| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2012 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Integration using reciprocal identities |
| Difficulty | Standard +0.3 This is a structured multi-part question with clear signposting ('show that', 'deduce', 'hence') that guides students through each step. Part (i) is routine quotient rule differentiation, (ii) is standard algebraic manipulation (multiplying by conjugate), (iii) follows directly from squaring the result of (ii), and (iv) applies the derivative from (i) to integrate term-by-term with straightforward evaluation at the given limits. While it requires familiarity with reciprocal trig functions and careful algebraic manipulation, the scaffolding makes it slightly easier than a typical A-level question. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05p Proof involving trig: functions and identities1.07l Derivative of ln(x): and related functions1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | ||
| Use correct quotient or chain rule | M1 | |
| Obtain the given answer correctly having shown sufficient working | A1 | [2] |
| (ii) | ||
| Use a valid method, e.g. multiply numerator and denominator by \(\sec x + \tan x\), and a version of Pythagoras to justify the given identity | B1 | [1] |
| (iii) | ||
| Substitute, expand \((\sec x + \tan x)^2\) and use Pythagoras once | M1 | |
| Obtain given identity | A1 | [2] |
| (iv) | ||
| Obtain integral \(2\tan x - x + 2\sec x\) | B1 | |
| Use correct limits correctly in an expression of the form \(a\tan x + bx + c\sec x\), or equivalent, where \(abc \neq 0\) | M1 | |
| Obtain the given answer correctly | A1 | [3] |
**(i)** | |
Use correct quotient or chain rule | M1 |
Obtain the given answer correctly having shown sufficient working | A1 | [2]
**(ii)** | |
Use a valid method, e.g. multiply numerator and denominator by $\sec x + \tan x$, and a version of Pythagoras to justify the given identity | B1 | [1]
**(iii)** | |
Substitute, expand $(\sec x + \tan x)^2$ and use Pythagoras once | M1 |
Obtain given identity | A1 | [2]
**(iv)** | |
Obtain integral $2\tan x - x + 2\sec x$ | B1 |
Use correct limits correctly in an expression of the form $a\tan x + bx + c\sec x$, or equivalent, where $abc \neq 0$ | M1 |
Obtain the given answer correctly | A1 | [3]5 (i) By differentiating $\frac { 1 } { \cos x }$, show that if $y = \sec x$ then $\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x$.\\
(ii) Show that $\frac { 1 } { \sec x - \tan x } \equiv \sec x + \tan x$.\\
(iii) Deduce that $\frac { 1 } { ( \sec x - \tan x ) ^ { 2 } } \equiv 2 \sec ^ { 2 } x - 1 + 2 \sec x \tan x$.\\
(iv) Hence show that $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 } { ( \sec x - \tan x ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 4 } ( 8 \sqrt { } 2 - \pi )$.
\hfill \mbox{\textit{CAIE P3 2012 Q5 [8]}}