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Current Question (ID: 17553)

Question:

$\text{Two point light sources are 24 cm apart. Where should a convex lens of focal length 9 cm be put in between them from one source so that the images of both the sources are formed at the same place?}$

Options:

1. $6 \text{ cm}$
2. $9 \text{ cm}$
3. $12 \text{ cm}$
4. $15 \text{ cm}$

Solution:

$\text{The given condition will be satisfied only if one source } (S_1) \text{ placed on one side such that } u < f \text{ (i.e., it lies under the focus). The other source } (S_2) \text{ is placed on the other side of the lens such that } u > f \text{ (i.e. it lies beyond the focus).}$ $\text{If } S_1 \text{ is the object for lens then}$ $\frac{1}{f} = \frac{1}{-y} - \frac{1}{-x}$ $\Rightarrow \frac{1}{y} = \frac{1}{x} - \frac{1}{f} \quad \text{....(i)}$ $\text{If } S_2 \text{ is the object for lens then}$ $\frac{1}{f} = \frac{1}{+y} - \frac{1}{(24-x)} \Rightarrow \frac{1}{y} = \frac{1}{f} - \frac{1}{(24-x)} \quad \text{....(ii)}$ $\text{From equation (i) and (ii)}$ $\frac{1}{x} - \frac{1}{f} = \frac{1}{f} - \frac{1}{(24-x)} \Rightarrow \frac{1}{x} + \frac{1}{(24-x)} = \frac{2}{f} = \frac{2}{9}$ $\Rightarrow x^2 - 24x + 108 = 0. \text{ After solving the equation } x = 18\text{cm, } 6\text{cm}$

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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
}