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The set of all values of \( \lambda \) for which the equation \( \cos^{2}2x-2\sin^{4}x-2\cos^{2}x=\lambda \) has a real solutions \( x \), is:

If \(\tan 15^\circ + \dfrac{1}{\tan 75^\circ} + \dfrac{1}{\tan 105^\circ} + \tan 195^\circ = 2a\), Then the value of \(\left(a + \dfrac{1}{a}\right)\) is:

Let \( f(\theta) = 3\Big(\sin^4\!\left(\dfrac{3\pi}{2}-\theta\right) + \sin^4(3\pi+\theta)\Big) - 2(1-\sin^2 2\theta) \) and \( S = \left\{ \theta \in [0,\pi] : f'(\theta) = -\dfrac{\sqrt{3}}{2} \right\} \). If \( 4\beta = \ \displaystyle \sum_{\theta \in S} \theta \), then \( f(\beta) \) is equal to:

If \( m \) and \( n \) respectively are the numbers of positive and negative values of \( \theta \) in the interval \( [-\pi,\pi] \) that satisfy the equation \( \cos 2\theta \cos \frac{\theta}{2} = \cos 30 \cos \frac{90}{2} \), then \( mn \) is equal to ____.

If the solution of the equation \(\log_{\cos x}\cot x + 4\log_{\sin x}\tan x = 1,\; x \in \left(0,\dfrac{\pi}{2}\right),\) is \(\sin^{-1}\!\left(\dfrac{\alpha + \sqrt{\beta}}{2}\right),\) where \(\alpha, \beta\) are integers, then \(\alpha + \beta\) is equal to:

Let \( S = \{\theta \in [0, 2\pi): \tan(\pi \cos \theta) + \tan(\pi \sin \theta) = 0\} \). Then \( \displaystyle \sum_{\theta \in S} \sin^{2}\!\left(\theta + \frac{\pi}{4}\right) \) is equal to __________.

In a triangle \( ABC \), if \( \cos A + 2 \cos B + \cos C = 2 \) and the lengths of the sides opposite to the angles \( A \) and \( C \) are \( 3 \) and \( 7 \) respectively, then \( \cos A - \cos C \) is equal to:

\( 96\cos\dfrac{\pi}{33}\cos\dfrac{2\pi}{33}\cos\dfrac{4\pi}{33}\cos\dfrac{8\pi}{33}\cos\dfrac{16\pi}{33} \) is equal to:

In the figure,\( \theta_{1} + \theta_{2} = \dfrac{\pi}{2} \) and \( \sqrt{3}(BE) = 4(AB) \). If the area of \( \triangle CAB \) is \( 2\sqrt{3} - 3 \) unit\( ^{2} \), when \( \theta_{2}/\theta_{1} \) is the largest, then the perimeter (in unit) of \( \triangle CED \) is equal to ______.

The value of \( 36(4\cos^{2}9^{\circ}-1)(4\cos^{2}27^{\circ}-1)(4\cos^{2}81^{\circ}-1)(4\cos^{2}243^{\circ}-1) \) is:

The number of elements in the set \( S = \left\{ \theta \in [0, 2\pi] : 3\cos^4 \theta - 5\cos^2 \theta - 2\sin^6 \theta + 2 = 0 \right\} \) is:

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JEE Main 2023 30th Jan Morning Shift

If \(\tan 15^\circ + \dfrac{1}{\tan 75^\circ} + \dfrac{1}{\tan 105^\circ} + \tan 195^\circ = 2a\), Then the value of \(\left(a + \dfrac{1}{a}\right)\) is: