| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2014 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Find gradient at specific point |
| Difficulty | Moderate -0.3 Part (i) is straightforward differentiation using chain rule and standard trig derivatives, then substitution. Part (ii) requires implicit differentiation which is slightly more involved but still a standard technique. Both are routine applications with clear methods, making this slightly easier than average for A-level. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Differentiate to obtain form \(k_1\sin 2x + k_2\cos x\) | M1 | |
| Obtain correct \(-6\sin 2x - 5\cos x\) | A1 | |
| Substitute \(\frac{1}{6}\pi\) to obtain \(-\frac{11}{2}\sqrt{3}\) or exact equivalent | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Obtain \(6y + 6x\dfrac{dy}{dx}\) as derivative of \(6xy\) | B1 | |
| Obtain \(3y^2\dfrac{dy}{dx}\) as derivative of \(y^3\) | B1 | |
| Obtain \(3x^2 + 6y + 6x\dfrac{dy}{dx} + 3y^2\dfrac{dy}{dx} = 0\) or equivalent | B1 | |
| Substitute \(1\) and \(2\) to find value of gradient dependent on at least one B1 | M1 | |
| Obtain gradient \(-\dfrac{15}{18}\) or \(-\dfrac{5}{6}\) | A1 | [5] |
## Question 4(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Differentiate to obtain form $k_1\sin 2x + k_2\cos x$ | M1 | |
| Obtain correct $-6\sin 2x - 5\cos x$ | A1 | |
| Substitute $\frac{1}{6}\pi$ to obtain $-\frac{11}{2}\sqrt{3}$ or exact equivalent | A1 | [3] |
## Question 4(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Obtain $6y + 6x\dfrac{dy}{dx}$ as derivative of $6xy$ | B1 | |
| Obtain $3y^2\dfrac{dy}{dx}$ as derivative of $y^3$ | B1 | |
| Obtain $3x^2 + 6y + 6x\dfrac{dy}{dx} + 3y^2\dfrac{dy}{dx} = 0$ or equivalent | B1 | |
| Substitute $1$ and $2$ to find value of gradient dependent on at least one B1 | M1 | |
| Obtain gradient $-\dfrac{15}{18}$ or $-\dfrac{5}{6}$ | A1 | [5] |
---4 For each of the following curves, find the exact gradient at the point indicated:\\
(i) $y = 3 \cos 2 x - 5 \sin x$ at $\left( \frac { 1 } { 6 } \pi , - 1 \right)$,\\
(ii) $x ^ { 3 } + 6 x y + y ^ { 3 } = 21$ at $( 1,2 )$.
\hfill \mbox{\textit{CAIE P2 2014 Q4 [8]}}