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JEE Mains | PYQ

The value of \( \dfrac{\sqrt{3}\,cosec 20^\circ - \sec 20^\circ}{\cos 20^\circ \cos 40^\circ \cos 60^\circ \cos 80^\circ} \) is equal to:

is equal to:

Let \( \tan A, \tan B \), where \( A, B \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \), be the roots of the quadratic equation \( x^2 - 2x - 5 = 0 \). Then \( 20\sin^2\!\left( \frac{A + B}{2} \right) \) is equal to:

Let \( \vec{a}_{k}=(\tan\theta_{k})\,\hat{i}+\hat{j} \) and \( \vec{b}_{k}=\hat{i}-(\cot\theta_{k})\,\hat{j} \), where \(\theta_k = \frac{2^{k-1}\pi}{2^{n} + 1}\), for some \( n\in \mathbb{N},\, n>5 \). Then the value of \( \dfrac{\sum_{k=1}^{n}|\vec{a}_{k}|^{2}}{\sum_{k=1}^{n}|\vec{b}_{k}|^{2}} \) is ________.

Let \( P=\{\theta\in[0,4\pi]:\tan^2\theta=1\} \) and \( S=\{a\in\mathbb{Z}: 2(\cos^8\theta-\sin^8\theta)\sec 2\theta=a^2,\; \theta\in P\} \). Then \( n(S) \) is:

Number of solutions of \( \sqrt{3}\cos 2\theta + 8\cos \theta + 3\sqrt{3} = 0, \ \theta \in [-3\pi, 2\pi] \) is:

The value of \( \ cosec 10^\circ - \sqrt{3}\sec 10^\circ \) is equal to:

If \( \dfrac{\tan(A-B)}{\tan A} + \dfrac{\sin^{2} C}{\sin^{2} A} = 1, \ A,B,C \in \left(0, \dfrac{\pi}{2}\right), \) then:

The sum of all the integral values of \( p \) such that the equation \( 3\sin^{2}x + 12\cos x - 3 = p,\, x \in \mathbb{R} \), has at least one solution, is:

Let \( S=\{x\in[-\pi,\pi]:\sin x(\sin x+\cos x)=a,\; a\in \mathbb{Z}\} \). Then \( n(S) \) is equal to:

If \( S=\left\{\theta\in[-\pi,\pi]:\cos\theta\cos\dfrac{50}{2}=\cos70\cos\dfrac{70}{2}\right\} \), then \( n(S) \) is equal to ______________ .

If \( \sin\!\left(\dfrac{\pi}{18}\right)\sin\!\left(\dfrac{5\pi}{18}\right)\sin\!\left(\dfrac{7\pi}{18}\right)=k \), then the value of \( \sin\!\left(\dfrac{10k\pi}{3}\right) \) is:

The number of elements in the set \( \{\, x \in [0,180^\circ] : \tan(x+100^\circ) = \tan(x+50^\circ)\tan x \tan(x-50^\circ) \,\} \) is _______________.