| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2020 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find stationary points coordinates |
| Difficulty | Standard +0.3 This question requires applying the quotient rule to find dy/dx, solving f'(x)=0 for stationary points, then polynomial division and integration. While it involves multiple steps and techniques, each component is standard A-level procedure with no novel insight required. The quotient rule application and solving the resulting equation are routine for P2 level, making this slightly easier than average. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Differentiate using the quotient rule (or product rule) | *M1 | |
| Obtain \(\frac{(2x-1)(12x^2+8) - 2(4x^3+8x-4)}{(2x-1)^2}\) | A1 | OE |
| Equate first derivative to zero and attempt solution | DM1 | |
| Obtain \((0, 4)\) | A1 | Allow if given separately; Allow A1 if both \(x\)-coordinates are given but \(y\) coordinates are omitted |
| Obtain \(\left(\frac{3}{4}, \frac{59}{8}\right)\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Carry out division to obtain quotient of form \(2x^2 + kx + m\) | M1 | For non-zero constants \(k\), \(m\) |
| Obtain correct quotient \(2x^2 + x + \frac{9}{2}\) | A1 | |
| Obtain remainder \(\frac{1}{2}\) | A1 | |
| Integrate to obtain at least \(k_1x^3\) and \(k_2\ln(2x-1)\) terms | M1 | For non-zero constants \(k_1\), \(k_2\) |
| Obtain \(\frac{2}{3}x^3 + \frac{1}{2}x^2 + \frac{9}{2}x + \frac{1}{4}\ln(2x-1)\) as final answer | A1 | Condone absence of \(\ldots + c\) and modulus signs |
## Question 8(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Differentiate using the quotient rule (or product rule) | *M1 | |
| Obtain $\frac{(2x-1)(12x^2+8) - 2(4x^3+8x-4)}{(2x-1)^2}$ | A1 | OE |
| Equate first derivative to zero and attempt solution | DM1 | |
| Obtain $(0, 4)$ | A1 | Allow if given separately; Allow A1 if both $x$-coordinates are given but $y$ coordinates are omitted |
| Obtain $\left(\frac{3}{4}, \frac{59}{8}\right)$ | A1 | |
## Question 8(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Carry out division to obtain quotient of form $2x^2 + kx + m$ | M1 | For non-zero constants $k$, $m$ |
| Obtain correct quotient $2x^2 + x + \frac{9}{2}$ | A1 | |
| Obtain remainder $\frac{1}{2}$ | A1 | |
| Integrate to obtain at least $k_1x^3$ and $k_2\ln(2x-1)$ terms | M1 | For non-zero constants $k_1$, $k_2$ |
| Obtain $\frac{2}{3}x^3 + \frac{1}{2}x^2 + \frac{9}{2}x + \frac{1}{4}\ln(2x-1)$ as final answer | A1 | Condone absence of $\ldots + c$ and modulus signs |8 A curve has equation $y = f ( x )$ where $f ( x ) = \frac { 4 x ^ { 3 } + 8 x - 4 } { 2 x - 1 }$.
\begin{enumerate}[label=(\alph*)]
\item Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and hence find the coordinates of each of the stationary points of the curve $y = \mathrm { f } ( x )$.
\item Divide $4 x ^ { 3 } + 8 x - 4$ by ( $2 x - 1$ ), and hence find $\int \mathrm { f } ( x ) \mathrm { d } x$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2020 Q8 [10]}}