Import Question JSON

Current Question (ID: 10789)

Question:

$\text{Hot coffee in a mug cools from } 90°\text{C to } 70°\text{C in } 4.8 \text{ minutes. The room temperature is } 20°\text{C. Applying Newton's law of cooling, the time needed to cool it further by } 10°\text{C should be nearly:}$

Options:

1. $4.2 \text{ minute}$
2. $3.8 \text{ minute}$
3. $3.2 \text{ minute}$ (Correct)
4. $2.4 \text{ minute}$

Solution:

$\text{Hint: Rate of fall of temp. is directly proportional to the temperature difference.}$ $\text{Step 1: Write down Newton's law of cooling}$ $-\frac{\Delta T}{\Delta t} = k\left[\frac{T_1 + T_2}{2} - T_0\right]$ $\text{Step 2: Apply law for both cases:}$ $\frac{20}{4.8} = k\left[\frac{90 + 70}{2} - 20\right] \ldots\ldots (1)$ $\frac{10}{t} = k\left[\frac{70 + 60}{2} - 20\right] \ldots\ldots (2)$ $\text{Step 3: Eliminate 'k' by dividing equations:}$ $\frac{2 \times t}{4.8} = \frac{60}{45}$ $t = 3.2 \text{ min}$ $\text{Therefore, the time needed to cool further by } 10°\text{C is } 3.2 \text{ minutes.}$

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
}