Given n1,n2,n3 are roots of the equation E1:x3+x2+lx+n=0....(i) ∴ Sum of roots x1+x2+x3=−1.....(ii) Now, E2:x3+ax2+bx+c=0 is reciprocal equation of class one ⇒c=1 and a=b ∴E2:x3+ax2+ax+1=0....(iii) Given, Eq. (iii) have roots
x1−1
2
,
x2−1
2
,
x3−1
2
⇒
x1−1
2
+
x2−1
2
+
x3−1
2
=−a ⇒
x1+x2+x3
2
−
3
2
=−a From Eq. (ii), x1+x2+x3=−1
−1
2
−
3
2
=−a⇒a=2 ∴ Eq. (iii), x3+2x2+2x+1=0
(x+1)(x2+x+1)=0∴x=−1,x2+x+1=0
⇒x=
−1±√3i
2
∴ Given,
x1−1
2
=−1⇒x1=−1
x2−1
2
=
−1+√3i
2
⇒x2=√3i
x3−1
2
=
−1−√3i
2
⇒x3=−√3i
∴ Roots of these two equation excluding common roots are