| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find dy/dx at a point |
| Difficulty | Standard +0.8 This is a solid implicit differentiation question requiring multiple steps: differentiate implicitly (standard), find two specific tangent lines by substituting x=0 and y=0 into the curve equation, then use the angle-between-lines formula. The final part requires careful algebraic manipulation and knowledge of the tan(angle between lines) formula, elevating it above routine implicit differentiation exercises. |
| Spec | 1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \(3y^2\frac{dy}{dx}\) as derivative of \(y^3\) | B1 | |
| State or imply \(2y + 2x\frac{dy}{dx}\) as derivative of \(2xy\) | B1 | |
| Complete differentiation and equate attempted derivative to zero and solve for \(\frac{dy}{dx}\) | M1 | |
| Obtain answer \(-\frac{3x^2+2y}{3y^2+2x}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Find gradient at either \((0,-2)\) or \((-2,0)\) | M1 | |
| Obtain answers \(\frac{1}{3}\) and \(3\) | A1 A1 | |
| Use \(\tan(A\pm B)\) formula to find \(\tan\alpha\) | M1 | |
| Obtain answer \(\tan\alpha = \frac{4}{3}\) | A1 |
## Question 8:
### Part 8(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $3y^2\frac{dy}{dx}$ as derivative of $y^3$ | B1 | |
| State or imply $2y + 2x\frac{dy}{dx}$ as derivative of $2xy$ | B1 | |
| Complete differentiation and equate attempted derivative to zero and solve for $\frac{dy}{dx}$ | M1 | |
| Obtain answer $-\frac{3x^2+2y}{3y^2+2x}$ | A1 | |
### Part 8(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Find gradient at either $(0,-2)$ or $(-2,0)$ | M1 | |
| Obtain answers $\frac{1}{3}$ and $3$ | A1 A1 | |
| Use $\tan(A\pm B)$ formula to find $\tan\alpha$ | M1 | |
| Obtain answer $\tan\alpha = \frac{4}{3}$ | A1 | |
---8 The equation of a curve is $x ^ { 3 } + y ^ { 3 } + 2 x y + 8 = 0$.
\begin{enumerate}[label=(\alph*)]
\item Express $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.\\
The tangent to the curve at the point where $x = 0$ and the tangent at the point where $y = 0$ intersect at the acute angle $\alpha$.
\item Find the exact value of $\tan \alpha$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q8 [9]}}