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

The density of the Sun is: 

(Given, mass of the Sun = 2.0 x 10³⁰ kg and radius of the Sun = 7.0 x 10⁸m)

(1) 2.9 x 10³ kg m^-3

(2) 1.4 x 10³ kg m^-3

(3) 5.6 x 10³ kg m^-3 

(4) 1.9 x 10³ kg m^-3

It is claimed that two caesium clocks if allowed to run for 100 years, free from any disturbance, may differ by only about 0.02 sec. What does this imply for the accuracy of the standard caesium clock in measuring a time interval of 1 sec?

$\left(1\right)<mspace\; width="1em"></mspace>{10}^{12}<mspace\; width="1em"></mspace>\mathrm{to}<mspace\; width="1em"></mspace>{10}^{13}$
$\left(2\right)<mspace\; width="1em"></mspace>{10}^7<mspace\; width="1em"></mspace>\mathrm{to}<mspace\; width="1em"></mspace>{10}^8$
$\left(3\right)<mspace\; width="1em"></mspace>{10}^{10}<mspace\; width="1em"></mspace>\mathrm{to}<mspace\; width="1em"></mspace>{10}^{13}$
$\left(4\right)<mspace\; width="1em"></mspace>{10}^{11}<mspace\; width="1em"></mspace>\mathrm{to}<mspace\; width="1em"></mspace>{10}^{12}$

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

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

A LASER is a source of very intense, monochromatic, and a unidirectional beam of light. These properties of laser light can be exploited to measure long distances. The distance of the Moon from the Earth has been already determined very precisely using a laser as a source of light. A laser light beamed at the Moon takes 2.56 s to return after reflection at the Moon’s surface. How much is the radius of the lunar orbit around the Earth?

(1) $2.14\times {10}^8<mspace\; width="1em"></mspace>m$

(2) $2.84\times {10}^8<mspace\; width="1em"></mspace>m$

(3) $3.84\times {10}^8<mspace\; width="1em"></mspace>m$

(4) $3.14\times {10}^8<mspace\; width="1em"></mspace>m$

A SONAR (sound navigation and ranging) uses ultrasonic waves to detect and locate objects underwater. In a submarine equipped with a SONAR, the time delay between the generation of a probe wave and the reception of its echo after reflection from an enemy submarine is found to be 77.0 sec. What is the distance of the enemy submarine? (Speed of sound in water = 1450 m/s )

(1) 60.32 km

(2) 57.70 km

(3) 44.23 km

(4) 55.80 km

The farthest objects in our Universe discovered by modern astronomers are so distant that light emitted by them takes billions of years to reach the Earth. These objects (known as quasars) have many puzzling features, which have not yet been satisfactorily explained. What is the distance in km of a quasar from which light takes 3.0 billion years to reach us? 

$\left(1\right)<mspace\; width="1em"></mspace>3.2\times {10}^{22}<mspace\; width="1em"></mspace>km$
$\left(2\right)<mspace\; width="1em"></mspace>3.7\times {10}^{22}<mspace\; width="1em"></mspace>km$
$\left(3\right)<mspace\; width="1em"></mspace>2.8\times {10}^{22}<mspace\; width="1em"></mspace>km$
$\left(4\right)<mspace\; width="1em"></mspace>2.1\times {10}^{22}<mspace\; width="1em"></mspace>km$