JEE PYQ: Units & Measurements - Question ID 5e3158890137 (JEE Main 2023)

In the chapter Units & Measurements, we study the systems of units used to quantify physical quantities. For astronomical distances—distances between celestial bodies—three specialized units are commonly employed:

• Astronomical Unit (Au): The average distance between the Earth and the Sun. By definition, $1 \text{ Au} = 1.496 \times 10^{11} \text{ m}$. It is the smallest of the three.
• Light Year (ly): The distance light travels in one year. Since light travels at $c = 3.00 \times 10^{8} \text{ m/s}$, we have $1 \text{ ly} = c \times \text{seconds in one year} = 3.00 \times 10^{8} \times 365.25 \times 24 \times 3600 \approx 9.461 \times 10^{15} \text{ m}.$
• Parsec (Pc): A parsec is defined as the distance at which one astronomical unit subtends an angle of one arcsecond (1/3600 of a degree). Using trigonometry, $1 \text{ Pc} = \frac{1 \text{ Au}}{\tan(1'')} \approx \frac{1.496 \times 10^{11}}{\tan(4.848 \times 10^{-6} \text{ rad})} \approx 3.086 \times 10^{16} \text{ m}.$

The key concept is that these three units are all valid for measuring astronomical distances, but their magnitudes differ significantly.

1. Verify Statement I:
   Statement I claims that Astronomical unit (Au), Parsec (Pc), and Light year (ly) are units for measuring astronomical distances.
   From the definitions above, all three units are indeed designed to measure distances on astronomical scales.
   Conclusion: Statement I is correct.
2. Verify Statement II:
   Statement II asserts the inequality $\mathrm{Au} < \mathrm{Parsec} (\mathrm{Pc}) < \mathrm{ly}$.
   We compare the numerical values:
   • $1 \text{ Au} \approx 1.496 \times 10^{11} \text{ m}$
   • $1 \text{ ly} \approx 9.461 \times 10^{15} \text{ m}$
   • $1 \text{ Pc} \approx 3.086 \times 10^{16} \text{ m}$
   Clearly, $1 \text{ Au} < 1 \text{ ly} < 1 \text{ Pc}$. The order given in Statement II is reversed: it states $\mathrm{Pc} < \mathrm{ly}$, which is incorrect.
   Conclusion: Statement II is incorrect.
3. Match with Options:
   Since Statement I is correct and Statement II is incorrect, the correct choice is: D: Statement I is correct but Statement II is incorrect.

Students often confuse the relative sizes of Parsec and Light year. A frequent mistake is to assume that because "light year" sounds larger, it must be bigger than a parsec. In reality, $1 \text{ Pc} \approx 3.26 \text{ ly}$, so a parsec is larger than a light year. Always rely on the numerical definitions rather than intuitive naming.

Exam Tip: When comparing astronomical units, convert all units to meters or memorize the conversion factors: $1 \text{ Pc} \approx 3.26 \text{ ly}, \quad 1 \text{ ly} \approx 63,241 \text{ Au}.$ This ensures you never mix up their relative magnitudes.