Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) PDF Download

Q1: Consider the complex function Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)The coefficient of z5 in the Taylor series expansion of f(z) about the origin is _____ (rounded off to 1 decimal place).       (2024)
(a) 0
(b) 0.5
(c) 0.8
(d) 0.2
Ans:
(a)
Sol: Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)It is series is of even powers.
āˆ“ Coefficient of z5 = 0.

Q2: Which of the following complex functions is/are analytic on the complex plane?        (2024)
(a) f(z) = jRe(z)
(b) f(z) = Im(z)
(c) š‘“(š‘§)=š‘’āˆ£š‘§āˆ£f(z) = eāˆ£zāˆ£
(d) š‘“(š‘§)=š‘§2āˆ’š‘§f(z) = z2āˆ’z  
Ans:
(d)
Sol: Let us take (a), f(z) = z2āˆ’z
Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Now by Cāˆ’R equation,
Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)i.e. C-R equation are satisfied.

Q3: Let R be a region in the first quadrant of the xy plane enclosed by a closed curve C considered in counter-clockwise direction. Which of the following expressions does not represent the area of the region R?      (2022)
Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)(a) Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

(b) Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(c) Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(d) Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Ans: (c)
Sol: Using green theorem?s
Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Check all the options:
Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Hence, Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) is not represent the area of the region.

Q4: Let Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) be the vertices of a rectangle C in the complex plane. Assuming that C is traversed in counter-clockwise direction, the value of the contour integral Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)      (2021)
(a) jĻ€/2

(b) 0
(c) -jĻ€/2

(d) jĻ€/16
Ans: (c)
Sol: Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Singularities are given by z2(zāˆ’4) = 0
ā‡’ z = 0, 4
z = 0 is pole of order m = 2 lies inside contour 'c'
z = 4 is pole of order m = 1 lies outside 'c'
Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)By CRT
Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q5: Let  p(z) = z+ (1 + j)z+ (2 + j)z + 3, where z is a complex number.
Which one of the following is true?     (2021)
(a) conjugate {p(z)} = p(conjugate {z}) for all z
(b) The sum of the roots of p(z) = 0 is a real number
(c) The complex roots of the equation p(z) = 0 come in conjugate pairs
(d) All the roots cannot be real
Ans:
(d)
Sol: Since sum of the roots is a complex number
ā‡’ absent one root is complex
So all the roots cannot be real.

Q6: The real numbers, x and y with y = 3x+ 3x + 1the maximum and minimum value of y for š‘„āˆˆ[āˆ’2,0]x āˆˆ [āˆ’2, 0] are respectively ________       (2020)
(a) 7 and 1/4
(b) 7 and 1
(c) -2 and -1/2
(d) 1 and 1/4
Ans:
(a)
Sol: Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Maximum value of y in [-2, 0] is maximum {f(-2), f(0)}
max{7, 1} = 7
Minimum value of y in [-2, 0]
Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Maximum value 7, minimum value 1/4.

Q7: The value of the following complex integral, with C representing the unit circle centered at origin in the counterclockwise sense, is:      (2020)
Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)(a) 8Ļ€i
(b) -8Ļ€i
(c) -Ļ€i
(d) Ļ€i
Ans: 
(c)
Sol: Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Poles are at z = 0 and 2 but only z = 0 lies inside the unit circle.
Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Using equation (i)
Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q8: Which of the following is true for all possible non-zero choices of integers m, n; m ā‰  n, or all possible non-zero choices of real numbers p, q; p ā‰  q,  as applicable?      (2020)
(a) Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

(b) Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(c) Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(d) Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Ans: (c)
Sol: Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q9: ax+ bx+ cx + d is a polynomial on real x over real coefficients š‘Ž,š‘,š‘,š‘‘a, b, c, d wherein a ā‰  0. Which of the following statements is true?      (2020)
(a) d can be chosen to ensure that x = 0 is a root for any given set a, b, c.
(b) No choice of coefficients can make all roots identical.
(c) a, b, c, d can be chosen to ensure that all roots are complex.
(d) c alone cannot ensure that all roots are real.
Ans:
(a, d)
Sol: Given Polynomial Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Option (A):
If d = 0, then the polynomial equation becomesPrevious Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)d can be choosen to ensure x = 0 is a root of given polynomial.
Hence, Option (A) is correct.
Option B:
A third degree polynomial equation with all root equal is given by
(x+Ī±)3 = 0
Thus, by selecting suitable values of a, b, c and d we can have all roots identical.
Hence, option (B) is incorrect.
Option (C): Complex roots always occurs in pairs,
So, the given polynomial will have maximum of 2 complex roots and 1 real root.
Hence, option (C) is incorrect.
Option (D): Nature or roots depends on other coefficients also apart from coefficient 'c'.
Hence, option (D) is correct.
Hence, the correct options are (A) and (D).

Q10: The closed loop line integral Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)evaluated counter-clockwise, is      (2019)
(a) +8jĻ€
(b) -8jĻ€
(c) -4jĻ€
(d) +4jĻ€
Ans:
(a)
Sol: Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q11: Which one of the following functions is analytic in the region |zāˆ£ ā‰¤ 1?        (2019)
(a) Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

(b) Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(c) Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(d) Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Ans: (b)
Sol: Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)By Cauchy integral theorem,
Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Therefore, Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) is analytic in the region āˆ£zāˆ£ ā‰¤ 1.

Q12: If C is a circle |z| = 4 and Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)      (2018)
(a) 1
(b) 0
(c) -1
(d) -2
Ans: 
(b)
Sol: Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)By residue theorem, Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

Q13: The value of the integral Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)in counter clockwise direction around a circle C of radius 1 with center at the point z = -2 is      (2018)
(a) Ļ€i/2
(b) 2Ļ€i
(c) -(Ļ€i/2)
(d) -2Ļ€i
Ans:
(a)
Sol: Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q14: The value of the contour integral in the complex-plane Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Along the contour |Z| = 3, taken counter-clockwise is      (SET-2 (2017))
(a) -18Ļ€i
(b) 0
(c) 14Ļ€i
(d) 48Ļ€i
Ans:
(c)
Sol: Pole, z = 2 lies inside |z| = 3
Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)By Cauche redisue theorem,
I = 2Ļ€i(7) = 14Ļ€i

The document Previous Year Questions- Complex Variables - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Engineering Mathematics for Electrical Engineering.
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FAQs on Previous Year Questions- Complex Variables - 1 - Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

1. What are the Cauchy-Riemann equations in complex variables?
Ans. The Cauchy-Riemann equations are a set of two partial differential equations that must be satisfied by a complex-valued function in order for it to be holomorphic.
2. How do you determine if a complex function is analytic?
Ans. A complex function is analytic if it is holomorphic, meaning it satisfies the Cauchy-Riemann equations and has continuous partial derivatives.
3. What is the significance of the residue theorem in complex variables?
Ans. The residue theorem is a powerful tool in complex analysis that allows for the evaluation of certain integrals along closed curves by analyzing the residues of the function within the curve.
4. How do you find the poles of a complex function?
Ans. The poles of a complex function are the points where the function becomes infinite. They can be found by setting the denominator of the function equal to zero and solving for the complex variable.
5. How are complex variables used in electrical engineering applications?
Ans. Complex variables are commonly used in electrical engineering for analyzing and designing circuits, signal processing, and control systems due to their ability to simplify mathematical expressions and solve complex problems efficiently.
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