| Exam Board | WJEC |
|---|---|
| Module | Unit 1 (Unit 1) |
| Year | 2022 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Tangent meets curve/axis — further geometry |
| Difficulty | Standard +0.3 This is a straightforward multi-part calculus question requiring standard techniques: finding a tangent equation (differentiation + point-slope form), calculating area between curve and x-axis (definite integration with factored cubic), and finding where f'(x) > 0 (solving a quadratic inequality). All parts are routine A-level procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.07m Tangents and normals: gradient and equations1.07o Increasing/decreasing: functions using sign of dy/dx1.08e Area between curve and x-axis: using definite integrals |
The diagram below shows a sketch of the curve $y = f(x)$, where $f(x) = 10x + 3x^2 - x^3$. The curve intersects the $x$-axis at the origin $O$ and at the points $A(-2, 0)$, $B(5, 0)$. The tangent to the curve at the point $C(2, 24)$ intersects the $y$-axis at the point $D$.
\includegraphics{figure_11}
\begin{enumerate}[label=(\alph*)]
\item Find the coordinates of $D$. [5]
\item Find the area of the shaded region. [6]
\item Determine the range of values of $x$ for which $f(x)$ is an increasing function. [4]
\end{enumerate}
\hfill \mbox{\textit{WJEC Unit 1 2022 Q11 [15]}}