| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | March |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find vertical tangent points |
| Difficulty | Standard +0.8 This question requires implicit differentiation (standard P3 technique) but part (b) demands conceptual understanding that vertical tangents occur when dx/dy = 0, requiring students to invert the derivative and solve a coupled system with the original curve equation. The algebraic manipulation is non-trivial, making this harder than routine implicit differentiation exercises but not requiring exceptional insight. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| State or imply \(3y^2 + 6xy\,\frac{dy}{dx}\) as derivative of \(3xy^2\) | B1 | |
| State or imply \(3y^2\,\frac{dy}{dx}\) as derivative of \(y^3\) | B1 | |
| Equate attempted derivative of LHS to zero and solve for \(\frac{dy}{dx}\) | M1 | Need to see \(\frac{dy}{dx}\) factorised out prior to AG |
| Obtain the given answer correctly | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Equate denominator to zero | *M1 | |
| Obtain \(y=2x\), or equivalent | A1 | |
| Obtain an equation in \(x\) or \(y\) | DM1 | |
| Obtain the point \((1,2)\) | A1 | |
| State the point \(\left(\sqrt[3]{5},0\right)\) | B1 | Alternatively \((1.71, 0)\) |
## Question 7(a):
| State or imply $3y^2 + 6xy\,\frac{dy}{dx}$ as derivative of $3xy^2$ | B1 | |
| State or imply $3y^2\,\frac{dy}{dx}$ as derivative of $y^3$ | B1 | |
| Equate attempted derivative of LHS to zero and solve for $\frac{dy}{dx}$ | M1 | Need to see $\frac{dy}{dx}$ factorised out prior to AG |
| Obtain the given answer correctly | A1 | AG |
## Question 7(b):
| Equate denominator to zero | *M1 | |
| Obtain $y=2x$, or equivalent | A1 | |
| Obtain an equation in $x$ or $y$ | DM1 | |
| Obtain the point $(1,2)$ | A1 | |
| State the point $\left(\sqrt[3]{5},0\right)$ | B1 | Alternatively $(1.71, 0)$ |
---7 The equation of a curve is $x ^ { 3 } + 3 x y ^ { 2 } - y ^ { 3 } = 5$.
\begin{enumerate}[label=(\alph*)]
\item Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } + y ^ { 2 } } { y ^ { 2 } - 2 x y }$.
\item Find the coordinates of the points on the curve where the tangent is parallel to the $y$-axis.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q7 [9]}}