Import Question JSON

Current Question (ID: 19365)

Question:

$\text{The position vector of a 1 kg object is } \vec{r} = (3\hat{i} + \hat{j}) \text{ m and its velocity vector is } \vec{v} = (3\hat{j} + \hat{k}) \text{ ms}^{-1}. \text{ If the magnitude of its angular momentum is } \sqrt{x} \text{ N-ms, then the value of } x \text{ will be:}$

Options:

1. $67$
2. $91$
3. $43$
4. $66$

Solution:

$\text{Hint: Use } \vec{L} = \vec{r} \times \vec{p}$ $\text{Step: Find the magnitude of its angular momentum and find } x.$ $\text{The angular momentum of the particle is given by:}$ $\vec{L} = \vec{r} \times \vec{p}$ $\Rightarrow \vec{L} = \vec{r} \times \vec{p} = \vec{r} \times m \vec{v}, [m = 1 \text{ kg}]$ $\Rightarrow \vec{L} = (3\hat{i} + \hat{j}) \times (3\hat{j} + \hat{k}) = (\hat{i} - 3\hat{j} + 9\hat{k}) \text{ N-ms}$ $\Rightarrow |\vec{L}| = \sqrt{x} \text{ N-ms} = \sqrt{91} \text{ N-ms}$ $\text{Therefore, the value of } x \text{ is } 91.$ $\text{Hence, option (2) is the correct answer.}$

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
}