question_answer

At what time between 4 and 5 o'clock will the hands of a clock be at right angle?

A) 30 minutes past 4          

B) \[16\frac{3}{4}\] minutes past 4

C) \[38\frac{2}{11}\] minutes past 4

D) 33 minutes past 4

E) None of these

Correct Answer: C

Solution :

Explanation Option [c] correct. Between 4 and 5 o?clock the hands of the clock will be at right angle twice. First situation will occur when minute hand is 15 minute space behind the hour hand and second when minute hand is 15 minute ahead of the hour hand. Figure (ii) shows the position when minute hand is 15 minute behind the hour hand. To come at this position minute hand has travelled 5 minute space from the position at 4 o?clock. Now, 55 minute are gained in 60 minute. 5 minute will be gained in \[\frac{60}{55}\times 35=\frac{420}{11}\,\min ute\,\,=\,35\frac{2}{11}\,\,\min ute\] It means that hands of the clock will be at right angle at \[5\frac{5}{11}\] minute past 5. Fig. (iii) shows the position when minute hand is 15 minute ahead the hour hand. To come at this position minute hand has travelled 35 minute spaces from the position at 4 o?clock. Now, 55 minute are gained in 60 minute 35 minute will be gainer in \[\frac{60}{55}\times 35=\frac{420}{11}\operatorname{minute}=35\frac{2}{11}\min ute\] It means that second position will come at \[38\frac{2}{11}\] minute past 4. Now, in the options \[38\frac{2}{11}\] minute past 4 is available as option (C). Hence, it is the correct answer.