InterviewSolution
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InterviewSolution
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If `alpha and beta` are roots of the equation `x^(2)-2x+2=0,` then the least value of n for which `((alpha)/beta)^(n)=1` isA. 2B. 5C. 4D. 3 |
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Answer» Correct Answer - C Given, `alpha and beta` are the roots of the quadratic equation `x^(2)-2x+2=0` `implies(x-1)^(2)+1=0` `implies(x-1)^(2)=-1` `implies x-a=+-i" "["where i"=sqrt-1]` `impliesx=(1+i)or (1-i)` Clearly, if `alpha =1 +i, then beta=1-i` According to the equation `((alpha)/(beta))^(n)=1` `implies((1+i)/(1-i))^(n)=1` `implies(((1+i)(1+i))/((1-i)(1-i)))^(n)=1" "["by rationalization"]` `implies((1+i^(2)+2i)/(1-i^(2)))=1implies((2i)/(2))^(n)=1impliesi^(n)=1` So, minimum value of n is `4." "[becausei^(4)=1]` |
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