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The least value of \( \big(\cos^{2}\theta - 6\sin\theta \cos\theta + 3\sin^{2}\theta + 2\big) \) is:

If \( \frac{\cos^2 48^\circ-\sin^2 12^\circ}{\sin^2 24^\circ-\sin^2 6^\circ}=\frac{\alpha+\beta\sqrt{5}}{2} \), where \( \alpha, \beta \in \)(\(\mathbb{R}\)) , then \( \alpha+\beta \) is equal to ______________.

If \( \cot x = \dfrac{5}{12} \) for some \( x \in \left({\pi}, \dfrac{3\pi}{2}\right) \), then \( \sin 7x \left( \cos \dfrac{13x}{2} + \sin \dfrac{13x}{2} \right) + \cos 7x \left( \cos \dfrac{13x}{2} - \sin \dfrac{13x}{2} \right) \) is equal to:

Let \( \dfrac{\pi}{2} < \theta < \pi \) and \( \cot\theta = -\dfrac{1}{2\sqrt{2}} \). Then the value of \( \sin\!\left(\dfrac{15\theta}{2}\right)(\cos 8 \theta+\sin{8} \theta)+\cos\!\left(\dfrac{15\theta}{2}\right)(\cos 8 \theta-\sin 8 \theta) \) is equal to:

Let \( \alpha \) and \( \beta \) respectively be the maximum and the minimum values of the function \( f(\theta) = 4\left[\sin^{4}\!\left(\frac{7\pi}{2}-\theta\right) + \sin^{4}(11\pi+\theta)\right] - 2\left[\sin^{6}\!\left(\frac{3\pi}{2}-\theta\right) + \sin^{6}(9\pi-\theta)\right],\ \theta \in \mathbb{R}. \) Then \( \alpha + 2\beta \) is equal to:

Let \( S=\{\theta \in (-2\pi,2\pi): \cos\theta+1=\sqrt{3}\sin\theta\} \). Then \( \sum_{\theta\in S}\theta \) is equal to:

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:

If \( A=\dfrac{\sin 3^{\circ}}{\cos 9^{\circ}}+\dfrac{\sin 9^{\circ}}{\cos 27^{\circ}}+\dfrac{\sin 27^{\circ}}{\cos 81^{\circ}} \) and \( B=\tan 81^{\circ}-\tan 3^{\circ} \), then \( \dfrac{B}{A} \) is equal to ________.

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 _______________.

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JEE Main 2026 5th April Morning Shift

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: