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Current Question (ID: 9586)

Question:

$\text{A river is flowing from } W \text{ to } E \text{ with a speed of 5 m/min. A man can swim in still water with a velocity of 10 m/min. In which direction should the man swim so as to take the shortest possible path to go to the south:}$

Options:

1. $30^\circ \text{ with downstream}$
2. $60^\circ \text{ with downstream}$
3. $120^\circ \text{ with downstream}$
4. $\text{South}$

Solution:

\text{For shortest possible path man should swim with an angle } (90° + \theta) \text{ with downstream.} \text{From the figure, we can see that the river flows from W to E with velocity } v_R = 5 \text{ m/min and the man swims with velocity } v_m = 10 \text{ m/min in still water.} \text{For the shortest path, the man needs to compensate for the river's flow. Using the vector diagram:} \sin \theta = \frac{v_R}{v_m} = \frac{5}{10} = \frac{1}{2} \therefore \theta = 30° \text{So angle with downstream } = 90° + 30° = 120°

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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
}