If cot (2 x)/3+tan x/3=cosec (k x)/3, then the value of k is

Correct answer: (B) 2. Given, cot (2 x)/3+tan x/3=cosec (k x)/3 Let θ=x/3, then ∴ cot 2 θ+tan θ=cosec k θ ⇒ (cos 2 θ)/(sin 2 θ)+(sin θ)/(cos θ)=cosec k θ ⇒ frac2 cos^2 θ-12 sin θ cos θ+(sin θ)/(cos θ)=cosec k θ ⇒ frac2 cos^2 θ-1+2 sin^2 θ2 sin θ cos θ=cosec k θ ⇒ 1/(2 sin θ cos θ)=cosec k θ ⇒ cosec 2 θ=cosec k θ ⇒ k=2

Q53 / 160 · WBJEE 2015 · Mathematics

If $\cot \frac{2 x}{3}+\tan \frac{x}{3}=\operatorname{cosec} \frac{k x}{3},$ then the value of $k$ is

A
1
Correct answer
B
2
C
3
D
-1

Solution

Given, $\cot \frac{2 x}{3}+\tan \frac{x}{3}=\operatorname{cosec} \frac{k x}{3}$ Let $\theta=\frac{x}{3},$ then $\therefore \quad \cot 2 \theta+\tan \theta=\operatorname{cosec} k \theta$ $\Rightarrow \quad \frac{\cos 2 \theta}{\sin 2 \theta}+\frac{\sin \theta}{\cos \theta}=\operatorname{cosec} k \theta$ $\Rightarrow \frac{2 \cos ^{2} \theta-1}{2 \sin \theta \cos \theta}+\frac{\sin \theta}{\cos \theta}=\operatorname{cosec} k \theta$ $\Rightarrow \frac{2 \cos ^{2} \theta-1+2 \sin ^{2} \theta}{2 \sin \theta \cos \theta}=\operatorname{cosec} k \theta$ $\Rightarrow \quad \frac{1}{2 \sin \theta \cos \theta}=\operatorname{cosec} k \theta$ $\Rightarrow \quad \operatorname{cosec} 2 \theta=\operatorname{cosec} k \theta$ $\Rightarrow \quad k=2$

Hence, the correct answer is option: 'B'.

If $\cot \frac{2 x}{3}+\tan \frac{x}{3}=\operatorname{cosec} \frac{k x}{3},$ then the value of $k$ is

Solution

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