| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find stationary points and nature |
| Difficulty | Moderate -0.3 This is a straightforward application of the product rule to find dy/dx, setting it to zero to find the stationary point, and using the second derivative test. The exponential never equals zero, making the algebra simple. Slightly easier than average due to being a standard textbook exercise with clean arithmetic. |
| Spec | 1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use correct product rule | M1 | |
| Obtain correct derivative in any form | A1 | \(\frac{dy}{dx} = e^{1-2x} - 2xe^{1-2x}\) |
| Equate derivative to zero and solve for \(x\) | M1 | |
| Obtain \(x = \frac{1}{2}\) and \(y = \frac{1}{2}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use a correct method for determining the nature of a stationary point | M1 | e.g. \(\frac{d^2y}{dx^2} = -2e^{1-2x} - 2(1-2x)e^{1-2x}\) |
| Show that it is a maximum point | A1 |
## Question 3(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct product rule | M1 | |
| Obtain correct derivative in any form | A1 | $\frac{dy}{dx} = e^{1-2x} - 2xe^{1-2x}$ |
| Equate derivative to zero and solve for $x$ | M1 | |
| Obtain $x = \frac{1}{2}$ and $y = \frac{1}{2}$ | A1 | |
## Question 3(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use a correct method for determining the nature of a stationary point | M1 | e.g. $\frac{d^2y}{dx^2} = -2e^{1-2x} - 2(1-2x)e^{1-2x}$ |
| Show that it is a maximum point | A1 | |3 The curve with equation $y = x \mathrm { e } ^ { 1 - 2 x }$ has one stationary point.
\begin{enumerate}[label=(\alph*)]
\item Find the coordinates of this point.
\item Determine whether the stationary point is a maximum or a minimum.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q3 [6]}}