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Current Question (ID: 15987)

Question:

$\text{A string of length } l \text{ is fixed at one end and free at the other. If it resonates in different modes, then the ratio of frequencies is:}$ $1.\ 1 : 2 : 3 : \ldots$ $2.\ 1 : 3 : 5 : 7 : \ldots$ $3.\ 1 : 2 : 4 : 8 : \ldots$ $4.\ 1 : 3 : 9 : \ldots$

Options:

1. $1 : 2 : 3 : \ldots$
2. $1 : 3 : 5 : 7 : \ldots$
3. $1 : 2 : 4 : 8 : \ldots$
4. $1 : 3 : 9 : \ldots$

Solution:

$\text{Hint: The frequency of the wave in the string (one end is free) is}$ $f = \left(2n + 1\right) \frac{v}{4l}.$ $\text{Step 1: Find the frequency, putting } n = 0, 1, 2, 3, 4, \ldots$ $\text{For different values of } n \text{ it will give different frequencies.}$ $n=0,\ f = \frac{v}{4l}$ $n=1,\ f = \frac{3v}{4l}$ $n=2,\ f = \frac{5v}{4l}$ $n=3,\ f = \frac{7v}{4l}$ $n=4,\ f = \frac{9v}{4l}$ $\text{Step 2: Find the ratio of frequencies.}$ $\text{Thus, the ratio of frequencies is } 1:3:5:7: \ldots$ $\text{Hence, option (2) is the correct answer.}$

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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
}