| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Find indefinite integral of polynomial/power |
| Difficulty | Easy -1.3 This is a straightforward C2 question testing basic index laws and standard integration of powers. Part (a) requires only recall of index law rules, while part (b) applies the standard power rule for integration with simple fractional powers. No problem-solving or insight required—purely routine manipulation. |
| Spec | 1.02a Indices: laws of indices for rational exponents1.08b Integrate x^n: where n != -1 and sums1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| 1(a)(i) | \(x^2\) | B1 |
| 1(a)(ii) | \(\frac{1}{x^2} = \sqrt{x}\) | B1 |
| 1(a)(iii) | \(x^3\) | B1 |
| 1(b)(i) | \(\int 3x^2 \, dx = \frac{3}{2}x^2 \{+ c\}\) | M1, A1 |
| \(= 2x^3 + c\) | A1 | 3 marks |
| 1(b)(ii) | \(\int_1^9 3x^2 \, dx = (2 \times 9^2) - (2 \times 1^2)\) | M1 |
| \(= 52\) | A1ft | 2 marks |
**1(a)(i)** | $x^2$ | B1 | 1 mark |
**1(a)(ii)** | $\frac{1}{x^2} = \sqrt{x}$ | B1 | 1 mark | Accept either form |
**1(a)(iii)** | $x^3$ | B1 | 1 mark |
**1(b)(i)** | $\int 3x^2 \, dx = \frac{3}{2}x^2 \{+ c\}$ | M1, A1 | | Index raised by 1; Simplification not yet required |
| | $= 2x^3 + c$ | A1 | 3 marks | Need simplification **and** the $+ c$ OE |
**1(b)(ii)** | $\int_1^9 3x^2 \, dx = (2 \times 9^2) - (2 \times 1^2)$ | M1 | | $F(9) - F(1)$, where $F(x)$ is candidate's answer to (b)(i) [or clear recovery] |
| | $= 52$ | A1ft | 2 marks | Ft on (b)(i) answer of form $kx^{1.5}$ i.e. $26k$ |
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\begin{enumerate}[label=(\alph*)]
\item Simplify:
\begin{enumerate}[label=(\roman*)]
\item $x ^ { \frac { 3 } { 2 } } \times x ^ { \frac { 1 } { 2 } }$;
\item $x ^ { \frac { 3 } { 2 } } \div x$;
\item $\left( x ^ { \frac { 3 } { 2 } } \right) ^ { 2 }$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Find $\int 3 x ^ { \frac { 1 } { 2 } } \mathrm {~d} x$.
\item Hence find the value of $\int _ { 1 } ^ { 9 } 3 x ^ { \frac { 1 } { 2 } } \mathrm {~d} x$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C2 2007 Q1 [8]}}