| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Standard +0.3 This is a standard two-part harmonic form question requiring routine application of R cos(x - α) transformation followed by solving a trigonometric equation. The techniques are well-practiced at A-level, though the half-angle substitution and finding solutions in the given range adds minor complexity beyond the most basic examples. Slightly easier than average due to its predictable structure. |
| 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 |
|---|---|---|
| (i) State answer \(R = 3\) | B1 | |
| Use trig formula to find | M1 | |
| Obtain \(\alpha = 41.81°\) with no errors seen | A1 | [3] |
| (ii) Evaluate \(\cos^{-1}(0.4)\) to at least 1 d.p. (66.42° to 2 d.p.) | B1 | ✓ |
| Carry out an appropriate method to find a value of \(x\) in the given range | M1 | |
| Obtain answer 216.5° only | A1 | [3] |
**(i)** State answer $R = 3$ | B1 |
Use trig formula to find | M1 |
Obtain $\alpha = 41.81°$ with no errors seen | A1 | [3]
**(ii)** Evaluate $\cos^{-1}(0.4)$ to at least 1 d.p. (66.42° to 2 d.p.) | B1 | ✓ |
Carry out an appropriate method to find a value of $x$ in the given range | M1 |
Obtain answer 216.5° only | A1 | [3]
*[Ignore answers outside the given interval.]*
---3 (i) Express $( \sqrt { } 5 ) \cos x + 2 \sin x$ in the form $R \cos ( x - \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$, giving the value of $\alpha$ correct to 2 decimal places.\\
(ii) Hence solve the equation
$$( \sqrt { } 5 ) \cos \frac { 1 } { 2 } x + 2 \sin \frac { 1 } { 2 } x = 1.2$$
for $0 ^ { \circ } < x < 360 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P3 2016 Q3 [6]}}