Question

Statement-1: If $x=\frac{a^{m} \cdot b^{n}}{c^{p} \cdot d^{q}},$ then the percentage
error in the measurement of $x$
$\frac{\Delta x}{x}=m\left(\frac{\Delta a}{a}\right)+n\left(\frac{\Delta b}{b}\right)+p\left(\frac{\Delta c}{c}\right)+q\left(\frac{\Delta d}{d}\right)$
Statement-2: The above is true for all values of $\Delta a, \Delta b$, $\Delta c$ and $\Delta d$

• A. Statement $-1$ is true, Statement $-2 $ is false
• B. Statement $-1 $ is false, Statement $-2$ is true
• C. Statement $-1 $ is true, Statement $-2$ is true and Statement $-2$ is a correct explanation for Statement $-1.$
• D. Statement $-1$ is true, Statement $-2$ is true but Statement $-2$ is not a correct explanation for Statement $-1.$

Answer 1

Answer: (A)

Statement $-1 $ is true, Statement $-2$ is false.
$\log x=m \log a+n \log b-p \log c-q \log d$
$\frac{d x}{x}=m\left(\frac{d a}{a}\right)+n\left(\frac{d b}{b}\right)-p\left(\frac{d c}{c}\right)-q\left(\frac{d d}{d}\right)$
We are doing a differential approximation here, by putting $d x=\Delta x, d a=\Delta a, d b=\Delta b, d c=\Delta c$ and $d d=\Delta d$. So $\Delta a$ $\Delta b, \Delta c, \Delta d$ must be small. Also the errors could be $+$ ve or $-ve, $ so for maximum possible error we put everyone as $ +ve.$ So,
$\frac{\Delta x}{x}=m\left(\frac{\Delta a}{a}\right)+n\left(\frac{\Delta b}{b}\right)+p\left(\frac{\Delta c}{c}\right)+q\left(\frac{\Delta d}{d}\right) $