The universal gravitational constant is dimensionally represented as:

1. ${\mathrm{ML}}^2{\mathrm{T}}^{-1}$

2. ${\mathrm{M}}^{-2}{\mathrm{L}}^3{\mathrm{T}}^{-2}$

3. ${\mathrm{M}}^{-2}{\mathrm{L}}^2{\mathrm{T}}^{-1}$

4. ${\mathrm{M}}^{-1}{\mathrm{L}}^3{\mathrm{T}}^{-2}$

The angle of \(1^\circ\) (degree) will be equal to:
(Use \(360^\circ=2\pi\) rad)
1. \(1.034\times10^{-3}\) rad
2. \(1.745\times10^{-2}\) rad
3. \(1.524\times10^{-2}\) rad
4. \(1.745\times10^{3}\) rad

Each side of a cube is measured to be \(7.203\) \(\mathrm{m}.\) What are the total surface area and the volume respectively of the cube to appropriate significant figures?

If \(y = a\sin(bt-cx)\), where \(y\) and \(x\) represent length and \(t\) represents time, then which of the following has the same dimensions as that of \(\frac{ab^2}{c}\)$\frac{\mathrm{}}{}$?
1. \((\text{speed})^2\)
2. \(\text{momentum}\)
3. \(\text{angle}\)
4. \(\text{acceleration}\)

The dimensions of \((\mu_0\varepsilon_0)^{\frac{-1}{2}}\) are:
1. \(\left[L^{-1}T\right]\)
2. \(\left[LT^{-1}\right]\)
3. \(\left[L^{{-1/2}}T^{{1/2}}\right]\)
4. \(\left[L^{{-1/2}}T^{{-1/2}}\right]\)

An object is moving through a liquid. The viscous damping force acting on it is proportional to the velocity. Then the dimensions of the constant of proportionality are:
1. \(\left[ML^{-1}T^{-1}\right]\)
2. \(\left[MLT^{-1}\right]\)
3. \(\left[M^0LT^{-1}\right]\)
4. \(\left[ML^{0}T^{-1}\right]\)

A screw gauge gives the following readings when used to measure the diameter of a wire:
Main scale reading: \(0\) mm
Circular scale reading: \(52\) divisions 
Given that \(1\) mm on the main scale corresponds to \(100\) divisions on the circular scale, the diameter of the wire that can be inferred from the given data is:
1. \(0.26\) cm 
2. \(0.052\) cm 
3. \(0.52\) cm 
4. \(0.026\) cm 

The pitch of the screw gauge is \(1\) mm and there are \(100\) divisions on the circular scale. When nothing is put in between the jaws, the zero of the circular scale lies \(8\) divisions below the reference line. When a wire is placed between the jaws, the first linear scale division is clearly visible while the 72^nd division on a circular scale coincides with the reference line. The radius of the wire is:
1. \(1.64\) mm
2. \(0.82\) mm
3. \(1.80\) mm
4. \(0.90\) mm

The density of a material in a CGS system of units is \(4~\text{g/cm}^3\). In a system of units in which the unit of length is \(10~\text{cm}\) and the unit of mass is \(100~\text{g}\), the value of the density of the material will be:
1. \( 0.04 \)
2. \( 0.4 \)
3. \( 40 \)
4. \(400\)

On the basis of dimensions, decide which of the following relations for the displacement of a particle undergoing simple harmonic motion is/are not correct.
a. \(y = a\sin \left(2\pi t / T\right)\)
b. \(y = a\sin(vt)\)
c. \(y = \left({a \over T}\right) \sin \left({t \over a}\right)\)
d. \(y = a \sqrt 2 \left(\sin \left({2 \pi t \over T}\right) - \cos \left({2 \pi t \over T}\right)\right)\)
(Symbols have their usual meanings.)
Choose the correct option:
1. (a), (c)
2. (a), (b)
3. (b), (c)
4. (a), (d)