| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Integration with algebraic manipulation |
| Difficulty | Moderate -0.8 This is a straightforward C2 integration question with clear scaffolding. Part (a) is routine application of the power rule, part (b) guides students through expanding a binomial, and part (c) connects these to evaluate a definite integral. The 'hence' structure removes any problem-solving challenge, making this easier than average for A-level. |
| 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 |
|---|---|---|
| \(\int\left(1 + 3x^{\frac{1}{2}} + x^{\frac{3}{2}}\right)dx = x + 2x^{\frac{3}{2}} + \frac{2}{5}x^{\frac{5}{2}} + c\) | M1 | At least one term integrated correctly |
| A1 | Two correct terms | |
| A1 | All correct including \(+c\) |
| Answer | Marks |
|---|---|
| \(n = 3\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \((1+\sqrt{x})^3 = 1 + 3\sqrt{x} + 3x + x\sqrt{x}\) | B1 | Accept \(1 + 3x^{\frac{1}{2}} + 3x + x^{\frac{3}{2}}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int_0^1(1+\sqrt{x})^3\,dx = \left[x + 2x^{\frac{3}{2}} + \frac{3}{2}x^2 + \frac{2}{5}x^{\frac{5}{2}}\right]_0^1\) | M1 | Integration of their expansion |
| \(= 1 + 2 + \frac{3}{2} + \frac{2}{5}\) | A1 | Correct integration |
| \(= \frac{49}{10}\) | A1 | Exact value |
# Question 2:
## Part (a):
| $\int\left(1 + 3x^{\frac{1}{2}} + x^{\frac{3}{2}}\right)dx = x + 2x^{\frac{3}{2}} + \frac{2}{5}x^{\frac{5}{2}} + c$ | M1 | At least one term integrated correctly |
| | A1 | Two correct terms |
| | A1 | All correct including $+c$ |
## Part (b)(i):
| $n = 3$ | B1 | |
## Part (b)(ii):
| $(1+\sqrt{x})^3 = 1 + 3\sqrt{x} + 3x + x\sqrt{x}$ | B1 | Accept $1 + 3x^{\frac{1}{2}} + 3x + x^{\frac{3}{2}}$ |
## Part (c):
| $\int_0^1(1+\sqrt{x})^3\,dx = \left[x + 2x^{\frac{3}{2}} + \frac{3}{2}x^2 + \frac{2}{5}x^{\frac{5}{2}}\right]_0^1$ | M1 | Integration of their expansion |
| $= 1 + 2 + \frac{3}{2} + \frac{2}{5}$ | A1 | Correct integration |
| $= \frac{49}{10}$ | A1 | Exact value |
---2
\begin{enumerate}[label=(\alph*)]
\item Find $\int \left( 1 + 3 x ^ { \frac { 1 } { 2 } } + x ^ { \frac { 3 } { 2 } } \right) \mathrm { d } x$.
\item \begin{enumerate}[label=(\roman*)]
\item The expression $( 1 + y ) ^ { 3 }$ can be written in the form $1 + 3 y + n y ^ { 2 } + y ^ { 3 }$. Write down the value of the constant $n$.
\item Hence, or otherwise, expand $( 1 + \sqrt { x } ) ^ { 3 }$.
\end{enumerate}\item Hence find the exact value of $\int _ { 0 } ^ { 1 } ( 1 + \sqrt { x } ) ^ { 3 } \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2014 Q2 [8]}}