Import Question JSON

Current Question (ID: 19081)

Question:

$\text{The trajectory of a projectile near the surface of the earth is given as}$ $y = 2x - 9x^2.$ $\text{If it were launched at an angle } \theta_0 \text{ with speed } v_0 \text{ then:}$ $(g = 10 \text{ ms}^{-2})$

Options:

1. $\theta_0 = \cos^{-1}\left(\frac{1}{\sqrt{5}}\right) \text{ and } v_0 = \frac{5}{3} \text{ ms}^{-1}$
2. $\theta_0 = \cos^{-1}\left(\frac{2}{\sqrt{5}}\right) \text{ and } v_0 = \frac{3}{5} \text{ ms}^{-1}$
3. $\theta_0 = \sin^{-1}\left(\frac{2}{\sqrt{5}}\right) \text{ and } v_0 = \frac{3}{5} \text{ ms}^{-1}$
4. $\theta_0 = \sin^{-1}\left(\frac{1}{\sqrt{5}}\right) \text{ and } v_0 = \frac{5}{3} \text{ ms}^{-1}$

Solution:

$\text{Hint: Compare the given equation with equation of projectile.}$ $y = x \tan \theta \left(1 - \frac{x}{R}\right) \ldots (i)$ $\text{given equation of trajectory: } y = 2x - 9x^2$ $\text{Comparing equation } \tan \theta = 2 \text{ and } R = \frac{2}{9}$ $\cos \theta = \frac{1}{\sqrt{5}} = \theta = \cos^{-1}\left(\frac{1}{\sqrt{5}}\right)$ $u^2 \sin 2\theta = \frac{2}{9}$ $u^2 \left[\frac{4}{5}\right] = \frac{2}{9} \times 10$ $u^2 = \frac{2}{9} \times \frac{10 \times 5}{4}$ $u = \frac{10}{6} = \frac{5}{3} \text{ m/s}$

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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
}