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Current Question (ID: 20006)

Question:

$\text{A transverse wave is represented by } y = 2 \sin(\omega t - kx) \text{ cm. The value of wavelength (in cm) for which the wave velocity becomes equal to the maximum particle velocity, will be:}$ $1. \ 4\pi$ $2. \ 2\pi$ $3. \ \pi$ $4. \ 2$

Options:

1. $4\pi$
2. $2\pi$
3. $\pi$
4. $2$

Solution:

$\text{Hint: The maximum particle velocity, } (v_p)_{\text{max}} = A\omega$ $\text{Step 1: Find the maximum particle velocity and wave velocity.}$ $\Rightarrow y = 2 \sin(\omega t - kx)$ $\text{Compare, the above equation with } y = A \sin(\omega t - kx) \text{ we get;}$ $\Rightarrow A = 2 \text{ cm}$ $\text{According to the given conditions-}$ $\Rightarrow A\omega = \frac{\omega}{k}$ $\Rightarrow k = A$ $\text{Step 2: Find the value of wavelength (in cm).}$ $\text{The wavelength of the wave is given by;}$ $\Rightarrow k = \frac{\lambda}{2\pi}$ $\Rightarrow \lambda = 4\pi \text{ cm}$ $\text{Hence, option (1) is the correct answer.}$

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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
}