Given `alphaa n dbeta` are the roots of the quadratic equation `x^2-4x+k=0(k!=0)dot` If `alphabeta,alphabeta^2,alpha^3+beta^3` are in geometric progression, then the value of `7k//2` equals__________.

Answer

16

Explanation

Given `alphabeta,alphabeta(alpha+beta),alpha^(3)+beta^(3)` are in G.P.
`alpha+beta=4,alphabeta=k,alphabeta^(2)+alpha^(2)beta=alphabeta(alpha+beta)=4k`
`alpha^(3)+beta^(3)=(alpha+beta)^(3)-3alphabeta(alpha+beta)`
`=64-3k(4)=4(16-3k)`
`:.k,4k,4(16-3k)` are in G.P.
`16k^(2)=4k(16-3k)`
`4k(4k-16+3k)=0`
`k=0,k=(16)/(7)`