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 \(c, G,~\text{and}~\dfrac{e^2}{4\pi\varepsilon_0}\)is [\(c\) is the velocity of light, \(G\) is the universal constant of gravitation and \(e\) is charge]:
1. \(c^2\left[G \dfrac{e^2}{4 \pi \varepsilon_0}\right]^{\dfrac{1}{2}}\)
2. \(\dfrac{1}{c^2}\left[\dfrac{e^2}{4 G \pi \varepsilon_0}\right]^{\dfrac{1}{2}}\)
3. \(\dfrac{1}{c} G \dfrac{e^2}{4 \pi \varepsilon_0}\)
4. \(\dfrac{1}{c^2}\left[G \dfrac{e^2}{4 \pi \varepsilon_0}\right]^{\dfrac{1}{2}}\)