| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Differentiation of reciprocal functions |
| Difficulty | Standard +0.3 Part (a) requires quotient rule differentiation and simplifying using trig identities to show the derivative is always negative—straightforward but multi-step. Part (b) involves recognizing an integration technique (likely substitution with u=1+sin x) and evaluating definite integral limits. Standard P3/C3 calculus with no novel insight required, slightly easier than average due to clear structure. |
| Spec | 1.07l Derivative of ln(x): and related functions1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use quotient or product rule | M1 | |
| Obtain derivative in any correct form e.g. \(\dfrac{-\sin x(1+\sin x) - \cos x(\cos x)}{(1+\sin x)^2}\) | A1 | |
| Use Pythagoras to simplify the derivative | M1 | |
| Justify the given statement | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State integral of the form \(a\ln(1+\sin x)\) | *M1 | |
| State correct integral \(\ln(1+\sin x)\) | A1 | |
| Use limits correctly | DM1 | |
| Obtain answer \(\ln\dfrac{4}{3}\) | A1 |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use quotient or product rule | M1 | |
| Obtain derivative in any correct form e.g. $\dfrac{-\sin x(1+\sin x) - \cos x(\cos x)}{(1+\sin x)^2}$ | A1 | |
| Use Pythagoras to simplify the derivative | M1 | |
| Justify the given statement | A1 | |
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## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State integral of the form $a\ln(1+\sin x)$ | *M1 | |
| State correct integral $\ln(1+\sin x)$ | A1 | |
| Use limits correctly | DM1 | |
| Obtain answer $\ln\dfrac{4}{3}$ | A1 | |
---7 Let $\mathrm { f } ( x ) = \frac { \cos x } { 1 + \sin x }$.
\begin{enumerate}[label=(\alph*)]
\item Show that $\mathrm { f } ^ { \prime } ( x ) < 0$ for all $x$ in the interval $- \frac { 1 } { 2 } \pi < x < \frac { 3 } { 2 } \pi$.
\item Find $\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 2 } \pi } \mathrm { f } ( x ) \mathrm { d } x$. Give your answer in a simplified exact form.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q7 [8]}}