| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Integration using reciprocal identities |
| Difficulty | Standard +0.3 This is a straightforward two-part question requiring expansion of a squared expression using reciprocal identities, then integration using standard results. Part (a) involves algebraic manipulation with sec x and the double angle formula (routine for P2), while part (b) applies standard integrals of sec²x and cos 2x with simple limits. The 'hence' structure guides students through the method, making this 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 | Mark | Guidance |
| Expand to obtain integrand of form \(\sec^2 x + k_1 + k_2\cos2x\) | M1 | With \(k_1k_2 \neq 0\) and \(k_1 \neq 2\) |
| Obtain correct \(\sec^2 x + \dfrac{5}{2} + \dfrac{1}{2}\cos2x\) | A1 | OE, allow unsimplified |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Integrate to obtain at least terms of form \(k_3\tan x\) and \(k_4\sin2x\) | *M1 | With \(k_3k_4 \neq 0\); Allow in terms of \(a\) and \(b\); Condone omission of term in \(x\) |
| Obtain correct \(\tan x + \dfrac{5}{2}x + \dfrac{1}{4}\sin2x\) | A1 | OE, allow unsimplified |
| Apply limits correctly to integral involving at least two terms | DM1 | Need to see at least 2 correct substitutions using *their* integral |
| Obtain \(\dfrac{5}{4} + \dfrac{5}{8}\pi\) or exact equivalent | A1 | CWO |
## Question 3(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Expand to obtain integrand of form $\sec^2 x + k_1 + k_2\cos2x$ | M1 | With $k_1k_2 \neq 0$ and $k_1 \neq 2$ |
| Obtain correct $\sec^2 x + \dfrac{5}{2} + \dfrac{1}{2}\cos2x$ | A1 | OE, allow unsimplified |
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## Question 3(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate to obtain at least terms of form $k_3\tan x$ and $k_4\sin2x$ | *M1 | With $k_3k_4 \neq 0$; Allow in terms of $a$ and $b$; Condone omission of term in $x$ |
| Obtain correct $\tan x + \dfrac{5}{2}x + \dfrac{1}{4}\sin2x$ | A1 | OE, allow unsimplified |
| Apply limits correctly to integral involving at least two terms | DM1 | Need to see at least 2 correct substitutions using *their* integral |
| Obtain $\dfrac{5}{4} + \dfrac{5}{8}\pi$ or exact equivalent | A1 | CWO |
---3
\begin{enumerate}[label=(\alph*)]
\item Show that $( \sec x + \cos x ) ^ { 2 }$ can be expressed as $\sec ^ { 2 } x + a + b \cos 2 x$, where $a$ and $b$ are constants to be determined.
\item Hence find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } ( \sec x + \cos x ) ^ { 2 } \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2021 Q3 [6]}}