A physical quantity P is related to four observables a, b, c and d as $P=\sqrt{ab}.d^{\alpha }/\sqrt{c}$ ( $\alpha$ is constant) The percentage errors is a, b, c and d are 0.5% in each. If the percentage error in P is 2%, then $\alpha$ is

As, $P=\frac{a^{1/2}.b^{1/2}.d^{\alpha }}{c^{1/2}}$  
So % error in $P=\frac{\Delta P}{P}\times 100=\frac{1}{2}\left(\frac{\Delta a}{a}\right)\times 100+\frac{1}{2}(\frac{\Delta b}{b}\times 100)+\alpha (\frac{\Delta d}{d}\times 100)+\frac{1}{2}(\frac{\Delta c}{c}\times 100)$  
$\Rightarrow$ % error in $P=\frac{1}{2}\times \%$ error in a + $\frac{1}{2}\times \%$ error in b + $\frac{1}{2}\times \%$ error in c + $\alpha \times \%$ error in d
Now given, 
% Error in P = 2%
% Error in a = % error in b
= % Error in c = % error in d = 0.5 %
Substituting these values in eq. (i), we get 
$2=\frac{1}{2}\times 0.5+\frac{1}{2}\times 0.5+\frac{1}{2}\times 0.5+\alpha \times 0.5$ 
$\Rightarrow 2=\frac{1}{2}(\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\alpha )\Rightarrow 4=\frac{3}{2}+\alpha$ or $\alpha =4-\frac{3}{2}=\frac{5}{2}$  
$\therefore \alpha$ is 5/2.