Import Question JSON

Current Question (ID: 17523)

Question:

$\text{A concave mirror of the focal length } f_1 \text{ is placed at a distance of } d \text{ from a convex lens of focal length } f_2. \text{ A beam of light coming from infinity and falling on this convex lens-concave mirror combination returns to infinity. The distance } d \text{ must be equal to:}$

Options:

1. $f_1 + f_2$
2. $-f_1 + f_2$
3. $2f_1 + f_2$
4. $-2f_1 + f_2$

Solution:

$\text{Given,}$ $\text{The focal length of the concave mirror } = f_1$ $\text{The focal length of the convex lens } = f_2$ $\text{A beam of light coming from infinity and falling on this convex lens-concave mirror combination returns to infinity.}$ $\text{Here, firstly, rays after passing through the convex lens intersect at } f_2 \text{ distance from the lens. Now, these rays will act as the object for the mirror, if the final rays coming out of lens is again parallel, then image formed from the mirror must be at the same point (i.e., } f_2 \text{ distance from the lens) here the condition is image and object are at one point for the mirror so it must be center of curvature (i.e., } 2f_1 \text{ from mirror). Hence, } d = -2f_1 + f_2.$

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Upload a JSON file containing LaTeX/MathJax formatted question, options, and solution.

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
}