If and are roots of the equation, x^2-4 2 k x+2 e^ 4 k -1=0 for some k , and ^2+ ^2=66 , then ^3+ ^3 is equal to:

Step-by-step solution

x^2-4 2 k x+2 e^ 4 k -1=0 or, x^2-4 2 k x+2 k^4-1=0 + =4 2 k and =2 k^4-1 Squaring both sides, we get aligned &( + )^2=(4 2 k)^2 & ^2+ ^2+2 =32 k^2 &66+2 =32 k^2 &66+2 (2 k^4-1 )=32 k^2 &66+4 k^4-2=32 k^2 & 4 k^4-32 k^2+64=0 & or, k^4-8 k^2+16=0 & (k^2 )^2-8 k^2+16=0 & (k^2-4 ) (k^2-4 )=0 & k^2=4, k^2=4 & k= 2 aligned Now, ^3+ ^3=( + ) ( ^2+ ^2- ) ^3+ ^3=(4 2 k) [66- (2 k^4-1 ) ] Putting k=-2,(k=+2 cannot be taken because it does not satisfy the above equation) ^3+ ^3=(4 2 (-2)) [66-2(-2)^4-1 ] ^3+ ^3=(-8 2 )(66-32