InterviewSolution
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InterviewSolution
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If `alpha !=1` is an `n^(th)` root of unity and `n in N` such thatfirst three terms in the expansion of `(alpha + x)^n` are `1, alpha and (n - 1)/(2n) bar a^2`, then the value of x, isA. `(1)//(n)`B. `(2)//(n)`C. `1//2`D. `(1)//(4)` |
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Answer» Correct Answer - a Let `alpha = e ^((i2rpi)/(n))`, where 0 `lt r lt n`. Then, `alpha^(n-1) = e ^((i2rpi(n-1))/(n)) = e^((-i2rpi)/(2))= bar(alpha) and alpha^(n-2) = bar(alpha)^(2)` Now, `(alpha + x)^(n) = ""^(n)C_(0) alpha^(n) x^(0) + ""^(n)C_(1) alpha^(n-1) x + ""^(n)C_(2)alpha^b=(n -2) x^(2) +....` `rArr (alpha + x)^(n) = 1 + n bar(alpha) x + (n(n-1))/(2) (bar(alpha))^(2) x^(2) [because alpha^(n) = 1]` `therefore alpha nx = 1 and (n(n-1))/(2) x^(2) = (n-1)/(2n)` `rArr x = (1)/(n)` |
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