x2+10x-1320 quadratic eqhations
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11th
Maths
Complex Numbers and Quadratic Equations
Algebra of Complex Numbers
Let α = -1+i√(3)2 . If a = ...
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Asked on May 01, 2020 by
Navneet Swrgy
Let α=
2
−1+i
3
. If a=(1+α)∑
k=0
100
α
2k
;b=∑
k=0
100
α
3k
, then a and b are the roots of the quadratic equation:
A
x
2
−102x+101=0
B
x
2
−101x+100=0
C
x
2
+102x+101=0
D
x
2
+101x+100=0
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ANSWER
Given
α=
2
−1+i
3
=ω← one of the roots of unity
Now
a=(1+α)∑
k=0
100
α
2k
=(1+ω)(1+ω
2
+ω
4
+ω
6
+.....ω
200
)
=(1+ω)
1−ω
2
1−ω
202
=
1−ω
2
(1+ω)(1−ω)
=1
b=∑
k=0
100
α
3k
=∑
k=0
100
((ω)
3
)
k
=∑
k=0
100
1=101
So equation will be
x
2
−(101+1)x+101×1
x
2
−102x+101=0
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