| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Integration with differentiation context |
| Difficulty | Standard +0.3 Part (a) requires applying the product rule to find the derivative, setting it to zero, and solving a linear equation—straightforward differentiation. Part (b) involves integration by parts (twice) which is standard P3 technique but requires careful execution. Overall slightly easier than average as both parts follow well-practiced procedures without requiring novel insight. |
| Spec | 1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07n Stationary points: find maxima, minima using derivatives1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use correct product or quotient rule | *M1 | |
| Obtain correct derivative in any form, e.g. \(\frac{dy}{dx} = -e^{-\frac{x}{3}} - \frac{1}{3}(3-x)e^{-\frac{x}{3}}\) | A1 | |
| Equate their derivative to zero and solve for \(x\) | DM1 | |
| Obtain \(x=6\) | A1 | |
| Obtain \(y = -3e^{-2}\) | A1 | Or exact equivalent |
| Total | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Commence integration and reach \(a(3-x)e^{-\frac{x}{3}} + b\int e^{-\frac{x}{3}}\,dx\), where \(ab\neq 0\) | *M1 | |
| Obtain \(-3(3-x)e^{-\frac{x}{3}} - 3\int e^{-\frac{x}{3}}\,dx\), or equivalent | A1 | |
| Complete integration and obtain \(3xe^{-\frac{x}{3}}\), or equivalent | A1 | \(-3e^{-\frac{x}{3}}(3-x) + 9e^{-\frac{x}{3}}\) |
| Substitute limits \(x=0\) and \(x=3\), having integrated twice | DM1 | |
| Obtain answer \(\frac{9}{e}\), or exact equivalent | A1 | |
| Total | 5 |
## Question 9(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct product or quotient rule | *M1 | |
| Obtain correct derivative in any form, e.g. $\frac{dy}{dx} = -e^{-\frac{x}{3}} - \frac{1}{3}(3-x)e^{-\frac{x}{3}}$ | A1 | |
| Equate their derivative to zero and solve for $x$ | DM1 | |
| Obtain $x=6$ | A1 | |
| Obtain $y = -3e^{-2}$ | A1 | Or exact equivalent |
| **Total** | **5** | |
## Question 9(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Commence integration and reach $a(3-x)e^{-\frac{x}{3}} + b\int e^{-\frac{x}{3}}\,dx$, where $ab\neq 0$ | *M1 | |
| Obtain $-3(3-x)e^{-\frac{x}{3}} - 3\int e^{-\frac{x}{3}}\,dx$, or equivalent | A1 | |
| Complete integration and obtain $3xe^{-\frac{x}{3}}$, or equivalent | A1 | $-3e^{-\frac{x}{3}}(3-x) + 9e^{-\frac{x}{3}}$ |
| Substitute limits $x=0$ and $x=3$, having integrated twice | DM1 | |
| Obtain answer $\frac{9}{e}$, or exact equivalent | A1 | |
| **Total** | **5** | |9\\
\includegraphics[max width=\textwidth, alt={}, center]{98001cfe-46a1-4c8f-9230-c140ebff6176-14_535_1082_274_520}
The diagram shows part of the curve $y = ( 3 - x ) \mathrm { e } ^ { - \frac { 1 } { 3 } x }$ for $x \geqslant 0$, and its minimum point $M$.
\begin{enumerate}[label=(\alph*)]
\item Find the exact coordinates of $M$.
\item Find the area of the shaded region bounded by the curve and the axes, giving your answer in terms of e.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q9 [10]}}