| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2018 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Double angle with reciprocal functions |
| Difficulty | Standard +0.8 This question requires proving a non-trivial reciprocal trig identity using double angle formulas, solving a transcendental equation involving both reciprocal and standard trig functions, and integrating a composite function with reciprocal trig terms. Part (i) requires algebraic manipulation with multiple identities; part (ii) combines the identity with numerical solving; part (iii) requires recognizing the integral structure from part (i). The multi-step reasoning and combination of reciprocal/double angle identities elevates this above standard A-level questions, though it remains within Further Maths P2 scope. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Express \(\text{cosec}^2 2x\) as \(\frac{1}{4\sin^2 x \cos^2 x}\) | B1 | |
| Attempt to express LHS in terms of \(\sin x\) and \(\cos x\) only | M1 | Must be using correct working for M1 |
| Obtain \(\frac{2 \times 2\sin^2 x}{4\sin^2 x \cos^2 x}\) or equivalent and hence \(\sec^2 x\) | A1 | AG; necessary detail needed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Express equation as \(1 + \tan^2 x = \tan x + 21\) | B1 | |
| Solve 3-term quadratic equation for \(\tan x\) | M1 | |
| Obtain \(\tan x = 5\) and hence \(x = 1.37\) | A1 | Or greater accuracy 1.3734... |
| Obtain \(\tan x = -4\) and hence \(x = 1.82\) | A1 | Or greater accuracy 1.8157... |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use \(x = 2y + 1\) | B1 | |
| Identify integral as of form \(\int \sec^2(ay+b)\, dy\) | M1 | Condone absence of or error with \(dy\) |
| Obtain \(\frac{1}{2}\tan(2y+1) + c\) | A1 |
## Question 7(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Express $\text{cosec}^2 2x$ as $\frac{1}{4\sin^2 x \cos^2 x}$ | B1 | |
| Attempt to express LHS in terms of $\sin x$ and $\cos x$ only | M1 | Must be using correct working for M1 |
| Obtain $\frac{2 \times 2\sin^2 x}{4\sin^2 x \cos^2 x}$ or equivalent and hence $\sec^2 x$ | A1 | AG; necessary detail needed |
## Question 7(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Express equation as $1 + \tan^2 x = \tan x + 21$ | B1 | |
| Solve 3-term quadratic equation for $\tan x$ | M1 | |
| Obtain $\tan x = 5$ and hence $x = 1.37$ | A1 | Or greater accuracy 1.3734... |
| Obtain $\tan x = -4$ and hence $x = 1.82$ | A1 | Or greater accuracy 1.8157... |
## Question 7(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use $x = 2y + 1$ | B1 | |
| Identify integral as of form $\int \sec^2(ay+b)\, dy$ | M1 | Condone absence of or error with $dy$ |
| Obtain $\frac{1}{2}\tan(2y+1) + c$ | A1 | |7 (i) Show that $2 \operatorname { cosec } ^ { 2 } 2 x ( 1 - \cos 2 x ) \equiv \sec ^ { 2 } x$.\\
(ii) Solve the equation $2 \operatorname { cosec } ^ { 2 } 2 x ( 1 - \cos 2 x ) = \tan x + 21$ for $0 < x < \pi$, giving your answers correct to 3 significant figures.\\
(iii) Find $\int \left[ 2 \operatorname { cosec } ^ { 2 } ( 4 y + 2 ) - 2 \operatorname { cosec } ^ { 2 } ( 4 y + 2 ) \cos ( 4 y + 2 ) \right] \mathrm { d } y$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\hfill \mbox{\textit{CAIE P2 2018 Q7 [10]}}