| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Find value where max/min occurs |
| Difficulty | Standard +0.3 This is a standard harmonic form question with straightforward compound angle expansion and routine application to find a minimum. The expansion of cos(x-60°) is direct, combining terms to get Rcos(x-α) follows a well-practiced algorithm, and finding the minimum is a simple application of the result. Slightly easier than average due to the guided structure and standard techniques. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use \(\cos(A-B)\) formula and obtain an expression in terms of \(\sin x\) and \(\cos x\) | M1 | |
| Collect terms and reach \(2\cos x + \sqrt{3}\sin x\) | A1 | |
| State \(R = \sqrt{7}\) | A1 | |
| Use trig formula to find \(\alpha\) | M1 | |
| Obtain \(\alpha = 40.89°\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use correct method to find \(x\) | M1 | |
| Obtain answer \(x = 220.9°\) | A1 |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use $\cos(A-B)$ formula and obtain an expression in terms of $\sin x$ and $\cos x$ | M1 | |
| Collect terms and reach $2\cos x + \sqrt{3}\sin x$ | A1 | |
| State $R = \sqrt{7}$ | A1 | |
| Use trig formula to find $\alpha$ | M1 | |
| Obtain $\alpha = 40.89°$ | A1 | |
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct method to find $x$ | M1 | |
| Obtain answer $x = 220.9°$ | A1 | |
---6
\begin{enumerate}[label=(\alph*)]
\item By first expanding $\cos \left( x - 60 ^ { \circ } \right)$, show that the expression
$$2 \cos \left( x - 60 ^ { \circ } \right) + \cos x$$
can be written in the form $R \cos ( x - \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$. Give the exact value of $R$ and the value of $\alpha$ correct to 2 decimal places.
\item Hence find the value of $x$ in the interval $0 ^ { \circ } < x < 360 ^ { \circ }$ for which $2 \cos \left( x - 60 ^ { \circ } \right) + \cos x$ takes its least possible value.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q6 [7]}}