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Current Question (ID: 19346)

Question:

$\text{A force } \vec{F} = \hat{i} + 2\hat{j} + 3\hat{k} \text{ N acts at a point } 4\hat{i} + 3\hat{j} - \hat{k} \text{ m. Let the magnitude of the torque about the point } \hat{i} + 2\hat{j} + \hat{k} \text{ m be } \sqrt{x} \text{ N-m.}$ $\text{The value of } x \text{ is:}$

Options:

1. $145$
2. $195$
3. $245$
4. $295$

Solution:

$\text{Hint: } \vec{\tau} = \left( \vec{r} \times \vec{F} \right)$ $\text{Step 1: Find the position vector of the point of application of force.}$ $\vec{r} = \left( 4\hat{i} + 3\hat{j} - \hat{k} \right) - \left( \hat{i} + 2\hat{j} + \hat{k} \right)$ $= 3\hat{i} + \hat{j} - 2\hat{k}$ $\text{Step 2: Find the torque.}$ $\vec{\tau} = \left( \vec{r} \times \vec{F} \right)$ $= \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 1 & -2 \\ 1 & 2 & 3 \end{vmatrix}$ $= (3 + 4)\hat{i} - 11\hat{j} + 5\hat{k}$ $\text{Step 3: Find the magnitude of the torque.}$ $|\vec{\tau}| = \sqrt{7^2 + (-11)^2 + 5^2}$ $= \sqrt{195}$ $\text{Hence, option (2) is the correct answer.}$

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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
}