Question

The Common roots of the equations z3 + 2z2 + 2z + 1 = 0 and z1985 + z100 + 1 = 0 are
(a) ω2, ω3        
(b) ω, ω3
(c) ω, ω2         
(d) None of these​

Answer 1

Answer:

Solution -

z  

3

+2z  

2

+2z+1=0 ___ (1)

z  

1985

+z  

100

+1=0 ___ (2)

We know that 1+ω+ω  

2

=0,ω  

3

=1

check to be root,

(1) ω  

3

+2ω  

2

+2ω+1

=1+ω+ω  

2

+1+ω+ω  

2

=0+0=0

(2) ω  

1985

+ω  

100

+1

ω  

2

+ω+1=0

ω is a common root

check ω  

2

 to be a root

(1) ω  

6

+2ω  

4

+2ω  

2

+1

=(ω  

3

)  

2

+2ω(ω  

3

)+2ω  

2

+1

=1+ω+ω  

2

+1+ω+ω  

2

=0

(2) =ω  

3970

+ω  

200

+1

=ω+ω  

2

+1=0

ω  

2

 is a common root

A is correct

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