1
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 __________.

2
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 ______________.

3
JEE Main 2022 (Online) 29th June Evening Shift
Numerical
+4
-1

The number of solutions of the equation $$\sin x = {\cos ^2}x$$ in the interval (0, 10) is _________.

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

The number of elements in the set $$S = \{ \theta \in [ - 4\pi ,4\pi ]:3{\cos ^2}2\theta + 6\cos 2\theta - 10{\cos ^2}\theta + 5 = 0\}$$ is __________.

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