1
JEE Main 2023 (Online) 29th January Evening Shift
MCQ (Single Correct Answer)
+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
$$\left[ { - 1, - {1 \over 2}} \right]$$
2
JEE Main 2023 (Online) 29th January Morning Shift
MCQ (Single Correct Answer)
+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
$$\frac{11}{8}$$
3
JEE Main 2022 (Online) 25th July Evening Shift
MCQ (Single Correct Answer)
+4
-1

$$2 \sin \left(\frac{\pi}{22}\right) \sin \left(\frac{3 \pi}{22}\right) \sin \left(\frac{5 \pi}{22}\right) \sin \left(\frac{7 \pi}{22}\right) \sin \left(\frac{9 \pi}{22}\right)$$ is equal to :

A
$$\frac{3}{16}$$
B
$$\frac{1}{16}$$
C
$$\frac{1}{32}$$
D
$$\frac{9}{32}$$
4
JEE Main 2022 (Online) 28th June Evening Shift
MCQ (Single Correct Answer)
+4
-1

If cot$$\alpha$$ = 1 and sec$$\beta$$ = $$- {5 \over 3}$$, where $$\pi < \alpha < {{3\pi } \over 2}$$ and $${\pi \over 2} < \beta < \pi$$, then the value of $$\tan (\alpha + \beta )$$ and the quadrant in which $$\alpha$$ + $$\beta$$ lies, respectively are :

A
$$- {1 \over 7}$$ and IVth quadrant
B
7 and Ist quadrant
C
$$-$$7 and IVth quadrant
D
$${1 \over 7}$$ and Ist quadrant
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