For the expression, \(10^{(at+3)}\), 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:

The pitch of a screw gauge is \(1~\)mm 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?

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

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. \(1,1,1\)