Import Question JSON

Current Question (ID: 15990)

Question:

$\text{An air column, closed at one end and open at the other, resonates with a tuning fork when the smallest length of the column is } 50 \text{ cm.}$ $\text{The next larger length of the column resonating with the same tuning fork will be:}$

Options:

1. $100 \text{ cm}$
2. $150 \text{ cm}$
3. $200 \text{ cm}$
4. $66.7 \text{ cm}$

Solution:

$\text{Hint: } \nu = \frac{v}{4L}$ $\text{Step: Find the next higher length of the air column.}$ $\text{Since the smallest length of the air column is associated with the fundamental mode of vibration as shown in the figure:}$ $L_{\text{min}} = \frac{\lambda}{4}$ $\Rightarrow 50 = \frac{\lambda}{4}$ $\Rightarrow \lambda = 200 \text{ cm}$ $\text{The next higher length of the air column is;}$ $L = \frac{\lambda}{4} + \frac{\lambda}{2} = \frac{\lambda + 2\lambda}{4}$ $L = \frac{3\lambda}{4} = \frac{3}{4} \times 200$ $\Rightarrow L = 150 \text{ cm}$ $\text{Therefore, the next higher length of the air column is } 150 \text{ cm.}$ $\text{Hence, option (2) is the correct answer.}$

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
}