The dimensional formula for thermal conductivity is: (here \(K\) denotes the temperature)
1. \( \left[M L T^{-2} K\right] \)
2. \( \left[M L T^{-3} K\right] \)
3. \( \left[M L T^{-3} K^{-1}\right] \)
4. \( \left[M L T^{-2} K^{-2}\right]\)

A quantity \(x\) is given by \(\left(\frac{IFv^2}{WL^4}\right) \) in terms of moment of inertia \(I\), force \(F\), velocity \(v\), work \(W\), and Length \(L\). The dimensional formula for \(x\) is the same as that of:

If \(e\) is the electronic charge, \(c\) is the speed of light in free space and \(h\) is Planck's constant, the quantity \(\frac{1}{4\pi \varepsilon_0} \frac{|e^2|}{hc}\) has dimensions of:
1. \(\left[M^0L^0T^0\right]\)
2. \(\left[LT^{-1}\right]\)
3. \(\left[MLT^{-1}\right]\)
4. \(\left[MLT^{0}\right]\)

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