Raised Fist0

Given Angle Between Hands

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Introduction

कई clock प्रश्नों में समय या overlap पूछने की बजाय कोई दिया हुआ कोण - जैसे 45°, 90°, 135° - दिया जाता है और पूछा जाता है कि hour और minute हाथ किस समय वह कोण बनाते हैं। यह पैटर्न इसलिए ज़रूरी है क्योंकि इसमें घड़ी के दोनों हाथों की relative speed और angular distance का उपयोग होता है।

Pattern: Given Angle Between Hands

Pattern: Given Angle Between Hands

मुख्य फॉर्मूला: जब हाथ θ° (θ = 0°, 45°, 90°, 135°, 180° आदि) का कोण बनाते हैं:

Angle = |30H - (11/2)M|

इसे M (घंटे के बाद मिनट) के लिए हल करने पर:

  • M = (60/11) × (H ± θ/30)

“±” का मतलब है दो संभावित समय - एक जब हाथ θ° अलग होते हैं और एक जब विपरीत दिशा में वही कोण बनता है।

Step-by-Step Example

Question

2 और 3 बजे के बीच किस समय घड़ी के हाथ 45° का कोण बनाएँगे?

Solution

  1. Step 1: फॉर्मूला लिखें

    M = (60/11) × (H ± θ/30)

  2. Step 2: H = 2, θ = 45° रखें

    M = (60/11) × (2 ± 45/30) = (60/11) × (2 ± 1.5)

  3. Step 3: दोनों मान निकालें

    + के लिए → (60/11) × 3.5 = 210/11 = 19 1/11 मिनट - के लिए → (60/11) × 0.5 = 30/11 = 2 8/11 मिनट

  4. Step 4: दोनों समय लिखें

    हाथ 45° पर होंगे: 2:02 8/11 और 2:19 1/11.

  5. Final Answer:

    2:02 8/11 और 2:19 1/11

  6. Quick Check:

    वापस रखकर देखें → |30×2 - (11/2)×(2.73)| ≈ 45° ✅

Quick Variations

1. θ = 90° रखने पर right angle और θ = 180° रखने पर opposite-hand समय मिलते हैं।

2. acute और obtuse दोनों कोणों के लिए “+” और “-” अलग-अलग निकालें।

3. यदि M ≥ 60 हो, तो 60 घटाकर अगले घंटे में बदल दें।

Trick to Always Use

  • Step 1 → M = (60/11)(H ± θ/30) का उपयोग करें।
  • Step 2 → दोनों + और - निकालकर दो समय प्राप्त करें।
  • Step 3 → यदि M > 60 निकले तो उसे अगली hour में कन्वर्ट करें।

Summary

  • फॉर्मूला: M = (60/11)(H ± θ/30)
  • हर कोण के लिए सामान्यतः दो समय मिलते हैं (कुछ boundary को छोड़कर)।
  • M को 0-60 की रेंज में जाँचें - तभी समय valid होगा।
  • Reflex या बड़े कोण भी (360 - θ) लगाकर निकाले जा सकते हैं।

याद रखने योग्य उदाहरण:
2 बजे पर 45° → 2:02 8/11 और 2:19 1/11

Practice

(1/5)
1. Find the times between 3 and 4 o’clock when the hands are 45° apart.
easy
A. 3:08 2/11 and 3:24 6/11
B. 3:10 10/11 and 3:49 1/11
C. 3:12 2/11 and 3:48 2/11
D. 3:15 and 3:45

Solution

  1. Step 1: Write formula

    M = (60/11) × (H ± θ/30) where θ = 45°.

  2. Step 2: Substitute H = 3, θ = 45°

    M = (60/11) × (3 ± 1.5) = (60/11) × {4.5, 1.5}.

  3. Step 3: Calculate minutes

    For 4.5 → m = 270/11 = 24 6/11 minutes → 3:24 6/11.
    For 1.5 → m = 90/11 = 8 2/11 minutes → 3:08 2/11.

