| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find stationary points and nature |
| Difficulty | Standard +0.3 This is a slightly above-average C3 question requiring multiple standard techniques: finding x-intercepts (straightforward), differentiation using product rule, finding normal equations, and locating stationary points. Part (c) requires solving dy/dx=0 which involves some algebraic manipulation but follows standard procedures. The multi-part structure and combination of techniques elevates it slightly above average difficulty. |
| Spec | 1.06d Natural logarithm: ln(x) function and properties1.07l Derivative of ln(x): and related functions1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \((4, 0)\) | B1 |
| (b) | \(\frac{dy}{dx} = \frac{3}{2}x^{1/2} \times \ln\frac{4}{5} + x^{1/2} \times \frac{1}{5} = \frac{1}{5}x^{1/2}(5\ln\frac{4}{5} + 2)\) | M1 A1 |
| Grad \(= 8\), grad of normal \(= -\frac{1}{8}\) | A1 | |
| \(y - 0 = -\frac{1}{8}(x-4)\) | M1 | |
| At \(Q\), \(x = 0\): \(y = \frac{1}{2}\) | A1 | |
| Area \(= \frac{1}{2} \times \frac{1}{2} \times 4 = 1\) | A1 | |
| (c) | \(\frac{1}{5}x^{1/2}(5\ln\frac{4}{5} + 2) = 0\) | |
| \(\ln\frac{4}{5} = -\frac{2}{5}\) | M1 | |
| \(x = 4e^{-\frac{4}{5}}\) | M1 A1 | |
| (10) |
(a) | $(4, 0)$ | B1 |
(b) | $\frac{dy}{dx} = \frac{3}{2}x^{1/2} \times \ln\frac{4}{5} + x^{1/2} \times \frac{1}{5} = \frac{1}{5}x^{1/2}(5\ln\frac{4}{5} + 2)$ | M1 A1 |
| Grad $= 8$, grad of normal $= -\frac{1}{8}$ | A1 |
| $y - 0 = -\frac{1}{8}(x-4)$ | M1 |
| At $Q$, $x = 0$: $y = \frac{1}{2}$ | A1 |
| Area $= \frac{1}{2} \times \frac{1}{2} \times 4 = 1$ | A1 |
(c) | $\frac{1}{5}x^{1/2}(5\ln\frac{4}{5} + 2) = 0$ | |
| $\ln\frac{4}{5} = -\frac{2}{5}$ | M1 |
| $x = 4e^{-\frac{4}{5}}$ | M1 A1 |
| | (10) |4. The curve with equation $y = x ^ { \frac { 5 } { 2 } } \ln \frac { x } { 4 } , x > 0$ crosses the $x$-axis at the point $P$.
\begin{enumerate}[label=(\alph*)]
\item Write down the coordinates of $P$.
The normal to the curve at $P$ crosses the $y$-axis at the point $Q$.
\item Find the area of triangle $O P Q$ where $O$ is the origin.
The curve has a stationary point at $R$.
\item Find the $x$-coordinate of $R$ in exact form.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q4 [10]}}