| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2002 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Moderate -0.3 This is a standard harmonic form question requiring routine application of the R cos(θ - α) formula, solving a trigonometric equation, and identifying a maximum from the harmonic form. All three parts follow textbook procedures with no novel problem-solving required, making it slightly easier than average for A-level. |
| 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 \(R = \sqrt{13}\) | B1 | |
| Use trig formula to find \(\alpha\) | M1 | |
| Obtain answer \(\alpha = 33.7°\) | A1 | 3 |
| (ii) Carry out, or indicate need for, evaluation of \(\cos^{-1}(3.5/\sqrt{13})\) (\(= 13.9°\)) | M1 | |
| Obtain answer \(47.6°\) | A1 | |
| Carry out correct method for second answer | M1 | |
| Obtain second answer \(19.8°\) | A1 | 4 |
| (iii) State coordinates \((33.7, \sqrt{13})\), or equivalent | A1 | 1 |
**(i)** State answer $R = \sqrt{13}$ | B1 |
Use trig formula to find $\alpha$ | M1 |
Obtain answer $\alpha = 33.7°$ | A1 | 3
**(ii)** Carry out, or indicate need for, evaluation of $\cos^{-1}(3.5/\sqrt{13})$ ($= 13.9°$) | M1 |
Obtain answer $47.6°$ | A1 |
Carry out correct method for second answer | M1 |
Obtain second answer $19.8°$ | A1 | 4
**(iii)** State coordinates $(33.7, \sqrt{13})$, or equivalent | A1 | 1
---4 (i) Express $3 \cos \theta + 2 \sin \theta$ in the form $R \cos ( \theta - \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$, stating the exact value of $R$ and giving the value of $\alpha$ correct to 1 decimal place.\\
(ii) Solve the equation
$$3 \cos \theta + 2 \sin \theta = 3.5$$
giving all solutions in the interval $0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }$.\\
(iii) The graph of $y = 3 \cos \theta + 2 \sin \theta$, for $0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }$, has one stationary point. State the coordinates of this point.\\
\includegraphics[max width=\textwidth, alt={}, center]{9b103197-7ba0-427a-b983-34edb51b6cca-3_421_823_299_662}
The diagram shows the curve $y = 2 x \mathrm { e } ^ { - x }$ and its maximum point $P$. Each of the two points $Q$ and $R$ on the curve has $y$-coordinate equal to $\frac { 1 } { 2 }$.\\
\hfill \mbox{\textit{CAIE P2 2002 Q4 [8]}}