| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | October |
| Marks | 8 |
| Topic | Product & Quotient Rules |
| Type | Show derivative equals given algebraic form |
| Difficulty | Standard +0.3 This is a straightforward quotient rule differentiation followed by a tangent line calculation. Part (a) requires applying the quotient rule with chain rule for trigonometric functions and algebraic simplification, while part (b) involves substituting a value and finding a line equation. Both are standard A-level techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07m Tangents and normals: gradient and equations1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
The curve $C$ has equation
$$y = \frac{3 + \sin 2x}{2 + \cos 2x}$$
\begin{enumerate}[label=(\alph*)]
\item Show that
$$\frac{dy}{dx} = \frac{6\sin 2x + 4\cos 2x + 2}{(2 + \cos 2x)^2}$$ [4]
\item Find an equation of the tangent to $C$ at the point on $C$ where $x = \frac{\pi}{2}$.
Write your answer in the form $y = ax + b$, where $a$ and $b$ are exact constants. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q5 [8]}}