| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Solve equation with tan(θ ± α) |
| Difficulty | Standard +0.3 This is a standard A-level question on compound angle formulae requiring systematic algebraic manipulation. Part (i) is a 'show that' requiring application of tan(x-60°) formula and algebraic simplification—routine but multi-step. Part (ii) is straightforward quadratic solving once part (i) is complete. The techniques are well-practiced at this level with no novel insight required, making it slightly easier than average. |
| 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 |
|---|---|---|
| Use \(\tan(A \pm B)\) formula and obtain an equation in \(\tan x\) | M1 | |
| Using \(\tan 60° = \sqrt{3}\), obtain a horizontal equation in \(\tan x\) in any correct form | A1 | |
| Reduce the equation to the given form | A1 | 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Solve the given quadratic for \(\tan x\) | M1 | |
| Obtain a correct answer, e.g. \(x = 21.6°\) | A1 | |
| Obtain a second answer, e.g. \(x = 128.4°\), and no others | A1 | 3 marks |
| [Ignore answers outside the given interval. Treat answers in radians as a misread (0.377, 2.24).] |
**(i)**
Use $\tan(A \pm B)$ formula and obtain an equation in $\tan x$ | M1 |
Using $\tan 60° = \sqrt{3}$, obtain a horizontal equation in $\tan x$ in any correct form | A1 |
Reduce the equation to the given form | A1 | 3 marks
**(ii)**
Solve the given quadratic for $\tan x$ | M1 |
Obtain a correct answer, e.g. $x = 21.6°$ | A1 |
Obtain a second answer, e.g. $x = 128.4°$, and no others | A1 | 3 marks
[Ignore answers outside the given interval. Treat answers in radians as a misread (0.377, 2.24).] | |3 (i) Show that the equation
$$\tan \left( x - 60 ^ { \circ } \right) + \cot x = \sqrt { } 3$$
can be written in the form
$$2 \tan ^ { 2 } x + ( \sqrt { } 3 ) \tan x - 1 = 0$$
(ii) Hence solve the equation
$$\tan \left( x - 60 ^ { \circ } \right) + \cot x = \sqrt { } 3$$
for $0 ^ { \circ } < x < 180 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P3 2014 Q3 [6]}}