AP EAPCET 20-Aug-2021 Shift 2 Paper

Roots of cubic equations are 1 and 2+√5.
Since, 2+√5 is root, then its conjugate also become roots, i.e. 2−√5 is root of cubic equation.
∴ Roots =1,2+√5,2−√5 i.e. α,β,γ
Let cubic equation be x3+bx2+cx+d=0
Then,
Sum of Roots =−b=1+2+√5+2−√5=5
⇒b=−5
Product of Roots =−d=(1)(2+√5)(2−√5) =4−5=−1
⇒d=1
∴αβ+βγ+γα=c
(1)(2+√5)+(2+√5)(2−√5)+(2−√5)(1)=c
⇒2+√5+2−√5+4−5=6
⇒3=c
∴ Cubic equation is x3−5x2+3x+1=0