When it is noon along 82° 30' E longitude along what degree of longitude it will be 6:30 am ?

Answer: (c) 0° E or W

6:30 am is 5 h 30 min behind; 5.5 × 15 = 82.5°, so 82°30′E − 82°30′ = 0° (Greenwich).

The idea

Local solar time changes continuously with longitude. A place with an earlier clock reading is west of the reference meridian under normal rotation. Convert the hours and minutes into an angular difference, then apply that difference to the reference longitude.

Noon minus 6:30 am gives 5.5 hours → Multiply 5.5 by 15° → Earlier time puts the target west → Subtract 82.5° from the reference 82.5°E → The target meridian is 0°

Why the others are wrong

(a) East would be ahead.

(d) 82°30′W would be 11 hours behind.

The trap

Just 15° per hour.

Also know

Greenwich is the zero-longitude reference, and IST is UTC + 5:30. GMT and UTC serve nearly the same clock-reference purpose in elementary questions, but UTC is based on atomic time. Solar noon is when the Sun crosses the local meridian, which need not be 12:00 on an official clock.