Import Question JSON

Current Question (ID: 17526)

Question:

$\text{A plane-convex lens fits exactly into a plano-concave lens. Their plane surfaces are parallel to each other. If lenses are made of different materials of refractive indices } \mu_1 \text{ and } \mu_2 \text{ and } R \text{ is the radius of curvature of the curved surface of the lenses, then the focal length of the combination is:}$

Options:

1. $\frac{R}{2(\mu_1 + \mu_2)}$
2. $\frac{R}{2(\mu_1 - \mu_2)}$
3. $\frac{R}{(\mu_1 - \mu_2)}$
4. $\frac{2R}{(\mu_2 - \mu_1)}$

Solution:

$\text{The focal length of the combination:}$ $\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2}$ $\text{We have, } f_1 = \frac{R}{(\mu_1 - 1)} \text{ and } f_2 = \frac{-R}{(\mu_2 - 1)}$ $\text{Putting these values in the equation, we get:}$ $\frac{1}{f} = (\mu_1 - 1)/R - (\mu_2 - 1)/R$ $\frac{1}{f} = (\mu_1 - 1 - \mu_2 + 1)/R = (\mu_1 - \mu_2)/R$

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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
}