| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Session | Specimen |
| Marks | 12 |
| Topic | Differential equations |
| Type | Geometric curve properties |
| Difficulty | Challenging +1.2 Part (a) involves standard differentiation using quotient rule and trigonometric identities—routine A-level work. Part (b) requires translating geometric conditions into a differential equation using tan v = dy/dx and tan u = y/x, then solving a separable equation. While multi-step, the techniques are standard and the geometric setup guides the solution clearly. Slightly above average due to the geometric interpretation required, but well within typical Further Maths capability. |
| Spec | 1.07l Derivative of ln(x): and related functions1.08k Separable differential equations: dy/dx = f(x)g(y) |
**(a)(i)** Use of the quotient rule with $u = 1$ and $v = \cos x$: M1
$\frac{dy}{dx} = \frac{\cos x \times 0 - 1 \times (-\sin x)}{\cos^2 x}$ A1
$= \sec x \tan x$ **AG** **[2]**
**(a)(ii)** $= \sec x\sqrt{\sec^2 x - 1}$ M1
$= y\sqrt{y^2 - 1}$ **AG** A1 **[2]**
**(b)(i)** Interpretation of at least one tangent: M1
$\tan u = \frac{y}{x}$, $\tan v = \frac{-1}{\frac{dy}{dx}}$ A1
Attempt at a substitution into a relationship: M1
$\frac{dy}{dx} = -\frac{x^2}{y^2}$ A1 **[4]**
**(b)(ii)** Attempt at separation of variables in integral form: M1
$\int y^2\,dy = -\int x^2\,dx$
Integrate both sides, giving $\frac{1}{3}y^3 = -\frac{1}{3}x^3$ $(+ C)$ A1
Substitute $(2, 1)$: M1
Obtain $y^3 + x^3 = 9$ or equivalent A1 **[4]**10
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item By writing $\sec x = \frac { 1 } { \cos x }$, prove that
$$\frac { \mathrm { d } } { \mathrm {~d} x } ( \sec x ) = \sec x \tan x .$$
\item Deduce that $y = \sec x$ satisfies the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = y \sqrt { y ^ { 2 } - 1 } , \quad 0 \leq x < \frac { 1 } { 2 } \pi .$$
\end{enumerate}\item A curve lies in the first quadrant of the cartesian plane with origin $O$ as shown in the diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{85043199-527d-4105-aa0b-c913dec0e35b-4_707_698_845_685}
The normal to the curve at the point $P ( x , y )$ meets the $x$-axis at the point $Q$. The angle between $O P$ and the $x$-axis is $u$, and the angle between $Q P$ and the $x$-axis is $v$.
\begin{enumerate}[label=(\roman*)]
\item If
$$\tan v = \tan ^ { 2 } u$$
obtain a differential equation satisfied by the curve.
\item The curve passes through the point $( 2,1 )$. By solving the differential equation, find an equation for the curve in the implicit form
$$\mathrm { F } ( x , y ) = C ,$$
where $C$ is a constant that should be determined.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 Q10 [12]}}