| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Derive triple angle then solve equation |
| Difficulty | Standard +0.3 This is a structured multi-part question requiring the addition formula for tan(2x+x) and double angle formula, followed by algebraic manipulation and solving a quadratic in tan x. Part (i) is guided derivation, part (ii) is routine solving, and part (iii) requires recognizing when tan²x would be negative. While it involves multiple steps, each step follows standard techniques with clear guidance, making it slightly easier than average. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use \(\tan(A+B)\) and \(\tan 2A\) formulae to obtain an equation in \(\tan x\) | M1 | |
| Obtain a correct equation in \(\tan x\) in any form | A1 | |
| Obtain an expression of the form \(a\tan^2 x = b\) | M1 | |
| Obtain the given answer | A1 | [4] |
| (ii) Substitute \(k=4\) in the given expression and solve for \(x\) | M1 | |
| Obtain answer, e.g. \(x = 16.8°\) | A1 | |
| Obtain second answer, e.g. \(x = 163.2°\), and no others in the given interval | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| (iii) Substitute \(k = -2\), show \(\tan^2 x < 0\) and justify given statement correctly | B1 | [1] |
**(i)** Use $\tan(A+B)$ and $\tan 2A$ formulae to obtain an equation in $\tan x$ | M1 |
Obtain a correct equation in $\tan x$ in any form | A1 |
Obtain an expression of the form $a\tan^2 x = b$ | M1 |
Obtain the given answer | A1 | [4]
**(ii)** Substitute $k=4$ in the given expression and solve for $x$ | M1 |
Obtain answer, e.g. $x = 16.8°$ | A1 |
Obtain second answer, e.g. $x = 163.2°$, and no others in the given interval | A1 | [3]
[Ignore answers outside the given interval. Treat answers in radians as a misread and deduct A1 from the marks for the angles.]
**(iii)** Substitute $k = -2$, show $\tan^2 x < 0$ and justify given statement correctly | B1 | [1]
---6 It is given that $\tan 3 x = k \tan x$, where $k$ is a constant and $\tan x \neq 0$.\\
(i) By first expanding $\tan ( 2 x + x )$, show that
$$( 3 k - 1 ) \tan ^ { 2 } x = k - 3$$
(ii) Hence solve the equation $\tan 3 x = k \tan x$ when $k = 4$, giving all solutions in the interval $0 ^ { \circ } < x < 180 ^ { \circ }$.\\
(iii) Show that the equation $\tan 3 x = k \tan x$ has no root in the interval $0 ^ { \circ } < x < 180 ^ { \circ }$ when $k = 2$.
\hfill \mbox{\textit{CAIE P3 2012 Q6 [8]}}