| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2020 |
| Session | March |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Multiple angle equations |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring standard techniques: solving a quadratic in sin x, differentiating using chain rule, and integrating a trigonometric expression. All steps are routine A-level procedures 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 formulae1.05o Trigonometric equations: solve in given intervals1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Solve equation \(y = 0\) to find value of \(x\) | M1 | |
| Obtain \(\frac{7}{6}\pi\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt first derivative using chain rule | M1 | OE |
| Obtain \(\frac{dy}{dx} = 8\sin x\cos x + 8\cos x\) | A1 | OE |
| Substitute value from part (a) to find gradient \(-2\sqrt{3}\) | A1 | Or exact equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Express integrand in the form \(k_1 + k_2\cos 2x + k_3\sin x\) | \*M1 | |
| Obtain correct \(5 - 2\cos 2x + 8\sin x\) | A1 | OE. Allow unsimplified |
| Integrate to obtain \(5x - \sin 2x - 8\cos x\) | A1 | |
| Apply limits 0 and *their* value from part (a) correctly | DM1 | |
| Obtain \(\frac{35}{6}\pi + \frac{7}{2}\sqrt{3} + 8\) or exact equivalent | A1 |
## Question 7(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Solve equation $y = 0$ to find value of $x$ | M1 | |
| Obtain $\frac{7}{6}\pi$ | A1 | |
## Question 7(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt first derivative using chain rule | M1 | OE |
| Obtain $\frac{dy}{dx} = 8\sin x\cos x + 8\cos x$ | A1 | OE |
| Substitute value from part (a) to find gradient $-2\sqrt{3}$ | A1 | Or exact equivalent |
## Question 7(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Express integrand in the form $k_1 + k_2\cos 2x + k_3\sin x$ | \*M1 | |
| Obtain correct $5 - 2\cos 2x + 8\sin x$ | A1 | OE. Allow unsimplified |
| Integrate to obtain $5x - \sin 2x - 8\cos x$ | A1 | |
| Apply limits 0 and *their* value from part (a) correctly | DM1 | |
| Obtain $\frac{35}{6}\pi + \frac{7}{2}\sqrt{3} + 8$ or exact equivalent | A1 | |7\\
\includegraphics[max width=\textwidth, alt={}, center]{78a9b100-c3bd-4054-b539-ec8304440063-10_551_641_260_751}
The diagram shows part of the curve with equation
$$y = 4 \sin ^ { 2 } x + 8 \sin x + 3 ,$$
where $x$ is measured in radians. The curve crosses the $x$-axis at the point $A$ and the shaded region is bounded by the curve and the lines $x = 0$ and $y = 0$.
\begin{enumerate}[label=(\alph*)]
\item Find the exact $x$-coordinate of $A$.
\item Find the exact gradient of the curve at $A$.
\item Find the exact area of the shaded region.\\
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 Q7 [10]}}