| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Solve equation with tan(θ ± α) |
| Difficulty | Moderate -0.3 This is a straightforward application of the tan addition formula followed by solving a quadratic equation. Part (i) is routine algebraic manipulation after applying tan(A+B), and part (ii) requires only solving a quadratic and finding angles from tan values. The question is slightly easier than average because it's highly structured (the hard work of forming the quadratic is done in part i) and uses standard techniques with no conceptual surprises. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use \(\tan(A + B)\) formula to obtain an equation in \(\tan x\) | M1 | |
| Use \(\tan 45° = 1\) and obtain a correct equation in any form | A1 | |
| Obtain the given equation correctly | A1 | [3] |
| (ii) Solve the given quadratic in \(\tan x\) and evaluate an inverse tangent | M1 | |
| Obtain a correct answer, e.g. \(18.4°\) | A1 | |
| Obtain second answer, e.g. \(26.6°\), and no others in the given interval | A1 | [3] |
**(i)** Use $\tan(A + B)$ formula to obtain an equation in $\tan x$ | M1 |
Use $\tan 45° = 1$ and obtain a correct equation in any form | A1 |
Obtain the given equation correctly | A1 | [3]
**(ii)** Solve the given quadratic in $\tan x$ and evaluate an inverse tangent | M1 |
Obtain a correct answer, e.g. $18.4°$ | A1 |
Obtain second answer, e.g. $26.6°$, and no others in the given interval | A1 | [3]
[Treat the giving of answers in radians as a misread. Ignore answers outside the given interval.]3 (i) Show that the equation $\tan \left( x + 45 ^ { \circ } \right) = 6 \tan x$ can be written in the form
$$6 \tan ^ { 2 } x - 5 \tan x + 1 = 0$$
(ii) Hence solve the equation $\tan \left( x + 45 ^ { \circ } \right) = 6 \tan x$, for $0 ^ { \circ } < x < 180 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P2 2010 Q3 [6]}}