| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area between curve and line |
| Difficulty | Standard +0.3 This is a structured multi-part question covering standard C1 integration and differentiation techniques. While it requires multiple steps (finding areas, integration, differentiation, tangent equations, and solving inequalities), each individual part uses routine methods with clear guidance. The 'hence' structure provides scaffolding, making it slightly easier than a typical C1 question that would score 0.0. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.08b Integrate x^n: where n != -1 and sums1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| \(3 - 6x - 15x - 10 > 0\) | M1 | Expanding correctly |
| \(-21x - 7 > 0 \Rightarrow x < -\frac{1}{3}\) | A1 | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| \(6x^2 - x - 12 \leq 0\) | M1 | Rearranging to zero |
| \((2x-3)(3x+4) = 0\), \(x = \frac{3}{2}\) or \(x = -\frac{4}{3}\) | M1 | Factorising and finding roots |
| Correct method for inequality (parabola/sign diagram) | M1 | |
| \(-\frac{4}{3} \leq x \leq \frac{3}{2}\) | A1 | Correct final answer |
# Question 8:
## Part (a):
| $3 - 6x - 15x - 10 > 0$ | M1 | Expanding correctly |
| $-21x - 7 > 0 \Rightarrow x < -\frac{1}{3}$ | A1 | Correct answer |
## Part (b):
| $6x^2 - x - 12 \leq 0$ | M1 | Rearranging to zero |
| $(2x-3)(3x+4) = 0$, $x = \frac{3}{2}$ or $x = -\frac{4}{3}$ | M1 | Factorising and finding roots |
| Correct method for inequality (parabola/sign diagram) | M1 | |
| $-\frac{4}{3} \leq x \leq \frac{3}{2}$ | A1 | Correct final answer |8 The diagram shows the curve with equation $y = 3 x ^ { 2 } - x ^ { 3 }$ and the line $L$.\\
\includegraphics[max width=\textwidth, alt={}, center]{b83c4e3a-36a6-4ca9-b44f-489676ca86d4-06_469_802_411_603}
The points $A$ and $B$ have coordinates $( - 1,0 )$ and $( 2,0 )$ respectively. The curve touches the $x$-axis at the origin $O$ and crosses the $x$-axis at the point $( 3,0 )$. The line $L$ cuts the curve at the point $D$ where $x = - 1$ and touches the curve at $C$ where $x = 2$.
\begin{enumerate}[label=(\alph*)]
\item Find the area of the rectangle $A B C D$.
\item \begin{enumerate}[label=(\roman*)]
\item Find $\int \left( 3 x ^ { 2 } - x ^ { 3 } \right) \mathrm { d } x$.
\item Hence find the area of the shaded region bounded by the curve and the line $L$.
\end{enumerate}\item For the curve above with equation $y = 3 x ^ { 2 } - x ^ { 3 }$ :
\begin{enumerate}[label=(\roman*)]
\item find $\frac { \mathrm { d } y } { \mathrm {~d} x }$;
\item hence find an equation of the tangent at the point on the curve where $x = 1$;
\item show that $y$ is decreasing when $x ^ { 2 } - 2 x > 0$.
\end{enumerate}\item Solve the inequality $x ^ { 2 } - 2 x > 0$.
\end{enumerate}
\hfill \mbox{\textit{AQA C1 Q8 [6]}}