| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2011 |
| Session | June |
| Marks | 9 |
| Topic | Composite & Inverse Functions |
| Type | Determine if inverse exists |
| Difficulty | Moderate -0.3 This question tests standard A-level concepts: determining invertibility (one-to-one test), finding range by completing the square, applying a double angle identity, and sketching a transformed trigonometric function. All parts are routine applications of well-practiced techniques with no novel problem-solving required, making it slightly easier than average but not trivial due to the multi-part nature and need for clear reasoning. |
| Spec | 1.02m Graphs of functions: difference between plotting and sketching1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.02w Graph transformations: simple transformations of f(x)1.05l Double angle formulae: and compound angle formulae |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Not invertible | B1 | |
| Not 1–1 or equivalent | B1 | [2] |
| (b) (Minimum value of –1 at \(x = 1\)) | B1 | |
| \(-1 \le f(x)\) | B1 | |
| [B1 for correct interval; B1 for correct inequality] | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(gh(x) = \sin^2 x\) | B1 | |
| Obtain \(\frac{1}{2}(1 - \cos 2x)\) with some working | AG | B1 |
| (b) Sine wave | M1 | |
| Period of \(\pi\) | A1 | |
| Completely correct | A1 | [3] |
**Part (i)**
**(a)** Not invertible | B1
Not 1–1 or equivalent | B1 | [2]
**(b)** (Minimum value of –1 at $x = 1$) | B1
$-1 \le f(x)$ | B1
[B1 for correct interval; B1 for correct inequality] | [2]
**Part (ii)**
**(a)** $gh(x) = \sin^2 x$ | B1
Obtain $\frac{1}{2}(1 - \cos 2x)$ with some working | AG | B1 | [2]
**(b)** Sine wave | M1
Period of $\pi$ | A1
Completely correct | A1 | [3]Functions f, g and h are defined for $x \in \mathbb{R}$ by
$$f : x \mapsto x^2 - 2x,$$
$$g : x \mapsto x^2,$$
$$h : x \mapsto \sin x.$$
\begin{enumerate}[label=(\roman*)]
\item \begin{enumerate}[label=(\alph*)]
\item State whether or not f has an inverse, giving a reason. [2]
\item Determine the range of the function f. [2]
\end{enumerate}
\item \begin{enumerate}[label=(\alph*)]
\item Show that gh(x) can be expressed as $\frac{1}{2}(1 - \cos 2x)$. [2]
\item Sketch the curve C defined by $y = \text{gh}(x)$ for $0 \leqslant x \leqslant 2\pi$. [3]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2011 Q7 [9]}}