Import Question JSON

Current Question (ID: 19171)

Question:

$\text{A particle of mass } 500 \text{ g is moving with a velocity given by:}$ $\vec{v} = (2t\hat{i} + 3t^2\hat{j}) \text{ m/s.}$ $\text{If the force acting on the particle at } t = 1 \text{ s is } \vec{F} = (\hat{i} + x\hat{j}) \text{ N,}$ $\text{then the value of } x \text{ is:}$

Options:

1. $6$
2. $1$
3. $2$
4. $3$

Solution:

$\text{Hint: } \vec{F} = m\vec{a}$ $\text{Step 1: Find the acceleration of the particle at } t = 1 \text{ s.}$ $\vec{a} = \frac{d\vec{v}}{dt}$ $\Rightarrow \vec{a} = \frac{d}{dt}(2t\hat{i} + 3t^2\hat{j})$ $\Rightarrow \vec{a} = 2\hat{i} + 6t\hat{j}$ $\text{at } t = 1 \text{ s;} \Rightarrow \vec{a} = 2\hat{i} + 6\hat{j}$ $\text{Step 2: Find the value of } x.$ $\vec{F} = m\vec{a}$ $\Rightarrow \vec{F} = 0.5(2\hat{i} + 6\hat{j})$ $\Rightarrow \hat{i} + x\hat{j} = \hat{i} + 3\hat{j}$ $\Rightarrow x = 3$ $\text{Hence, option (4) is the correct answer.}$

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
}