Question:
Two numbers differ by 10. If the sum of their squares is 500, find the larger number.
दो संख्याओं का अंतर 10 है। यदि उनके वर्गों का योग 500 है, तो बड़ी संख्या ज्ञात कीजिए।
Options:
- 20
- 15
- 25
- 30
- 20
- 15
- 25
- 30
Correct Answer: a)
Explanation:
Let the numbers be \(x\) and \(y\) (with \(x > y\)).
\(x – y = 10 \implies y = x – 10\).
\(x^2 + y^2 = 500 \implies x^2 + (x – 10)^2 = 500 \implies 2x^2 – 20x + 100 = 500 \implies 2x^2 – 20x – 400 = 0 \implies x^2 – 10x – 200 = 0\).
\((x – 20)(x + 10) = 0 \implies x = 20\).
The larger number is 20 (smaller is 10; check: \(20^2 + 10^2 = 400 + 100 = 500\)).
माना संख्याएँ \(x\) और \(y\) हैं।
\(x – y = 10 \implies y = x – 10\)।
\(x^2 + (x – 10)^2 = 500 \implies 2x^2 – 20x – 400 = 0 \implies x^2 – 10x – 200 = 0\)।
\((x – 20)(x + 10) = 0 \implies x = 20\)।
अतः बड़ी संख्या 20 है।