| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2007 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Area under curve |
| Difficulty | Moderate -0.8 This is a straightforward C1 question testing basic integration, differentiation, and coordinate geometry. All parts involve routine procedures: finding y-intercept, integrating a polynomial, calculating areas using the standard 'curve minus line' technique, finding gradient by differentiation, and writing a tangent equation. No problem-solving insight required, just methodical application of standard techniques. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.08e Area between curve and x-axis: using definite integrals1.08f Area between two curves: using integration |
| Answer | Marks | Guidance |
|---|---|---|
| Area \(AOB = \frac{1}{2} \times 1 \times 5 = 2\frac{1}{2}\) | B1, M1, A1 | Condone slip in number or a minus sign; 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| (may have \(+ c\) or not) | M1, A1, A1 | Raise one power by 1; One term correct; All correct unsimplified; 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Area under curve \(= \int_{-1}^{0} f(x) dx\) | B1 | Correctly written or \(F(0) - F(-1)\) correct |
| \([0] - [-\frac{1}{2} + 1 - 5]\) Area under curve \(= 3\frac{1}{2}\) | M1, A1 | Attempt to sub limit(s) of \(-1\) (and 0); Must have integrated; CSO (no fudging) |
| Area of shaded region \(= 3\frac{1}{2} - 2\frac{1}{2} = 1\) | B1✓ | FT their integral and triangle (very generous); 4 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dy}{dx} = 15x^4 + 2\) | M1, A1 | One term correct; All correct (no \(+c\) etc); 3 marks |
| when \(x = -1\), gradient \(= 17\) | A1 | cso |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = \text{"their gradient"}(x + 1)\) | B1✓ | Must be finding tangent – not normal any form e.g. \(y = 17x + 17\); 1 mark |
## 6(a)(i)
$B (0,5)$
Area $AOB = \frac{1}{2} \times 1 \times 5 = 2\frac{1}{2}$ | B1, M1, A1 | Condone slip in number or a minus sign; 3 marks
## 6(a)(ii)
$\frac{3x^6}{6} - \frac{2x^3}{2} + 5x$ or $\frac{x^6}{2} + x^3 + 5x$
(may have $+ c$ or not) | M1, A1, A1 | Raise one power by 1; One term correct; All correct unsimplified; 3 marks
## 6(a)(iii)
Area under curve $= \int_{-1}^{0} f(x) dx$ | B1 | Correctly written or $F(0) - F(-1)$ correct
$[0] - [-\frac{1}{2} + 1 - 5]$ Area under curve $= 3\frac{1}{2}$ | M1, A1 | Attempt to sub limit(s) of $-1$ (and 0); Must have integrated; CSO (no fudging)
Area of shaded region $= 3\frac{1}{2} - 2\frac{1}{2} = 1$ | B1✓ | FT their integral and triangle (very generous); 4 marks
## 6(b)(i)
$\frac{dy}{dx} = 15x^4 + 2$ | M1, A1 | One term correct; All correct (no $+c$ etc); 3 marks
when $x = -1$, gradient $= 17$ | A1 | cso
## 6(b)(ii)
$y = \text{"their gradient"}(x + 1)$ | B1✓ | Must be finding **tangent** – not normal any form e.g. $y = 17x + 17$; 1 mark
**Total: 14**
---6 The curve with equation $y = 3 x ^ { 5 } + 2 x + 5$ is sketched below.\\
\includegraphics[max width=\textwidth, alt={}, center]{33da89e2-f74f-4d5a-8bbd-ceaa728b6c34-5_428_563_372_740}
The curve cuts the $x$-axis at the point $A ( - 1,0 )$ and cuts the $y$-axis at the point $B$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item State the coordinates of the point $B$ and hence find the area of the triangle $A O B$, where $O$ is the origin.
\item Find $\int \left( 3 x ^ { 5 } + 2 x + 5 \right) \mathrm { d } x$.
\item Hence find the area of the shaded region bounded by the curve and the line $A B$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Find the gradient of the curve with equation $y = 3 x ^ { 5 } + 2 x + 5$ at the point $A ( - 1,0 )$.
\item Hence find an equation of the tangent to the curve at the point $A$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C1 2007 Q6 [14]}}