Previous Year Questions- Complex Variables - 1 - Engineering for Electrical Engineering

Q1: Consider the complex function 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: It is series is of even powers.
∴ 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) $𝑓\left(𝑧\right)=𝑒\mid 𝑧\mid$f(z) = e^∣z∣
(d) ^{$𝑓\left(𝑧\right)=𝑧2-𝑧$}f(z) = z²−z  
Ans: (d)
Sol: Let us take (a), f(z) = z²−z
Now by C−R equation,
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)
(a)
(b)
(c)
(d)
Ans: (c)
Sol: Using green theorem?s
Check all the options:
Hence,  is not represent the area of the region.

Q4: Let  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       (2021)
(a) jπ/2
(b) 0
(c) -jπ/2
(d) jπ/16
Ans: (c)
Sol: Singularities are given by z²(z−4) = 0
⇒ 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'
By CRT

Q5: Let  p(z) = z³ + (1 + j)z² + (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
⇒ absent one root is complex
So all the roots cannot be real.

Q6: The real numbers, x and y with y = 3x² + 3x + 1, the maximum and minimum value of y for $𝑥\in [-2,0]$x ∈ [−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: Maximum value of y in [-2, 0] is maximum {f(-2), f(0)}
max{7, 1} = 7
Minimum value of y in [-2, 0]
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)
(a) 8πi
(b) -8πi
(c) -πi
(d) πi
Ans: (c)
Sol: Poles are at z = 0 and 2 but only z = 0 lies inside the unit circle.
Using equation (i)

Q8: Which of the following is true for all possible non-zero choices of integers m, n; m ≠ n, or all possible non-zero choices of real numbers p, q; p ≠ q,  as applicable?      (2020)
(a)
(b)
(c)
(d)
Ans: (c)
Sol: 
Q9: ax³ + bx² + cx + d is a polynomial on real x over real coefficients $𝑎,𝑏,𝑐,𝑑$a, b, c, d wherein a ≠ 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
Option (A):
If d = 0, then the polynomial equation becomesd 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+α)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 evaluated counter-clockwise, is      (2019)
(a) +8jπ
(b) -8jπ
(c) -4jπ
(d) +4jπ
Ans: (a)
Sol: 
Q11: Which one of the following functions is analytic in the region |z∣ ≤ 1?        (2019)
(a)
(b)
(c)
(d)
Ans: (b)
Sol: By Cauchy integral theorem,
Therefore,  is analytic in the region ∣z∣ ≤ 1.

Q12: If C is a circle |z| = 4 and       (2018)
(a) 1
(b) 0
(c) -1
(d) -2
Ans: (b)
Sol: By residue theorem,

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