| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2003 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find stationary points |
| Difficulty | Standard +0.3 This question involves standard differentiation of a product (requiring the product rule), finding stationary points by setting the derivative to zero, and finding where a tangent passes through a given point. All techniques are routine for P2 level with no novel problem-solving required, though part (iii) requires setting up and solving an equation involving the tangent line, making it slightly above pure recall but still straightforward application. |
| Spec | 1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation |
| Answer | Marks |
|---|---|
| State \(A\) is \((4, 0)\) | B1 |
| State \(B\) is \((0, 4)\) | B1 |
| Answer | Marks |
|---|---|
| Use the product rule to obtain the first derivative | M1(dep) |
| Obtain derivative \((4-x)e^x - e^x\), or equivalent | A1 |
| Equate derivative to zero and solve for \(x\) | M1(dep) |
| Obtain answer \(x = 3\) only | A1 |
| Answer | Marks |
|---|---|
| Attempt to form an equation in \(p\) e.g. by equating gradients of \(OP\) and the tangent at \(P\), or by substituting \((0, 0)\) in the equation of the tangent at \(P\) | M1 |
| Obtain equation in any correct form e.g. \(\frac{4-p}{p} = 3 - p\) | A1 |
| Obtain 3-term quadratic \(p^2 - 4p + 4 = 0\), or equivalent | A1 |
| Attempt to solve a quadratic equation in \(p\) | M1 |
| Obtain answer \(p = 2\) only | A1 |
### (i)
State $A$ is $(4, 0)$ | B1 |
State $B$ is $(0, 4)$ | B1 |
**Total: [2]**
### (ii)
Use the product rule to obtain the first derivative | M1(dep) |
Obtain derivative $(4-x)e^x - e^x$, or equivalent | A1 |
Equate derivative to zero and solve for $x$ | M1(dep) |
Obtain answer $x = 3$ only | A1 |
**Total: [4]**
### (iii)
Attempt to form an equation in $p$ e.g. by equating gradients of $OP$ and the tangent at $P$, or by substituting $(0, 0)$ in the equation of the tangent at $P$ | M1 |
Obtain equation in any correct form e.g. $\frac{4-p}{p} = 3 - p$ | A1 |
Obtain 3-term quadratic $p^2 - 4p + 4 = 0$, or equivalent | A1 |
Attempt to solve a quadratic equation in $p$ | M1 |
Obtain answer $p = 2$ only | A1 |
**Total: [5]**\includegraphics{figure_6}
The diagram shows the curve $y = (4 - x)e^x$ and its maximum point $M$. The curve cuts the $x$-axis at $A$ and the $y$-axis at $B$.
\begin{enumerate}[label=(\roman*)]
\item Write down the coordinates of $A$ and $B$. [2]
\item Find the $x$-coordinate of $M$. [4]
\item The point $P$ on the curve has $x$-coordinate $p$. The tangent to the curve at $P$ passes through the origin $O$. Calculate the value of $p$. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2003 Q6 [11]}}