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Current Question (ID: 17529)

Question:

$\text{Two convex lenses of focal lengths 10 cm and 30 cm are kept at a separation of 20 cm. Then the correct statement is:}$

Options:

1. $\text{the effective focal length is 15 cm.}$
2. $\text{the chromatic aberration is minimized.}$
3. $\text{the combination behaves like a convergent lens.}$
4. $\text{all of these.}$

Solution:

$\text{Hint: The effective focal length } f_{\text{eff}} = \frac{f_1 f_2}{f_1 + f_2 - d}$ $\text{Step: Identify the correct statements.}$ $\text{The effective focal length is given by:}$ $f_{\text{eff}} = \frac{f_1 f_2}{f_1 + f_2 - d}$ $f = \frac{10 \times 30}{10 + 30 - 20}$ $f = \frac{300}{20} \Rightarrow 15 \text{ cm}$ $\text{For minimum chromatic aberration:}$ $d = \frac{f_1 + f_2}{2}$ $d = \frac{10 + 30}{2} \Rightarrow 20 \text{ cm}$ $\text{As } f_{\text{eff}} \text{ is positive, the combination is like a convergent lens.}$ $\text{Hence, option (4) is the correct answer.}$

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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
}