1

### JEE Main 2019 (Online) 11th January Morning Slot

The force of interaction between two atoms is given by F = $\alpha$$\beta$exp $\left( { - {{{x^2}} \over {\alpha kt}}} \right)$; where x is the distance, k is the Boltzmann constant and T is temperature and $\alpha$ and $\beta$ are two constants. The dimension of $\beta$ is :
A
M2L2T$-$2
B
M2LT$-$4
C
MLT$-$4
D
M0L2LT$-$4

## Explanation

$F = \alpha \beta {e^{\left( {{{ - {x^2}} \over {\alpha KT}}} \right)}}$

$\left[ {{{{x^2}} \over {\alpha KT}}} \right] = {M^o}{L^o}{T^o}$

${{{L^2}} \over {\left[ \alpha \right]M{L^2}{T^{ - 2}}}}$ $=$ ${M^o}{L^o}{T^o}$

$\Rightarrow$  $\left[ \alpha \right] = {M^{ - 1}}{T^2}$

$\left[ F \right] = \left[ \alpha \right]\left[ \beta \right]$

MLT$-$2 = M$-$1T2[$\beta$]

$\Rightarrow$  [$\beta$] = M2LT$-$4
2

### JEE Main 2019 (Online) 11th January Evening Slot

If speed (V), acceleration (A) and force (F) are considered as fundamental units, the dimension of Young,s modulus will be:
A
V$-$2A2F2
B
V$-$4A$-$2F
C
V$-$4A2F
D
V$-$2A2F$-$2

## Explanation

We know,

Young's modulus (Y) = ${{{F \over A}} \over {{{\Delta l} \over l}}}$

$\therefore$ [Y] = ${{\left[ {ML{T^{ - 2}}} \right]} \over {\left[ {{L^2}} \right]}}$ = [ ML-1T-2]

Let [Y] = [V]x [A]y [F]z

$\therefore$ [ ML-1T-2] =

[LT-1]x [LT-2]y [MLT-2]z

$\Rightarrow$ [ ML-1T-2] =

[ Mz Lx + y + z T-x -2y - 2z

For dimensional balance, the dimension on both sides should be same.

So, z = 1

x + y + z = -1

$\Rightarrow$ x + y = -2 ........(1)

and -x -2y - 2z = -2

$\Rightarrow$ x + 2y = 0 ...........(2)

By solving those two equations we get,

x = -4 and y = 2

$\therefore$ [Y] = V$-$4A2F1
3

### JEE Main 2019 (Online) 12th January Morning Slot

The least count of the main scale of a screw gauge is 1 mm. The minimum number of divisions on its circular scale required to measure 5 $\mu$m diameter of a wire is :
A
500
B
100
C
200
D
50

## Explanation

Least count = ${{Pitch} \over {Number\,\,of\,\,division\,\,on\,\,circular\,scale}}$

5 $\times$ 10$-$6 = ${{{{10}^{ - 3}}} \over N}$

N = 200
4

### JEE Main 2019 (Online) 12th January Evening Slot

Let $\ell$, r, C and V represent inductance, resistance, capacitance and voltage, respectively. The dimension of ${\ell \over {rCV}}$ in SI units will be :
A
[A–1]
B
[LTA]
C
[LA–2]
D
[LT2]

## Explanation

$\left[ {{\ell \over r}} \right] =$ T

[CV] $=$ AT

So,   $\left[ {{\ell \over {rCV}}} \right]$ = ${T \over {AT}}$ = [A$-$1]