Import Question JSON

Current Question (ID: 19967)

Question:

$\text{A wire of length } 2L \text{ is formed by joining two wires, } A \text{ and } B, \text{ each of the same length but with different radii, } r \text{ and } 2r, \text{ respectively, and made of the same material. The wire vibrates at a frequency such that the joint between the two wires forms a node. If the number of antinodes in wire } A \text{ is } p \text{ and in wire } B \text{ is } q, \text{ the ratio } p : q \text{ is:}$

Options:

1. $3 : 5$
2. $4 : 9$
3. $1 : 2$
4. $1 : 4$

Solution:

$\text{Hint: } f = \frac{N}{2L} \sqrt{\frac{T}{\rho \pi r^2}}$ $\text{Step: Find the ratio number of antinodes in wire } A \text{ and } B.$ $\text{The frequency of the vibration wave is given by:}$ $f = \frac{N}{2L} \sqrt{\frac{T}{\rho \pi r^2}}$ $\Rightarrow N \propto r$ $\text{The ratio number of antinodes in wire } A \text{ and } B \text{ is given by:}$ $\frac{p}{q} = \frac{r_A}{r_B} = \frac{1}{2}$ $\Rightarrow p : q = 1 : 2$ $\text{Hence, option (3) is the correct answer.}$

Import JSON File

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
}