Import Question JSON

Current Question (ID: 15991)

Question:

$\text{The equation of a stationary wave is } y = 0.8 \cos\left(\frac{\pi x}{20}\right) \sin 200(\pi t), \text{ where } x \text{ is in cm and } t \text{ is in sec. The separation between consecutive nodes will be:}$

Options:

1. $20 \text{ cm}$
2. $10 \text{ cm}$
3. $40 \text{ cm}$
4. $30 \text{ cm}$

Solution:

$\text{Hint: The separation between two consecutive notes } = \frac{\lambda}{2}$ $\text{Step: Find the separation between two consecutive nodes.}$ $\text{We know that the equation of stationary wave is given by the relation:}$ $y = A \cos\left(\frac{2\pi x}{\lambda}\right) \sin\left(\frac{2\pi vt}{\lambda}\right)$ $\text{By comparing the given equation and the general equation we get,}$ $y = 0.8 \cos\left(\frac{\pi x}{20}\right) \sin 200(\pi t)$ $\Rightarrow \frac{2\pi}{\lambda} = \frac{\pi}{20} \Rightarrow \lambda = 40 \text{ cm}$ $\text{The separation between two consecutive nodes is given by;}$ $\text{The separation between two consecutive nodes } = \frac{\lambda}{2} = \frac{40}{2} = 20 \text{ cm.}$ $\text{Hence, option (1) is the correct answer.}$

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
}