Question:
The number of non-zero integral solutions of the equation \(|1 – 2i|^x = 5^x\) is:
समीकरण \(|1 – 2i|^x = 5^x\) के गैर-शून्य पूर्णांक समाधानों की संख्या है:
Options:
- Zero (No solution)
- One
- Two
- Three
- शून्य (कोई समाधान नहीं)
- एक
- दो
- तीन
Correct Answer: a)
Explanation:
Evaluate the modulus of \(1 – 2i\):
\[|1 – 2i| = \sqrt{1^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} = 5^{1/2}\]
The equation becomes:
\[(5^{1/2})^x = 5^x \implies 5^{x/2} = 5^x \implies \frac{x}{2} = x \implies x = 0\]
The only integral solution is \(x = 0\).
Since the question asks for the number of non-zero integral solutions, there are none (zero).
Hence, the correct option is (a).
मापांक: \(|1 – 2i| = \sqrt{1 + 4} = 5^{1/2}\)।
समीकरण:
\[5^{x/2} = 5^x \implies \frac{x}{2} = x \implies x = 0\]
एकमात्र पूर्णांक हल \(x = 0\) है।
चूँकि गैर-शून्य (non-zero) पूर्णांक हल पूछा गया है, अतः ऐसे हलों की संख्या 0 (शून्य) है।
अतः सही विकल्प (a) है।