| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2011 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Standard +0.3 This is a standard C3 inverse function question with routine steps: finding range of a cosine function, deriving inverse using arccos, solving a simple equation, composing with absolute value, sketching a transformed trig graph, and describing transformations. All techniques are textbook exercises requiring careful execution but no novel insight. Slightly easier than average due to straightforward function forms. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02u 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)1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(-3 \leq f(x) \leq 3\) | B1 B1 | B1 for each correct bound; accept \([-3, 3]\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x = 3\cos\frac{1}{2}y \Rightarrow \cos\frac{1}{2}y = \frac{x}{3}\) | M1 | Attempt to rearrange |
| \(\frac{1}{2}y = \cos^{-1}\left(\frac{x}{3}\right)\) | M1 | Applying \(\cos^{-1}\) |
| \(f^{-1}(x) = 2\cos^{-1}\left(\frac{x}{3}\right)\) | A1 | Correct final answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(2\cos^{-1}\left(\frac{1}{3}\right) = 1\) — solve \(\cos^{-1}\left(\frac{1}{3}\right) = \frac{1}{2}\) | M1 | Setting up equation |
| \(x = 3\cos\frac{1}{2}\) | A1 | Accept exact form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\text{gf}(x) = \left | 3\cos\frac{1}{2}x\right | \) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Correct shape of \(3 | \cos\frac{1}{2}x | \) for \(0 \leq x \leq 2\pi\) |
| Touches \(x\)-axis at \(x = \pi\) | B1 | |
| Ends at \((2\pi, 3)\) with correct W/V shape | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Stretch parallel to \(x\)-axis, scale factor 2 | B1 M1 | |
| Stretch parallel to \(y\)-axis, scale factor 3 | A1 | Both transformations required for full marks |
# Question 4:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $-3 \leq f(x) \leq 3$ | B1 B1 | B1 for each correct bound; accept $[-3, 3]$ |
## Part (b)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $x = 3\cos\frac{1}{2}y \Rightarrow \cos\frac{1}{2}y = \frac{x}{3}$ | M1 | Attempt to rearrange |
| $\frac{1}{2}y = \cos^{-1}\left(\frac{x}{3}\right)$ | M1 | Applying $\cos^{-1}$ |
| $f^{-1}(x) = 2\cos^{-1}\left(\frac{x}{3}\right)$ | A1 | Correct final answer |
## Part (b)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $2\cos^{-1}\left(\frac{1}{3}\right) = 1$ — solve $\cos^{-1}\left(\frac{1}{3}\right) = \frac{1}{2}$ | M1 | Setting up equation |
| $x = 3\cos\frac{1}{2}$ | A1 | Accept exact form |
## Part (c)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\text{gf}(x) = \left|3\cos\frac{1}{2}x\right|$ | B1 | Correct expression |
## Part (c)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Correct shape of $3|\cos\frac{1}{2}x|$ for $0 \leq x \leq 2\pi$ | B1 | Starts at $(0,3)$ |
| Touches $x$-axis at $x = \pi$ | B1 | |
| Ends at $(2\pi, 3)$ with correct W/V shape | B1 | |
## Part (d)
| Answer | Mark | Guidance |
|--------|------|----------|
| Stretch parallel to $x$-axis, scale factor 2 | B1 M1 | |
| Stretch parallel to $y$-axis, scale factor 3 | A1 | Both transformations required for full marks |
---4 The functions f and g are defined with their respective domains by
$$\begin{array} { l l }
\mathrm { f } ( x ) = 3 \cos \frac { 1 } { 2 } x , & \text { for } 0 \leqslant x \leqslant 2 \pi \\
\mathrm {~g} ( x ) = | x | , & \text { for all real values of } x
\end{array}$$
\begin{enumerate}[label=(\alph*)]
\item Find the range of f .
\item The inverse of f is $\mathrm { f } ^ { - 1 }$.
\begin{enumerate}[label=(\roman*)]
\item Find $\mathrm { f } ^ { - 1 } ( x )$.
\item Solve the equation $\mathrm { f } ^ { - 1 } ( x ) = 1$, giving your answer in an exact form.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Write down an expression for $\mathrm { gf } ( x )$.
\item Sketch the graph of $y = \operatorname { gf } ( x )$ for $0 \leqslant x \leqslant 2 \pi$.
\end{enumerate}\item Describe a sequence of two geometrical transformations that maps the graph of $y = \cos x$ onto the graph of $y = 3 \cos \frac { 1 } { 2 } x$.
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2011 Q4 [14]}}