Import Question JSON

Current Question (ID: 8819)

Question:

$\text{A particle moves with a velocity } v = \alpha t^3 \text{ along a straight line. The average speed in time interval } t = 0 \text{ to } t = T \text{ will be:}$

Options:

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

Solution:

$\text{Hint: Find the total distance travelled using the formula } x = \int v dt$ $\text{Step 1: Use a general formula for speed}$ $v = \frac{dx}{dt}$ $\text{Step 2: Integrate velocity to obtain distance}$ $dx = v dt$ $x = \int_0^T \alpha t^3 dt = \frac{\alpha T^4}{4}$ $\text{Step 3: Use the formula for average velocity}$ $\langle v \rangle = \frac{\int_0^T v dt}{\int dt} = \frac{\frac{\alpha T^4}{4}}{T}$ $\Rightarrow \langle v \rangle = \frac{\text{Total distance}}{\text{total time}} = \frac{\alpha T^4}{4T} = \frac{\alpha T^3}{4}$ $\text{Hence option (4) is the correct answer.}$

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
}