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Current Question (ID: 17297)

Question:

$\text{A transistor-oscillator using a resonant circuit with an inductance } \mathcal{L} \text{ (of negligible resistance) and a capacitance } \mathcal{C} \text{ has a frequency } f. \text{ If } \mathcal{L} \text{ is doubled and } \mathcal{C} \text{ is changed to } 4\mathcal{C}, \text{ the frequency will be:}$

Options:

1. $\frac{f}{4}$
2. $8f$
3. $\frac{f}{2\sqrt{2}}$
4. $\frac{f}{2}$

Solution:

$\text{Given,}$ $\mathcal{L}_2 = 2\mathcal{L}$ $\mathcal{C}_2 = 4\mathcal{C}$ $\text{In a series } LC \text{ circuit, frequency of } LC \text{ oscillations is given by:}$ $f = \frac{1}{2\pi\sqrt{LC}} \propto \frac{1}{\sqrt{LC}}$ $\text{or}$ $\frac{f_1}{f_2} = \sqrt{\frac{\mathcal{L}_2\mathcal{C}_2}{\mathcal{L}_1\mathcal{C}_1}}$ $\text{Given, } \mathcal{L}_1 = \mathcal{L}$ $f_1 = f$ $\therefore \frac{f_1}{f_2} = \sqrt{\frac{2\mathcal{L} \times 4\mathcal{C}}{\mathcal{L}\mathcal{C}}} = \sqrt{8}$ $\Rightarrow f_2 = \frac{f}{2\sqrt{2}}$

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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
}