| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2012 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Standard binomial expansion |
| Difficulty | Moderate -0.8 This is a straightforward C2 binomial expansion question with routine transformations. Part (a) involves basic function transformations (stretch and translation) requiring simple substitution. Part (b) is direct application of the binomial theorem formula to find coefficients—purely procedural with no problem-solving required. Easier than average A-level questions. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Stretch (I) in \(x\)-direction (II), scale factor \(\frac{1}{6}\) (III) | M1 | Need (I) and either (II) or (III) |
| A1 (Total: 2) | Need (I) and (II) and (III) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(g(x) = \left(1+\frac{x-3}{3}\right)^6\) | M1 | OE Replaces \(\frac{x}{3}\) by \(\frac{x-3}{3}\) |
| \(= \left(\frac{x}{3}\right)^6\) or \(\frac{x^6}{3^6}\) or \(\frac{x^6}{729}\) | A1 (Total: 2) | Must be simplified |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\left(1+\frac{x}{3}\right)^6 = 1+\binom{6}{1}\frac{x}{3}+\binom{6}{2}\left(\frac{x}{3}\right)^2+\binom{6}{3}\left(\frac{x}{3}\right)^3\ldots\) \(= (1+) 2x\) | B1 | \(a=2\). Condone '\(2x\)' |
| \(+\frac{6!}{4!2!}\left(\frac{x}{3}\right)^2 + \frac{6!}{3!3!}\left(\frac{x}{3}\right)^3\) | M1 | Either \((1\ 6\ 15\ 20)\) seen or \(\binom{6}{2}, \binom{6}{3}\) written in terms of factorials OE |
| \(= (1+2x) + \frac{15}{9}x^2 + \frac{20}{27}x^3\) | ||
| \(b = \frac{5}{3},\ c = \frac{20}{27}\) | A1 | \(b=\frac{5}{3}\) (or \(1\frac{2}{3}\)). Condone \(\ldots+\frac{5}{3}x^2\) |
| A1 (Total: 4) | \(c=\frac{20}{27}\). Condone \(\ldots+\frac{20}{27}x^3\). Accept equivalent recurring decimals. Ignore terms with higher powers of \(x\). SC If A0A0 award A1 for either \(+15\frac{x^2}{9},\ +20\frac{x^3}{27}\) seen or \(+\frac{15x^2}{9},\ +\frac{20x^3}{27}\) seen |
## Question 5:
**Part (a)(i):**
| Working | Mark | Guidance |
|---------|------|----------|
| Stretch (I) in $x$-direction (II), scale factor $\frac{1}{6}$ (III) | M1 | Need (I) and either (II) or (III) |
| | A1 (Total: 2) | Need (I) and (II) and (III) |
**Part (a)(ii):**
| Working | Mark | Guidance |
|---------|------|----------|
| $g(x) = \left(1+\frac{x-3}{3}\right)^6$ | M1 | OE Replaces $\frac{x}{3}$ by $\frac{x-3}{3}$ |
| $= \left(\frac{x}{3}\right)^6$ or $\frac{x^6}{3^6}$ or $\frac{x^6}{729}$ | A1 (Total: 2) | Must be simplified |
**Part (b):**
| Working | Mark | Guidance |
|---------|------|----------|
| $\left(1+\frac{x}{3}\right)^6 = 1+\binom{6}{1}\frac{x}{3}+\binom{6}{2}\left(\frac{x}{3}\right)^2+\binom{6}{3}\left(\frac{x}{3}\right)^3\ldots$ $= (1+) 2x$ | B1 | $a=2$. Condone '$2x$' |
| $+\frac{6!}{4!2!}\left(\frac{x}{3}\right)^2 + \frac{6!}{3!3!}\left(\frac{x}{3}\right)^3$ | M1 | Either $(1\ 6\ 15\ 20)$ seen or $\binom{6}{2}, \binom{6}{3}$ written in terms of factorials OE |
| $= (1+2x) + \frac{15}{9}x^2 + \frac{20}{27}x^3$ | | |
| $b = \frac{5}{3},\ c = \frac{20}{27}$ | A1 | $b=\frac{5}{3}$ (or $1\frac{2}{3}$). Condone $\ldots+\frac{5}{3}x^2$ |
| | A1 (Total: 4) | $c=\frac{20}{27}$. Condone $\ldots+\frac{20}{27}x^3$. Accept equivalent recurring decimals. Ignore terms with higher powers of $x$. **SC** If A0A0 award A1 for either $+15\frac{x^2}{9},\ +20\frac{x^3}{27}$ seen or $+\frac{15x^2}{9},\ +\frac{20x^3}{27}$ seen |5
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Describe the geometrical transformation that maps the graph of $y = \left( 1 + \frac { x } { 3 } \right) ^ { 6 }$ onto the graph of $y = ( 1 + 2 x ) ^ { 6 }$.
\item The curve $y = \left( 1 + \frac { x } { 3 } \right) ^ { 6 }$ is translated by the vector $\left[ \begin{array} { l } 3 \\ 0 \end{array} \right]$ to give the curve $y = \mathrm { g } ( x )$. Find an expression for $\mathrm { g } ( x )$, simplifying your answer.
\end{enumerate}\item The first four terms in the binomial expansion of $\left( 1 + \frac { x } { 3 } \right) ^ { 6 }$ are $1 + a x + b x ^ { 2 } + c x ^ { 3 }$. Find the values of the constants $a , b$ and $c$, giving your answers in their simplest form.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2012 Q5 [8]}}