Question 1

z = (2 - 3i)/(-5 + i) can be expressed as

Accepted Answer

-0.5 + 0.5i. To simplify the expression, multiply the numerator and denominator by the conjugate of the denominator, which results in a real denominator and allows for the separation of real and imaginary parts, leading to the correct result of -0.5 + 0.5i.

Question 2

If the semi-circular contour D of radius 2 is as shown in the figure, then the value of the integral ∫_D 1/(s² - 1) ds is

Accepted Answer

jπ. The integral evaluates to jπ because the integrand has simple poles at s = ±1, and the contour encloses the pole at s = 1, contributing to the integral's value through the residue theorem.

Question 3

The equation sin(z) = 10 has

Accepted Answer

an infinite number of complex solutions. The sine function is periodic and can take on any complex value, leading to an infinite number of complex solutions for the equation sin(z) = 10, as it can be expressed in terms of its periodic nature.

Question 4

The residue of the function f(z) = 1 / ((z + 2)² (z - 2)²) at z = 2 is

Accepted Answer

(A) -1/32. At the double pole z=2, residue = d/dz[(z-2)^2 f(z)] = d/dz[1/(z+2)^2] = -2/(z+2)^3. Evaluated at z=2: -2/(4)^3 = -2/64 = -1/32.

Question 5

If f(z)=c₀+c₁ z⁻¹, then ∮ (1+f(z))/z dz is given by unit circle

Accepted Answer

(D) 2πj(1+c₀). The integral evaluates the residue of the function (1 + f(z))/z at the pole z=0. The term (1 + f(z)) contributes a constant term of (1 + c₀) to the residue, and since the integral is taken over the unit circle, it results in 2πj times this residue.

Question 6

An AC source of RMS voltage 20 V with internal impedance Zs = (1 + 2j) Ω feeds a load of impedance ZL = (7 + 4j) Ω in the figure below. The reactive power consumed by the load is

Accepted Answer

16 VAR. Total impedance = Zs + ZL = (1+2j)+(7+4j) = 8+6j, |Z| = 10 ohm. RMS current |I| = 20/10 = 2 A. Reactive power in the load = |I|^2 * X_L = 4 * 4 = 16 VAR, not 32 VAR.

Question 7

The residues of a complex function X(z)=(1-2z)/(z(z-1)(z-2)) at its poles are

Accepted Answer

(1)/(2),1 and -(3)/(2). For X(z)=(1-2z)/(z(z-1)(z-2)): residue at 0 = (1)/((-1)(-2))=1/2; at 1 = (1-2)/((1)(-1))=1; at 2 = (1-4)/((2)(1))=-3/2. So residues are 1/2, 1, -3/2 (index 2), not the stored option 3.

Question 8

The value of the integral ∮_c (-3z + 4)/(z² + 4z + 5) dz where c is the circle |z| = 1 is given by

Accepted Answer

0. The integral evaluates to zero because the integrand is analytic inside the contour defined by |z| = 1, and by Cauchy's integral theorem, the integral of an analytic function over a closed curve is zero.

Question 9

A system with transfer function G(s) = ((s² + 9)(s + 2))/((s + 1)(s + 3)(s + 4)) is excited by sin(ωt). The steady-state output of the system is zero at

Accepted Answer

ω = 3 rad/s. The steady-state output of a system excited by a sinusoidal input is zero when the frequency of the input matches the natural frequency of the system's poles. In this case, the transfer function has a zero at ω = 3 rad/s, which leads to a cancellation of the output, resulting in a steady-state output of zero.

Question 10

C is a closed path in the z-plane given by |z| = 3. The value of the integral ∮_C ((z² − z + 4j)/(z + 2j)) dz is

Accepted Answer

4π(1 − j2). The integral evaluates to 4π(1 − j2) because the integrand has a simple pole at z = -2j, which lies within the contour |z| = 3. By applying the residue theorem, we find the residue at this pole and multiply it by 2πi, leading to the final result.

Question 11

The real part of an analytic function f(z) where z = x + jy is given by e^(−y)cos(x). The imaginary part of f(z) is

Accepted Answer

e^(−y)sin(x). With u=e^(-y)cos x, Cauchy-Riemann give v_x=-u_y=e^(-y)cos x and v_y=u_x=-e^(-y)sin x, integrating to v=e^(-y)sin x. Then f(z)=e^(-y)cos x + i e^(-y)sin x = e^(iz). The imaginary part is +e^(-y)sin(x), so the stored '-e^(-y)sin x' is wrong.

