| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2020 |
| Session | Specimen |
| 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 straightforward P2 integration question requiring standard techniques: proving an identity using basic trig identities (tan²x = sec²x - 1, cos²x = (1+cos2x)/2), then integrating using the simplified form, followed by a volume of revolution using the given curve. All steps are routine applications of A-level formulas with no novel problem-solving 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^21.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Replace \(\tan^2 x\) by \(\sec^2 x - 1\) | B1 | |
| Express \(\cos^2 x\) in the form \(\pm\frac{1}{2} \pm \frac{1}{2}\cos 2x\) | M1 | |
| Obtain given answer \(\sec^2 x + \frac{1}{2}\cos 2x - \frac{1}{2}\) correctly | A1 | AG – Showing necessary detail |
| Attempt integration of expression | M1 | |
| Obtain \(\tan x + \frac{1}{4}\sin 2x - \frac{1}{2}x\) | A1 | |
| Use limits correctly for integral involving at least \(\tan x\) and \(\sin 2x\) | M1 | |
| Obtain \(\frac{5}{4} - \frac{1}{8}\pi\) or exact equivalent | A1 | |
| Total | 7 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply volume is \(\int \pi(\tan x + \cos x)^2\, dx\) | B1 | Presence of \(\pi\) implied by its appearance later if not shown initially |
| Attempt expansion and simplification | M1 | |
| Integrate to obtain one additional term of form \(k\cos x\) | M1 | |
| Obtain \(\pi\left(\frac{5}{4} - \frac{1}{8}\pi\right) + \pi(2 - \sqrt{2})\) or equivalent | A1 | |
| Total | 4 |
# Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Replace $\tan^2 x$ by $\sec^2 x - 1$ | B1 | |
| Express $\cos^2 x$ in the form $\pm\frac{1}{2} \pm \frac{1}{2}\cos 2x$ | M1 | |
| Obtain given answer $\sec^2 x + \frac{1}{2}\cos 2x - \frac{1}{2}$ correctly | A1 | AG – Showing necessary detail |
| Attempt integration of expression | M1 | |
| Obtain $\tan x + \frac{1}{4}\sin 2x - \frac{1}{2}x$ | A1 | |
| Use limits correctly for integral involving at least $\tan x$ and $\sin 2x$ | M1 | |
| Obtain $\frac{5}{4} - \frac{1}{8}\pi$ or exact equivalent | A1 | |
| **Total** | **7** | |
# Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply volume is $\int \pi(\tan x + \cos x)^2\, dx$ | B1 | Presence of $\pi$ implied by its appearance later if not shown initially |
| Attempt expansion and simplification | M1 | |
| Integrate to obtain one additional term of form $k\cos x$ | M1 | |
| Obtain $\pi\left(\frac{5}{4} - \frac{1}{8}\pi\right) + \pi(2 - \sqrt{2})$ or equivalent | A1 | |
| **Total** | **4** | |7 (a) Show that $\tan ^ { 2 } x + \cos ^ { 2 } x \equiv \sec ^ { 2 } x + \frac { 1 } { 2 } \cos 2 x - \frac { 1 } { 2 }$ and hence find the exact value of
$$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \tan ^ { 2 } x + \cos ^ { 2 } x \right) \mathrm { d } x$$
(b)\\
\includegraphics[max width=\textwidth, alt={}, center]{0af2714b-d3eb-4112-a869-eda5cf266cd8-13_535_771_274_648}
The region enclosed by the curve $y = \tan x + \cos x$ and the lines $x = 0 , x = \frac { 1 } { 4 } \pi$ and $y = 0$ is shown in the diagram.
Find the exact volume of the solid produced when this region is rotated completely about the $x$-axis.\\
\hfill \mbox{\textit{CAIE P2 2020 Q7 [11]}}