1
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 __________.

2
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 __________.

3
JEE Main 2022 (Online) 26th July Evening Shift
Numerical
+4
-1

If the sum of solutions of the system of equations $$2 \sin ^{2} \theta-\cos 2 \theta=0$$ and $$2 \cos ^{2} \theta+3 \sin \theta=0$$ in the interval $$[0,2 \pi]$$ is $$k \pi$$, then $$k$$ is equal to __________.

4
JEE Main 2022 (Online) 30th June Morning Shift
Numerical
+4
-1

Let $${S_1} = \{ x \in [0,12\pi ]:{\sin ^5}x + {\cos ^5}x = 1\}$$

and $${S_2} = \{ x \in [0,8\pi ]:{\sin ^7}x + {\cos ^7}x = 1\}$$

Then $$n({S_1}) - n({S_2})$$ is equal to ______________.

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