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

1. \(7.62\)
2. \(7.63\)
3. \(7.64\)
4. \(7.65\)

Two resistors \(R_1 = (3.0\pm0.3)~\Omega\) and \(R_2 = (5.0 \pm0.1)~\Omega\) are connected in parallel. The equivalent resistance, \(R_{eq}\), will be:
Hint: \({1 \over R_{eq}} = {1 \over R_{1}} + {1 \over R_{2}} \)

The pitch of a screw gauge is \(1~\)mm 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

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:

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:

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: