If `alpha,beta` are the roots of the equation `lamda(x^(2)-x)+x+5=0` and if `lamda_(1) and lamda_(2)` are two values of `lamda` obtained from `(alpha)/(beta)+(beta)/(alpha)=4`, then `(lamda_(1))/(lamda_(2)^(2))+(lamda_(2))/(lamda_(1)^(2))` equals.

Options

A. 4192
B. 4144
C. 4096
D. 4048

Answer

D

Explanation

Since `alpha and beta` are the roots of the equation `lambda x^(2) + x(1-lambda)+5 = 0`.
`therefore" "alpha + beta = (lambda -1)/(lambda) and alpha beta =(5)/(lambda)`
Now, `(alpha)/(beta)+(beta)/(alpha)=(4)/(5)`
`rArr" "5(alpha + beta)^(2) = 14 alpha beta`
`5(lambda -1)^(2) = 70 lambda rArr lambda^(2) - 16 lambda + 1 = 0`
Since `lambda_(1) and lambda_(2)` are roots of this equation.
`therefore" "lambda_(1) + lambda_(2) = 16 and lambda_(1) lambda_(2) = 1`
Hence, `(lambda_(1))/(lambda_(2)^(2))+(lambda_(2))/(lambda_(1)^(2))=(lambda_(1)^(3)+lambda_(2)^(3))/((lambda_(1)lambda_(2)))=((lambda_(1)+ lambda_(2))^(3)-3lambda_(1) lambda_(2)(lambda_(1)+lambda_(2)))/((lambda_(1) lambda_(2))^(2))=16^(3) - 3 xx 16 = 4048`