Import Question JSON

Current Question (ID: 10479)

Question:

$\text{A water tank is kept at a height H above the ground and the tank contains depth H of water as shown. Find the maximum possible range of water current, if there exists a small hole on the sidewall of the tank.}$

Options:

1. $2H$
2. $\frac{3H}{2}$
3. $\frac{5H}{2}$
4. $H$

Solution:

\text{To find the maximum range of the water current:} \text{1. Let the hole be at depth } x \text{ from the water surface.} \text{2. Velocity of efflux: } v = \sqrt{2gx} \text{3. Height of the hole above the ground: } h_{ground} = (2H - x) \text{4. Time to fall: } t = \sqrt{\frac{2(2H-x)}{g}} \text{5. Horizontal range: } R = v \times t = \sqrt{2gx} \times \sqrt{\frac{2(2H-x)}{g}} = 2\sqrt{x(2H - x)} \text{6. To maximize R, maximize } f(x) = x(2H - x)\text{. This quadratic is maximized when } x = H \text{7. Substitute } x = H \text{ into the range equation: } R_{max} = 2\sqrt{H(2H - H)} = 2\sqrt{H^2} = 2H \text{The maximum possible range is } 2H

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
}