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.
$\log x=m\log a+n\log b-p\log c-q\log d$
$\frac{dx}{x}=m\left(\frac{da}{a}\right)+n\left(\frac{db}{b}\right)-p\left(\frac{dc}{c}\right)-q\left(\frac{dd}{d}\right)$
We are doing a differential approximation here, by putting $dx=\Delta x,da=\Delta a,db=\Delta b,dc=\Delta c$ and $dd=\Delta d$. 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,
$\frac{\Delta x}{x}=m\left(\frac{\Delta a}{a}\right)+n\left(\frac{\Delta b}{b}\right)+p\left(\frac{\Delta c}{c}\right)+q\left(\frac{\Delta d}{d}\right)$