| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2016 |
| Session | March |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Double angle with reciprocal functions |
| Difficulty | Standard +0.8 Part (i) is a straightforward identity proof using double angle formulas. Part (ii)(a) requires recognizing that the expression simplifies to a form involving cos(2x) and finding its minimum using the double angle formula again. Part (ii)(b) involves manipulating the integrand using the identity from (i), requiring insight to rewrite cosec(4x)tan(2x) appropriately and then integrate. The multi-step nature, need for strategic substitution/manipulation, and integration with specific limits elevate this above routine exercises. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State \(2\sin x\cos x \cdot \frac{\cos x}{\sin x}\) | B1 | |
| Simplify to confirm \(2\cos^2 x\) | B1 | [2] |
| (ii) (a) Use \(\cos 2x = 2\cos^2 x - 1\) | B1 | |
| Express in terms of \(\cos x\) | M1 | |
| Obtain \(16\cos^2 x + 3\) or equivalent | A1 | |
| State 3, following their expression of form \(a\cos^2 x + b\) | A1 | [4] |
| (b) Obtain integrand as \(\frac{1}{2}\sec^2 2x\) | B1 | |
| Integrate to obtain form \(k\tan 2x\) | M1* | |
| Obtain correct \(\frac{1}{4}\tan 2x\) | A1 | |
| Apply limits correctly | dep M1* | |
| Obtain \(\frac{1}{4}\sqrt{3} - \frac{1}{4}\) or exact equivalent | A1 | [5] |
(i) State $2\sin x\cos x \cdot \frac{\cos x}{\sin x}$ | B1 |
Simplify to confirm $2\cos^2 x$ | B1 | [2]
(ii) (a) Use $\cos 2x = 2\cos^2 x - 1$ | B1 |
Express in terms of $\cos x$ | M1 |
Obtain $16\cos^2 x + 3$ or equivalent | A1 |
State 3, following their expression of form $a\cos^2 x + b$ | A1 | [4]
(b) Obtain integrand as $\frac{1}{2}\sec^2 2x$ | B1 |
Integrate to obtain form $k\tan 2x$ | M1* |
Obtain correct $\frac{1}{4}\tan 2x$ | A1 |
Apply limits correctly | dep M1* |
Obtain $\frac{1}{4}\sqrt{3} - \frac{1}{4}$ or exact equivalent | A1 | [5]8 (i) Show that $\sin 2 x \cot x \equiv 2 \cos ^ { 2 } x$.\\
(ii) Using the identity in part (i),
\begin{enumerate}[label=(\alph*)]
\item find the least possible value of
$$3 \sin 2 x \cot x + 5 \cos 2 x + 8$$
as $x$ varies,
\item find the exact value of $\int _ { \frac { 1 } { 8 } \pi } ^ { \frac { 1 } { 6 } \pi } \operatorname { cosec } 4 x \tan 2 x \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2016 Q8 [11]}}