A physical quantity X is given by X = (a²b³)/c√d .If the percentage errors of measurement in a,b,c and d are 4%, 2%, 3% and 1% respectively, then calculate the percentage error in X.

The correct option is 17.5%.

To find the percentage error in the physical quantity $X$, we use the principle of error propagation for combined mathematical operations.

The relation for $X$ is given by:
$X=\frac{a^2{b}^3}{c\sqrt{d}}$
This can also be written in exponential form as:
$X=a^2{b}^3{c}^{-1}{d}^{-1/2}$

When propagating errors, the relative error of a quantity raised to a power is the relative error of that quantity multiplied by the absolute value of the power. Furthermore, errors always add up to find the maximum possible error. Therefore, the maximum fractional error in $X$ is:
$\frac{\Delta X}{X}=2\left(\frac{\Delta a}{a}\right)+3\left(\frac{\Delta b}{b}\right)+1\left(\frac{\Delta c}{c}\right)+\frac{1}{2}\left(\frac{\Delta d}{d}\right)$

Expressing this relation in terms of percentage error by multiplying the entire equation by 100:
$\left(\frac{\Delta X}{X}\times 100\right)=2\left(\frac{\Delta a}{a}\times 100\right)+3\left(\frac{\Delta b}{b}\times 100\right)+1\left(\frac{\Delta c}{c}\times 100\right)+\frac{1}{2}\left(\frac{\Delta d}{d}\times 100\right)$

We are given the individual percentage errors:
• Percentage error in $a$ = 4%
• Percentage error in $b$ = 2%
• Percentage error in $c$ = 3%
• Percentage error in $d$ = 1%

Substitute these values into the equation:
$\text{Percentage error in }X=2(4\%)+3(2\%)+1(3\%)+\frac{1}{2}(1\%)$
$\text{Percentage error in }X=8\%+6\%+3\%+0.5\%$
$\text{Percentage error in }X=17.5\%$

Thus, the total percentage error in $X$ is 17.5%.