Import Question JSON

Current Question (ID: 10302)

Question:

$\text{The position of a particle is given by } \vec{r} = \hat{i} + 2\hat{j} - \hat{k} \text{ and momentum } \vec{P} = (3\hat{i} + 4\hat{j} - 2\hat{k}). \text{ The angular momentum is perpendicular to:}$

Options:

1. $\text{X-axis}$
2. $\text{Y-axis}$
3. $\text{Z-axis}$
4. $\text{Line at equal angles to all the three axes}$

Solution:

\text{(a) } \vec{L} = \vec{r} \times \vec{p} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & -1 \\ 3 & 4 & -2 \end{vmatrix} = -\hat{j} - 2\hat{k} \text{ and the X-axis is given by } \hat{i} + 0\hat{j} + 0\hat{k}. \text{ The dot product of these two vectors is zero, i.e., angular momentum is perpendicular to X-axis.}

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
}