| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2019 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Standard +0.8 Part (a) is a standard harmonic form question requiring routine application of R-sin(θ+α) formula and solving, typical of P2. Part (b) is more challenging, requiring manipulation of cot 2x and tan x using double angle formulas, algebraic rearrangement into a solvable form, and careful consideration of the domain - this elevates it above average difficulty but remains within standard A-level techniques. |
| Spec | 1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State \(R = \sqrt{32}\) or equivalent or 5.657… | B1 | |
| Use appropriate trigonometry to find \(\alpha\) | M1 | |
| Obtain \(\alpha = 45\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Carry out correct process to find one value of \(\theta\) | M1 | |
| Obtain 17.1 | A1 | Ignore other positive values greater than 17.1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use or imply \(\cot 2x = \frac{1}{\tan 2x}\) | B1 | |
| Use identity of form \(\tan 2x = \frac{\pm 2\tan x}{1 \pm \tan^2 x}\) to obtain equation in \(\tan x\) | M1 | |
| Obtain \(6\tan^2 x + 10\tan x - 4 = 0\) or equivalent | A1 | |
| Attempt solution of 3-term quadratic equation for \(\tan x\) | M1 | |
| Obtain \(\tan x = \frac{1}{3}\) and hence 0.32 | A1 | Allow greater accuracy |
| Obtain \(\tan x = -2\) and hence 2.03 and no others between 0 and \(\pi\) | A1 | Allow greater accuracy |
## Question 7(a)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $R = \sqrt{32}$ or equivalent or 5.657… | B1 | |
| Use appropriate trigonometry to find $\alpha$ | M1 | |
| Obtain $\alpha = 45$ | A1 | |
---
## Question 7(a)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Carry out correct process to find one value of $\theta$ | M1 | |
| Obtain 17.1 | A1 | Ignore other positive values greater than 17.1 |
---
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use or imply $\cot 2x = \frac{1}{\tan 2x}$ | B1 | |
| Use identity of form $\tan 2x = \frac{\pm 2\tan x}{1 \pm \tan^2 x}$ to obtain equation in $\tan x$ | M1 | |
| Obtain $6\tan^2 x + 10\tan x - 4 = 0$ or equivalent | A1 | |
| Attempt solution of 3-term quadratic equation for $\tan x$ | M1 | |
| Obtain $\tan x = \frac{1}{3}$ and hence 0.32 | A1 | Allow greater accuracy |
| Obtain $\tan x = -2$ and hence 2.03 and no others between 0 and $\pi$ | A1 | Allow greater accuracy |7
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Express $4 \sin \theta + 4 \cos \theta$ in the form $R \sin ( \theta + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.
\item Hence find the smallest positive value of $\theta$ satisfying the equation $4 \sin \theta + 4 \cos \theta = 5$.
\end{enumerate}\item Solve the equation
$$4 \cot 2 x = 5 + \tan x$$
for $0 < x < \pi$, showing all necessary working and giving the answers correct to 2 decimal places.\\
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 P2 2019 Q7 [11]}}