If the dimensions of a physical quantity are given by ${\mathrm{}}^{}$\(M^aL^bT^c\), 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. \(2\%\)

The pitch of a screw gauge is 1mm and there are 100 divisions on the circular scale. While measuring the diameter of a wire, the linear scale reads 1 mm and 47^th division on the circular scale coincides with the reference line. The length of the wire is 5.6 cm. Curved surface area (in cm²) of the wire in appropriate number of significant figures will be

1.  2.4 cm²

2.  2.56 cm²

3.  2.6 cm²

4.  2.8 cm²

Find the thickness of the wire. The least count is 0.01 mm. The main scale reads in mm.
            

1. 7.62 mm
2. 7.63 mm
3. 7.64 mm
4. 7.65 mm

Two resistors R₁ = (3.0 ± 0.3) Ω  and R₂ = (5.0 ± 0.1) Ω are connected in parallel. The equivalent resistance, R_eq, will be
Hint: \({1 \over R_{eq}} = {1 \over R_{1}} + {1 \over R_{2}} \)
1. 1.9 ± 0.07 Ω
2. 1.9 ± 0.1 Ω
3. 2.9 ± 0.2 Ω
4. 2.9 ± 0.3 Ω

If energy (\(E\)), velocity (\(v\)) and time (\(T\)) are chosen as the fundamental quantities, the dimensional formula of surface tension will be:
1. \( {\left[E v^{-2} T^{-1}\right]} \)
2. \( {\left[E v^{-1} T^{-2}\right]} \)
3. \( {\left[E v^{-2} T^{-2}\right]} \)
4. \({\left[E^{-2} v^{-1} T^{-3}\right]}\)

 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 quantities \(A\) and \(B\) are related by the relation, \(m= \frac{A}{B}\), where \(m\) is the linear density and \(A\) is the force. The dimensions of \(B\) are of:

The frequency of vibration \(f\) of a mass \(m\) suspended from a spring of spring constant \(k\) is given by a relation of type \(f= Cm^{x}k^{y}\); where \(C\) is a dimensionless quantity. The values of \(x\) and \(y\) will be:
1. \(x=\frac{1}{2},~y= \frac{1}{2}\)
2. \(x=-\frac{1}{2},~y= -\frac{1}{2}\)
3. \(x=\frac{1}{2},~y= -\frac{1}{2}\)
4. \(x=-\frac{1}{2},~y= \frac{1}{2}\)

If \(\int \frac{d x}{\sqrt{a^2-x^2}}=a^n \sin ^{-1} \frac{x}{a}\) is dimensionally correct, then the value of \(n\) will be:
1. \(1\)
2. \(\text{zero}\)
3. \(\text-1\)
4. \(2\)