| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 7 |
| 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: expand a binomial with fractional powers, integrate term-by-term using the power rule, then evaluate definite integral. All steps are routine with no problem-solving required beyond following the 'hence' structure. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.08b Integrate x^n: where n != -1 and sums1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| \(\left(x^{\frac{3}{2}}\right)^2 - 2x^{\frac{3}{2}} + 1 = x^3 - 2x^{\frac{3}{2}} + 1\) | B2,1,0 | B2 for \(x^3 - 2x^{\frac{3}{2}}+1\) or \(x^3-2x\sqrt{x}+1\); B1 fully correct unsimplified expression; B1 for either \(x^3-2x^{\frac{3}{2}}\ldots\) OE seen |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int\left(x^{\frac{3}{2}}-1\right)^2\,dx = \frac{x^4}{4} - \frac{2x^{\frac{5}{2}}}{2.5} + x\ (+c)\) \(\{= 0.25x^4 - 0.8x^{2.5} + x\ (+c)\}\) | B1F, M1, A1F | Ft on correct integration of all non \(x^{\frac{3}{2}}\) terms (at least two) in c's expression in (a); integration of \(kx^{\frac{3}{2}}\) as \(\lambda x^{\frac{5}{2}}\) (power correct); correct integration of c's \(x^{\frac{3}{2}}\) terms ACF |
| Answer | Marks | Guidance |
|---|---|---|
| \(\left\{=\frac{212}{5}-\frac{9}{20}=42.4-0.45\right\}=41.95\) | M1, A1 | \(F(4)-F(1)\) attempted following integration; 41.95 OE e.g. \(\frac{839}{20}\); Since 'Hence' NMS scores 0/2 |
# Question 3:
## Part (a)
$\left(x^{\frac{3}{2}}\right)^2 - 2x^{\frac{3}{2}} + 1 = x^3 - 2x^{\frac{3}{2}} + 1$ | B2,1,0 | B2 for $x^3 - 2x^{\frac{3}{2}}+1$ or $x^3-2x\sqrt{x}+1$; B1 fully correct unsimplified expression; B1 for either $x^3-2x^{\frac{3}{2}}\ldots$ OE seen
## Part (b)
$\int\left(x^{\frac{3}{2}}-1\right)^2\,dx = \frac{x^4}{4} - \frac{2x^{\frac{5}{2}}}{2.5} + x\ (+c)$ $\{= 0.25x^4 - 0.8x^{2.5} + x\ (+c)\}$ | B1F, M1, A1F | Ft on correct integration of all non $x^{\frac{3}{2}}$ terms (at least two) in c's expression in (a); integration of $kx^{\frac{3}{2}}$ as $\lambda x^{\frac{5}{2}}$ (power correct); correct integration of c's $x^{\frac{3}{2}}$ terms ACF
## Part (c)
$\int_1^4\left(x^{\frac{3}{2}}-1\right)^2\,dx = \left(\frac{4^4}{4}-\frac{2(4^{\frac{5}{2}})}{2.5}+4\right)-\left(\frac{1}{4}-\frac{2}{2.5}+1\right)$
$\left\{=\frac{212}{5}-\frac{9}{20}=42.4-0.45\right\}=41.95$ | M1, A1 | $F(4)-F(1)$ attempted following integration; 41.95 OE e.g. $\frac{839}{20}$; Since 'Hence' NMS scores 0/2
---3
\begin{enumerate}[label=(\alph*)]
\item $\quad$ Expand $\left( x ^ { \frac { 3 } { 2 } } - 1 \right) ^ { 2 }$.
\item Hence find $\int \left( x ^ { \frac { 3 } { 2 } } - 1 \right) ^ { 2 } \mathrm {~d} x$.
\item Hence find the value of $\int _ { 1 } ^ { 4 } \left( x ^ { \frac { 3 } { 2 } } - 1 \right) ^ { 2 } \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2012 Q3 [7]}}