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Current Question (ID: 19636)

Question:

$\text{A unit scale is to be prepared whose length does not change with temperature and remains } 20 \text{ cm, using a bimetallic strip made of brass and iron, each of different lengths. The length of both components would change in such a way that the difference between their lengths remains constant. If the length of the brass is } 40 \text{ cm, then the length of the iron will be:}$ $\left( \alpha_{\text{iron}} = 1.2 \times 10^{-5} \text{ K}^{-1} \text{ and } \alpha_{\text{brass}} = 1.8 \times 10^{-5} \text{ K}^{-1} \right).$

Options:

1. $20 \text{ cm}$
2. $40 \text{ cm}$
3. $60 \text{ cm}$
4. $80 \text{ cm}$

Solution:

$\text{Hint: } \Delta L = L_0(1 + \alpha \Delta T)$ $\text{Step: Find the length of the iron strip.}$ $l_B - l_I = l'_B - l'_I$ $\Rightarrow l_B - l_I = l_B(1 + \alpha_B \Delta T) - l_I(1 + \alpha_I \Delta T)$ $\Rightarrow l_B - l_I = l_B + l_B \alpha_B \Delta T - l_I - l_I \alpha_I \Delta T$ $\Rightarrow 0 = l_B \alpha_B \Delta T - l_I \alpha_I \Delta T$ $\Rightarrow l_B \alpha_B = l_I \alpha_I$ $\Rightarrow 40 \times 1.8 \times 10^{-5} = l_I \times 1.2 \times 10^{-5}$ $\Rightarrow l_I = 60 \text{ cm}$ $\text{Hence, option (3) is the correct answer.}$

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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
}