Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) PDF Download

Q15: Consider the line integral Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) where z = x + iy. The line C is shown in the figure below. 
Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)The value of I is      (SET-1 (2017))

(a) 1/2(i)
(b) 2/3(i)
(c) 3/4 (i)
(d) 4/5 (i)
Ans: 
(b)
Sol: From the diagram C is y = x
Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q16: For a complex number z, Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) is      (SET-1 (2017))
(a) -2i
(b) -i
(c) i
(d) 2i
Ans:
(d)
Sol: Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q17: Consider the function f(z) = z + z where z is a complex variable and z* denotes its complex conjugate. Which one of the following is TRUE?       (SET-2 (2016))
(a) f(z) is both continuous and analytic
(b) f(z) is continuous but not analytic
(c) f(z) is not continuous but is analytic
(d) f(z) is neither continuous nor analytic
Ans:
(b)
Sol: Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)C.R. equation not satisfied.
Therefore, no where analytic.

Q18: Given f(z) = g(z) + h(z), where f, g, h are complex valued functions of a complex variable z. Which one of the following statements is TRUE?      (SET-2(2015))
(a) If f(z) is differentiable at z0, then g(z) and g(z) are also differentiable at z0.
(b) If g(z) and h(z) are differentiable at z0, then f(z) is also differentiable at z0.
(c) If f(z) is continuous at z0, then it is differentiable at z0.
(d) If f(z) is differentiable at z0, then so are its real and imaginary parts.
Ans:
(b)

Q19: All the values of the multi-valued complex function 1i, where i = √-1, are      (SET-2 (2014))
(a) purely imaginary
(b) real and non-negative
(c) on the unit circle
(d) equal in real and imaginary parts
Ans: 
b
Sol: Let Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
z = 1 which is purly real and non negative.

Q20: Let S be the set of points in the complex plane corresponding to the unit circle. (That is, S = {z:|z| = 1}). Consider the function f(z) = zz* where z* denotes the complex conjugate of z. The f(z) maps S to which one of the following in the complex plane      (SET-1 (2014))
(a) unit circle
(b) horizontal axis line segment from origin to (1, 0)
(c) the point (1, 0)
(d) the entire horizontal axis
Ans:
(c)
Sol: Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)which is equal to (1) always as givenPrevious Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)


Q21: Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) evaluated anticlockwise around the circle |z - i| = 2, where Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)is      (2013)
(a) -4π
(b) 0
(c) 2 + π
(d) 2 + 2i
Ans:
(a)
Sol: Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Poles at 2i and -2i i.e. (0, 2i) and (0, -2i)
Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)From figure of |Z - i| = 2, we see that pole, is insice C.
while pole, -2i is outside of C.
Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q22: Square roots of -i , where i = √-1, are      (2013)
𝑖=1
(a) i, -i
(b) Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(c) Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(d) Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Ans: (b)
Sol: Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q23: Given Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) If C is a counter clockwise path in the z-plane such that |z+1|=1, the value of Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) is      (2012)
(a) -2
(b) -1
(c) 1
(d) 2
Ans:
(c)
Sol: Given:
Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Poles are at -1 and -3 i.e. (-1, 0) and (-3, 0).
From figure below of |Z + 1| = 1,
we see that (-1, 0) is inside the circle and (-3, 0) is outside the circle.
Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Residue theorem says,
Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Residue of those poles which are inside C.
So the required integral Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) is given by the residue of function at poles (-1, 0) ( which is inside the circle).
This residue is Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

Q24: If Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) then the value of xx is       (2012)
(a) e-π/2
(b) eπ/2
(c) x
(d) 1
Ans:
(a)
Sol: Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q25: A point Z has been plotted in the complex plane, as shown in figure below.
Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
The plot of the complex number 𝑦 = 1/z is     (2011)
(a) Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)(b) Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)(c) Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)(d) Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Ans:
(d)
Sol: Let Z = a + bi
Since Z is shown inside the unit circle in I quadrant, a and b are both +ve and Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)So (1/Z) is outside the unit circle is IV quadrant.

Q26: The period of the signal x(t) = 8sin(0.8πt + π/4) is      (2010)
(a) 0.4 π s
(b) 0.8 π s
(c) 1.25 s
(d) 2.5 s
Ans: 
(d)
Sol: Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q27: The value of Previous Year Questions- Complex Variables - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) where C is the contour |(z-i)/2| = 1 is      (2007)
(a) 2πi
(b) π
(c) tan-1z
(d) πitan-1z
Ans:
(b)

The document Previous Year Questions- Complex Variables - 2 | 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 - 2 - Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

1. How do you find the Laurent series of a function in complex variables?
Ans. To find the Laurent series of a function, you first express the function as a sum of a Taylor series and a series of Laurent coefficients. The Laurent series includes both positive and negative powers of the complex variable, unlike a Taylor series which only includes non-negative powers.
2. What is the residue of a function in complex variables?
Ans. The residue of a function at a singular point is the coefficient of the term with a negative power of $(z-z_0)$ in the Laurent series expansion of the function around that point. Residues play a crucial role in evaluating complex integrals using the residue theorem.
3. How is the concept of poles related to complex variables?
Ans. Poles of a function are the points where the function becomes infinite or undefined. They are closely related to the residues of the function and are essential for understanding the behavior of functions in the complex plane.
4. Can you explain the Cauchy integral formula in the context of complex variables?
Ans. The Cauchy integral formula states that for a function that is analytic inside a simple closed curve, the value of the function at any point within the curve can be expressed as a contour integral around the curve. This formula is a fundamental result in complex analysis.
5. How can one determine if a function is analytic in a given region of the complex plane?
Ans. A function is analytic in a region of the complex plane if it has continuous partial derivatives with respect to both the real and imaginary parts of the complex variable in that region. This condition ensures that the function satisfies the Cauchy-Riemann equations and can be expanded as a Taylor series.
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