| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2017 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Find normal equation at parameter |
| Difficulty | Standard +0.8 This question requires finding dy/dx using the chain rule for parametric equations, performing polynomial division to simplify the result, then solving for a specific parameter value where the normal has a given gradient. The polynomial division and connecting the normal gradient to dy/dx adds complexity beyond standard parametric differentiation exercises. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Differentiate \(x\) and \(y\) and form \(\frac{dy}{dx}\) | M1 | |
| Obtain \(\dfrac{4t^3 - 6t^2 + 8t - 12}{3t^2 + 6}\) | A1 | First 2 marks may be implied by an attempt at division |
| Carry out division at least as far as \(kt\) or equivalent | M1 | For M1, it must be division by a quadratic factor. Allow attempt at factorisation with same conditions as for division |
| Obtain \(\frac{4}{3}t\) | A1 | |
| Obtain \(\frac{4}{3}t - 2\) with complete division shown and no errors seen | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply gradient of straight line is \(\frac{1}{2}\) | B1 | Allow B1 if \(y = \frac{1}{2}x + \frac{9}{2}\) is seen |
| Attempt value of \(t\) from their \(\frac{dy}{dx}\) = their negative reciprocal of gradient of line | M1 | |
| Obtain \(t = 0\) and hence \((1, 5)\) | A1 |
## Question 7(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Differentiate $x$ and $y$ and form $\frac{dy}{dx}$ | M1 | |
| Obtain $\dfrac{4t^3 - 6t^2 + 8t - 12}{3t^2 + 6}$ | A1 | First 2 marks may be implied by an attempt at division |
| Carry out division at least as far as $kt$ or equivalent | M1 | For M1, it must be division by a quadratic factor. Allow attempt at factorisation with same conditions as for division |
| Obtain $\frac{4}{3}t$ | A1 | |
| Obtain $\frac{4}{3}t - 2$ with complete division shown and no errors seen | A1 | |
**Total: 5**
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## Question 7(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply gradient of straight line is $\frac{1}{2}$ | B1 | Allow B1 if $y = \frac{1}{2}x + \frac{9}{2}$ is seen |
| Attempt value of $t$ from their $\frac{dy}{dx}$ = their negative reciprocal of gradient of line | M1 | |
| Obtain $t = 0$ and hence $(1, 5)$ | A1 | |
**Total: 3**
---7 The parametric equations of a curve are
$$x = t ^ { 3 } + 6 t + 1 , \quad y = t ^ { 4 } - 2 t ^ { 3 } + 4 t ^ { 2 } - 12 t + 5$$
(i) Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and use division to show that $\frac { \mathrm { d } y } { \mathrm {~d} x }$ can be written in the form $a t + b$, where $a$ and $b$ are constants to be found.\\
(ii) The straight line $x - 2 y + 9 = 0$ is the normal to the curve at the point $P$. Find the coordinates of $P$.\\
\hfill \mbox{\textit{CAIE P2 2017 Q7 [8]}}