In SI units, the dimensions of \(\sqrt{\frac{\varepsilon_0}{\mu_0}} \) is:
1. \( A T^{-3} M L^{3 / 2} \)
2. \( A^{-1} T M L^3 \)
3. \( A T^2 M^{-1} L^{-1} \)
4. \( A^2 T^3 M^{-1} L^{-2}\)

In a simple pendulum experiment for determination of acceleration due to gravity (\(g\)), time taken for \(20\) oscillations is measured by using a watch of \(1\) second least count. The mean value of time taken comes out to be \(30~\text{s}\). The length of pendulum is measured by using a meter scale of least count \(1~\text{mm}\) and the value obtained is \(55.0~\text{cm}\). The percentage error in the determination of \(g\) is close to:
1. \(0.7\%\)
2. \(6.8\%\)
3. \(0.2\%\)
4. \(3.5\%\)

If surface tension \((S),\) Moment of Inertia \((I)\) and Plank's constant \((h),\) were to be taken as the fundamental units, the dimensional formula for linear momentum would be:
1. \( S^{3 / 2} I^{1 / 2} h^0 \)
2. \( S^{1 / 2} I^{1 / 2} h^{-1} \)
3. \( S^{1 / 2} I^{3 / 2} h^{-1} \)
4. \( S^{1 / 2} I^{1 / 2} h^0 \)

In the formula \(X=5YZ^2\), \(X\) and \(Z\) have dimensions of capacitance and magnetic field, respectively. What are the dimensions of \(Y\) in SI units?
1. \(\left[M^{-3} L^{-2} T^8 A^4\right]\)
2. \(\left[M^{-2} L^{-2} T^6 A^3\right]\)
3. \(\left[M^{-1} L^{-2} T^4 A^2\right]\)
4. \(\left[M^{-2} L^0 T^{-4} A^{-2}\right]\)

Which of the following combinations has the dimension of electrical resistance (\(\varepsilon_0\) is the permittivity of the vacuum and \(\mu_0\) is the permeability of vacuum)?
1. \(\sqrt{\frac{\varepsilon_0}{\mu_0}}\)
2. \(\frac{\varepsilon_0}{\mu_0}\)
3. \(\frac{\mu_0}{\varepsilon_0}\)
4. \(\sqrt{\frac{\mu_0}{\varepsilon_0}}\)

The area of a square is \(5.29~\text{cm}^2. \) The area of \(7\) such squares taking into account the significant figures is:
1. \(37.03~\text{cm}^2\)
2. \(37.0~\text{cm}^2\)
3. \(37~\text{cm}^2\)
4. \(37.030~\text{cm}^2\)

The least count of the main scale of a vernier callipers is \(1~\mathrm{mm}\). Its vernier scale is divided into \(10\) divisions and coincide with \(9\) divisions of the main scale. When jaws are touching each other, the \(7^{th}\) division of vernier scale coincides with a division of main scale and the zero of vernier scale is lying right side of the zero of main scale. When this vernier is used to measure length of a cylinder the zero of the vernier scale between \(3.1~\mathrm{cm}\) and \(3.2~\mathrm{cm}\) and \(4^{th}\) VSD coincides with a main scale division. The length of the cylinder is : (VSD is vernier scale division)
1. \(3.21~\text{cm}\)
2. \(2.99 ~\text{cm}\) 
3. \(3.2~\text{cm}\)
4. \(3.07~\text{cm}\)

If speed \(v\), area \(A\) and force \(F\) are chosen as fundamental units, then the dimension of Young's modulus will be:
1. \(\left[FA^{-1} v^0 \right]\)
2. \(\left[FA^2 v^{-1} \right]\)
3. \( \left[FA^2 {v}^{-3}\right]\)
4. \(\left[FA^2 v^{-2}\right] \)

If momentum \((P), \) area \((A) ,\) and time \((T)\) are taken as the fundamental physical quantities, then the dimensional formula of energy in terms of \(P,\) \(A,\) and \(T\) is:

Using a screw gauge with pitch \(0.1 ~\text{cm}\) and \(50\) divisions on its circular scale, the thickness of an object is measured. It should be accurately recorded as:
1. \(2.124~\text{cm}\)
2. \(2.121~\text{cm}\)
3. \(2.125~\text{cm}\)
4. \(2.123~\text{cm}\)