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Let \( S \) be the sum of all solutions (in radians) of the equation \( \sin^4\theta+\cos^4\theta-\sin\theta\cos\theta=0 \) in \( [0,\,4\pi] \). Then \( \dfrac{8S}{\pi} \) is equal to ______.

The number of solutions of the equation \( 32^{\tan^{2}x} + 32^{\sec^{2}x} = 81, \; 0 \le x \le \frac{\pi}{4} \) is:

The value of \( \cot \dfrac{\pi}{24} \) is:

The sum of solutions of the equation \( \dfrac{\cos x}{1+\sin x} = |\tan 2x| \), \( x \in \left( -\dfrac{\pi}{2}, \dfrac{\pi}{2} \right) - \left\{ \dfrac{\pi}{4}, -\dfrac{\pi}{4} \right\} \) is:

Let in a right angled triangle, the smallest angle be \( \theta \). If a triangle formed by taking the reciprocal of its sides is also a right angled triangle, then \( \sin \theta \) is equal to:

A pole stands vertically inside a triangular park ABC. Let the angle of elevation of the top of the pole from each corner of the park be \( \dfrac{\pi}{3} \). If the radius of the circumcircle of \( \triangle ABC \) is \( 2 \), then the height of the pole is equal to:

If \( 15\sin^4\alpha + 10\cos^4\alpha = 6 \), for some \( \alpha \in \mathbb{R} \), then the value of \( 27\sec^6\alpha + 8\,\text{cosec}^6\alpha \) is equal to:

If \( n \) is the number of solutions of the equation \( 2 \cos x \left( 4 \sin\!\left(\frac{\pi}{4} + x\right) \sin\!\left(\frac{\pi}{4} - x\right) - 1 \right) = 1 \), \( x \in [0, \pi] \) and \( S \) is the sum of all these solutions, then the ordered pair \( (n, S) \) is:

A vertical pole fixed to the horizontal ground is divided in the ratio \( 3:7 \) by a mark on it with lower part shorter than the upper part. If the two parts subtend equal angles at a point on the ground \( 18 \, \text{m} \) away from the base of the pole, then the height of the pole (in meters) is:

Two poles, \( AB \) of length \( a \) meters and \( CD \) of length \( a+b \) (\( b\neq a \)) meters are erected at the same horizontal level with bases at \( B \) and \( D \). If \( BD=x \) and \( \tan \angle ACB=\frac{1}{2} \), then:

The value of \( 2 \sin\!\left(\frac{\pi}{8}\right)\sin\!\left(\frac{2\pi}{8}\right)\sin\!\left(\frac{3\pi}{8}\right)\sin\!\left(\frac{5\pi}{8}\right)\sin\!\left(\frac{6\pi}{8}\right)\sin\!\left(\frac{7\pi}{8}\right) \) is:

The number of solutions of \(\displaystyle \sin^7 x + \cos^7 x = 1, \, x \in [0, 4\pi]\) is equal to:

The sum of all values of \( x \) in \( [0,2\pi] \) for which \( \sin x+\sin 2x+\sin 3x+\sin 4x=0 \) is equal to:

The number of solutions of the equation \( x + 2 \tan x = \dfrac{\pi}{2} \) in the internal \( [0, 2\pi] \) is :

The number of solutions of the equation \( |\cot x| = \cot x + \frac{1}{\sin x} \) in the interval \( [0, 2\pi] \) is ______________ .

If \( \sin \theta + \cos \theta = \frac{1}{2} \), then \( 16(\sin (2\theta) + \cos (4\theta) + \sin (6\theta)) \) is equal to:

A \( 10 \) inches long pencil \( AB \) with mid point \( C \) and a small eraser \( P \) are placed on the horizontal top of a table such that \( PC = \sqrt{5} \) inches and \( \angle PCB = \tan^{-1}(2) \). The acute angle through which the pencil must be rotated about \( C \) so that the perpendicular distance between eraser and pencil becomes exactly \( 1 \) inch is:

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JEE Main 2021 18th March Morning Shift

The number of solutions of the equation \( |\cot x| = \cot x + \frac{1}{\sin x} \) in the interval \( [0, 2\pi] \) is ______________ .