Previous Year Questions- Complex Variables - 2 - Engineering for Electrical Engineering

Q15: Consider the line integral  where z = x + iy. The line C is shown in the figure below. 
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

Q16: For a complex number z,  is      (SET-1 (2017))
(a) -2i
(b) -i
(c) i
(d) 2i
Ans: (d)
Sol: 
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: 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
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: 

which is equal to (1) always as given

Q21:  evaluated anticlockwise around the circle |z - i| = 2, where is      (2013)
(a) -4π
(b) 0
(c) 2 + π
(d) 2 + 2i
Ans: (a)
Sol: Poles at 2i and -2i i.e. (0, 2i) and (0, -2i)
From figure of |Z - i| = 2, we see that pole, is insice C.
while pole, -2i is outside of C.

Q22: Square roots of -i , where i = √-1, are      (2013)
$𝑖=-1$(a) i, -i
(b) 
(c)
(d)
Ans: (b)
Sol: 
Q23: Given  If C is a counter clockwise path in the z-plane such that |z+1|=1, the value of  is      (2012)
(a) -2
(b) -1
(c) 1
(d) 2
Ans: (c)
Sol: Given:
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.
Residue theorem says,
Residue of those poles which are inside C.
So the required integral  is given by the residue of function at poles (-1, 0) ( which is inside the circle).
This residue is

Q24: If  then the value of x^x is       (2012)
(a) e^-π/2
(b) e^π/2
(c) x
(d) 1
Ans: (a)
Sol: 
Q25: A point Z has been plotted in the complex plane, as shown in figure below.
The plot of the complex number 𝑦 = 1/z is     (2011)
(a) (b) (c) (d) 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 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: 
Q27: The value of  where C is the contour |(z-i)/2| = 1 is      (2007)
(a) 2πi
(b) π
(c) tan^-1z
(d) πitan^-1z
Ans: (b)