A small steel ball of radius \(r\) is allowed to fall under gravity through a column of a viscous liquid of coefficient of viscosity \(\eta\). After some time the velocity of the ball attains a constant value known as terminal velocity \(v_T\). The terminal velocity depends on \((\text{i})\) the mass of the ball \(m\) \((\text{ii})\) \(\eta\) \((\text{iii})\) \(r\) and \((\text{iv})\) acceleration due to gravity \(g\). Which of the following relations is dimensionally correct:
1. \(v_T \propto \frac{mg}{\eta r}\)
2. \(v_T \propto \frac{\eta r}{mg}\)
3. \(v_T \propto \eta rmg\)
4. \(v_T \propto \frac{mgr}{\eta }\)

The quantity $X=\frac{{\epsilon }_{0}LV}{t}:{\epsilon }_0$ is the permittivity of free space, L is length, V is the potential difference and t is time. The dimensions of X are the same as that of  

(1) Resistance

(2) Charge

(3) Voltage

(4) Current

The dimensions of physical quantity X in the equation Force $=\frac{X}{\text{Density}}$ is given by 

(1) $M^1{L}^4{T}^{-2}$

(2) $M^2{L}^{-2}{T}^{-1}$

(3) $M^2{L}^{-2}{T}^{-2}$

(4) $M^1{L}^{-2}{T}^{-1}$

Two quantities A and B have different dimensions. Which mathematical operation given below is physically meaningful 

(1) A/B

(2) A + B

(3) A – B

(4) None

A force F is given by F = at + bt², where t is time. What are the dimensions of a and b 

(1) $ML{T}^{-3}$ and $M{L}^2{T}^{-4}$

(2) $ML{T}^{-3}$ and $ML{T}^{-4}$

(3) $ML{T}^{-1}$ and $ML{T}^0$

(4) $ML{T}^{-4}$ and $ML{T}^1$

The decimal equivalent of 1/20 upto three significant figures is

(1) 0.0500

(2) 0.05000

(3) 0.0050

(4) 5.0 × 10^-2

Accuracy of measurement is determined by

(1) Absolute error

(2) Percentage error

(3) Both

(4) None of these

The number of significant figures in all the given numbers 25.12, 2009, 4.156 and $1.217\times {10}^{-4}$ is 

1. 1

2. 2

3. 3

4. 4

Assertion: Out of three measurements l = 0.7 m; l = 0.70 m and l = 0.700 m, the last one is most accurate.

Reason: In every measurement, only the last significant digit is not accurately known.

Choose any one of the following four responses :

(1) If both assertion and reason are true and the reason is the correct explanation of the assertion.

(2) If both assertion and reason are true but the reason is not the correct explanation of the assertion.

(3) If the assertion is true but the reason is false.

(4) If the assertion and reason both are false.

Assertion: Avogadro number is the number of atoms in one gram mole.

Reason: Avogadro number is a dimensionless constant.