| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2012 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Differentiation of reciprocal functions |
| Difficulty | Standard +0.3 This is a structured multi-part question requiring quotient rule differentiation, product rule for the second derivative, and integration using standard results. Part (i) is routine application of quotient rule, part (ii) follows directly using product rule and Pythagorean identity, and part (iii) uses the identity 1+tan²θ=sec²θ and recognizes the derivative from part (i). All steps are standard techniques with clear guidance, making it slightly easier than average. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.07l Derivative of ln(x): and related functions1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Differentiate using chain or quotient rule | M1 | |
| Obtain derivative in any correct form | A1 | |
| Obtain given answer correctly | A1 | [3] |
| (ii) Differentiate using product rule | M1 | |
| State derivative of \(\tan \theta = \sec^2 \theta\) | B1 | |
| Use trig identity \(1 + \tan^2 \theta = \sec^2 \theta\) correctly | M1 | |
| Obtain \(2\sec^3 \theta - \sec \theta\) | A1 | [4] |
| (iii) Use \(\tan^2 x = \sec^2 x - 1\) to integrate \(\tan^2 x\) | M1 | |
| Obtain \(3\sec \theta\) from integration of \(3\sec \theta \tan \theta\) | B1 | |
| Obtain \(\tan \theta - 3\sec \theta\) | A1 | |
| Attempt to substitute limits, using exact values | M1 | |
| Obtain answer \(4 - 3\sqrt{2}\) | A1 | [5] |
(i) Differentiate using chain or quotient rule | M1 |
Obtain derivative in any correct form | A1 |
Obtain given answer correctly | A1 | [3]
(ii) Differentiate using product rule | M1 |
State derivative of $\tan \theta = \sec^2 \theta$ | B1 |
Use trig identity $1 + \tan^2 \theta = \sec^2 \theta$ correctly | M1 |
Obtain $2\sec^3 \theta - \sec \theta$ | A1 | [4]
(iii) Use $\tan^2 x = \sec^2 x - 1$ to integrate $\tan^2 x$ | M1 |
Obtain $3\sec \theta$ from integration of $3\sec \theta \tan \theta$ | B1 |
Obtain $\tan \theta - 3\sec \theta$ | A1 |
Attempt to substitute limits, using exact values | M1 |
Obtain answer $4 - 3\sqrt{2}$ | A1 | [5]8 (i) By differentiating $\frac { 1 } { \cos \theta }$, show that if $y = \sec \theta$ then $\frac { \mathrm { d } y } { \mathrm {~d} \theta } = \tan \theta \sec \theta$.\\
(ii) Hence show that
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} \theta ^ { 2 } } = a \sec ^ { 3 } \theta + b \sec \theta$$
giving the values of $a$ and $b$.\\
(iii) Find the exact value of
$$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( 1 + \tan ^ { 2 } \theta - 3 \sec \theta \tan \theta \right) d \theta$$
\hfill \mbox{\textit{CAIE P2 2012 Q8 [12]}}