A calorie is a unit of heat (energy in transit) and it equals about \(4.2~\text{J}\) where \(1~\text{J}= 1~\text{kg-m}^2\text{s}^{-2}\). Suppose we employ a system of units in which the unit of mass equals \(\alpha~\text{kg}\) the unit of length equals \(\beta~\text{m}\), and the unit of time is \(\gamma~\text{s}\) then the magnitude of calories in terms of new units is:
1. \(4.2\alpha^{-2}\beta^{-2}\gamma^{2}\)
2. \(4.2\alpha^{2}\beta^{-2}\gamma^{2}\)
3. \(4.2\alpha^{-1}\beta^{-2}\gamma^{2}\)
4. \(4.2\alpha^{-1}\beta^{2}\gamma^{-2}\)

A new unit of length is chosen such that the speed of light in the vacuum is unity. What is the distance between the Sun and the Earth in terms of the new unit if light takes \(8\) min and \(20\) secs to cover this distance?
1. \(500\) new units of length.
2. \(100\) new units of length.
3. \(300\) new units of length.
4. \(200\) new units of length.

Which one of the following formulas is dimensionally incorrect?

$\left(1\right)<mspace\; width="1em"></mspace>\mathrm{y}=\mathrm{asin}\frac{2\mathrm{\pi t}}{\mathrm{T}}$
$\left(2\right)<mspace\; width="1em"></mspace>\mathrm{y}=\frac{\mathrm{a}}{\sqrt{2}}\left(\sin \frac{2\mathrm{\pi t}}{\mathrm{T}}+\cos \frac{2\mathrm{\pi t}}{\mathrm{T}}\right)$
$\left(3\right)<mspace\; width="1em"></mspace>\mathrm{y}=\mathrm{asinvt}$
$\left(4\right)<mspace\; width="1em"></mspace>\mathrm{y}=\mathrm{asin}\left(\frac{2\mathrm{\pi t}}{\mathrm{T}}+\mathrm{\pi }\right)$

The unit of length convenient on the atomic scale is known as an angstrom and is denoted by $\stackrel{\mathrm{o}}{\mathrm{A}}$ $\left(1<mspace\; width="1em"></mspace>\stackrel{\mathrm{o}}{\mathrm{A}}={10}^{-10}<mspace\; width="1em"></mspace>\mathrm{m}\right).$ The size of a hydrogen atom is about 0.5 $\stackrel{o}{A}$. What is the total atomic volume in $m^3$ of a mole of hydrogen atoms?

$\left(1\right)<mspace\; width="1em"></mspace>3\times {10}^{-7}<mspace\; width="1em"></mspace>m^3$
$\left(2\right)<mspace\; width="1em"></mspace>2\times {10}^{-7}<mspace\; width="1em"></mspace>m^3$
$\left(3\right)<mspace\; width="1em"></mspace>3\times {10}^{-6}<mspace\; width="1em"></mspace>m^3$
$\left(4\right)<mspace\; width="1em"></mspace>2\times {10}^{-6}<mspace\; width="1em"></mspace>m^3$

One mole of an ideal gas at standard temperature and pressure occupies 22.4 L (molar volume). What is the ratio of molar volume to the atomic volume of a mole of hydrogen? (Take the size of the hydrogen molecule to be about 1 Å).

$\left(1\right)<mspace\; width="1em"></mspace>7.08\times {10}^4$
$\left(2\right)<mspace\; width="1em"></mspace>22.4\times {10}^3$
$\left(3\right)<mspace\; width="1em"></mspace>11.2\times {10}^5$
$\left(4\right)<mspace\; width="1em"></mspace>8.0\times {10}^3$

The nearest star to our solar system is \(4.29\) light-years away. How much is this distance in terms of parsecs?
1. \(1.11\) parsec
2. \(1.22\) parsec
3. \(1.32\) parsec
4. \(1.39\) parsec

What will be the average mass density of a sodium atom (assuming its size to be about $2.5<mspace\; width="1em"></mspace>\stackrel{0}{A}$)?

$\left(1\right)<mspace\; width="1em"></mspace>5.67\times {10}^3<mspace\; width="1em"></mspace>kg<mspace\; width="1em"></mspace>m^{-3}$
$\left(2\right)<mspace\; width="1em"></mspace>4.67\times {10}^3<mspace\; width="1em"></mspace>kg<mspace\; width="1em"></mspace>m^{-3}$
$\left(3\right)<mspace\; width="1em"></mspace>4.67\times {10}^4<mspace\; width="1em"></mspace>kg<mspace\; width="1em"></mspace>m^{-3}$
$\left(4\right)<mspace\; width="1em"></mspace>5.67\times {10}^4<mspace\; width="1em"></mspace>kg<mspace\; width="1em"></mspace>m^{-3}$