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The number of solutions of the equation \( 2x + 3 \tan x = \pi \), \( x \in [ -2\pi, 2\pi ] - \left\{ \pm \frac{\pi}{2}, \pm \frac{3\pi}{2} \right\} \), is:

The value of \( (\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1) \) is:

\(\displaystyle \sin x + \sin^2 x = 1,\, x \in \left(0, \frac{\pi}{2}\right) \), then \( (\cos^{12} x + \tan^{12} x) + 3(\cos^{10} x + \tan^{10} x + \cos^{8} x + \tan^{8} x) + (\cos^{6} x + \tan^{6} x) \) is equal to:

If \( \displaystyle \sum_{r=1}^{13} \left\{ \frac{1}{\displaystyle \sin\!\left( \frac{\pi}{4} + (r-1)\frac{\pi}{6} \right)\,\sin\!\left( \frac{\pi}{4} + \frac{r\pi}{6} \right)} \right\} = a\sqrt{3} + b,\; a,b\in\mathbb{Z} \), then \( a^{2}+b^{2} \) is equal to:

If \( \theta \in [-2\pi, 2\pi] \), then the number of solutions of \( 2\sqrt{2}\cos^{2}\theta + (2-\sqrt{6})\cos\theta - \sqrt{3} = 0 \), is equal to:

If $10 \sin^4 \theta + 15 \cos^4 \theta = 6$, then the value of $\frac{27 cosec^6 \theta + 8 \sec^6 \theta}{16 \sec^8 \theta}$ is:

Let the distance between two parallel lines be \( 5 \) units and a point \( P \) lie between the lines at a unit distance from one of them. An equilateral triangle \( PQR \) is formed such that \( Q \) lies on one of the parallel lines, while \( R \) lies on the other. Then \( (QR)^{2} \) is equal to ____________.

The sum of all values of \( \theta \in [0, 2\pi] \) satisfying \( 2\sin^2\theta = \cos 2\theta \) and \( 2\cos^2\theta = 3\sin\theta \) is:

The number of solutions of the equation \( (4-\sqrt{3})\sin x-2\sqrt{3}\cos^{2}x=-\dfrac{4}{1+\sqrt{3}},\; x\in\left[ -2\pi,\;\dfrac{5\pi}{2}\right] \) is:

If for \( \theta \in \left[ -\dfrac{\pi}{3}, 0 \right] \), the points \( (x, y) = \left( 3 \tan \left( \theta + \dfrac{\pi}{3} \right), \, 2 \tan \left( \theta + \dfrac{\pi}{6} \right) \right) \) lie on \( xy + \alpha x + \beta y + \gamma = 0 \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to:

The number of solutions of the equation \( \cos 2\theta \cos \dfrac{\theta}{2}+\cos \dfrac{5\theta}{2}=2\cos^{3}\dfrac{5\theta}{2} \) in \( \left[ -\dfrac{\pi}{2},\dfrac{\pi}{2} \right] \) is:

Let the range of function \( f(x) = 6 + 16 \cos x \cdot \cos\!\left(\frac{\pi}{3} - x\right) \cdot \cos\!\left(\frac{\pi}{3} + x\right) \cdot \sin 3x \cdot \cos 6x, \; x \in R \) be \( [\alpha,\beta] \). Then the distance of the point \( (\alpha,\beta) \) from the line \( 3x + 4y + 12 = 0 \) is:

Let \( \alpha_{\theta} \) and \( \beta_{\theta} \) be the distinct roots of \( 2x^{2}+(\cos \theta)x-1=0 , \; \theta \in (0,2\pi) \). If \( m \) and \( M \) are the minimum and the maximum values of \( \alpha_{\theta}^{4}+\beta_{\theta}^{4} \), then \( 16{(M+m)} \) equals: