Complex NumbersHard
Question
If i = √-1, then 4 + 5 
is equal to
is equal toOptions
A.1 - i√3
B.-1 + i√3
C.i√3
D.- i√3
Solution
If in a complex number a + ib, the ratio a : b isv √3 , then it always convert the complex number in ω
Since, ω = -
∴ 4 + 5
= 4 + 5 ω334 + 3ω365
= 4 + 5 (ω3)111 ω + 3.(ω2)121 .ω2
= 4 + 5 ω + 3ω2 (∵ ω3 = 1)
= 1 + 3 + 2ω + 3ω + 3ω2
= 1 + 2ω + 3(1 + ω + ω2) = 1 + 2ω + 3 × 0 (∵ 1 + ω + ω2 = 0)
= 1 + ( -1 + √3i) = √3i.
Since, ω = -
∴ 4 + 5
= 4 + 5 ω334 + 3ω365
= 4 + 5 (ω3)111 ω + 3.(ω2)121 .ω2
= 4 + 5 ω + 3ω2 (∵ ω3 = 1)
= 1 + 3 + 2ω + 3ω + 3ω2
= 1 + 2ω + 3(1 + ω + ω2) = 1 + 2ω + 3 × 0 (∵ 1 + ω + ω2 = 0)
= 1 + ( -1 + √3i) = √3i.
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