Planck's constant (h),speed of light in vacuum (c) and Newton's gravitational constant (G) are three fundamental constants. Which of the following combinations of these has the dimensions of length?

(1) $\frac{\sqrt{hG}}{c^{3/2}}$

(2) $\frac{\sqrt{hG}}{c^{5/2}}$

(3) $\sqrt{\frac{hc}{G}}$

(4)$\sqrt{\frac{Gc}{h^{3/2}}}$

If energy (E), velocity (v) and time (T) are chosen as the fundamental quantities, the dimensional formula of surface tension will be

(1) $\left[\begin{array}{cc}E& v^{-2}{T}^{-1}\end{array}\right]$

(2) [$E{v}^{-1}{T}^{-2}$]

(3) $\left[\begin{array}{cc}E{v}^{-2}& T^{-2}\end{array}\right]$

(4) [$E^{-2}{v}^{-1_T-3}$]
 

In an experiment four quantities a, b, c and d are measured with percentage error 1%,2%,3% and 4% respectively. Quantity P is calculated as follows P=a³b²/cd %, Error in P is

1. 14%

2. 10%

3. 7%

4. 4%

The density of material in the CGS system of units is $4g/c{m}^3.$ In a system of units in which unit of length is 10cm and unit of mass is 100g when the value of density of material in this system will be:

(1) 0.4

(2) 40

(3) 400

(4) 0.04 

A student measures the distance traversed in free fall of a body, initially at rest in a given time. He uses this data to estimate g, the acceleration due to gravity. If the maximum percentage errors in measurement of the distance and the time are ${\mathrm{e}}_1$ and ${\mathrm{e}}_2$ respectively, the percentage error in the estimation of g is 

1. ${\mathrm{e}}_2-{\mathrm{e}}_1$

2. ${\mathrm{e}}_1+2{\mathrm{e}}_2$

3. ${\mathrm{e}}_1+{\mathrm{e}}_2$ 

4. ${\mathrm{e}}_1-2{\mathrm{e}}_2$

If the dimension of a physical quantity are given by ${\mathrm{M}}^{\mathrm{a}}{\mathrm{L}}^{\mathrm{b}}{\mathrm{T}}^{\mathrm{c}},$ then the physical quantity will be

(1) pressure if a=1, b=-1, c=-2

(2) velocity if a=1, b=0, c=-1

(3) acceleration if a=1, b=1,c=-2

(4) force if a=0, b=-1, c=-2

If the error in the measurement of the radius of a sphere is 2%, then the error in the determination of the volume of the sphere will be:

1. 4%                                                   2. 6%

3. 8%                                                   4. 2%

Which two of the following five physical parameters have the same dimensions ?

(1) energy density 

(2) refractive index 

(3) dielectric constant 

(4) Young's modulus 

(5) magnetic field 

1. 2 and 4

2. 3 and 5 

3. 1 and 4 

4. 1 and 5

A physical quantity of the dimensions of length that can be formed out of \(c, G,~\text{and}~\frac{e^2}{4\pi\varepsilon_0}\)is [\(c\) is the velocity of light, \(G\) is the universal constant of gravitation and \(e\) is charge]:
\(1. ~c^2\left[G \frac{e^2}{4 \pi \varepsilon_0}\right]^{\frac{1}{2}}\\ 2.~ \frac{1}{c^2}\left[\frac{e^2}{4 G \pi \varepsilon_0}\right]^{\frac{1}{2}}\\ 3. ~\frac{1}{c} G \frac{e^2}{4 \pi \varepsilon_0}\\ 4. ~\frac{1}{c^2}\left[G \frac{e^2}{4 \pi \varepsilon_0}\right]^{\frac{1}{2}}\)

A student measured the diameter of a small steel ball using a screw gauge of least count \(0.001\) cm. The main scale reading is \(5\) mm and zero of circular scale division coincides with \(25\) divisions above the reference level. If the screw gauge has a zero error of \(-0.004\) cm, the correct diameter of the ball is:
1. \(0.521\) cm
2. \(0.525\) cm
3. \(0.053\) cm
4. \(0.529\) cm