| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2002 |
| Session | November |
| 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 requiring standard technique to find dy/dx, then solving y=2x simultaneously with the curve equation. While it involves multiple steps, the methods are routine for P2 level with no novel insight required, making it slightly easier than average. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State \(6y \frac{dy}{dx}\) as the derivative of \(3y^2\) | B1 | |
| State \(\pm 2x \frac{dy}{dx} \pm 2y\) as the derivative of \(-2xy\) (allow any combination of signs here) | B1 | |
| Equate attempted derivative of LHS to 0 (or 10) and solve for \(\frac{dy}{dx}\) | M1 | |
| Obtain the given answer correctly | A1 | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) State or imply the points lie on \(y - 2x = 0\) | B1 | ⊕ |
| Carry out complete method for finding one coordinate of a point of intersection of \(y = kx\) with the given curve | M1 | |
| Obtain \(10x^2 = 10\) or \(2 \frac{1}{2}y^2 = 10\) or 2-term equivalent | A1 | |
| Obtain one correct point e.g. \((1, 2)\) | A1 | ⊕ |
| Obtain a second correct point e.g. \((-1, -2)\) | A1 | 5⊕ |
**(i)** State $6y \frac{dy}{dx}$ as the derivative of $3y^2$ | B1 |
State $\pm 2x \frac{dy}{dx} \pm 2y$ as the derivative of $-2xy$ (allow any combination of signs here) | B1 |
Equate attempted derivative of LHS to 0 (or 10) and solve for $\frac{dy}{dx}$ | M1 |
Obtain the given answer correctly | A1 | 4 |
[The M1 is dependent on at least one of the B marks being earned.]
**(ii)** State or imply the points lie on $y - 2x = 0$ | B1 | ⊕ |
Carry out complete method for finding one coordinate of a point of intersection of $y = kx$ with the given curve | M1 |
Obtain $10x^2 = 10$ or $2 \frac{1}{2}y^2 = 10$ or 2-term equivalent | A1 |
Obtain one correct point e.g. $(1, 2)$ | A1 | ⊕ | [2 or 3 values of $x$ (or $y$)]
Obtain a second correct point e.g. $(-1, -2)$ | A1 | 5⊕ |7 The equation of a curve is
$$2 x ^ { 2 } + 3 y ^ { 2 } - 2 x y = 10 .$$
(i) Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - 2 x } { 3 y - x }$.\\
(ii) Find the coordinates of the points on the curve where the tangent is parallel to the $x$-axis.
\hfill \mbox{\textit{CAIE P2 2002 Q7 [9]}}