If `ax^(2)+bx+c =0` where `ane0` is satisfied by `alpha,beta,alpha^(2)andbeta^(2)` where `alphabetane0`. Let set S be the set of all possible unordered pairs `(alpha,beta)`.
Then match the following lists:

Options

A. `{:(,a,b,c,d),((1),q,s,s,r):}`
B. `{:(,a,b,c,d),((2),r,s,q,p):}`
C. `{:(,a,b,c,d),((3),q,s,r,p):}`
D. `{:(,a,b,c,d),((4),r,s,p,q):}`

Answer

1

Explanation

Equation is satisfied by `alpha,beta,alpha^(2)andbeta^(2)`. So, we have following possibilities:
(1) Let `alpha^(2)=alphaandbeta^(2)=beta`.
`implies(alpha,beta)-=(1,1)`
(ii) `alpha^(2)=betaandbeta^(2)=alpha`
`impliesalpha^(4)=alpha`
`impliesalpha(alpha^(3)-1)=0`
`impliesalpha=0,1,(-1pmsqrt3i)/(2)`
`implies(alpha,beta)-=((-1-sqrt3i)/(2),(-1+sqrt3i)/(2))`
(iii) `alpha^(2)betaandbeta^(2)=beta(oralpha^(2)=alphaandbeta^(2)=alpha)`
`impliesalpha^(2)=beta^(2)impliesalpha=pmbeta`
`implies(alpha,beta)=(-1,1),(1,1)` Thus, possible unordered pairs `(alpha,beta)` such that that `alphabetane0` is
`(1,1),(-1,1)or((-1-sqrt3i)/(2),(-1+sqrt3i)/(2))`.