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Current Question (ID: 19974)

Question:

$\text{For a transverse wave travelling along a straight line, the distance between two peaks (crests) is } 5 \text{ m, while the distance between one crest and one trough is } 1.5 \text{ m. The possible wavelengths (in m) of the waves are:}$ $1. \ 1, 2, 3, \ldots$ $2. \ \frac{1}{1}, \frac{1}{3}, \frac{1}{5}, \ldots$ $3. \ \frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \ldots$ $4. \ 1, 3, 5, \ldots$

Options:

1. $1, 2, 3, \ldots$
2. $\frac{1}{1}, \frac{1}{3}, \frac{1}{5}, \ldots$
3. $\frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \ldots$
4. $1, 3, 5, \ldots$

Solution:

$\text{Wavelength is the distance between consecutive crest and trough.}$ $\text{Given T to C} = 1.5 \text{ m}$ $\text{C to C} = 5 \text{ m}$ $\text{T to C} = (2n_1 + 1) \frac{\lambda}{2}$ $\text{C to C} = n_2 \lambda$ $\frac{1.5}{5} = \frac{(2n_1+1)}{2n_2} \Rightarrow 3n_2 = 10n_1 + 5$ $n_1 = 1 ; n_2 = 5 \rightarrow \lambda = 1$ $n_1 = 4 ; n_2 = 15 \rightarrow \lambda = 1/3$ $n_1 = 7 ; n_2 = 25 \rightarrow \lambda = 1/5$

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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
}