| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Compound angle with reciprocal functions |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring standard techniques: (i) uses the identity sec²θ = 1 + tan²θ to form a quadratic in tan θ, then (ii) applies memorized compound angle and double angle formulas. All steps are routine applications of known identities with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use \(\sec^2 \theta = 1 + \tan^2 \theta\) | B1 | |
| Attempt solution of quadratic equation in \(\tan \theta\) | M1 | |
| Obtain \(\tan^2 \theta - 12 \tan \theta + 36 = 0\) or equivalent and hence \(\tan \theta = 6\) | A1 | [3] |
| (ii) (a) Attempt use of \(\tan(A - B)\) formula | M1 | |
| Obtain \(\frac{3}{4}\) following their value of \(\tan \theta\) | A1∨ | [2] |
| (ii) (b) Attempt use of \(\tan 2\theta\) formula | M1 | |
| Obtain \(-\frac{12}{35}\) | A1 | [2] |
**(i)** Use $\sec^2 \theta = 1 + \tan^2 \theta$ | B1 |
Attempt solution of quadratic equation in $\tan \theta$ | M1 |
Obtain $\tan^2 \theta - 12 \tan \theta + 36 = 0$ or equivalent and hence $\tan \theta = 6$ | A1 | [3]
**(ii) (a)** Attempt use of $\tan(A - B)$ formula | M1 |
Obtain $\frac{3}{4}$ following their value of $\tan \theta$ | A1∨ | [2]
**(ii) (b)** Attempt use of $\tan 2\theta$ formula | M1 |
Obtain $-\frac{12}{35}$ | A1 | [2]4 (i) Given that $35 + \sec ^ { 2 } \theta = 12 \tan \theta$, find the value of $\tan \theta$.\\
(ii) Hence, showing the use of an appropriate formula in each case, find the exact value of
\begin{enumerate}[label=(\alph*)]
\item $\tan \left( \theta - 45 ^ { \circ } \right)$,
\item $\tan 2 \theta$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2012 Q4 [7]}}