The density of a material in the shape of a cube is determined by measuring three sides of the cube and its mass. If the relative errors in measuring the mass and length are respectively \(1.5\%\) and \(1\%\), the maximum error in determining the density is:
1. \(2.5\%\)
2. \(3.5\%\)
3. \(4.5\%\)
4. \(6\%\)

The work done by a gas molecule in an isolated system is given by, \(\mathrm{W}=\alpha \beta^2 \mathrm{e}^{-\frac{\mathrm{x}^2}{\alpha k \mathrm{~T}}}\), where \(x\) is the displacement, \(k\) is the Boltzmann constant and \(T\) is the temperature, \(\alpha\) and \(\beta\) are constants. Then the dimensions of \(\beta\) will be:

In a typical combustion engine the work done by a gas molecule is given \(W=\alpha^2 \beta e^{\frac{-\beta x^2}{k T}}\). where \(x\) is the displacement, \(k\) is the Boltzmann constant and \(T\) is the temperature. If \(\alpha\) and \(\beta\) are constants, dimensions of \(\alpha\) will be:
1. \( {\left[\mathrm{MLT}^{-2}\right]} \)
2. \( {\left[\mathrm{M}^0 \mathrm{LT}^0\right]} \)
3. \( {\left[\mathrm{M}^2 \mathrm{LT}^{-2}\right]} \)
4. \( {\left[\mathrm{MLT}^{-1}\right]}\)

A student measuring the diameter of a pencil of circular cross- section with the help of a vernier scale records the following four readings  \(5.50 ~\text{mm}\), \(5.55~\text{mm}\), \(5.45~\text{mm}\), \(5.65~\text{mm}\). The average of these four readings is \(5.5375~\text{mm}\) and the standard deviation of the data is \(0.07395~\text{mm}\). The average diameter of the pencil should therefore be recorded as: 

The period of oscillation of a simple pendulum \(T=2\pi\sqrt{\dfrac{L}{g}} \). Measured value of '\(L\)' is \(1.0\text{ m}\) from meter scale having a minimum division of \(1\text{ mm}\) and time of one complete oscillation is \(1.95\text{ s}\) measured from stopwatch of \(0.01\text{ s}\) resolution. The percentage error in the determination of '\(g\)' will be :
1. \(1.13\%\)
2. \(1.03\%\)
3. \(1.33\%\)
4. \(1.30\%\)

The pitch of the screw gauge is \(1\) mm and there are \(100\) divisions on the circular scale. When nothing is put in between the jaws, the zero of the circular scale lies \(8\) divisions below the reference line. When a wire is placed between the jaws, the first linear scale division is clearly visible while the \(72\)^nd division on a circular scale coincides with the reference line. The radius of the wire is:
1. \(1.64\) mm
2. \(0.82\) mm
3. \(1.80\) mm
4. \(0.90\) mm

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, \(0.1~\mu\text{m}\)

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 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. \( 92 \pm 3~\text{s} \)