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JEE Main 2022 · Physics · Units and Measurements · easy

A physical quantity is given by X=(a^3√(b))/(c^2d). The percentage errors in measuring a,b,c,d are 1%, 4%, 2%, 3% respectively. The maximum…

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A physical quantity is given by X=a3bc2dX=\dfrac{a^{3}\sqrt{b}}{c^{2}d}. The percentage errors in measuring a,b,c,da,b,c,d are 1%, 4%, 2%, 3%1\%,\,4\%,\,2\%,\,3\% respectively. The maximum percentage error in XX is
  1. A. 9%9\%
  2. B. 12%12\%
  3. C. 14%14\%
  4. D. 15%15\%
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Answer: (B) 12%12\%

For a product or quotient of powers, the fractional errors add after multiplying each by its power: ΔXX=3Δaa+12Δbb+2Δcc+Δdd\frac{\Delta X}{X}=3\frac{\Delta a}{a}+\frac12\frac{\Delta b}{b}+2\frac{\Delta c}{c}+\frac{\Delta d}{d}. Substituting, 3(1)+12(4)+2(2)+3=3+2+4+3=12%3(1)+\frac12(4)+2(2)+3=3+2+4+3=12\%. Errors always add in the maximum-error estimate, even for quantities in the denominator, so 9%9\% (subtracting the denominator terms) is wrong.
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