Given below are two statements: Statement I: In every hour, the minute hand…
2023
Given below are two statements:
Statement I: In every hour, the minute hand and hour hand coincide once.
Statement II: In every hour, the minute hand and hour hand are twice at right angles.
In the light of the above statements, choose the correct answer from the options given below:
Answer: B. Both Statement I and Statement II are false — ConceptOn an ideal clock, the minute hand moves at 6 degrees per minute and the hour hand at 0.5 degrees per minute, so their relative speed is 5.5 degrees…
- A.
Both Statement I and Statement II are true
- B.
Both Statement I and Statement II are false
- C.
Statement I is true but Statement II is false
- D.
Statement I is false but Statement II is true
Show answer & explanation
Correct answer: B
Concept
On an ideal clock, the minute hand moves at 6 degrees per minute and the hour hand at 0.5 degrees per minute, so their relative speed is 5.5 degrees per minute. Coincidence occurs when the relative angle is a multiple of 360 degrees; a right angle occurs when it is 90 degrees or 270 degrees.
Application
In 12 hours, the minute hand gains 11 complete revolutions on the hour hand. Therefore, consecutive coincidences are 360 divided by 5.5, or 720/11 minutes, apart, giving 11 coincidences in 12 hours rather than one in each of the twelve civil hours. For example, there is no coincidence from 11:00 up to but not including 12:00.
For right angles, each relative revolution has two positions: 90 degrees and 270 degrees. Thus there are 22 right-angle events in 12 hours, not the 24 events that would be required for exactly two in every one of the twelve civil hours. More directly, between 2:00 and 3:00 the hands are at a right angle only at 2:27 3/11.
Statement I and Statement II both fail because the event counts are not uniform across every civil hour.
Cross-check
A universal claim about every hour is disproved by one counterexample interval: 11:00 to 12:00 contains no coincidence before its endpoint.
The 2:00 to 3:00 interval contains only one right-angle event, which also contradicts a count of two in every hour.
Therefore, both Statement I and Statement II are false.