InterviewSolution
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InterviewSolution
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If `alpha` and `beta` are the roots of the equation `x^2-ax+b=0` and `A_n=alpha^n+beta^n`,A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.C. Statement-1 is True, Statement-2 is False.D. Statement-1 is False, Statement-2 is True. |
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Answer» Correct Answer - D We have, `alpha + beta = a and alpha beta = b` `therefore" "alpha^(n+1) + beta^(n+1) = (alpha^(n) + beta^(n))(alpha + beta) - alpha beta(alpha^(n-1) + beta^(n-1))` `rArr" "V_(n+1) = a V_(n) - b V_(n-1)` So, statement-2 is true. Now, `V_(n+1) = a V_(n) - V_(n-1)` `rArr" "V_(2)=a V_(1) -b V_(0)=a(alpha+beta)-b(alpha^(0)+beta^(0))=a^(2)-2b` `" "V_(3)=a V_(2) -b V_(1)=a(a^(2)-2b)-ab=a^(3)-3ab` `" "V_(4)=a V_(3) -b V_(2)=a(a^(3)-3ab)-b(a^(2)-2b)=a^(4)-4a^(2)b+2b^(2)` and, `V_(5)=a V_(4)-b V_(3) = a(a^(4) -4a^(2)b + 2b^(2))-b(a^(3)-3ab)=a^(5)-5a^(3)b + 5ab^(2)` So, statement-1 is false. |
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