Import Question JSON

Current Question (ID: 8837)

Question:

$\text{The position } x \text{ of a particle varies with time } t \text{ as } x = at^2 - bt^3. \text{ The acceleration of the particle will be zero at time } t \text{ equal to:}$

Options:

1. $\frac{a}{b}$
2. $\frac{2a}{3b}$
3. $\frac{a}{3b}$
4. $\text{zero}$

Solution:

$\text{Hint: The acceleration of the particle is given by } a = \frac{d^2x}{dt^2}$ $\text{Step: Find the time at which acceleration is zero.}$ $\text{First, find velocity by differentiating position:}$ $\frac{dx}{dt} = 2at - 3bt^2$ $\text{Then, find acceleration by differentiating velocity:}$ $\frac{d^2x}{dt^2} = 2a - 6bt = 0$ $\text{Solving for when acceleration equals zero:}$ $2a - 6bt = 0 \Rightarrow t = \frac{a}{3b}$ $\text{Hence, option (3) is the correct answer.}$

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
}