| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | March |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find stationary points |
| Difficulty | Challenging +1.2 This is a standard implicit differentiation problem requiring product rule and chain rule application, followed by solving for a stationary point. While it involves multiple steps (differentiating implicitly, rearranging for dy/dx, setting equal to zero, and solving), these are well-practiced techniques at A-level. The algebraic manipulation is moderately involved but follows predictable patterns for this topic. |
| Spec | 1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) EITHER: State correct derivative of \(\sin y\) with respect to \(x\) | B1 | |
| Use product rule to differentiate the LHS | M1 | |
| Obtain correct derivative of the LHS | A1 | |
| Obtain a complete and correct derived equation in any form | A1 | |
| Obtain a correct expression for \(\frac{dy}{dx}\) in any form | A1 | |
| OR: State correct derivative of \(\sin y\) with respect to \(x\) | B1 | |
| Rearrange the given equation as \(\sin y = x/(ln x + 2)\) and attempt to differentiate both sides | B1 | |
| Use quotient or product rule to differentiate the RHS | M1 | |
| Obtain correct derivative of the RHS | A1 | |
| Obtain a correct expression for \(\frac{dy}{dx}\) in any form | A1 | [5] |
| (ii) Equate \(\frac{dy}{dx}\) to zero and obtain a horizontal equation in \(x\) or \(\sin y\) | M1 | |
| Solve for \(\ln x\) | M1 | |
| Obtain final answer \(x = 1/e\), or exact equivalent | A1 | [3] |
(i) **EITHER:** State correct derivative of $\sin y$ with respect to $x$ | B1 |
Use product rule to differentiate the LHS | M1 |
Obtain correct derivative of the LHS | A1 |
Obtain a complete and correct derived equation in any form | A1 |
Obtain a correct expression for $\frac{dy}{dx}$ in any form | A1 |
**OR:** State correct derivative of $\sin y$ with respect to $x$ | B1 |
Rearrange the given equation as $\sin y = x/(ln x + 2)$ and attempt to differentiate both sides | B1 |
Use quotient or product rule to differentiate the RHS | M1 |
Obtain correct derivative of the RHS | A1 |
Obtain a correct expression for $\frac{dy}{dx}$ in any form | A1 | [5]
(ii) Equate $\frac{dy}{dx}$ to zero and obtain a horizontal equation in $x$ or $\sin y$ | M1 |
Solve for $\ln x$ | M1 |
Obtain final answer $x = 1/e$, or exact equivalent | A1 | [3]6 A curve has equation
$$\sin y \ln x = x - 2 \sin y$$
for $- \frac { 1 } { 2 } \pi \leqslant y \leqslant \frac { 1 } { 2 } \pi$.\\
(i) Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.\\
(ii) Hence find the exact $x$-coordinate of the point on the curve at which the tangent is parallel to the $x$-axis.
\hfill \mbox{\textit{CAIE P3 2016 Q6 [8]}}