JEE Advanced · Mathematics · 8. Trigonometric Equations

Let $\theta, \varphi \in[0,2 \pi]$ be such that $2 \cos \theta(1-\sin \varphi)$ $=\sin ^{2} \theta ~\times$ $\left(\tan \frac{\theta}{2}+\cot \frac{\theta}{2}\right) \cos \varphi-1, \tan$ $(2 \pi-\theta)>0$ and $-1 < \sin \theta < -\frac{\sqrt{3}}{2}$, then $\varphi$ cannot satisfy

A $0 < \varphi < \frac{\pi}{2}$
B $\frac{\pi}{2} < \varphi < \frac{4 \pi}{3}$
C $\frac{4 \pi}{3} < \varphi < \frac{3 \pi}{2}$
D $\frac{3 \pi}{2} < \varphi < 2 \pi$

Verified Solution

Answer & Solution

Correct Answer

(A) $0 < \varphi < \frac{\pi}{2}$

Step-by-step Solution

Detailed explanation

As $\tan (2 \pi-\theta)>0$ and $-1 < \sin \theta < -\frac{\sqrt{3}}{2}, \theta \in[0,2 \pi]$
Hence $\frac{3 \pi}{2} < \theta < \frac{5 \pi}{3}$
Now $2 \cos \theta(1-\sin \varphi)=\sin ^{2} \theta\left(\tan \frac{\theta}{2}+\cot \frac{\theta}{2}\right) \cos \varphi-1$
$\Rightarrow 2 \cos \theta(1-\sin \varphi)=2 \sin \theta \cos \varphi-1$
$\Rightarrow 2 \cos \theta+1=2 \sin (\theta+\varphi)$
As $\quad \theta \in\left(\frac{3 \pi}{2}, \frac{5 \pi}{3}\right), 1 < 2 \sin (\theta+\varphi) < 2$
As $\theta+\varphi \in\left(\frac{\pi}{6}, \frac{5 \pi}{6}\right)$ or $(\theta+\varphi) \in\left(\frac{13 \pi}{6}, \frac{17 \pi}{6}\right)$
We have $\varphi \in\left(-\frac{3 \pi}{2},-\frac{2 \pi}{3}\right) \cup\left(\frac{2 \pi}{3}, \frac{7 \pi}{6}\right)$

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Let

f (x) = x + \log_{e} x - x \log_{e} x, x \in (0, \infty)

•

Column 1 contains information about zeros of

f (x)

f' (x)

and

f'' (x)

•

Column 2 contains information about the limiting behaviour of

f (x)

f' (x)

and

f'' (x)

at infinity.

•

Column 3 contains information about increasing-decreasing nature of

f (x)

and

f' (x)

Column 1	Column 2	Column 3
(I) $f (x) = 0$ for some $x \in (1, e^{2})$	(i) $\lim_{x \to \infty} f (x) = 0$	(P) $f$ is increasing in $(0, 1)$
(II) $f^{'} (x) = 0$ for some $x i n (1, e)$	(ii) $\lim_{x \to \infty} f (x) = - \infty$	(Q) $f$ is decreasing in $(e, e^{2})$
(III) $f^{'} (x) = 0$ for some $x \in (0, 1)$	(iii) $\lim_{x \to \infty} f^{'} (x) = - \infty$	(R) $f^{'}$ is increasing in $(0, 1)$
(IV) $f^{''} (x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x \to \infty} f^{''} (x) = 0$	(S) $f^{'}$ is decreasing in $(e, e^{2})$

JEE Advanced 2017 Hard

	List- I		List- II
$(I)$	$2 h + k$	$(P)$	$6$
$(I I)$	$\frac{L e n g t h o f Z W}{L e n g t h o f X Y}$	$(Q)$	$\sqrt{6}$
$(I I I)$	$\frac{A r e a o f t r i a n g l e M Z N}{A r e a o f t r i a n g l e Z M W}$	$(R)$	$\frac{5}{4}$
$(I V)$	$α$	$(S)$	$\frac{21}{5}$
		$(T)$	$2 \sqrt{6}$
		$(U)$	$\frac{10}{3}$

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Let \(\theta, \varphi \in[0,2 \pi]\) be such that \(2 \cos \theta(1-\sin \varphi)\) \(=\sin ^{2} \theta ~\times\) \(\left(\tan \frac{\theta}{2}+\cot \frac{\theta}{2}\right) \cos \varphi-1, \tan\) \((2 \pi-\theta)>0\) and \(-1 < \sin \theta < -\frac{\sqrt{3}}{2}\), then \(\varphi\) cannot satisfy

Answer & Solution

Step-by-step Solution

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