If `alpha and beta ` are roots of the equation `x^2-8x+p=0`, then `α^2+β^2=40`, find the value of `p`.

Answer

A, B, C, D

Explanation

`atoq,btor,ctos,dtop.`
We have, `alpha+beta=8 and alphabeta=4`.
a. If `alpha,a_(1),a_(2),a_(3),a_(4),a_(5),beta` are in A.P., then
`sum_(i=1)^(5)a_(i)=5((alpha+beta)/2)=5(8/2)=20`
b. If `alpha,a_(1),a_(2),a_(3),a_(4),a_(5),beta` are in G.P., then
`sum_(i=1)^(5)log_(2)a_(i)=log_(2)a_(1)+log_(2)a_(2)+log_(2)a_(3)+log_(2)a_(4)+log_(2)a_(5)`
`=log_(2)(a_(1)a_(2)a_(3)a_(4)a_(5))`
`=log_(2)(sqrt(alphabeta))^(5)=5log_(2)sqrt4=5`
c. If `alpha^(2),(a_(1))^(2),(a_(3))^(2),(a_(4))^(2),(a_(5))^(2),beta^(2)` are in A.P., then
`sum_(i=1)^(5)a_(i)^(2)=5((alpha^(2)+beta^(2))/2)`
`=5(((alpha+beta)^(2)-2alphabeta)/2)`
`=5((8^(2)-2xx4)/2)=140`
d. If `alpha,a_(1),a_(2),a_(3),a_(4),a_(5),beta` are in H.P., then
`sum_(i=1)^(5)1/a_(1)=1/5((1/alpha+1/beta)/2)=(alpha+beta)/(10alphabeta)=8/(10xx4)=1/5`
`therefore(1/(underset(i=1)overset(5)(Sigma)1/(a_(i)))+1)=5+1=6`