Let `alpha,beta in Rdot` If `alpha,beta^2` are the roots of quadratic equation `x^2-p x+1=0a n dalpha^2,beta` is the roots of quadratic equation `x^2-q x+8=0` , then the value of `rifr/8` is the arithmetic mean of `pa n dq ,` is `(83)/2` b. `83` c. `(83)/8` d. `(83)/4`

Options

A. `83/2`
B. 83
C. `83/8`
D. `83/4`

Answer

B

Explanation

For the equation `x^(2)-px+1=0`
the product of roots,`alphabeta^(2)`=1
and for the equation `x^(2)-qx+8=0`
the product of roots`alpha^(2)beta=8`
Hence, `(alphabeta^(2))(alpha^(2)beta)=8`
`rArralpha^(3)beta^(3)=8rArralphabeta=2`
`therefore` From `alphabeta^(2)`=1, we have `beta=1/2` and form `alpha^(2).beta=8`, we have `alpha=4`
Hence, from sum of roots=`-b/a`, we have
`p=alpha+beta^(2)=4+1/4=17/4`
and `q=alpha^(2)+beta=16+1/2=33/2`
`r/8` is arthimatic mean of p and q
`thereforer/8=(p+q)/2`
`rArrr=4(p+q)=4(17/4+33/2)=17+66=83`