| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2023 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Integration using reciprocal identities |
| Difficulty | Standard +0.3 This is a structured multi-part question with standard techniques: (a) proving a trig identity using double angle formulas and basic manipulation, (b) direct substitution using part (a), and (c) routine integration of cos(2x) after finding critical points via differentiation. All steps are textbook procedures with no novel insight 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 formulae1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt to express LHS in terms of \(\sin x\) and \(\cos x\) only | M1* | |
| Obtain \(2\cos^2 x+6\sin^2 x\) | A1 | OE |
| Use relevant identities to express in terms of \(\cos 2x\) only | DM1 | |
| Confirm given result \(4-2\cos 2x\) with sufficient detail | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Substitute \(x=\frac{1}{12}\pi\) and attempt exact value of \(\frac{4-2\cos 2\left(\frac{\pi}{12}\right)}{\sin 2\left(\frac{\pi}{12}\right)}\) | M1 | |
| Obtain \(2(4-2\cos\frac{1}{6}\pi)\) or equivalent and hence \(8-2\sqrt{3}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Differentiate to obtain \(4\sin 2x\) | B1 | B2 if both limits obtained by symmetry or observation |
| Equate to 4 and obtain limits \(\frac{1}{4}\pi\) and \(\pi\) | B1 | |
| Integrate to obtain \(4x-\sin 2x\) | B1 | |
| Use limits \(a\) and \(b\) to find exact value for integral of form \(k_1x+k_2\sin 2x\) | M1 | Angles must be in radians in terms of \(\pi\) |
| Obtain \(4\pi-\sin 2\pi-(\pi-\sin\frac{1}{2}\pi)\) and confirm given result \(3\pi+1\) with sufficient detail | A1 | AG |
## Question 7(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt to express LHS in terms of $\sin x$ and $\cos x$ only | M1* | |
| Obtain $2\cos^2 x+6\sin^2 x$ | A1 | OE |
| Use relevant identities to express in terms of $\cos 2x$ only | DM1 | |
| Confirm given result $4-2\cos 2x$ with sufficient detail | A1 | AG |
## Question 7(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Substitute $x=\frac{1}{12}\pi$ and attempt exact value of $\frac{4-2\cos 2\left(\frac{\pi}{12}\right)}{\sin 2\left(\frac{\pi}{12}\right)}$ | M1 | |
| Obtain $2(4-2\cos\frac{1}{6}\pi)$ or equivalent and hence $8-2\sqrt{3}$ | A1 | |
## Question 7(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Differentiate to obtain $4\sin 2x$ | B1 | B2 if both limits obtained by symmetry or observation |
| Equate to 4 and obtain limits $\frac{1}{4}\pi$ and $\pi$ | B1 | |
| Integrate to obtain $4x-\sin 2x$ | B1 | |
| Use limits $a$ and $b$ to find exact value for integral of form $k_1x+k_2\sin 2x$ | M1 | Angles must be in radians in terms of $\pi$ |
| Obtain $4\pi-\sin 2\pi-(\pi-\sin\frac{1}{2}\pi)$ and confirm given result $3\pi+1$ with sufficient detail | A1 | AG |7
\begin{enumerate}[label=(\alph*)]
\item Prove that $\sin 2 x ( \cot x + 3 \tan x ) \equiv 4 - 2 \cos 2 x$.
\item Hence find the exact value of $\cot \frac { 1 } { 12 } \pi + 3 \tan \frac { 1 } { 12 } \pi$.
\item \\
\includegraphics[max width=\textwidth, alt={}, center]{b104e2a7-06c8-4e2e-a4f9-5095ad56897a-13_796_789_278_708}
The diagram shows the curve with equation $y = 4 - 2 \cos 2 x$, for $0 < x < 2 \pi$. At the point $A$, the gradient of the curve is 4 . The point $B$ is a minimum point. The $x$-coordinates of $A$ and $B$ are $a$ and $b$ respectively.
Show that $\int _ { a } ^ { b } ( 4 - 2 \cos 2 x ) \mathrm { d } x = 3 \pi + 1$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2023 Q7 [11]}}