Synopsis IAS

Answer

Solution

UPSC Prelims

Mathematics

Theme:

Clocks and Calendars / घड़ी और कैलेंडर

Q57 2021, Clocks and Calendars

At which one of the following times, do the hour hand and the minute hand of the clock make an angle of 180° with each other?

निम्नलिखित में से किस समय, किसी घड़ी के घंटे की सुई तथा मिनट की सुई आपस में 180° का कोण बनाएँगी ?

Explanation

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Option D: Between 7:05 hours and 7:10 hours
To determine when the hour and minute hands of a clock make an angle of 180°, we need to calculate the positions of the hands at different times.
     ● Step 1: Calculate the angle moved by the hour hand.
           ○ The hour hand moves 360° in 12 hours, which is 30° per hour.
           ○ At 7:00, the hour hand is at 7 × 30° = 210°.
     ● Step 2: Calculate the angle moved by the minute hand.
           ○ The minute hand moves 360° in 60 minutes, which is 6° per minute.
           ○ At 7:00, the minute hand is at 0°.
     ● Step 3: Determine the time when the angle between the hour and minute hands is 180°.
           ○ Let the time be 7:x minutes.
           ○ The angle of the hour hand at 7:x is 210° + 0.5x° (since it moves 0.5° per minute).
           ○ The angle of the minute hand at 7:x is 6x°.
           ○ We need the absolute difference between these angles to be 180°:

| (210 + 0.5 x) - 6 x | = 180

$|(210 + 0.5x) - 6x| = 180$

| 210 - 5.5 x | = 180

$|210 - 5.5x| = 180$
● Step 4: Solve the equation.
○ Case 1:

210 - 5.5 x = 180

$210 - 5.5x = 180$

210 - 5.5 x = 180 ⟹ 5.5 x = 30 ⟹ x = \frac{30}{5.5} \approx 5.45

$210 - 5.5x = 180 \implies 5.5x = 30 \implies x = \frac{30}{5.5} \approx 5.45$
○ Case 2:

5.5 x - 210 = 180

$5.5x - 210 = 180$ (not possible as it would imply a negative x)
● Conclusion: The hands make an angle of 180° at approximately 7:05.45, which is between 7:05 and 7:10.
Therefore, the correct answer is Option D: Between 7:05 hours and 7:10 hours.

Option D: Between 7:05 hours and 7:10 hours To determine when the hour and minute hands of a clock make an angle of 180°, we need to calculate the positions of the hands at different times.      ● Step 1: Calculate the angle moved by the hour hand.
              ○ The hour hand moves 360° in 12 hours, which is 30° per hour.            ○ At 7:00, the hour hand is at 7 × 30° = 210°.      ● Step 2: Calculate the angle moved by the minute hand.
              ○ The minute hand moves 360° in 60 minutes, which is 6° per minute.            ○ At 7:00, the minute hand is at 0°.      ● Step 3: Determine the time when the angle between the hour and minute hands is 180°.
              ○ Let the time be 7:x minutes.            ○ The angle of the hour hand at 7:x is 210° + 0.5x° (since it moves 0.5° per minute).            ○ The angle of the minute hand at 7:x is 6x°.            ○ We need the absolute difference between these angles to be 180°:          <mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo stretchy="false">(</mo><mn>210</mn><mo>+</mo><mn>0.5</mn><mi>x</mi><mo stretchy="false">)</mo><mo>&#x2212;</mo><mn>6</mn><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>=</mo><mn>180</mn></math>" role="presentation" style="font-size: 121%; text-align: center; position: relative;">|(210+0.5x)−6x|=180<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo stretchy="false">(</mo><mn>210</mn><mo>+</mo><mn>0.5</mn><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mn>6</mn><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>=</mo><mn>180</mn></math><script type="math/tex; mode=display" id="MathJax-Element-1">          |(210 + 0.5x) - 6x| = 180          </script>          <mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mn>210</mn><mo>&#x2212;</mo><mn>5.5</mn><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>=</mo><mn>180</mn></math>" role="presentation" style="font-size: 121%; text-align: center; position: relative;">|210−5.5x|=180<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mn>210</mn><mo>−</mo><mn>5.5</mn><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>=</mo><mn>180</mn></math><script type="math/tex; mode=display" id="MathJax-Element-2">          |210 - 5.5x| = 180          </script>      ● Step 4: Solve the equation.
              ○ Case 1: <mn>210</mn><mo>&#x2212;</mo><mn>5.5</mn><mi>x</mi><mo>=</mo><mn>180</mn></math>" role="presentation" style="font-size: 121%; position: relative;">210−5.5x=180<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>210</mn><mo>−</mo><mn>5.5</mn><mi>x</mi><mo>=</mo><mn>180</mn></math><script type="math/tex" id="MathJax-Element-3">210 - 5.5x = 180</script>          <mn>210</mn><mo>&#x2212;</mo><mn>5.5</mn><mi>x</mi><mo>=</mo><mn>180</mn><mspace width="thickmathspace" /><mo stretchy="false">&#x27F9;</mo><mspace width="thickmathspace" /><mn>5.5</mn><mi>x</mi><mo>=</mo><mn>30</mn><mspace width="thickmathspace" /><mo stretchy="false">&#x27F9;</mo><mspace width="thickmathspace" /><mi>x</mi><mo>=</mo><mfrac><mn>30</mn><mn>5.5</mn></mfrac><mo>&#x2248;</mo><mn>5.45</mn></math>" role="presentation" style="font-size: 121%; text-align: center; position: relative;">210−5.5x=180⟹5.5x=30⟹x=305.5≈5.45<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mn>210</mn><mo>−</mo><mn>5.5</mn><mi>x</mi><mo>=</mo><mn>180</mn><mspace width="thickmathspace"></mspace><mo stretchy="false">⟹</mo><mspace width="thickmathspace"></mspace><mn>5.5</mn><mi>x</mi><mo>=</mo><mn>30</mn><mspace width="thickmathspace"></mspace><mo stretchy="false">⟹</mo><mspace width="thickmathspace"></mspace><mi>x</mi><mo>=</mo><mfrac><mn>30</mn><mn>5.5</mn></mfrac><mo>≈</mo><mn>5.45</mn></math><script type="math/tex; mode=display" id="MathJax-Element-4">          210 - 5.5x = 180 \implies 5.5x = 30 \implies x = \frac{30}{5.5} \approx 5.45          </script>            ○ Case 2: <mn>5.5</mn><mi>x</mi><mo>&#x2212;</mo><mn>210</mn><mo>=</mo><mn>180</mn></math>" role="presentation" style="font-size: 121%; position: relative;">5.5x−210=180<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5.5</mn><mi>x</mi><mo>−</mo><mn>210</mn><mo>=</mo><mn>180</mn></math><script type="math/tex" id="MathJax-Element-5">5.5x - 210 = 180</script> (not possible as it would imply a negative x)      ● Conclusion: The hands make an angle of 180° at approximately 7:05.45, which is between 7:05 and 7:10.
        Therefore, the correct answer is Option D: Between 7:05 hours and 7:10 hours.

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Q 57. 2021, Clocks and Calendars

Mathematics

निम्नलिखित में से किस समय, किसी घड़ी के घंटे की सुई तथा मिनट की सुई आपस में 180° का कोण बनाएँगी ?

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John Doe

Mathematics

निम्नलिखित में से किस समय, किसी घड़ी के घंटे की सुई तथा मिनट की सुई आपस में 180° का कोण बनाएँगी ?

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John Doe

Explanation

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