| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2009 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration using inverse trig and hyperbolic functions |
| Type | Derivative of inverse trig function |
| Difficulty | Standard +0.3 Part (i) is a standard FP2 proof requiring implicit differentiation of y = sin⁻¹x, which is bookwork. Part (ii) applies implicit differentiation to a given equation—straightforward application once you differentiate both inverse sine terms. The calculation at x=1/4 requires care with surds but follows directly. This is a routine FP2 question testing standard techniques without requiring novel insight, making it slightly easier than average overall. |
| Spec | 1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.07s Parametric and implicit differentiation |
\begin{enumerate}[label=(\roman*)]
\item Prove that the derivative of $\sin^{-1} x$ is $\frac{1}{\sqrt{1-x^2}}$. [3]
\item Given that
$$\sin^{-1} 2x + \sin^{-1} y = \frac{1}{2}\pi,$$
find the exact value of $\frac{dy}{dx}$ when $x = \frac{1}{4}$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR FP2 2009 Q3 [7]}}