KVPY 2019 SB SX Exam Previous Papers

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Question : 12 of 120

Marks: +1, -0

Let z = x + iy and w = u + iv be complex numbers on the unit circle such that

z^2 + w^2 = 1

. Then the number of ordered pairs (z, w) is

0
4
8
infinite

Solution:

Let

z = e^{\i\α}

and

w = e^{\i\β}

z^2 + w^2 = 1 ⇒ e^{2\i\α} + e^{2\i\β} =1

⇒

\cos 2α+ \cos 2β= 1

and

\sin 2α+ sin 2β= 0

⇒

2 \cos(α+ β)\cos(α-β) = 1

and

2\sin(α+ β) \cos(α-β) = 0

⇒

\sin(α+ β) = 0 ⇒ α + β= nπ

for

α+ β= 0

, we have

\cos 2α=1/2⇒4 \pairs (α, β) \for α∈ [0, 2π)

for

α+ β=π

, we have

\cos 2α=1/2⇒4 \pairs (α, β) \for α∈ [0, 2π)

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