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Current Question (ID: 15936)

Question:

$\text{The equation of a progressive wave is given by } y = 4 \sin \left\{ \pi \left( \frac{t}{5} - \frac{x}{9} \right) + \frac{\pi}{6} \right\}$ $\text{, where } x \text{ and } y \text{ are in metres and } t \text{ in seconds.}$ $\text{Which of the following is correct?}$

Options:

1. $v = 5 \text{ m/s}$
2. $\lambda = 18 \text{ m}$
3. $A = 0.04 \text{ m}$
4. $\nu = 50 \text{ Hz}$

Solution:

$\text{Hint: } y = a \sin(\omega t - kx)$ $\text{Step: Analyse each option one by one.}$ $\text{From the given equation, amplitude } a = 4 \text{ m}$ $\text{Frequency, } \nu = \frac{\text{Co-efficient of } t}{2\pi} = \frac{\pi/5}{2\pi} = \frac{1}{10} \text{ Hz}$ $\text{Wavelength, } \lambda = \frac{2\pi}{\text{Co-efficient of } x} = \frac{2\pi}{\pi/9} = 18 \text{ m.}$ $\text{Wave speed, } v = \frac{\text{Co-efficient of } t}{\text{Co-efficient of } x} = \frac{\pi/5}{\pi/9} = 1.8 \text{ m/s}$ $\text{Hence, option (2) is the correct answer.}$

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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
}