A physical quantity \(z\) depends on four observables \(a,b,c\) and \(d\), as \(z=\frac{a^2 b^{\frac{2}{3}}}{\sqrt{c }d^3}\) The percentage of error in the measurement of  \(a,b,c\) and \(d\) is  \(2\%\), \(1.5\%\), \(4\%\) and \(2.5\%\) respectively. The percentage of error in \(z\) is:
1. \(12.25\%\)
2. \(14.5\%\)
3. \(16.5\%\)
4. \(13.5\%\)

The quantities \(x=\frac{1}{\sqrt{\mu_0 \varepsilon_0}}, y=\frac{E}{B} \) and \(z=\frac{L}{C R}\) are defined where \(C\)-capacitance, \(R\)-Resistance, \(L\)-length, \(E\)-Electric field, \(B\)-magnetic field and \(\varepsilon_0, \mu_0\) free space permittivity and permeability respectively. Then:

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. \(1.30\%\)