Import Question JSON

Current Question (ID: 19088)

Question:

$\text{The trajectory of a projectile in a vertical plane is } y = \alpha x - \beta x^2, \text{ where } \alpha \text{ and } \beta \text{ are constants and } x \text{ and } y \text{ are respectively the horizontal and vertical distances of the projectile from the point of projection. The angle of projection } \theta \text{ and the maximum height attained } H \text{ are respectively given by:}$

Options:

1. $\tan^{-1} \alpha, \frac{\alpha^2}{4\beta}$
2. $\tan^{-1} \beta, \frac{\alpha^2}{2\beta}$
3. $\tan^{-1} \alpha, \frac{4\alpha^2}{\beta}$
4. $\tan^{-1} \left( \frac{\beta}{\alpha} \right), \frac{\alpha^2}{\beta}$

Solution:

$\text{Hint: Compare the given equation with trajectory equation.}$ $y = \alpha x - \beta x^2$ $\text{comparing with trajectory equation}$ $y = x \tan \theta - \frac{1}{2} \frac{gx^2}{u^2 \cos^2 \theta}$ $\tan \theta = \alpha \Rightarrow \theta = \tan^{-1} \alpha$ $\beta = \frac{1}{2} \frac{g}{u^2 \cos^2 \theta}$ $u^2 = \frac{g}{2\beta \cos^2 \theta}$ $\text{Maximum height: } H$ $H = \frac{u^2 \sin^2 \theta}{2g} = \frac{g}{2\beta \cos^2 \theta} \frac{\sin^2 \theta}{2g}$ $H = \frac{\tan^2 \theta}{4\beta} = \frac{\alpha^2}{4\beta}$

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
}