$$\eqalign{ & I = {I_0}{\sin ^2}\theta \Rightarrow \theta = {\sin ^{ - 1}}\sqrt {\frac{I}{{{I_0}}}} \cr & d\theta = \frac{1}{2}\frac{{\sqrt {{I_0}} }}{{\sqrt {{I_0} - I} }}\frac{{dI}}{{\sqrt {{I_0}} \sqrt I }} \cr & \frac{{d\theta }}{\theta } = \frac{1}{2}\frac{{dI}}{{\sqrt {I\left( {{I_0} - I} \right)} {{\sin }^{ - 1}}\sqrt {\frac{I}{{{I_0}}}} }} \cr & \therefore \% \,{\text{error}} = \frac{4}{\pi }\sqrt 3 \times {10^{ - 2}}\% \cr} $$

$$E = \sigma A{T^4}$$
$$E$$ is energy dissipated per second.
$$\sigma = \frac{E}{{A{T^4}}} = \frac{{{\text{Watt}}}}{{{m^2}{K^4}}}$$

From $$K = \frac{{Bx}}{{{x^2} + {A^2}}} = \frac{{Bx}}{{{x^2}}} = \frac{B}{x}$$
$$\eqalign{ & \therefore B = K \times x = K.E. \times {\text{distance}} \cr & = {\text{work}} \times {\text{distance}} \cr} $$

Relative density $$ = \frac{{{\text{Density of Substance}}}}{{{\text{Density of water}}}}$$
Hence no dimension.

A quantity which has dimensions and also has a constant value is called dimensional constant. Here, gravitational constant $$\left( G \right)$$  is a dimensional constant.

$$\eqalign{ & {\text{As}}\,{\text{density}}\,\rho = \frac{m}{V} = \frac{m}{{{l^3}}} \cr & \therefore \frac{{\Delta \rho }}{\rho } \times 100 = \pm \left( {\frac{{\Delta m}}{m} + 3\frac{{\Delta l}}{l}} \right) \times 100\% \cr & = \pm \left( {4 + 3 \times 3} \right) \cr & = \pm 13\% \cr} $$

$$\eqalign{ & {\text{According to Coulomb's law, the electrostatic force}} \cr & F = \frac{1}{{4\pi {\varepsilon _0}}} \times \frac{{{q_1}{q_2}}}{{{r^2}}} \cr & {q_1}\,{\text{and }}{q_2}\, = {\text{charges, }}r = {\text{distance between charges}} \cr & {\text{and}}\,{\varepsilon _0} = \,{\text{permittivity of free space}} \cr & \Rightarrow {\varepsilon _0} = \frac{1}{{4\pi }} \times \frac{{{q_1}{q_2}}}{{{r^2}F}} \cr & {\text{Substituting the units for }}q,r{\text{ and }}F,\,{\text{we obtain unit}}\,{\text{of}}\,{\varepsilon _0} \cr & = \frac{{{\text{coulomb}} \times {\text{coulomb}}}}{{{\text{newton}} - {{\left( {{\text{metre}}} \right)}^2}}} \cr & = \frac{{{{\left( {{\text{coulomb}}} \right)}^2}}}{{{\text{newton}} - {{\left( {{\text{metre}}} \right)}^2}}} \cr} $$

Surface tension, $$T = \frac{{rhg}}{2} \times {10^3}$$
Relative error in surface tension, $$\frac{{\Delta T}}{T} = \frac{{\Delta r}}{r} + \frac{{\Delta h}}{h} + 0$$     ($$\because $$ g, 2 and $${10^3}$$ are constant)
Percentage error
$$\eqalign{ & 100 \times \frac{{\Delta T}}{T} = \left( {\frac{{{{10}^{ - 2}} \times 0.01}}{{1.25 \times {{10}^{ - 2}}}} + \frac{{{{10}^{ - 2}} \times 0.01}}{{1.45 \times {{10}^{ - 2}}}}} \right)100 \cr & = \left( {0.8 + 0.689} \right) \cr & = \left( {1.489} \right) \cr & = 1.489\% \cong 1.5\% \cr} $$

Percentage error in $$A = \left( {2 \times 1 + 3 \times 3 + 1 \times 2 + \frac{1}{2} \times 2} \right)\% $$
$$ = 14\% $$

Here, $$b$$ and $${x^2} = {L^2}$$  have same dimensions
Also, $$a = \frac{{{x^2}}}{{E \times t}} = \frac{{{L^2}}}{{\left( {M{L^2}{T^{ - 2}}} \right)T}} = {M^{ - 1}}{T^1}$$
$$a \times b = \left[ {{M^{ - 1}}{L^2}{T^1}} \right]$$