| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2004 |
| 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 with routine steps: (i) uses the formula R=√(a²+b²) and tan α=b/a, (ii) applies the result to solve a straightforward equation, and (iii) requires recognizing that minimum occurs at R=-5, giving answer 2. All techniques are textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05g Exact trigonometric values: for standard angles1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State answer \(R = 5\) | B1 | |
| Use trigonometric formulae to find \(\alpha\) | M1 | |
| Obtain answer \(\alpha = 53.13°\) | A1 | Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Carry out, or indicate need for, calculation of \(\sin^{-1}(4.5/5)\) | M1 | |
| Obtain answer 11.0° | \(\text{A1}\sqrt{}\) | |
| Carry out correct method for the second root e.g. \(180° - 64.16° - 53.13°\) | M1 | |
| Obtain answer 62.7° and no others in the range | \(\text{A1}\sqrt{}\) | Total: 4 |
| Ignore answers outside the given range. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State least value is 2 | \(\text{B1}\sqrt{}\) | Total: 1 |
## Question 4:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State answer $R = 5$ | B1 | |
| Use trigonometric formulae to find $\alpha$ | M1 | |
| Obtain answer $\alpha = 53.13°$ | A1 | **Total: 3** |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Carry out, or indicate need for, calculation of $\sin^{-1}(4.5/5)$ | M1 | |
| Obtain answer 11.0° | $\text{A1}\sqrt{}$ | |
| Carry out correct method for the second root e.g. $180° - 64.16° - 53.13°$ | M1 | |
| Obtain answer 62.7° and no others in the range | $\text{A1}\sqrt{}$ | **Total: 4** |
| | | Ignore answers outside the given range. |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State least value is 2 | $\text{B1}\sqrt{}$ | **Total: 1** |
---4 (i) Express $3 \sin \theta + 4 \cos \theta$ 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.\\
(ii) Hence solve the equation
$$3 \sin \theta + 4 \cos \theta = 4.5$$
giving all solutions in the interval $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$, correct to 1 decimal place.\\
(iii) Write down the least value of $3 \sin \theta + 4 \cos \theta + 7$ as $\theta$ varies.
\hfill \mbox{\textit{CAIE P2 2004 Q4 [8]}}