| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2020 |
| Session | November |
| Marks | 8 |
| 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 stationary points question requiring basic differentiation of a sum including a quotient (or chain rule), setting the derivative to zero, and using the second derivative test. All steps are routine A-level techniques with no problem-solving insight needed, making it slightly easier than average. |
| Spec | 1.07d Second derivatives: d^2y/dx^2 notation1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks |
|---|---|
| \(\frac{dy}{dx} = [2]\left[-2(2x+1)^{-2}\right]\) | B1 B1 |
| \(\frac{d^2y}{dx^2} = 8(2x+1)^{-3}\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Set *their* \(\frac{dy}{dx} = 0\) and attempt solution | M1 | |
| \((2x+1)^2 = 1 \rightarrow 2x+1 = (\pm)1\) or \(4x^2 + 4x = 0 \rightarrow (4)x(x+1) = 0\) | M1 | Solving as far as \(x = ...\) |
| \(x = 0\) | A1 | WWW. Ignore other solution |
| \((0, 2)\) | A1 | One solution only. Accept \(x = 0\), \(y = 2\) only |
| \(\frac{d^2y}{dx^2} > 0\) from a solution \(x > -\frac{1}{2}\) hence minimum | B1 | Ignore other solution. Condone arithmetic slip in value of \(\frac{d^2y}{dx^2}\). *Their* \(\frac{d^2y}{dx^2}\) must be of the form \(k(2x+1)^{-3}\) |
## Question 8(a):
| $\frac{dy}{dx} = [2]\left[-2(2x+1)^{-2}\right]$ | B1 B1 | |
|---|---|---|
| $\frac{d^2y}{dx^2} = 8(2x+1)^{-3}$ | B1 | |
## Question 8(b):
| Set *their* $\frac{dy}{dx} = 0$ and attempt solution | M1 | |
|---|---|---|
| $(2x+1)^2 = 1 \rightarrow 2x+1 = (\pm)1$ or $4x^2 + 4x = 0 \rightarrow (4)x(x+1) = 0$ | M1 | Solving as far as $x = ...$ |
| $x = 0$ | A1 | WWW. Ignore other solution |
| $(0, 2)$ | A1 | One solution only. Accept $x = 0$, $y = 2$ only |
| $\frac{d^2y}{dx^2} > 0$ from a solution $x > -\frac{1}{2}$ hence minimum | B1 | Ignore other solution. Condone arithmetic slip in value of $\frac{d^2y}{dx^2}$. *Their* $\frac{d^2y}{dx^2}$ must be of the form $k(2x+1)^{-3}$ |
---8 The equation of a curve is $y = 2 x + 1 + \frac { 1 } { 2 x + 1 }$ for $x > - \frac { 1 } { 2 }$.
\begin{enumerate}[label=(\alph*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
\item Find the coordinates of the stationary point and determine the nature of the stationary point.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2020 Q8 [8]}}