| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2002 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Show equation reduces to tan form |
| Difficulty | Standard +0.3 This is a straightforward application of compound angle formulae followed by routine algebraic manipulation and solving. Part (i) is guided algebra, part (ii) is standard tan equation solving, and part (iii) applies double angle formula to a known value. Slightly above average due to the algebraic manipulation required, but all techniques are standard bookwork with no novel insight needed. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use relevant formulae for \(\cos(x - 30°)\) and \(\sin(x - 60°)\) | M1* | ⊕ |
| Use \(\sin 30° = \cos 60° = \frac{1}{2}\) and \(\sin 60° = \cos 30° = \frac{\sqrt{3}}{2}\) | M1(dep*) | |
| Collect terms and obtain given answer correctly | A1 | 3 |
| (ii) Carry out correct processes to evaluate a single trig ratio | M1 | |
| Obtain answer 73.9° | A1 | |
| Obtain second answer 233.9° and no others | A1 | 3 |
| (iii) State or imply that \(\cos^2 x = \frac{1}{13}\) or \(\sin^2 x = \frac{12}{13}\) | B1 | |
| Use a relevant trig formula to evaluate \(\cos 2x\) | M1 | |
| Obtain exact answer \(-\frac{11}{13}\) correctly | A1 | 3 |
**(i)** Use relevant formulae for $\cos(x - 30°)$ and $\sin(x - 60°)$ | M1* | ⊕ | [allow cwc sign error]
Use $\sin 30° = \cos 60° = \frac{1}{2}$ and $\sin 60° = \cos 30° = \frac{\sqrt{3}}{2}$ | M1(dep*) |
Collect terms and obtain given answer correctly | A1 | 3 |
**(ii)** Carry out correct processes to evaluate a single trig ratio | M1 |
Obtain answer 73.9° | A1 |
Obtain second answer 233.9° and no others | A1 | 3 |
**(iii)** State or imply that $\cos^2 x = \frac{1}{13}$ or $\sin^2 x = \frac{12}{13}$ | B1 |
Use a relevant trig formula to evaluate $\cos 2x$ | M1 |
Obtain exact answer $-\frac{11}{13}$ correctly | A1 | 3 |
[Use of only say $\cos x = \pm \frac{1}{\sqrt{13}}$, probably from a right triangle, can earn B1M1A0.]
---5 The angle $x$, measured in degrees, satisfies the equation
$$\cos \left( x - 30 ^ { \circ } \right) = 3 \sin \left( x - 60 ^ { \circ } \right)$$
(i) By expanding each side, show that the equation may be simplified to
$$( 2 \sqrt { } 3 ) \cos x = \sin x$$
(ii) Find the two possible values of $x$ lying between $0 ^ { \circ }$ and $360 ^ { \circ }$.\\
(iii) Find the exact value of $\cos 2 x$, giving your answer as a fraction.
\hfill \mbox{\textit{CAIE P2 2002 Q5 [9]}}