| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2012 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Two angles with tan relationships |
| Difficulty | Standard +0.3 Part (a) requires straightforward application of the tan addition formula and algebraic manipulation. Part (b) combines the tan subtraction formula with solving a trigonometric equation, requiring multiple steps but using standard techniques. Both parts are routine applications of addition formulae with no novel insight required, making this slightly easier than average. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks |
|---|---|
| (a) Use \(\tan(A + B)\) formula to obtain an equation in \(\tan B\) | M1 |
| State equation \(\frac{t + \tan B}{1 - t \tan B} = 4\), or equivalent | A1 |
| Solve to obtain \(\tan B = \frac{4 - t}{1 + 4t}\) | A1 |
| [3] | |
| (b) State equation \(2\left(\frac{\tan 45° - \tan x}{1 + \tan 45° \tan x}\right) = 3 \tan x\), or equivalent | B1 |
| Transform to a quadratic equation | M1 |
| Obtain \(3\tan^2 x + 5\tan x - 2 = 0\) (or equivalent) | A1 |
| Solve the quadratic and calculate one angle, or establish that \(\tan x = \frac{1}{3}, -2\) | M1 |
| Obtain one answer, e.g. \(x = 18.4°\) | A1 |
| Obtain other 3 answers \(116.6°, 198.4°, 296.6°\) and no others in range | A1 |
| [6] |
**(a)** Use $\tan(A + B)$ formula to obtain an equation in $\tan B$ | M1 |
State equation $\frac{t + \tan B}{1 - t \tan B} = 4$, or equivalent | A1 |
Solve to obtain $\tan B = \frac{4 - t}{1 + 4t}$ | A1 |
| | [3] |
**(b)** State equation $2\left(\frac{\tan 45° - \tan x}{1 + \tan 45° \tan x}\right) = 3 \tan x$, or equivalent | B1 |
Transform to a quadratic equation | M1 |
Obtain $3\tan^2 x + 5\tan x - 2 = 0$ (or equivalent) | A1 |
Solve the quadratic and calculate one angle, or establish that $\tan x = \frac{1}{3}, -2$ | M1 |
Obtain one answer, e.g. $x = 18.4°$ | A1 |
Obtain other 3 answers $116.6°, 198.4°, 296.6°$ and no others in range | A1 |
| | [6] |8
\begin{enumerate}[label=(\alph*)]
\item Given that $\tan A = t$ and $\tan ( A + B ) = 4$, find $\tan B$ in terms of $t$.
\item Solve the equation
$$2 \tan \left( 45 ^ { \circ } - x \right) = 3 \tan x$$
giving all solutions in the interval $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2012 Q8 [9]}}