| Exam Board | WJEC |
|---|---|
| Module | Unit 1 (Unit 1) |
| Year | 2019 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Find stationary point then sketch curve |
| Difficulty | Moderate -0.8 This is a straightforward multi-part calculus question requiring routine differentiation, solving quadratic equations, and curve sketching. Part (a) is simple substitution into the derivative formula, part (b) uses standard second derivative test, and part (c) is basic sketching. All techniques are standard AS-level procedures with no problem-solving insight required, making it easier than average. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives |
A curve $C$ has equation $y = \frac{1}{9}x^3 - kx + 5$. A point $Q$ lies on $C$ and is such that the tangent to $C$ at $Q$ has gradient $-9$. The $x$-coordinate of $Q$ is $3$.
\begin{enumerate}[label=(\alph*)]
\item Show that $k = 12$. [3]
\item Find the coordinates of each of the stationary points of $C$ and determine their nature. [6]
\item Sketch the curve $C$, clearly labelling the stationary points and the point where the curve crosses the $y$-axis. [2]
\end{enumerate}
\hfill \mbox{\textit{WJEC Unit 1 2019 Q13 [11]}}