1
JEE Main 2022 (Online) 27th July Evening Shift
Numerical
+4
-1
Change Language

Let f be a differentiable function satisfying $$f(x)=\frac{2}{\sqrt{3}} \int\limits_{0}^{\sqrt{3}} f\left(\frac{\lambda^{2} x}{3}\right) \mathrm{d} \lambda, x>0$$ and $$f(1)=\sqrt{3}$$. If $$y=f(x)$$ passes through the point $$(\alpha, 6)$$, then $$\alpha$$ is equal to _____________.

Your input ____
2
JEE Main 2022 (Online) 27th July Evening Shift
Numerical
+4
-1
Change Language

Let $$\overrightarrow a $$, $$\overrightarrow b $$, $$\overrightarrow c $$ be three non-coplanar vectors such that $$\overrightarrow a $$ $$\times$$ $$\overrightarrow b $$ = 4$$\overrightarrow c $$, $$\overrightarrow b $$ $$\times$$ $$\overrightarrow c $$ = 9$$\overrightarrow a $$ and $$\overrightarrow c $$ $$\times$$ $$\overrightarrow a $$ = $$\alpha$$$$\overrightarrow b $$, $$\alpha$$ > 0. If $$\left| {\overrightarrow a } \right| + \left| {\overrightarrow b } \right| + \left| {\overrightarrow c } \right| = {1 \over {36}}$$, then $$\alpha$$ is equal to __________.

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3
JEE Main 2022 (Online) 27th July Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

An expression of energy density is given by $$u=\frac{\alpha}{\beta} \sin \left(\frac{\alpha x}{k t}\right)$$, where $$\alpha, \beta$$ are constants, $$x$$ is displacement, $$k$$ is Boltzmann constant and t is the temperature. The dimensions of $$\beta$$ will be :

A
$$\left[\mathrm{ML}^{2} \mathrm{~T}^{-2} \theta^{-1}\right]$$
B
$$\left[\mathrm{M}^{0} \mathrm{~L}^{2} \mathrm{~T}^{-2}\right]$$
C
$$\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]$$
D
$$\left[\mathrm{M}^{0} \mathrm{~L}^{2} \mathrm{~T}^{0}\right]$$
4
JEE Main 2022 (Online) 27th July Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

A body of mass $$10 \mathrm{~kg}$$ is projected at an angle of $$45^{\circ}$$ with the horizontal. The trajectory of the body is observed to pass through a point $$(20,10)$$. If $$\mathrm{T}$$ is the time of flight, then its momentum vector, at time $$\mathrm{t}=\frac{\mathrm{T}}{\sqrt{2}}$$, is _____________.

[Take $$\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$$ ]

A
$$ 100 \hat{i}+(100 \sqrt{2}-200) \hat{j}$$
B
$$100 \sqrt{2} \hat{i}+(100-200 \sqrt{2}) \hat{j}$$
C
$$100 \hat{i}+(100-200 \sqrt{2}) \hat{j}$$
D
$$100 \sqrt{2} \hat{i}+(100 \sqrt{2}-200) \hat{j}$$
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