| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2013 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Standard +0.3 This is a standard harmonic form question with routine application of R sin(θ + α) transformation followed by straightforward equation solving and finding minimum values. Part (i) uses standard formulas (R = √(a² + b²), tan α = b/a), part (ii) is direct substitution and solving, and part (iii) requires recognizing that minimizing the denominator maximizes the expression. All techniques are textbook exercises with no novel insight required, making it slightly easier than average. |
| 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 \(R = \sqrt{29}\) | B1 | |
| Use trig formula to find \(\alpha\) | M1 | |
| Obtain \(\alpha = 21.80°\) with no errors seen | A1 | [3] |
| (ii) Carry out evaluation of \(\sin^{-1}\left(\frac{4}{R}\right)\) (\(\approx 47.97°\)) | M1 | |
| Carry out correct method for one correct answer | M1 | |
| Obtain one correct answer e.g. 13.1° | A1 | |
| Carry out correct method for a further answer | M1 | |
| Obtain remaining 3 answers \(55.1°, 193.1°, 235.1°\) and no others in the range | A1 | [5] |
| (iii) Greatest value of \(10\sin 2\theta + 4\cos 2\theta = 2\sqrt{29}\) | M1 | |
| \(\frac{1}{116}\) | A1 | [2] |
(i) State $R = \sqrt{29}$ | B1 |
Use trig formula to find $\alpha$ | M1 |
Obtain $\alpha = 21.80°$ with no errors seen | A1 | [3]
(ii) Carry out evaluation of $\sin^{-1}\left(\frac{4}{R}\right)$ ($\approx 47.97°$) | M1 |
Carry out correct method for one correct answer | M1 |
Obtain one correct answer e.g. 13.1° | A1 |
Carry out correct method for a further answer | M1 |
Obtain remaining 3 answers $55.1°, 193.1°, 235.1°$ and no others in the range | A1 | [5]
(iii) Greatest value of $10\sin 2\theta + 4\cos 2\theta = 2\sqrt{29}$ | M1 |
$\frac{1}{116}$ | A1 | [2]7 (i) Express $5 \sin 2 \theta + 2 \cos 2 \theta$ in the form $R \sin ( 2 \theta + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$, giving the exact value of $R$ and the value of $\alpha$ correct to 2 decimal places.
Hence\\
(ii) solve the equation
$$5 \sin 2 \theta + 2 \cos 2 \theta = 4$$
giving all solutions in the interval $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$,\\
(iii) determine the least value of $\frac { 1 } { ( 10 \sin 2 \theta + 4 \cos 2 \theta ) ^ { 2 } }$ as $\theta$ varies.
\hfill \mbox{\textit{CAIE P2 2013 Q7 [10]}}