Import Question JSON

Current Question (ID: 19420)

Question:

$\text{Three identical spheres, each of mass } M, \text{ are placed at the corners of a right-angled triangle with mutually perpendicular sides equal to } 3 \text{ m each.}$ $\text{Taking the point of intersection of mutually perpendicular sides as the origin, the magnitude of the position vector of the centre of mass of the system will be } \sqrt{x} \text{ m.}$ $\text{The value of } x \text{ is:}$

Options:

1. 4
2. 5
3. 2
4. 7

Solution:

$\text{The coordinates of the spheres are } (0,0), (3,0), (0,3).$ $\text{The center of mass } (x_{cm}, y_{cm}) \text{ is given by:}$ $x_{cm} = \frac{M \cdot 0 + M \cdot 3 + M \cdot 0}{3M} = 1$ $y_{cm} = \frac{M \cdot 0 + M \cdot 0 + M \cdot 3}{3M} = 1$ $\text{The magnitude of the position vector is:}$ $\sqrt{x_{cm}^2 + y_{cm}^2} = \sqrt{1^2 + 1^2} = \sqrt{2}$ $\text{Thus, } x = 2.$

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
}