Question

Column-I gives some incomplete statements with a parameter x. Column-II gives some statements which can complete the statements given in column-I. Match statements in column-I to all the statements in column-II which can possibly complete the statements in column-I

Answer 1

A: Q, R B: Q, R

Answer 2

Part (A): Two identical fuses rated (in Ampere) at x, when combined together may act as a fuse of rating

Fuses are designed to protect circuits by breaking when the current exceeds their rated value.

1. Fuses in Series: When two identical fuses, each rated at 'x' Amperes, are connected in series, the current flows through both fuses sequentially. If the current in the circuit exceeds 'x', either fuse (or both, if they blow simultaneously) will melt, thereby breaking the circuit. The effective current rating of the series combination is determined by the lowest rating among the fuses. Since both are identical and rated 'x', the effective rating is 'x'.

2. Fuses in Parallel: When two identical fuses, each rated at 'x' Amperes, are connected in parallel, the total current flowing into the combination divides between them. Assuming the fuses are truly identical, the current will split equally, with I/2 flowing through each fuse. Since each fuse can safely carry up to 'x' Amperes, the condition for not blowing is I/2 ≤ x, which implies I ≤ 2x. Therefore, the effective rating of the parallel combination is '2x'.

Based on this analysis, statement (A) can be completed by (Q) x and (R) 2x.

Part (B): Two identical non-ideal cells of emf x each, when combined together may act as a cell of e.m.f.

Non-ideal cells have an electromotive force (emf) and an internal resistance.

1. Cells in Series: When two identical non-ideal cells, each with an emf of 'x' and internal resistance 'r', are connected in series (positive terminal of one to the negative terminal of the other), their emfs add up. The total emf of the series combination will be x + x = 2x.

2. Cells in Parallel: When two identical non-ideal cells, each with an emf of 'x' and internal resistance 'r', are connected in parallel (positive terminal to positive terminal, negative terminal to negative terminal), the effective emf of the combination remains the same as that of a single cell. This can be mathematically derived using the formula for parallel combination of cells: $E_{eq} = \frac{E_1/r_1 + E_2/r_2}{1/r_1 + 1/r_2}$ For identical cells, $E_1 = E_2 = x$ and $r_1 = r_2 = r$. $E_{eq} = \frac{x/r + x/r}{1/r + 1/r} = \frac{2x/r}{2/r} = x$ Thus, the effective emf of the parallel combination is 'x'.

Based on this analysis, statement (B) can be completed by (Q) x and (R) 2x.

The option (P) 0.5x is not a standard outcome for combining two identical fuses or two identical cells in typical configurations.