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Current Question (ID: 12057)

Question:

A physical parameter a can be determined by measuring the parameters b, c, d and e using the relation $a=b^\alpha c^\beta /d^\gamma e^\delta$. If the maximum errors in the measurement of b, c, d and e are $b_1\%$, $c_1\%$, $d_1\%$, and $e_1\%$ respectively, then the maximum error in the value of a determined by the experiment is

Options:

1. $(b_1+c_1+d_1+e_1)\%$
2. $(\alpha b_1+\beta c_1+\gamma d_1+\delta e_1)\%$
3. $(\alpha b_1+\beta c_1-\gamma d_1-\delta e_1)\%$
4. $(\alpha b_1+\beta c_1+\gamma d_1+\delta e_1)\%$

Solution:

For a quantity $a=b^\alpha c^\beta d^\gamma e^\delta$, the maximum percentage error is calculated by summing the products of the powers and the individual percentage errors, regardless of the sign of the power. The formula is given by: $(\frac{\Delta a}{a}\times100)_{max} = |\alpha|(\frac{\Delta b}{b}\times100) + |\beta|(\frac{\Delta c}{c}\times100) + |\gamma|(\frac{\Delta d}{d}\times100) + |\delta|(\frac{\Delta e}{e}\times100)$. In this specific case, all exponents are positive in the error formula, so the correct expression is $(\alpha b_1+\beta c_1+\gamma d_1+\delta e_1)\%$.

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Expected JSON Format:

{
  "question": "The mass of carbon present in 0.5 mole of $\\mathrm{K}_4[\\mathrm{Fe(CN)}_6]$ is:",
  "options": [
    {
      "id": 1,
      "text": "1.8 g"
    },
    {
      "id": 2,
      "text": "18 g"
    },
    {
      "id": 3,
      "text": "3.6 g"
    },
    {
      "id": 4,
      "text": "36 g"
    }
  ],
  "solution": "\\begin{align}\n&\\text{Hint: Mole concept}\\\\\n&1 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\text{ moles of carbon atom}\\\\\n&0.5 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\times 0.5 \\text{ mol} = 3 \\text{ mol}\\\\\n&1 \\text{ mol of carbon} = 12 \\text{ g}\\\\\n&3 \\text{ mol carbon} = 12 \\times 3 = 36 \\text{ g}\\\\\n&\\text{Hence, 36 g mass of carbon present in 0.5 mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6].\n\\end{align}",
  "correct_answer": 4
}