1
JEE Main 2023 (Online) 25th January Evening Shift
Numerical
+4
-1

If m and n respectively are the numbers of positive and negative values of $$\theta$$ in the interval $$[-\pi,\pi]$$ that satisfy the equation $$\cos 2\theta \cos {\theta \over 2} = \cos 3\theta \cos {{9\theta } \over 2}$$, then mn is equal to ____________.

2
JEE Main 2023 (Online) 24th January Evening Shift
Numerical
+4
-1

Let $$\mathrm{S = \{ \theta \in [0,2\pi ):\tan (\pi \cos \theta ) + \tan (\pi \sin \theta ) = 0\}}$$. Then $$\sum\limits_{\theta \in S} {{{\sin }^2}\left( {\theta + {\pi \over 4}} \right)}$$ is equal to __________.

3
JEE Main 2022 (Online) 29th July Morning Shift
Numerical
+4
-1

Let $$S=\left\{\theta \in(0,2 \pi): 7 \cos ^{2} \theta-3 \sin ^{2} \theta-2 \cos ^{2} 2 \theta=2\right\}$$. Then, the sum of roots of all the equations $$x^{2}-2\left(\tan ^{2} \theta+\cot ^{2} \theta\right) x+6 \sin ^{2} \theta=0, \theta \in S$$, is __________.

4
JEE Main 2022 (Online) 28th July Evening Shift
Numerical
+4
-1

Let $$S=\left[-\pi, \frac{\pi}{2}\right)-\left\{-\frac{\pi}{2},-\frac{\pi}{4},-\frac{3 \pi}{4}, \frac{\pi}{4}\right\}$$. Then the number of elements in the set $$\mid A=\{\theta \in S: \tan \theta(1+\sqrt{5} \tan (2 \theta))=\sqrt{5}-\tan (2 \theta)\}$$ is __________.

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