| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2022 |
| Session | March |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Show derivative equals expression - algebraic/trigonometric identity proof |
| Difficulty | Standard +0.3 Part (a) is a routine application of chain rule with tan x, requiring knowledge that d/dx(tan x) = sec²x and the identity sec²x = 1 + tan²x. Part (b) cleverly uses the result from (a) but is still straightforward: recognizing that tan x + tan³x appears in the derivative allows direct integration, with tan²x handled via the identity. This is a standard 'show that' followed by a structured integration question, slightly easier than average due to the guided nature. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Differentiate to obtain \(2\tan x\sec^2 x\) | B1 | |
| Use \(\sec^2 x = 1 + \tan^2 x\) to confirm \(2\tan x + 2\tan^3 x\) | B1 | AG – necessary detail needed |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Attempt to use part (a) result to integrate \(\tan x + \tan^3 x\) | M1 | |
| Obtain \(\frac{1}{2}\tan^2 x\) | A1 | |
| Use relevant identity to integrate \(\tan^2 x\) | M1 | |
| Obtain \(\sec^2 x - 1\) and hence \(\tan x - x\) | A1 | |
| Use limits correctly for integrand of form \(k_1\tan^2 x + k_2\tan x + k_3 x\) | M1 | |
| Obtain \(\sqrt{3} - \frac{1}{12}\pi\) | A1 | Or exact equivalent |
| Total | 6 |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Differentiate to obtain $2\tan x\sec^2 x$ | B1 | |
| Use $\sec^2 x = 1 + \tan^2 x$ to confirm $2\tan x + 2\tan^3 x$ | B1 | AG – necessary detail needed |
| **Total** | **2** | |
## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt to use part (a) result to integrate $\tan x + \tan^3 x$ | M1 | |
| Obtain $\frac{1}{2}\tan^2 x$ | A1 | |
| Use relevant identity to integrate $\tan^2 x$ | M1 | |
| Obtain $\sec^2 x - 1$ and hence $\tan x - x$ | A1 | |
| Use limits correctly for integrand of form $k_1\tan^2 x + k_2\tan x + k_3 x$ | M1 | |
| Obtain $\sqrt{3} - \frac{1}{12}\pi$ | A1 | Or exact equivalent |
| **Total** | **6** | |5
\begin{enumerate}[label=(\alph*)]
\item Given that $y = \tan ^ { 2 } x$, show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \tan x + 2 \tan ^ { 3 } x$.
\item Find the exact value of $\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } \left( \tan x + \tan ^ { 2 } x + \tan ^ { 3 } x \right) \mathrm { d } x$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2022 Q5 [8]}}