 = 8 + 12x^{-2} + 6(x^{-2})^2 + (x^{-2})^3\) | M1 | A replacement of \(y\) by \(x^{-2}\) in c's (a)(i) working. PI |
| \(\{2 - x^{-2}\}^3 = 8 - 12x^{-2} + 6(x^{-2})^2 - (x^{-2})^3\) | A1F | Ft one incorrect coefficient in (a)(i) expansion. |
| \(\{2 + x^{-2}\}^3 + \{2 - x^{-2}\}^3 = 16 + 12x^{-4}\) | A1 | CSO Be convinced. SC2 for a fully correct solution, not using 'Hence' |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int \left[\{2 + x^{-2}\}^3 + \{2 - x^{-2}\}^3\right] dx = 16x - 4x^{-3}(+c)\) | M1 | Valid method to obtain the correct power of \(x\) after integrating \(qx^{-4}\). |
| A1F | 16x – 4x⁻³ or 16x – 4/x³ condone missing '+c'. Ft on c's \(p\) and \(q\) values. Coefficients and signs must be simplified | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int_{1}^{2} ............ dx = [16(2) - 4(2^{-3})] - [16-4]\) | M1 | F(2)–F(1) following integration (b)(i) |
| \(= 31.5 - 12 = 19.5\) | A1F | OE Ft on c's positive integer values of \(p\) and \(q\). Since 'Hence' NMS scores 0/2 |
| Total | 9 |
**3(a)(i)**
$\{(2+y)^3\} = 8 + 12y + 6y^2 + y^3$ | M1 | At least 3 terms simplified and correct
| A1 | All correct | 2
**3(a)(ii)**
$\{2 + x^{-2}\}^3 = 8 + 12x^{-2} + 6(x^{-2})^2 + (x^{-2})^3$ | M1 | A replacement of $y$ by $x^{-2}$ in c's (a)(i) working. PI
$\{2 - x^{-2}\}^3 = 8 - 12x^{-2} + 6(x^{-2})^2 - (x^{-2})^3$ | A1F | Ft one incorrect coefficient in (a)(i) expansion.
$\{2 + x^{-2}\}^3 + \{2 - x^{-2}\}^3 = 16 + 12x^{-4}$ | A1 | CSO Be convinced. SC2 for a fully correct solution, not using 'Hence'
| | | 3
**3(b)(i)**
$\int \left[\{2 + x^{-2}\}^3 + \{2 - x^{-2}\}^3\right] dx = 16x - 4x^{-3}(+c)$ | M1 | Valid method to obtain the correct power of $x$ after integrating $qx^{-4}$.
| A1F | 16x – 4x⁻³ or 16x – 4/x³ condone missing '+c'. Ft on c's $p$ and $q$ values. Coefficients and signs must be simplified | 2
**3(b)(ii)**
$\int_{1}^{2} ............ dx = [16(2) - 4(2^{-3})] - [16-4]$ | M1 | F(2)–F(1) following integration (b)(i)
$= 31.5 - 12 = 19.5$ | A1F | OE Ft on c's positive integer values of $p$ and $q$. Since 'Hence' NMS scores 0/2 | 2
Total | | 9
---3
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Using the binomial expansion, or otherwise, express $( 2 + y ) ^ { 3 }$ in the form $a + b y + c y ^ { 2 } + y ^ { 3 }$, where $a , b$ and $c$ are integers.
\item Hence show that $\left( 2 + x ^ { - 2 } \right) ^ { 3 } + \left( 2 - x ^ { - 2 } \right) ^ { 3 }$ can be expressed in the form $p + q x ^ { - 4 }$, where $p$ and $q$ are integers.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Hence find $\int \left[ \left( 2 + x ^ { - 2 } \right) ^ { 3 } + \left( 2 - x ^ { - 2 } \right) ^ { 3 } \right] \mathrm { d } x$.
\item Hence find the value of $\int _ { 1 } ^ { 2 } \left[ \left( 2 + x ^ { - 2 } \right) ^ { 3 } + \left( 2 - x ^ { - 2 } \right) ^ { 3 } \right] \mathrm { d } x$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C2 2013 Q3 [9]}}