GATE EC 2023 Set 1 — Question 13
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Engineering Mathematics → Complex Analysis
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Question
Let . Which of the following cannot be a value of ?
Correct answer
(A) 2e^(j2π/8)
Solution
We are given .
First, express in polar form:
Magnitude: .
Angle: The complex number lies on the positive imaginary axis, so its angle is radians.
Thus, for integer .
Now, .
To find , we take the fourth root:
Let's find the four distinct roots for :
For :
For :
For :
For : Now, let's check the given options:
(A) . This is not among the calculated roots.
(B) . This is one of the roots ().
(C) . This is one of the roots ().
(D) . This is one of the roots ().Therefore, option (A) cannot be a value of .
First, express in polar form:
Magnitude: .
Angle: The complex number lies on the positive imaginary axis, so its angle is radians.
Thus, for integer .
Now, .
To find , we take the fourth root:
Let's find the four distinct roots for :
For :
For :
For :
For : Now, let's check the given options:
(A) . This is not among the calculated roots.
(B) . This is one of the roots ().
(C) . This is one of the roots ().
(D) . This is one of the roots ().Therefore, option (A) cannot be a value of .
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