1
JEE Main 2022 (Online) 27th July Evening Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus
Change Language

The angle of elevation of the top P of a vertical tower PQ of height 10 from a point A on the horizontal ground is $$45^{\circ}$$. Let R be a point on AQ and from a point B, vertically above $$\mathrm{R}$$, the angle of elevation of $$\mathrm{P}$$ is $$60^{\circ}$$. If $$\angle \mathrm{BAQ}=30^{\circ}, \mathrm{AB}=\mathrm{d}$$ and the area of the trapezium $$\mathrm{PQRB}$$ is $$\alpha$$, then the ordered pair $$(\mathrm{d}, \alpha)$$ is :

A
$$(10(\sqrt{3}-1), 25)$$
B
$$\left(10(\sqrt{3}-1), \frac{25}{2}\right)$$
C
$$(10(\sqrt{3}+1), 25)$$
D
$$\left(10(\sqrt{3}+1), \frac{25}{2}\right)$$
2
JEE Main 2022 (Online) 27th July Evening Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus
Change Language

Let $$S=\left\{\theta \in\left(0, \frac{\pi}{2}\right): \sum\limits_{m=1}^{9} \sec \left(\theta+(m-1) \frac{\pi}{6}\right) \sec \left(\theta+\frac{m \pi}{6}\right)=-\frac{8}{\sqrt{3}}\right\}$$. Then

A
$$ S=\left\{\frac{\pi}{12}\right\} $$
B
$$ S=\left\{\frac{2 \pi}{3}\right\} $$
C
$$ \sum\limits_{\theta \in S} \theta=\frac{\pi}{2} $$
D
$$ \sum\limits_{\theta \in S} \theta=\frac{3\pi}{4} $$
3
JEE Main 2022 (Online) 27th July Evening Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus
Change Language

If the truth value of the statement $$(P \wedge(\sim R)) \rightarrow((\sim R) \wedge Q)$$ is F, then the truth value of which of the following is $$\mathrm{F}$$ ?

A
$$\mathrm{P} \vee \mathrm{Q} \rightarrow \,\sim \mathrm{R}$$
B
$$\mathrm{R} \vee \mathrm{Q} \rightarrow \,\sim \mathrm{P}$$
C
$$\sim(\mathrm{P} \vee \mathrm{Q}) \rightarrow \sim \mathrm{R}$$
D
$$\sim(\mathrm{R} \vee \mathrm{Q}) \rightarrow \,\sim \mathrm{P}$$
4
JEE Main 2022 (Online) 27th July Evening Shift
Numerical
+4
-1
Out of Syllabus
Change Language

Consider a matrix $$A=\left[\begin{array}{ccc}\alpha & \beta & \gamma \\ \alpha^{2} & \beta^{2} & \gamma^{2} \\ \beta+\gamma & \gamma+\alpha & \alpha+\beta\end{array}\right]$$, where $$\alpha, \beta, \gamma$$ are three distinct natural numbers.

If $$\frac{\operatorname{det}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A))))}{(\alpha-\beta)^{16}(\beta-\gamma)^{16}(\gamma-\alpha)^{16}}=2^{32} \times 3^{16}$$, then the number of such 3 - tuples $$(\alpha, \beta, \gamma)$$ is ____________.

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