| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2010 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Find curve equation from derivative (straightforward integration + point) |
| Difficulty | Moderate -0.8 This is a straightforward C2 integration question requiring basic index law manipulation (converting a root to fractional power), applying the standard power rule for integration, and finding the constant using a boundary condition. All steps are routine with no problem-solving insight needed. |
| Spec | 1.02a Indices: laws of indices for rational exponents1.08a Fundamental theorem of calculus: integration as reverse of differentiation1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\sqrt{x^5} = x^{\frac{5}{2}}\) | B1 | Accept \(k = 2.5\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int(7\sqrt{x^5}-4)\,dx = \frac{7}{3.5}x^{3.5} - 4x\ (+c)\) | M1, A1F | Index \(k\) raised by 1 in integrating \(x^k\); 1st term correct follow through on non-integer \(k\) |
| \(-4x\) as integral of \(-4\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y = 2x^{3.5} - 4x + c\) | B1F | \(y =\) c's answer to (b) with \(+c\); (\(y=\) PI by next line) |
| When \(x=1\), \(y=3 \Rightarrow 3 = 2-4+c\) | M1 | Subst. \((1,3)\) in attempt to find constant of integration |
| \(y = 2x^{3.5} - 4x + 5\) | A1 | Accept \(c=5\) after correct eqn; must include \(y=\); coefficients must be tidied |
## Question 2:
**Part (a)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\sqrt{x^5} = x^{\frac{5}{2}}$ | B1 | Accept $k = 2.5$ |
**Part (b)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int(7\sqrt{x^5}-4)\,dx = \frac{7}{3.5}x^{3.5} - 4x\ (+c)$ | M1, A1F | Index $k$ raised by 1 in integrating $x^k$; 1st term correct follow through on non-integer $k$ |
| $-4x$ as integral of $-4$ | B1 | |
**Part (c)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y = 2x^{3.5} - 4x + c$ | B1F | $y =$ c's answer to (b) with $+c$; ($y=$ PI by next line) |
| When $x=1$, $y=3 \Rightarrow 3 = 2-4+c$ | M1 | Subst. $(1,3)$ in attempt to find constant of integration |
| $y = 2x^{3.5} - 4x + 5$ | A1 | Accept $c=5$ after correct eqn; must include $y=$; coefficients must be tidied |
---2 At the point $( x , y )$ on a curve, where $x > 0$, the gradient is given by
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 7 \sqrt { x ^ { 5 } } - 4$$
\begin{enumerate}[label=(\alph*)]
\item Write $\sqrt { x ^ { 5 } }$ in the form $x ^ { k }$, where $k$ is a fraction.
\item Find $\int \left( 7 \sqrt { x ^ { 5 } } - 4 \right) \mathrm { d } x$.
\item Hence find the equation of the curve, given that the curve passes through the point $( 1,3 )$.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2010 Q2 [7]}}