JEE PYQ: Units & Measurements - Question ID ec7fece7aad0 (JEE Main 2026)

Core Formula & Concept:

In this problem, we deal with the fundamental relationship between distance, speed, and time. The key formula is:

$\text{Distance} = \text{Speed} \times \text{Time}$

Here, the speed of light in vacuum ($c$) is given a new unit of length, denoted as $\alpha$. This means: $1 \alpha = c$ where $c = 3 \times 10^8 \, \text{m/s}$ (the speed of light in vacuum).

The problem provides the time taken for light to travel from Venus to Earth (6 minutes 40 seconds) and asks for the distance in terms of $\alpha$.

Step-by-Step Derivation:

Step 1: Convert the given time into seconds.
The time taken is 6 minutes and 40 seconds. $6 \, \text{minutes} = 6 \times 60 = 360 \, \text{seconds}$ Total time ($t$) = $360 \, \text{s} + 40 \, \text{s} = 400 \, \text{s}$.

Step 2: Express the distance in meters using the speed of light.
Using the formula $\text{Distance} = \text{Speed} \times \text{Time}$: $\text{Distance} = c \times t = (3 \times 10^8 \, \text{m/s}) \times (400 \, \text{s}) = 1.2 \times 10^{11} \, \text{meters}.$

Step 3: Convert the distance into $\alpha$ units.
Since $1 \alpha = c = 3 \times 10^8 \, \text{m}$, we can write: $\text{Distance in } \alpha = \frac{\text{Distance in meters}}{c} = \frac{1.2 \times 10^{11} \, \text{m}}{3 \times 10^8 \, \text{m/}\alpha} = 400 \alpha.$

Step 4: Match the result with the given options.
The calculated distance is $400 \alpha$, which corresponds to Option B.

Common Traps & Exam Tip:

1. Incorrect Time Conversion: Students often make mistakes in converting minutes to seconds. For example, they might forget to multiply by 60 or miscount the seconds. Always double-check time conversions.

2. Misinterpreting the Unit $\alpha$: Some students confuse $\alpha$ as a unit of time or speed rather than length. Remember, $\alpha$ is defined as the distance light travels in 1 second (since $1 \alpha = c$).

3. Forgetting to Divide by $c$: When converting meters to $\alpha$, students might directly equate the distance in meters to $\alpha$ without dividing by $c$. Always ensure the units align correctly.

Exam Tip: In such problems, always write down the given quantities and their units clearly. Then, systematically apply the formula while keeping track of unit conversions. This minimizes errors and ensures clarity.