| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2003 |
| Session | June |
| Marks | 7 |
| 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 application of the tan addition formula requiring algebraic manipulation to reach a given form, followed by solving a quadratic equation in tan x. The 'show that' structure provides scaffolding, and the techniques are routine for P2 level, 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 |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use \(\tan(A \pm B)\) formula to obtain an equation in \(\tan x\) | M1 | |
| State equation \(\dfrac{\tan x + 1}{1 - \tan x} = 4\dfrac{(1-\tan x)}{1 + \tan x}\), or equivalent | A1 | |
| Transform to a 2- or 3-term quadratic equation | M1 | |
| Obtain given answer correctly | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Solve the quadratic and calculate one angle, or establish that \(t = \frac{1}{3}\), \(3\) (only) | M1 | |
| Obtain one answer, e.g. \(x = 18.4° \pm 0.1°\) | A1 | |
| Obtain second answer \(x = 71.6°\) and no others in the range | A1 | |
| Ignore answers outside the given range |
# Question 4(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use $\tan(A \pm B)$ formula to obtain an equation in $\tan x$ | M1 | |
| State equation $\dfrac{\tan x + 1}{1 - \tan x} = 4\dfrac{(1-\tan x)}{1 + \tan x}$, or equivalent | A1 | |
| Transform to a 2- or 3-term quadratic equation | M1 | |
| Obtain given answer correctly | A1 | |
**Total: [4]**
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# Question 4(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Solve the quadratic and calculate one angle, or establish that $t = \frac{1}{3}$, $3$ (only) | M1 | |
| Obtain one answer, e.g. $x = 18.4° \pm 0.1°$ | A1 | |
| Obtain second answer $x = 71.6°$ and no others in the range | A1 | |
| Ignore answers outside the given range | | |
**Total: [3]**
---4 (i) Show that the equation
$$\tan \left( 45 ^ { \circ } + x \right) = 4 \tan \left( 45 ^ { \circ } - x \right)$$
can be written in the form
$$3 \tan ^ { 2 } x - 10 \tan x + 3 = 0$$
(ii) Hence solve the equation
$$\tan \left( 45 ^ { \circ } + x \right) = 4 \tan \left( 45 ^ { \circ } - x \right)$$
for $0 ^ { \circ } < x < 90 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P2 2003 Q4 [7]}}