Evaluate: 31³ + 18³ − 37³ + 210
मान ज्ञात कीजिए: 31³ + 18³ − 37³ + 210
Options / विकल्प
- A. -36810
- B. -14820 ✅
- C. -45670
- D. -23450
Answer / उत्तर
−14820
Solution / समाधान
Shortcut Trick / शॉर्टकट ट्रिक
Use the Digital Sum (DS) method to eliminate wrong options quickly.
गलत विकल्पों को हटाने के लिए डिजिटल सम (DS) विधि का प्रयोग करें।
| DS(31³) | = DS(31)³ = DS(4³) = DS(64) = 1 |
| DS(18³) | = DS(18)³ = DS(9³) = DS(729) = 9 |
| DS(37³) | = DS(37)³ = DS(1³) = 1 |
| DS(210) | = 2 + 1 + 0 = 3 |
Therefore, DS = 1 + 9 − 1 + 3 = 12
DS = 1 + 9 - 1 + 3 = 12
DS = 3 (since 12 → 1 + 2 = 3)
Now check the options:
Option B: DS(−14820) = 1 + 4 + 8 + 2 + 0 = 15 → 1 + 5 = 6
Since the value is negative, −14820 ≡ −6 ≡ 3 (mod 9), which matches the expression.
Hence, Option B is the correct answer.
Verification / सत्यापन
Calculate each cube:
- 31³ = 29791
- 18³ = 5832
- 37³ = 50653
Now substitute:
31³ + 18³ − 37³ + 210 = 29791 + 5832 − 50653 + 210
= (29791 + 5832) − 50653 + 210
= 35623 − 50653 + 210
= −15030 + 210
= −14820
Final Answer / अंतिम उत्तर
−14820 ✅ Correct Answer / सही उत्तर
Quick Revision / Key Points
- Use the Digital Sum (DS) method to quickly eliminate incorrect options by checking congruence modulo 9.
- Calculate DS of each term and sum accordingly.
- Verify the answer by direct calculation of cubes and arithmetic operations.
- Negative values modulo 9 can be adjusted to match the DS of the expression.
For related problems, see: Evaluate: 7 1/4 − [ 5/6 ÷ { 1/3 − ( 1/2 × ( 3/4 − 1/2 ) ) } ]