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Current Question (ID: 9561)

Question:

$\text{If three coordinates of a particle change according to the equations } x = 3t^2, y = 2t, z = 4, \text{ then the magnitude of the velocity of the particle at time } t = 1 \text{ second will be:}$

Options:

1. $2\sqrt{11} \text{ unit}$
2. $\sqrt{34} \text{ unit}$
3. $40 \text{ unit}$
4. $2\sqrt{10} \text{ unit}$

Solution:

$\text{Hint: } \vec{v} = \frac{d\vec{r}}{dt}$ $\text{Step: Find the magnitude of the velocity of the particle.}$ $x = 3t^2 \Rightarrow v_x = \frac{dx}{dt} = 6t$ $y = 2t \Rightarrow v_y = \frac{dy}{dt} = 2$ $z = 4 \Rightarrow v_z = \frac{dz}{dt} = 0$ $v = \sqrt{v_x^2 + v_y^2 + v_z^2} = \sqrt{(6t)^2 + (2)^2}$ $\text{At } t = 1 \text{ sec, } v = \sqrt{(6)^2 + (2)^2}$ $v = \sqrt{36 + 4} = 2\sqrt{10} \text{ unit}$ $\text{Therefore, the magnitude of the velocity of the particle at } t = 1 \text{ s is } 2\sqrt{10} \text{ unit.}$ $\text{Hence, option (4) is the correct answer.}$

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
}