| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2016 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Solve double/multiple angle equation |
| Difficulty | Standard +0.3 This is a straightforward multi-part question on trigonometric functions covering standard techniques: finding range from amplitude, sketching a sine transformation, solving a basic trig equation, and finding the domain restriction for invertibility. All parts follow routine procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.05f Trigonometric function graphs: symmetries and periodicities1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| \(3 \leqslant f(x) \leqslant 7\) | B1 B1 | Identifying both 3 and 7 or correctly stating one inequality. Completely correct statement. NB \(3 \leqslant x \leqslant 7\) scores B1B0 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| [Graph] | B1* DB1 | One complete oscillation of a sinusoidal curve between \(0\) and \(\pi\). All correct, initially going downwards, all above \(f(x)=0\) [2] |
| Answer | Marks | Guidance |
|---|---|---|
| \(5 - 2\sin 2x = 6 \rightarrow \sin 2x = -\frac{1}{2}\) | M1 | Make \(\sin 2x\) the subject |
| Answer | Marks | Guidance |
|---|---|---|
| \(1.83^c\) or \(2.88^c\) | A1 A1✓ | ✓ for \(\frac{3\pi}{2} - 1\)st answer from \(\sin 2x = -\frac{1}{2}\) only, if in given range. SR A1A0 for both [3] |
| Answer | Marks | Guidance |
|---|---|---|
| \(k = \frac{\pi}{4}\) | B1 | [1] |
| Answer | Marks | Guidance |
|---|---|---|
| \(2\sin 2x = 5 - y \rightarrow \sin 2x = \frac{1}{2}(5-y)\) | M1 M1 | Makes \(\pm\sin 2x\) the subject soi by final answer. Correct order of operations including correctly dealing with "\(-\)" |
| \((g^{-1}(x)) = \frac{1}{2}\sin^{-1}\left(\frac{5-x}{2}\right)\) | A1 | Must be a function of \(x\) [3] |
# Question 10:
## Part (i):
$3 \leqslant f(x) \leqslant 7$ | B1 B1 | Identifying both 3 and 7 or correctly stating one inequality. Completely correct statement. **NB** $3 \leqslant x \leqslant 7$ scores B1B0 [2]
## Part (ii):
[Graph] | B1* DB1 | One complete oscillation of a sinusoidal curve between $0$ and $\pi$. All correct, initially going downwards, all above $f(x)=0$ [2]
## Part (iii):
$5 - 2\sin 2x = 6 \rightarrow \sin 2x = -\frac{1}{2}$ | M1 | Make $\sin 2x$ the subject
$\rightarrow 2x = \frac{7\pi}{6}$ or $\frac{11\pi}{6}$
$\rightarrow x = \frac{7\pi}{12}$ or $\frac{11\pi}{12}$
$0.583\pi$ or $0.917\pi$
$\frac{\pi + 0.524}{2}$ or $\frac{2\pi - 0.524}{2}$
$1.83^c$ or $2.88^c$ | A1 A1✓ | ✓ for $\frac{3\pi}{2} - 1$st answer from $\sin 2x = -\frac{1}{2}$ only, if in given range. SR A1A0 for both [3]
## Part (iv):
$k = \frac{\pi}{4}$ | B1 | [1]
## Part (v):
$2\sin 2x = 5 - y \rightarrow \sin 2x = \frac{1}{2}(5-y)$ | M1 M1 | Makes $\pm\sin 2x$ the subject soi by final answer. Correct order of operations including correctly dealing with "$-$"
$(g^{-1}(x)) = \frac{1}{2}\sin^{-1}\left(\frac{5-x}{2}\right)$ | A1 | Must be a function of $x$ [3]10 A function f is defined by $\mathrm { f } : x \mapsto 5 - 2 \sin 2 x$ for $0 \leqslant x \leqslant \pi$.\\
(i) Find the range of f .\\
(ii) Sketch the graph of $y = \mathrm { f } ( x )$.\\
(iii) Solve the equation $\mathrm { f } ( x ) = 6$, giving answers in terms of $\pi$.
The function g is defined by $\mathrm { g } : x \mapsto 5 - 2 \sin 2 x$ for $0 \leqslant x \leqslant k$, where $k$ is a constant.\\
(iv) State the largest value of $k$ for which g has an inverse.\\
(v) For this value of $k$, find an expression for $\mathrm { g } ^ { - 1 } ( x )$.
\hfill \mbox{\textit{CAIE P1 2016 Q10 [11]}}