Question:
If the sum of the coefficients in the expansion of \((\alpha^2 x^2 – 2\alpha x + 1)^{51}\) vanishes, then the value of \(\alpha\) is
यदि \((\alpha^2 x^2 – 2\alpha x + 1)^{51}\) के प्रसार में गुणांकों का योग शून्य हो जाता है, तो \(\alpha\) का मान क्या है?
Options:
- 2
- -2
- 1
- -1
- 2
- -2
- 1
- -1
Correct Answer: c)
Explanation:
Sum of coefficients is obtained by setting \(x = 1\):
\((\alpha^2(1)^2 – 2\alpha(1) + 1)^{51} = (\alpha^2 – 2\alpha + 1)^{51} = ((\alpha-1)^2)^{51} = (\alpha-1)^{102}\).
Since the sum of coefficients vanishes (equals 0): \((\alpha-1)^{102} = 0 \implies \alpha = 1\).
गुणांकों का योग \(x = 1\) रखने पर प्राप्त होता है:
\((\alpha^2 – 2\alpha + 1)^{51} = (\alpha-1)^{102}\)।
प्रश्नानुसार गुणांकों का योग शून्य हो जाता है: \((\alpha-1)^{102} = 0 \implies \alpha = 1\)।