| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2008 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Integration using reciprocal identities |
| Difficulty | Standard +0.8 This is a multi-part question requiring proof of trigonometric identities, differentiation from first principles, and integration using substitution. While each individual step uses standard techniques, the question requires connecting multiple results across parts and recognizing that the identity in part (i)(b) enables the integration in part (iii). The exact value calculation with limits 0 to π/4 adds computational complexity. This is above average difficulty but not exceptional for P2 level. |
| 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^21.05p Proof involving trig: functions and identities1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Use trig formulae and justify given result | B1 | |
| (b) Use \(1 - \sin^2 x = \cos^2 x\) | M1 | |
| Obtain given result correctly | A1 | [3] |
| (ii) Use quotient or chain rule | M1 | |
| Obtain correct derivative in any form | A1 | |
| Obtain given result correctly | A1 | [3] |
| (iii) Obtain integral \(\tan x + \sec x\) | B1 | |
| Substitute limits correctly | M1 | |
| Obtain exact answer \(\sqrt{2}\), or equivalent | A1 | [3] |
**(i)**
(a) Use trig formulae and justify given result | B1 |
(b) Use $1 - \sin^2 x = \cos^2 x$ | M1 |
Obtain given result correctly | A1 | [3]
**(ii)** Use quotient or chain rule | M1 |
Obtain correct derivative in any form | A1 |
Obtain given result correctly | A1 | [3]
**(iii)** Obtain integral $\tan x + \sec x$ | B1 |
Substitute limits correctly | M1 |
Obtain exact answer $\sqrt{2}$, or equivalent | A1 | [3]8
\begin{enumerate}[label=(\roman*)]
\item (a) Prove the identity
$$\sec ^ { 2 } x + \sec x \tan x \equiv \frac { 1 + \sin x } { \cos ^ { 2 } x }$$
(b) Hence prove that
$$\sec ^ { 2 } x + \sec x \tan x \equiv \frac { 1 } { 1 - \sin x }$$
\item By differentiating $\frac { 1 } { \cos x }$, show that if $y = \sec x$ then $\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x$.
\item Using the results of parts (i) and (ii), find the exact value of
$$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 } { 1 - \sin x } \mathrm {~d} x$$
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2008 Q8 [9]}}