Import Question JSON

Current Question (ID: 9697)

Question:

$\text{Two particles having position vectors } \vec{r_1} = (3\hat{i} + 5\hat{j}) \text{ metres and } \vec{r_2} = (-5\hat{i} - 3\hat{j}) \text{ metres are moving with velocities } \vec{v_1} = (4\hat{i} + 3\hat{j}) \text{ m/s and } \vec{v_2} = (\alpha\hat{i} + 7\hat{j}) \text{ m/s. If they collide after 2 seconds, the value of } \alpha \text{ is:}$

Options:

1. $2$
2. $4$
3. $6$
4. $8$

Solution:

\text{It is clear from figure that the displacement vector } \Delta\vec{r} \text{ between particles } p_1 \text{ and } p_2 \text{ is } \Delta\vec{r} = \vec{r}_2 - \vec{r}_1 = -8\hat{i} - 8\hat{j} \text{ ....(1)} \text{Now, as the particles are moving in the same direction (} \vec{v}_1 \text{ and } \vec{v}_2 \text{ are +ve),} \vec{v}_{\text{rel}} = \vec{v}_2 - \vec{v}_1 = (\alpha - 4)\hat{i} + 4\hat{j} \vec{v}_{\text{rel}} = \frac{\Delta\vec{r}}{t} \text{then on solving we get } \alpha = 8.

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
}