| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2020 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Integration using reciprocal identities |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard trigonometric identities and integration. Part (a) is routine algebraic manipulation using double angle and cotangent identities. Part (b) requires solving a quadratic equation in cos θ. Part (c) applies the identity from (a) with a substitution. All techniques are standard P2/C3 level with no novel insight required, making it slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use at least one of \(\sin 2\theta = 2\sin\theta\cos\theta\) and \(\cot\theta = \frac{\cos\theta}{\sin\theta}\) | B1 | |
| Use both and conclude \(6\cos^2\theta\) | B1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Attempt solution of \(\cos^2\theta = \frac{5}{6}\) to find at least one value | M1 | |
| Obtain \(0.421\) | A1 | |
| Obtain \(2.72\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Express integrand in form \(a + b\cos x\) | M1 | |
| Obtain correct integrand \(3 + 3\cos x\) | A1 | |
| Integrate to obtain \(px + q\sin x\) | \*M1 | |
| Apply limits correctly | DM1 | |
| Obtain \(\dfrac{3}{4}\pi + 3 - \dfrac{3}{\sqrt{2}}\) or exact equivalent | A1 |
## Question 8(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use at least one of $\sin 2\theta = 2\sin\theta\cos\theta$ and $\cot\theta = \frac{\cos\theta}{\sin\theta}$ | B1 | |
| Use both and conclude $6\cos^2\theta$ | B1 | AG |
## Question 8(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt solution of $\cos^2\theta = \frac{5}{6}$ to find at least one value | M1 | |
| Obtain $0.421$ | A1 | |
| Obtain $2.72$ | A1 | |
## Question 8(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Express integrand in form $a + b\cos x$ | **M1** | |
| Obtain correct integrand $3 + 3\cos x$ | **A1** | |
| Integrate to obtain $px + q\sin x$ | **\*M1** | |
| Apply limits correctly | **DM1** | |
| Obtain $\dfrac{3}{4}\pi + 3 - \dfrac{3}{\sqrt{2}}$ or exact equivalent | **A1** | |
**Total: 5 marks**8
\begin{enumerate}[label=(\alph*)]
\item Show that $3 \sin 2 \theta \cot \theta \equiv 6 \cos ^ { 2 } \theta$.
\item Solve the equation $3 \sin 2 \theta \cot \theta = 5$ for $0 < \theta < \pi$.
\item Find the exact value of $\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } 3 \sin x \cot \frac { 1 } { 2 } x \mathrm {~d} x$.\\
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 2020 Q8 [10]}}