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Current Question (ID: 16738)

Question:

$\text{In the figure shown, each resistor has resistance } R.$ $\text{Match Column-I and Column-II.}$ $\begin{array}{|c|c|} \hline \text{Column-I} & \text{Column-II} \\ \hline \text{(A) Equivalent resistance between } a \text{ and } b & \text{(P) } \frac{R}{2} \\ \text{(B) Equivalent resistance between } a \text{ and } c & \text{(Q) } \frac{5R}{8} \\ \text{(C) Equivalent resistance between } b \text{ and } d & \text{(R) } R \\ \hline \end{array}$

Options:

1. (A) \rightarrow (P), (B) \rightarrow (Q), (C) \rightarrow (R)
2. (A) \rightarrow (Q), (B) \rightarrow (P), (C) \rightarrow (R)
3. (A) \rightarrow (R), (B) \rightarrow (P), (C) \rightarrow (Q)
4. (A) \rightarrow (R), (B) \rightarrow (Q), (C) \rightarrow (P)

Solution:

$\text{Hint: Recall the combination of resistors.}$ $\text{Step: Find the equivalent resistance between given points.}$ $(A) \text{ Equivalent resistance between } 'a' \text{ and } 'b' \text{ is given by:}$ $R_{AB} = \left[ \left( R_{ad} + R_{cd} \right) \parallel \left( R_{ac} \right) + R_{bc} \right] \parallel R_{ab}$ $\Rightarrow R_{AB} = \left[ \left[ 2R \parallel R \right] + R \right] \parallel R$ $\Rightarrow R_{AB} = \left[ \frac{2R \times R}{2R + R} + R \right] \parallel R$ $\Rightarrow R_{AB} = \frac{5R}{3} \parallel R = \frac{5}{8} R$ $(B) \text{ Equivalent resistance between } 'a' \text{ and } 'c' \text{ is given by:}$ $R_{AC} = \left( R_{ad} + R_{cd} \right) \parallel R_{ac} \parallel \left( R_{ab} + R_{bc} \right)$ $\Rightarrow R_{AC} = (2R) \parallel R \parallel (2R) = \frac{R \times R}{R + R} = \frac{R}{2}$ $\Rightarrow R_{AC} = \frac{R}{2}$ $(C) \text{ Equivalent resistance between } 'b' \text{ and } 'd' \text{ is given by:}$ $R_{BD} = (R_{ab} + R_{ad}) \parallel (R_{bc} + R_{cd})$ $\Rightarrow R_{BD} = 2R \parallel 2R = \frac{2R \times 2R}{2R + 2R} \left[ \text{since } \frac{R_{ad}}{R_{bc}} = \frac{R_{ab}}{R_{bc}}, \text{ Balanced Wheatstone Bridge} \right]$ $\Rightarrow R_{BD} = R$ $\text{Therefore, the correct match is } A \rightarrow Q, B \rightarrow P, C \rightarrow R.$ $\text{Hence, option (2) is the correct answer.}$

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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
}