| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Expand then express in harmonic form |
| Difficulty | Standard +0.3 This is a standard two-part harmonic form question requiring expansion using compound angle formula, combining terms to find R and α, then solving a straightforward equation. While it involves multiple steps, the techniques are routine for P3 level with no novel insight required, making it slightly easier than average. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05n 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 |
| Use \(\sin(A - B)\) formula and obtain an expression in terms of \(\sin x\) and \(\cos x\) | M1 | |
| Collect terms and reach \(\sqrt{3}\sin x - 2\cos x\), or equivalent | A1 | |
| Obtain \(R = \sqrt{7}\) | A1 | |
| Use trig formula to find \(\alpha\) | M1 | |
| Obtain \(\alpha = 49.11°\) with no errors seen | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Evaluate \(\sin^{-1}(1/\sqrt{7})\) to at least 1 d.p. (\(22.21°\) to 2 d.p.) | B1 FT | |
| Use a correct method to find a value of \(x\) in the interval \(0° < x < 180°\) | M1 | |
| Obtain answer \(71.3°\) | A1 | Ignore answers outside given range |
## Question 8(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use $\sin(A - B)$ formula and obtain an expression in terms of $\sin x$ and $\cos x$ | M1 | |
| Collect terms and reach $\sqrt{3}\sin x - 2\cos x$, or equivalent | A1 | |
| Obtain $R = \sqrt{7}$ | A1 | |
| Use trig formula to find $\alpha$ | M1 | |
| Obtain $\alpha = 49.11°$ with no errors seen | A1 | |
## Question 8(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Evaluate $\sin^{-1}(1/\sqrt{7})$ to at least 1 d.p. ($22.21°$ to 2 d.p.) | B1 FT | |
| Use a correct method to find a value of $x$ in the interval $0° < x < 180°$ | M1 | |
| Obtain answer $71.3°$ | A1 | Ignore answers outside given range |8 (i) By first expanding $2 \sin \left( x - 30 ^ { \circ } \right)$, express $2 \sin \left( x - 30 ^ { \circ } \right) - \cos x$ in the form $R \sin ( x - \alpha )$, 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. [5]\\
(ii) Hence solve the equation
$$2 \sin \left( x - 30 ^ { \circ } \right) - \cos x = 1$$
for $0 ^ { \circ } < x < 180 ^ { \circ }$.\\
\hfill \mbox{\textit{CAIE P3 2017 Q8 [8]}}