| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Convert equation to quadratic form |
| Difficulty | Standard +0.3 This is a straightforward two-part question requiring standard techniques: (i) uses the identity sec²θ = 1 + tan²θ to form a quadratic in tanθ, then solve; (ii) applies the tan addition formula with a standard angle. Both parts are routine applications of core identities with no novel insight required, making it slightly easier than average. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.05l Double angle formulae: and compound angle formulae |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use identity \(\sec^2 \theta = 1 + \tan^2 \theta\); Solve three-term quadratic equation in \(\tan \theta\); Obtain at least \(\tan \theta = \frac{5}{2}\) | B1, M1, A1 | [3] |
| Substitute numerical values into \(\tan(A + B)\) identity; Obtain \(\frac{\frac{5}{3} + (-1)}{1 - \frac{5}{3}(-1)}\) or equivalent, following their positive answer from part (i); Obtain \(\frac{3}{7}\) or exact equivalent and no other answers | M1, A1✓, A1 | [3] |
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use identity $\sec^2 \theta = 1 + \tan^2 \theta$; Solve three-term quadratic equation in $\tan \theta$; Obtain at least $\tan \theta = \frac{5}{2}$ | B1, M1, A1 | [3] |
| Substitute numerical values into $\tan(A + B)$ identity; Obtain $\frac{\frac{5}{3} + (-1)}{1 - \frac{5}{3}(-1)}$ or equivalent, following their positive answer from part (i); Obtain $\frac{3}{7}$ or exact equivalent and no other answers | M1, A1✓, A1 | [3] |
---It is given that $\theta$ is an acute angle measured in degrees such that
$$2\sec^2\theta + 3\tan\theta = 22.$$
\begin{enumerate}[label=(\roman*)]
\item Find the value of $\tan\theta$. [3]
\item Use an appropriate formula to find the exact value of $\tan(\theta + 135°)$. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2015 Q3 [6]}}