| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Solve reciprocal trig equation |
| Difficulty | Standard +0.8 This is a multi-part question requiring manipulation of reciprocal trig identities, harmonic form conversion, and equation solving. Part (i) requires recognizing that cosec 2θ = 1/(2sinθcosθ) and algebraic manipulation through a common denominator. Part (ii) is standard harmonic form technique. Part (iii) combines these to solve R sin(θ+α) = 3, requiring careful consideration of the range. While systematic, it demands fluency with multiple techniques and careful algebraic work across several steps, placing it moderately above average difficulty. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use \(\sec\theta = \frac{1}{\cos\theta}\) and \(\cosec\theta = \frac{1}{\sin\theta}\) | B1 | |
| Use \(\sin 2\theta = 2\sin\theta\cos\theta\) and to form a horizontal equation in \(\sin\theta\) and \(\cos\theta\) or fractions with common denominator | M1 | |
| Obtain given equation \(2\sin\theta + 4\cos\theta = 3\) correctly | A1 | [3] |
| (ii) State or imply \(R = \sqrt{20}\) or \(4.47\) or equivalent | B1 | |
| Use correct trigonometry to find \(\alpha\) | M1 | |
| Obtain \(63.43\) or \(63.44\) with no errors seen | A1 | [3] |
| (iii) Carry out a correct method to find one value in given range | M1 | |
| Obtain \(74.4°\) (or \(338.7°\)) | A1 | |
| Carry out a correct method to find second value in given range | M1 | |
| Obtain \(338.7°\) (or \(74.4°\)) and no others between \(0°\) and \(360°\) | A1 | [4] |
**(i)** Use $\sec\theta = \frac{1}{\cos\theta}$ and $\cosec\theta = \frac{1}{\sin\theta}$ | B1 |
Use $\sin 2\theta = 2\sin\theta\cos\theta$ and to form a horizontal equation in $\sin\theta$ and $\cos\theta$ or fractions with common denominator | M1 |
Obtain given equation $2\sin\theta + 4\cos\theta = 3$ correctly | A1 | [3]
**(ii)** State or imply $R = \sqrt{20}$ or $4.47$ or equivalent | B1 |
Use correct trigonometry to find $\alpha$ | M1 |
Obtain $63.43$ or $63.44$ with no errors seen | A1 | [3]
**(iii)** Carry out a correct method to find one value in given range | M1 |
Obtain $74.4°$ (or $338.7°$) | A1 |
Carry out a correct method to find second value in given range | M1 |
Obtain $338.7°$ (or $74.4°$) and no others between $0°$ and $360°$ | A1 | [4]7 (i) Given that $\sec \theta + 2 \operatorname { cosec } \theta = 3 \operatorname { cosec } 2 \theta$, show that $2 \sin \theta + 4 \cos \theta = 3$.\\
(ii) Express $2 \sin \theta + 4 \cos \theta$ in the form $R \sin ( \theta + \alpha )$ where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$, giving the value of $\alpha$ correct to 2 decimal places.\\
(iii) Hence solve the equation $\sec \theta + 2 \operatorname { cosec } \theta = 3 \operatorname { cosec } 2 \theta$ for $0 ^ { \circ } < \theta < 360 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P3 2013 Q7 [10]}}