| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find stationary points |
| Difficulty | Standard +0.3 This is a straightforward implicit differentiation question with standard techniques. Part (i) requires routine application of implicit differentiation rules, and part (ii) involves setting dy/dx = 0 and solving the resulting system—both are textbook exercises requiring no novel insight, making it slightly easier than average. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply \(6xy + 3x^2\frac{dy}{dx}\) as derivative of \(3x^2 y\) | B1 | |
| State \(3y^2\frac{dy}{dx}\) as derivative of \(y^3\) | B1 | |
| Equate attempted derivative of LHS to zero and solve for \(\frac{dy}{dx}\) | M1 | |
| Obtain the given answer | A1 | |
| Total | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Equate numerator to zero | M1* | |
| Obtain \(x = 2y\), or equivalent | A1 | |
| Obtain an equation in \(x\) or \(y\) | DM1* | |
| Obtain the point \((-2, -1)\) | A1 | |
| State the point \((0, 1.44)\) | B1 | |
| Total | [5] |
## Question 7:
### Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply $6xy + 3x^2\frac{dy}{dx}$ as derivative of $3x^2 y$ | B1 | |
| State $3y^2\frac{dy}{dx}$ as derivative of $y^3$ | B1 | |
| Equate attempted derivative of LHS to zero and solve for $\frac{dy}{dx}$ | M1 | |
| Obtain the given answer | A1 | |
| **Total** | **[4]** | |
### Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Equate numerator to zero | M1* | |
| Obtain $x = 2y$, or equivalent | A1 | |
| Obtain an equation in $x$ or $y$ | DM1* | |
| Obtain the point $(-2, -1)$ | A1 | |
| State the point $(0, 1.44)$ | B1 | |
| **Total** | **[5]** | |
---7 The equation of a curve is $x ^ { 3 } - 3 x ^ { 2 } y + y ^ { 3 } = 3$.\\
(i) Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } - 2 x y } { x ^ { 2 } - y ^ { 2 } }$.\\
(ii) Find the coordinates of the points on the curve where the tangent is parallel to the $x$-axis.
\hfill \mbox{\textit{CAIE P3 2016 Q7 [9]}}