| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Sequence of transformations order |
| Difficulty | Standard +0.3 Part (a) requires understanding transformation order (horizontal stretch then translation, vertical stretch) which is a common source of confusion but follows standard rules. Part (b) is routine trigonometric equation solving with basic manipulation and finding solutions in a given range. Both parts are slightly above average due to the transformation ordering subtlety and exact value requirement, but remain standard A-level fare. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| EITHER (1){Translation}\(\begin{pmatrix}30°\\0\end{pmatrix}\) OR (2){Translation}\(\begin{pmatrix}60°\\0\end{pmatrix}\) | B2,1,0 | B2 for fully correct, B1 with two elements correct. \(\{\ \}\) indicates different elements. Accept angle in radians |
| (3){Stretch} {factor 2} {in \(x\)-direction} | B2,1,0 | B2 for fully correct, B1 with two elements correct |
| (4) Stretch factor 4 in \(y\)-direction and correct order | B1 | Stretch, \(y\)-direction and factor and correct order. Correct order is either (1) then (3) or (3) then (2). (4) can be anywhere in the sequence |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(4\sin\left(\frac{1}{2}x - 30°\right) = 2\sqrt{2} \Rightarrow \sin^{-1}\left(\frac{\sqrt{2}}{2}\right)[= 45°]\) | M1 | SOI |
| \(\frac{1}{2}x - 30 = 45\) or \(135 \Rightarrow x = 2(45+30)\) or \(x = 2(135+30)\) | M1 | SOI. The M marks are independent |
| \(x = 150°,\ x = 330°\) | A1 | Both exact values, condone \(\frac{5\pi}{6}, \frac{11\pi}{6}\). A0 if extra solutions in the interval. Ignore other solutions outside \([0°, 360°]\) |
## Question 8(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| **EITHER** (1){Translation}$\begin{pmatrix}30°\\0\end{pmatrix}$ **OR** (2){Translation}$\begin{pmatrix}60°\\0\end{pmatrix}$ | B2,1,0 | B2 for fully correct, B1 with two elements correct. $\{\ \}$ indicates different elements. Accept angle in radians |
| (3){Stretch} {factor 2} {in $x$-direction} | B2,1,0 | B2 for fully correct, B1 with two elements correct |
| (4) Stretch factor 4 in $y$-direction **and** correct order | B1 | Stretch, $y$-direction and factor **and** correct order. Correct order is **either** (1) then (3) **or** (3) then (2). (4) can be anywhere in the sequence |
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## Question 8(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $4\sin\left(\frac{1}{2}x - 30°\right) = 2\sqrt{2} \Rightarrow \sin^{-1}\left(\frac{\sqrt{2}}{2}\right)[= 45°]$ | M1 | SOI |
| $\frac{1}{2}x - 30 = 45$ or $135 \Rightarrow x = 2(45+30)$ or $x = 2(135+30)$ | M1 | SOI. The M marks are independent |
| $x = 150°,\ x = 330°$ | A1 | Both exact values, condone $\frac{5\pi}{6}, \frac{11\pi}{6}$. A0 if extra solutions in the interval. Ignore other solutions outside $[0°, 360°]$ |8
\begin{enumerate}[label=(\alph*)]
\item The curve $y = \sin x$ is transformed to the curve $y = 4 \sin \left( \frac { 1 } { 2 } x - 30 ^ { \circ } \right)$.\\
Describe fully a sequence of transformations that have been combined, making clear the order in which the transformations are applied.
\item Find the exact solutions of the equation $4 \sin \left( \frac { 1 } { 2 } x - 30 ^ { \circ } \right) = 2 \sqrt { 2 }$ for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q8 [8]}}