Import Question JSON

Current Question (ID: 19026)

Question:

$\text{A particle is moving with speed } v = b\sqrt{x} \text{ along the positive } x\text{-axis.}$ $\text{The speed of the particle at time } t = T \text{ is: (assume that the particle is at origin at } t = 0)$

Options:

1. $\frac{b^2 T}{\sqrt{2}}$
2. $b^2 T$
3. $\frac{b^2 T}{2}$
4. $\frac{b^2 T}{4}$

Solution:

$\text{Hint: Express velocity as a function of time.}$ $\text{Step: Find the velocity of particle at } t = T.$ $\text{We have given that, } v = b\sqrt{x}$ $\frac{dx}{dt} = b\sqrt{x} \Rightarrow \frac{dx}{\sqrt{x}} = b \cdot dt$ $\text{Applying integration both sides}$ $\Rightarrow \int \frac{dx}{\sqrt{x}} = b \int dt$ $\Rightarrow 2\sqrt{x} = bt + C$ $\text{At, } t = 0, x = 0$ $\Rightarrow C = 0$ $\text{Then, } x = \frac{b^2 t^2}{4}$ $\text{Now, at } t = T$ $v = b\sqrt{x} = b \cdot \frac{bT}{2} = \frac{b^2 T}{2}$ $\text{Hence, option (3) is the correct answer.}$

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
}