| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2020 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Transformations of functions |
| Difficulty | Moderate -0.3 This is a slightly below-average A-level question covering standard transformations of trigonometric functions. Parts (a)-(c) involve routine identification of max/min values, sketching, and counting intersections with lines. Parts (d)-(e) require describing function transformations, which is standard P1 content but can cause confusion with order of operations. Overall, it's straightforward application of well-practiced techniques with no novel problem-solving required. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.02w Graph transformations: simple transformations of f(x)1.05a Sine, cosine, tangent: definitions for all arguments1.05f Trigonometric function graphs: symmetries and periodicities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(5,\ -1\) | B1 B1 | Sight of each value |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| [curve sketch shown] | *B1 | Needs to be a curve, not straight lines. One complete cycle starting and finishing at *their* largest value. |
| DB1 | One complete cycle starting and finishing at \(y=5\) and going down to \(y=-1\) and starting to level off at least one end. | |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| 0 solutions | B1 | |
| 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| 2 solutions | B1 | |
| 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| 1 solution | B1 | |
| 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Stretch by scale factor \(\frac{1}{2}\), parallel to \(x\)-axis or in \(x\) direction (or horizontally) | B1 | |
| Translation of \(\begin{pmatrix}0\\4\end{pmatrix}\) | B1 | Accept translation/shift. Accept translation 4 units in positive \(y\)-direction. |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Translation of \(\begin{pmatrix}-\frac{\pi}{2}\\0\end{pmatrix}\) | B1 | Accept translation/shift. Accept translation \(-\frac{\pi}{2}\) units in \(x\)-direction. |
| Stretch by scale factor 2 parallel to \(y\)-axis (or vertically) | B1 | |
| 2 |
## Question 11(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $5,\ -1$ | B1 B1 | Sight of each value |
| | **2** | |
## Question 11(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| [curve sketch shown] | *B1 | Needs to be a curve, not straight lines. One complete cycle starting and finishing at *their* largest value. |
| | DB1 | One complete cycle starting and finishing at $y=5$ and going down to $y=-1$ and starting to level off at least one end. |
| | **2** | |
## Question 11(c)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| 0 solutions | B1 | |
| | **1** | |
## Question 11(c)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| 2 solutions | B1 | |
| | **1** | |
## Question 11(c)(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| 1 solution | B1 | |
| | **1** | |
## Question 11(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| Stretch by scale factor $\frac{1}{2}$, parallel to $x$-axis or in $x$ direction (or horizontally) | B1 | |
| Translation of $\begin{pmatrix}0\\4\end{pmatrix}$ | B1 | Accept translation/shift. Accept translation 4 units in positive $y$-direction. |
| | **2** | |
## Question 11(e):
| Answer | Mark | Guidance |
|--------|------|----------|
| Translation of $\begin{pmatrix}-\frac{\pi}{2}\\0\end{pmatrix}$ | B1 | Accept translation/shift. Accept translation $-\frac{\pi}{2}$ units in $x$-direction. |
| Stretch by scale factor 2 parallel to $y$-axis (or vertically) | B1 | |
| | **2** | |11 A curve has equation $y = 3 \cos 2 x + 2$ for $0 \leqslant x \leqslant \pi$.
\begin{enumerate}[label=(\alph*)]
\item State the greatest and least values of $y$.
\item Sketch the graph of $y = 3 \cos 2 x + 2$ for $0 \leqslant x \leqslant \pi$.
\item By considering the straight line $y = k x$, where $k$ is a constant, state the number of solutions of the equation $3 \cos 2 x + 2 = k x$ for $0 \leqslant x \leqslant \pi$ in each of the following cases.
\begin{enumerate}[label=(\roman*)]
\item $k = - 3$
\item $k = 1$
\item $k = 3$\\
Functions $\mathrm { f } , \mathrm { g }$ and h are defined for $x \in \mathbb { R }$ by
$$\begin{aligned}
& \mathrm { f } ( x ) = 3 \cos 2 x + 2 \\
& \mathrm {~g} ( x ) = \mathrm { f } ( 2 x ) + 4 \\
& \mathrm {~h} ( x ) = 2 \mathrm { f } \left( x + \frac { 1 } { 2 } \pi \right)
\end{aligned}$$
\end{enumerate}\item Describe fully a sequence of transformations that maps the graph of $y = \mathrm { f } ( x )$ on to $y = \mathrm { g } ( x )$.
\item Describe fully a sequence of transformations that maps the graph of $y = \mathrm { f } ( x )$ on to $y = \mathrm { h } ( x )$. [2]\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2020 Q11 [11]}}