Import Question JSON

Current Question (ID: 9688)

Question:

$\text{A boat moves with a speed of 5 km/h relative to water in a river flowing with a speed of 3 km/h. Width of the river is 1 km. The minimum time taken for a round trip will be:}$

Options:

1. $\text{5 min}$
2. $\text{60 min}$
3. $\text{20 min}$
4. $\text{30 min}$

Solution:

$\text{For minimum time in a round trip across a river, the boat should cross perpendicular to the river flow.}$ $\text{Given:}$ $\text{Boat speed relative to water: } v_b = 5 \text{ km/h}$ $\text{River flow speed: } v_r = 3 \text{ km/h}$ $\text{River width: } d = 1 \text{ km}$ $\text{Step 1: For perpendicular crossing}$ $\text{The boat should head at an angle upstream to compensate for river flow.}$ $\text{Effective speed perpendicular to river: }$ $v_{\text{eff}} = \sqrt{v_b^2 - v_r^2} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4 \text{ km/h}$ $\text{Step 2: Calculate time for one-way trip}$ $\text{Time to cross river = } \frac{\text{width}}{v_{\text{eff}}} = \frac{1 \text{ km}}{4 \text{ km/h}} = 0.25 \text{ hr} = 15 \text{ min}$ $\text{Step 3: Calculate round trip time}$ $\text{Total time = } 2 \times 15 \text{ min} = 30 \text{ min}$

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
}