  4. Final Answer:

    3:08 2/11 and 3:24 6/11 → Option A

  5. Quick Check:

    Substitute into |30H - 11m/2| gives ≈45° for both times ✅
Hint: Use H ± θ/30 inside (60/11) to get both roots.
Common Mistakes: Forgetting the smaller root (use ± both ways).
2. Find the time between 4 and 5 o’clock when the hands are 135° apart.
easy
A. 4:14 7/11
B. 4:46 4/11
C. 4:47 3/11
D. 4:13 1/11

Solution

  1. Step 1: Use formula

    M = (60/11)(H ± θ/30) with θ = 135° → θ/30 = 4.5.

  2. Step 2: Substitute H = 4

    M = (60/11)(4 ± 4.5) → (60/11)×8.5 = 510/11 = 46 4/11 minutes; the other root is negative (discard).

  3. Final Answer:

    4:46 4/11 → Option B

  4. Quick Check:

    |30×4 - (11/2)×46.3636| ≈ 135° ✅
Hint: If one root is negative, the valid time lies in the other root (or adjacent hour).
Common Mistakes: Including negative minute roots as times within the same hour.
3. Find the times between 6 and 7 o’clock when the hands are 90° apart.
easy
A. 6:16 4/11 and 6:49 1/11
B. 6:14 6/11 and 6:47 3/11
C. 6:20 and 6:50
D. 6:10 10/11 and 6:49 1/11

Solution

  1. Step 1: Formula

    M = (60/11)(H ± θ/30) with θ = 90° → θ/30 = 3.

  2. Step 2: Substitute H = 6

    M = (60/11)(6 ± 3) → (60/11)×9 = 540/11 = 49 1/11, and (60/11)×3 = 180/11 = 16 4/11.

  3. Final Answer:

    6:16 4/11 and 6:49 1/11 → Option A

  4. Quick Check:

    Both times give |30H - 11m/2| ≈ 90° ✅
Hint: For right angles use ±3 inside the (60/11) factor.
Common Mistakes: Dropping one of the ± roots.
4. Find the time between 8 and 9 o’clock when the hands are 135° apart.
medium
A. 8:19 1/11
B. 8:14 7/11
C. 8:47 3/11
D. 8:38 2/11

Solution

  1. Step 1: Formula

    M = (60/11)(H ± θ/30) with θ = 135° → θ/30 = 4.5.

  2. Step 2: Substitute H = 8

    m = (60/11)(8 - 4.5) = (60/11)×3.5 = 210/11 = 19 1/11 minutes. The (H + 4.5) root gives >60 (belongs to next hour).

  3. Final Answer:

    8:19 1/11 → Option A

  4. Quick Check:

    |30×8 - (11/2)×19.0909| ≈ 135° ✅
Hint: If one root exceeds 60, the valid time in the hour is the other root.
Common Mistakes: Not shifting >60 roots to the next hour when interpreting answers.
5. Find the times between 9 and 10 o’clock when the hands are 45° apart.
medium
A. 9:08 2/11 and 9:41 9/11
B. 9:10 10/11 and 9:49 1/11
C. 9:12 2/11 and 9:47 3/11
D. 9:40 10/11 and 9:57 3/11

Solution

  1. Step 1: Use formula

    M = (60/11)(H ± θ/30) with θ = 45° → θ/30 = 1.5.

  2. Step 2: Substitute H = 9

    m = (60/11)(9 ± 1.5) → (60/11)×10.5 = 630/11 = 57 3/11, and (60/11)×7.5 = 450/11 = 40 10/11 minutes.

  3. Final Answer:

    9:40 10/11 and 9:57 3/11 → Option D

  4. Quick Check:

    Both values satisfy |30H - 11m/2| ≈ 45° ✅
Hint: Evaluate both ± roots; normalize if one is near the hour boundary.
Common Mistakes: Swapping minute roots or using wrong hour when m > 60.