| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Derive triple angle then solve equation |
| Difficulty | Standard +0.8 This question requires deriving the triple angle formula using addition formulae (tan(2x+x)), algebraic manipulation to reach the given quartic form, then solving a quadratic-in-tan²x. While systematic, it demands multiple techniques (addition formula, double angle, algebraic rearrangement, quadratic formula) and careful manipulation across several steps, placing it moderately above average difficulty. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use \(\tan(A+B)\) formula to express the LHS in terms of \(\tan 2x\) and \(\tan x\) | M1 | |
| Using the \(\tan 2A\) formula, express the entire equation in terms of \(\tan x\) | M1 | |
| Obtain a correct equation in \(\tan x\) in any form | A1 | |
| Obtain the given form correctly | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use correct method to solve the given equation for \(x\) | M1 | |
| Obtain answer, e.g. \(x = 26.8°\) | A1 | |
| Obtain second answer, e.g. \(x = 73.7°\) and no other | A1 | Ignore answers outside the given interval |
## Question 4(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use $\tan(A+B)$ formula to express the LHS in terms of $\tan 2x$ and $\tan x$ | M1 | |
| Using the $\tan 2A$ formula, express the entire equation in terms of $\tan x$ | M1 | |
| Obtain a correct equation in $\tan x$ in any form | A1 | |
| Obtain the given form correctly | A1 | AG |
## Question 4(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct method to solve the given equation for $x$ | M1 | |
| Obtain answer, e.g. $x = 26.8°$ | A1 | |
| Obtain second answer, e.g. $x = 73.7°$ and no other | A1 | Ignore answers outside the given interval |4 (i) By first expanding $\tan ( 2 x + x )$, show that the equation $\tan 3 x = 3 \cot x$ can be written in the form $\tan ^ { 4 } x - 12 \tan ^ { 2 } x + 3 = 0$.\\
(ii) Hence solve the equation $\tan 3 x = 3 \cot x$ for $0 ^ { \circ } < x < 90 ^ { \circ }$.\\
\hfill \mbox{\textit{CAIE P3 2019 Q4 [7]}}