Import Question JSON

Current Question (ID: 9551)

Question:

$\text{A particle is moving on a circular path of radius } R. \text{ When the particle moves from point } A \text{ to } B \text{ (angle } \theta \text{), the ratio of the distance to that of the magnitude of the displacement will be:}$

Options:

1. $\frac{\theta}{\sin \frac{\theta}{2}}$
2. $\frac{\theta}{2 \sin \frac{\theta}{2}}$
3. $\frac{\theta}{2 \cos \frac{\theta}{2}}$
4. $\frac{\theta}{\cos \frac{\theta}{2}}$

Solution:

\text{Hint: Arc length (AB) is given by: } R\theta \text{Step: Find the ratio of the distance to that of the magnitude of the displacement.} \text{The motion of the particle on a circle is shown in the figure below:} \text{The distance covered } = R \cdot \theta \text{In } \triangle OAC \sin \frac{\theta}{2} = \frac{x}{R} \Rightarrow x = R \sin \frac{\theta}{2} \text{The displacement } = 2x = 2R \sin \frac{\theta}{2} \frac{\text{Distance}}{\text{Displacement}} = \frac{R \theta}{2R \sin \frac{\theta}{2}} = \frac{\theta}{2 \sin \frac{\theta}{2}} \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
}