For \(10^{(at+3)}\), the dimensions of \(a\) will be:
1. \(\left[M^0L^0T^{0}\right]\)
2. \(\left[M^0L^0T^{1}\right]\)
3. \(\left[M^0L^0T^{-1}\right]\)
4. None of these

If the dimensions of a physical quantity are given by ${\mathrm{}}^{}$\(M^aL^bT^c\), then the physical quantity will be:

If the error in the measurement of the radius of a sphere is \(2\%\), then the error in the determination of the volume of the sphere will be:
1. \(4\%\)
2. \(6\%\)
3. \(8\%\)                                                  
4. \(2\%\)

The pitch of a screw gauge is 1mm and there are 100 divisions on the circular scale. While measuring the diameter of a wire, the linear scale reads 1 mm and 47^th division on the circular scale coincides with the reference line. The length of the wire is 5.6 cm. Curved surface area (in cm²) of the wire in appropriate number of significant figures will be

1.  2.4 cm²

2.  2.56 cm²

3.  2.6 cm²

4.  2.8 cm²

The position of a particle at time \(t\) is given by the relation \({x}({t})=\left(\frac{{v}_0}{\alpha}\right)\left(1-{e}^{-\alpha {t}}\right)\), where \(v_0\)${\mathrm{}}_{}$ is a constant and \(\alpha >0\). The dimensions of \(v_0\) and \(\alpha\) are respectively:
1. \(\left[M^0L^{1}T^{-1}\right]~\text{and}~\left[T^{-1}\right]\)
2. \(\left[M^0L^{1}T^{0}\right]~\text{and}~\left[T^{-1}\right]\)
3. \(\left[M^0L^{1}T^{-1}\right]~\text{and}~\left[LT^{-1}\right]\)
4. \(\left[M^0L^{1}T^{-1}\right]~\text{and}~\left[T\right]\)

When units of mass, length, and time are taken as \(10~\text{kg}, 60~\text{m}~\text{and}~60~\text{s}\) respectively, the new unit of energy becomes \(x\) times the initial SI unit of energy. The value of \(x\) will be:
1. \(10\)
2. \(20\)
3. \(60\)
4. \(120\)

Which of the following relations is dimensionally wrong? [The symbols have their usual meanings]
1. \(s= ut+\frac{1}{6}at^2\)
2. \(v^2= u^2+\frac{2as^2}{\pi}\)
3. \(v= u-2at\)
4. All of these

In which of the following, the number of significant figures is different from that in the others?
1. \(2.303~\mathrm{kg}\)
2. \(12.23~\mathrm{m}\)
3. \(0.002\times10^{5}~\mathrm{m}\)
4. \(2.001\times10^{-3}~\mathrm{kg}\) 

If force (\(F\)), velocity (\(\mathrm{v}\)), and time (\(T\)) are taken as fundamental units, the dimensions of mass will be:
1. \( {\left[\mathrm{FvT}^{-1}\right]} \)
2. \({\left[\mathrm{FvT}^{-2}\right]} \)
3. \( {\left[\mathrm{Fv}^{-1} \mathrm{~T}^{-1}\right]} \)
4. \( {\left[\mathrm{Fv}^{-1} \mathrm{~T}\right]}\)

If dimensions of critical velocity \({v_c}\) of a liquid flowing through a tube are expressed as \(\eta^{x}\rho^yr^{z}\), where \(\eta, \rho~\text{and}~r\) are the coefficient of viscosity of the liquid, the density of the liquid, and the radius of the tube respectively, then the values of \({x},\) \({y},\) and \({z},\) respectively, will be:
1. \(1,-1,-1\)
2. \(-1,-1,1\)
3. \(-1,-1,-1\)
4. \(1,1,1\)