Option 4 : 38(2/11) past 4 o’clock
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10 Questions 10 Marks 8 Mins
When the hour hand and minute hand are perpendicular to each other, angle between them = 90°
The angle between the hour hand and the minute hand is computed as:
Angle = |(11/2)M – 30H|, where M = minutes & H = hours
We have, angle = 90° & H = 4
⇒ 90 = |(11/2)M – 120|
⇒ 90 + 120 = (11/2)M
⇒ M = (2/11) × 210 = 420/11 = 38(2/11) minutes
Hence, the hour hand and minute hand will be perpendicular to each other at 38(2/11) past 4 o’clockWith hundreds of Questions based on Clock and Calendar, we help you gain expertise on Logical Reasoning. All for free. Explore Testbook Learn to attain the subject expertise with us.
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If the minute hand has made $x$ revolutions around the clock ($60x$ minutes), the hour hand has made $\frac1{12}x$ revolutions; the difference between these two values represents an angle. Both hands start at 0 revolutions.
For (1), the relevant angle is $\frac14$ of the circle: $$x=\frac14+\frac1{12}x$$ Solving, we get $x=\frac3{11}$, which corresponds to a time of $\frac{180}{11}$ minutes after noon.
For (2) we replace $\frac14$ with 1, since the minute hand has lapped the hour hand. Then $$x=1+\frac1{12}x$$ Solving, we get $x=\frac{12}{11}$, i.e. $\frac{720}{11}$ minutes after noon.
Therefore, both your solutions are correct.