A screw gauge with a pitch of \(0.5~\text{mm}\) and a circular scale with \(50\) divisions is used to measure the thickness of a thin sheet of Aluminium. Before starting the measurement, it is found that when the two jaws of the screw gauge are brought in contact, the \(45^{\text{th}}\) division coincides with the main scale line and that the zero of the main scale is barely visible. What is the thickness of the sheet if the main scale reading is \(0.5~\text{mm}\) and the \(25^{\text{th}}\) division coincides with the main scale line?
1. \(0.75~\text{mm}\)
2. \(0.80~\text{mm}\)
3. \(0.70~\text{mm}\)
4. \(0.50~\text{mm}\)

A student measured the length of a rod and wrote its as \(3.50~\text{cm}\). Which instrument did he use to measure it?

The current-voltage relation of the diode is given by \(I=(e^{1000V/T}-1)~\text{mA}\), where the applied voltage \(V\) is in volts and the temperature \(T\) is in degree Kelvin. If a student makes an error measuring \(\pm ~0.01~\text{V}\) while measuring the current of \(5~\text{mA}\) at \(300~\text{K}\), what will be the error in the value of current in \(\text{mA}\)?
1. \(0.02~\text{mA}\)
2. \(0.5~\text{mA}\)
3. \(0.05~\text{mA}\)
4. \(0.2~\text{mA}\)

The period of oscillation on a simple pendulum is \(T=2\pi\sqrt{\frac{L}{g}}\). The measured value of \(L\) is \(20.0~\text{cm}\) known to have \(1~\text{mm}\) accuracy and the time for \(100\) oscillations of the pendulum is found to be \(90~\text{s}\) using a wristwatch of \(1~\text{s}\) resolution. The accuracy in the determination of \(g\) is:
1. \(2\%\)
2. \(3\%\)
3. \(1\%\)
4. \(5\%\)

If '\(C\)' and '\(V\)' represent capacity and voltage respectively then what are the dimensions of \(\lambda\), where \(\lambda =\frac{C}{V}\)?
1. \( {\left[\mathrm{M}^{-2} \mathrm{~L}^{-3} \mathrm{I}^2 \mathrm{~T}^6\right]}\)
2. \( {\left[\mathrm{M}^{-3} \mathrm{~L}^{-4} \mathrm{I}^3 \mathrm{~T}^7\right]} \)
3. \( {\left[\mathrm{M}^{-1} \mathrm{~L}^{-3} \mathrm{I}^{-2} \mathrm{~T}^{-7}\right]}\)
4. \( {\left[\mathrm{M}^{-2} \mathrm{~L}^{-4} \mathrm{I}^3 \mathrm{~T}^7\right]} \)

The following observations were taken to determine the surface tension \(T\) of water by the capillary method:
diameter of the capillary, \(D=1.25 \times 10^{-2} ~\text{m}\)
rise of water, \(h=1.45\times 10^{-2}~\text{m}\)
Using \(g= 9.80~\text{m/s}^2\) and the simplified relation, the possible error in surface tension is closest to: 
1. \(0.15\%\)
2. \(1.5\%\)
3. \(2.4\%\)
4. \(10\%\)

A student measures the time period of \(100\) oscillations of a simple pendulum four times. The data set is \(90~\text{s}, ~91~\text{s},~95~\text{s}~\text{and}~92~\text{s}.\) If the minimum division in the measuring clock is \(1~\text{s}\), then the reported mean time should be:
1. \( 92 \pm 2 ~\text{s} \)
2. \( 92 \pm 5.0 ~\text{s} \)
3. \( 92 \pm 1.8 ~\text{s} \)
4. \( 92 \pm 3~\text{s} \)

The density of a solid metal sphere is determined by measuring its mass and its diameter. The maximum error in the density of the sphere is \((\frac{x}{100})\%\). If the relative errors in measuring the mass and the diameter are \(6.0\%\) and \(1.5\%\) respectively, the value of \(x\) is:
1. \(503\)
2. \(1050\)
3. \(532\)
4. \(120\)

A screw gauge has \(50\) divisions on its circular scale. The circular scale is \(4\) units ahead of the pitch scale marking, prior to use. Upon one complete rotation of the circular scale, a displacement of \(0.5~\text{mm}\) is noticed on the pitch scale. The nature of the zero error involved and the last count of the screw gauge are respectively:
1. Positive, \(10~\mu\text{m}\)
2. Negative, \(2~\mu\text{m}\)
3. Positive, \(0.1~\mu\text{m}\)
4. Positive, \(0.1~\mu\text{m}\)