1
JEE Main 2023 (Online) 30th January Morning Shift
+4
-1

If $$\tan 15^\circ + {1 \over {\tan 75^\circ }} + {1 \over {\tan 105^\circ }} + \tan 195^\circ = 2a$$, then the value of $$\left( {a + {1 \over a}} \right)$$ is :

A
$$5 - {3 \over 2}\sqrt 3$$
B
$$4 - 2\sqrt 3$$
C
2
D
4
2
JEE Main 2023 (Online) 29th January Evening Shift
+4
-1

The set of all values of $$\lambda$$ for which the equation $${\cos ^2}2x - 2{\sin ^4}x - 2{\cos ^2}x = \lambda$$ has a real solution $$x$$, is

A
$$\left[ { - 2, - 1} \right]$$
B
$$\left[ { - {3 \over 2}, - 1} \right]$$
C
$$\left[ { - 2, - {3 \over 2}} \right]$$
D
$$\left[ { - 1, - {1 \over 2}} \right]$$
3
JEE Main 2023 (Online) 29th January Morning Shift
+4
-1

Let $$f(\theta ) = 3\left( {{{\sin }^4}\left( {{{3\pi } \over 2} - \theta } \right) + {{\sin }^4}(3\pi + \theta )} \right) - 2(1 - {\sin ^2}2\theta )$$ and $$S = \left\{ {\theta \in [0,\pi ]:f'(\theta ) = - {{\sqrt 3 } \over 2}} \right\}$$. If $$4\beta = \sum\limits_{\theta \in S} \theta$$, then $$f(\beta )$$ is equal to

A
$$\frac{9}{8}$$
B
$$\frac{3}{2}$$
C
$$\frac{5}{4}$$
D
$$\frac{11}{8}$$
4
JEE Main 2022 (Online) 29th July Evening Shift
+4
-1

The number of elements in the set $$S=\left\{x \in \mathbb{R}: 2 \cos \left(\frac{x^{2}+x}{6}\right)=4^{x}+4^{-x}\right\}$$ is :

A
1
B
3
C
0
D
infinite
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