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Current Question (ID: 8722)

Question:

\text{The force } F \text{ acting on a body as a function of position } (x) \text{ and time } (t) \text{ is expressed as:} F = A \sin(k_{1} x) + B \cos(k_{2} t) \text{From the given information, match Column I with Column II.} \text{Column I:} \text{(A) Dimensions of } A \text{(B) Dimensions of } k_{1} \text{(C) Dimensions of } k_{2} \text{(D) Dimensions of } k_{1} k_{2} \text{Column II:} \text{(P) } [M^{0} L^{0} T^{-1}] \text{(Q) } [M^{0} L^{-1} T^{-1}] \text{(R) } [MLT^{-2}] \text{(S) } [M^{0} L^{-1} T^{0}]

Options:

1. $A \rightarrow R, B \rightarrow S, C \rightarrow P, D \rightarrow Q$
2. $A \rightarrow P, B \rightarrow Q, C \rightarrow R, D \rightarrow S$
3. $A \rightarrow R, B \rightarrow P, C \rightarrow Q, D \rightarrow S$
4. $A \rightarrow S, B \rightarrow P, C \rightarrow Q, D \rightarrow R$

Solution:

\text{Hint: Trigonometric functions must be dimensionless.} \text{Step: Find the dimensions of given physical quantities.} [A] = [F] = [MLT^{-2}] [k_1] = \left[\frac{1}{x}\right] = [L^{-1}] [k_2] = \left[\frac{1}{t}\right] = [T^{-1}] [k_1 k_2] = [L^{-1} T^{-1}] \text{Therefore, the correct match is } A \to R, B \to S, C \to P, D \to Q \text{Hence, option (1) is the correct answer.}

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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
}