Question:medium

If $3\sin^{-1}\!\left(\dfrac{2x}{1+x^{2}}\right) - 4\cos^{-1}\!\left(\dfrac{1-x^{2}}{1+x^{2}}\right) + 2\tan^{-1}\!\left(\dfrac{2x}{1-x^{2}}\right) = \dfrac{\pi}{3}$, then $x$ equals

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Key identities ($|x| \le 1$): $\sin^{-1}\tfrac{2x}{1+x^2} = 2\tan^{-1}x$; $\cos^{-1}\tfrac{1-x^2}{1+x^2} = 2\tan^{-1}x$; $\tan^{-1}\tfrac{2x}{1-x^2} = 2\tan^{-1}x$.

Updated On: Apr 8, 2026

$\dfrac{1}{\sqrt{3}}$
$-\dfrac{1}{\sqrt{3}}$
$\sqrt{3}$
$-\dfrac{\sqrt{3}}{2}$

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The Correct Option is A

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Questions Asked in MET exam

If $f(x) = \begin{cases} x, & 0 \le x \le 1 \\ 2x - 1, & 1<x \end{cases}$, then

MET - 2019
Continuity

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Helicopters are very different from airplanes. They can do three things that airplanes cannot do. First, when airplanes move upward, they must also move forward, but helicopters can move straight up without moving ahead. Second, helicopters can fly backward, which airplanes cannot do. Third, helicopters can use their rotors to hover in the air, which is impossible for airplanes. Helicopters can perform actions that airplanes cannot, they are used for different tasks. Since, helicopters can take off without moving forward, they do not need a runway for take off. They are used in congested areas where there is no room for airplanes or in isolated areas, which do not have airports. Because they can hover, they are used on firefighting missions to drop water on fires. They are used in logging operations to lift trees out of forests. Helicopters are used as air ambulances to airlift patients out of situations, which are difficult to reach by conventional ambulances. The police used helicopters to follow suspects on the ground or to search for cars on the ground. Of course, helicopters have military uses because of their design and capabilities.