| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2013 |
| Session | June |
| Marks | 8 |
| Topic | Implicit equations and differentiation |
| Type | Find normal equation at point |
| Difficulty | Moderate -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 finding a normal (perpendicular gradient, then point-slope form). Both parts are mechanical with no conceptual challenges, making it slightly easier than average. |
| Spec | 1.07s Parametric and implicit differentiation |
(i) Differentiate implicitly, using product rule — M1; $y + x\frac{dy}{dx}$ — A1; final term $2y\frac{dy}{dx}$ — B1; complete $2x + y + x\frac{dy}{dx} + 2y\frac{dy}{dx} = 0$, and manipulate to given answer — A1 **[4]**
(ii) Substitute $x = 2$, $y = 3$: $\frac{dy}{dx} = -\frac{7}{8}$ — M1; Gradient of normal is $\frac{8}{7}$ — A1; Line through $(2, 3)$ with their $m$ — M1; Obtain $8x - 7y + 5 = 0$ — A1 **[4]**
**Total: [8]**5 The curve $C$ has equation $x ^ { 2 } + x y + y ^ { 2 } = 19$.\\
(i) Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 2 x - y } { x + 2 y }$.\\
(ii) Hence find the equation of the normal to $C$ at the point $( 2,3 )$ in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2013 Q5 [8]}}