Question 12

The values of the integral (1)/(2π j)∮_c (e^z)/(z-2) dz along a closed contour c in anti-clockwise direction for
(i) the point z0 = 2 inside the contour c, and
(ii) the point z0 = 2 outside the contour c,
respectively, are

Accepted Answer

(B) (i) 7.39, (ii) 0. By the Cauchy integral formula, (1/2 pi j)*contour-integral of e^z/(z-2) with z0=2 inside equals e^2 = 7.39. With z0=2 outside the contour the integrand is analytic inside, so the integral is 0. Hence (i) 7.39, (ii) 0.

Question 13

Which one of the following functions is analytic over the entire complex plane?

Accepted Answer

cos(z). The function cos(z) is entire, meaning it is analytic everywhere in the complex plane, as it can be expressed as a power series that converges for all z. In contrast, the other options have singularities or branch cuts that restrict their analyticity.

Question 14

Let w⁴ = 16j. Which of the following cannot be a value of w?

Accepted Answer

2e^(j2π/8). 16j = 16 e^{j pi/2}, so w = 2 e^{j(pi/8 + k pi/2)}: angles pi/8, 5pi/8, 9pi/8, 13pi/8. The angle 2pi/8 = pi/4 (option A) is not among them, so it cannot be a value. Stored answer (pi/8) is in fact a valid root, hence wrong; correct is 2e^{j2pi/8}.

Question 15

The value of the contour integral ∮_C ((z+2)/(z²+2z+2)) dz, where the contour C is {z: |z + 1 - 3j/2| = 1}, taken in the counter clockwise direction, is

Accepted Answer

π(1 + j). The integral evaluates to π(1 + j) because the integrand has a pole inside the contour defined by C, and by applying the residue theorem, we find that the residue at the pole contributes this value to the integral.

Question 16

Which of the following statements involving contour integrals (evaluated counter-clockwise) on the unit circle C in the complex plane is/are TRUE?

Accepted Answer

∮_C e^z dz = 0. The integral of the function e^z over a closed contour, such as the unit circle, is zero due to Cauchy's integral theorem, which states that the integral of an analytic function over a closed curve is zero if the function is analytic inside the curve.

Question 17

H(z) is a transfer function of a real system. When a signal x[n] = (1 + j)ⁿ is the input to such a system, the output is zero. Further, the Region Of Convergence (ROC) of (1 - 1/2 z⁻¹) H(z) is the entire Z-plane (except z = 0). It can then be inferred that H(z) can have a minimum of

Accepted Answer

one pole and two zeros. The output being zero for the input signal indicates that the system has a specific behavior with respect to the poles and zeros. Given that the ROC is the entire Z-plane except at z = 0, it suggests that H(z) must have at least one pole to ensure stability, and having two zeros allows for the cancellation of the input signal, leading to a zero output.

Question 18

Given X(z) = z / (z - a)² with |z| > a, the residue of X(z) z^-n-1 at z = a for n ≥ 0 will be

Accepted Answer

n a^(n-1). The residue of a function at a pole can be found using the formula for the residue at a pole of order greater than one. In this case, since the pole at z = a is of order 2, the residue is calculated as the coefficient of z^(-n-1) in the Laurent series expansion, which results in n a^(n-1).

Question 19

The voltage applied to a circuit is 100√2 cos(100πt) volts and the circuit draws a current of 10√2 sin(100πt + π/4) amperes. Taking the voltage as the reference phasor, the phasor representation of the current in amperes is

Accepted Answer

10∠−π/4. The current is given in sine form, which can be converted to phasor form by recognizing that a sine function can be represented as a cosine function with a phase shift of -π/2. Thus, the current phasor becomes 10√2 at an angle of -π/4, but when expressed in standard phasor form, it simplifies to 10 at an angle of -π/4.

Question 20

If x = √(-1), then the value of x^x is

Accepted Answer

e^(-π/2). When x = √(-1), or i, we can express x^x as iⁱ. Using Euler's formula, i can be represented as e^(iπ/2), and raising this to the power of i results in e^(-(π/2)), which corresponds to option A.

Question 21

Given f(z) = 1/(z+1) - z²/(z+3). If C is a counterclockwise path in the z-plane such that |z+1|=1, the value of (1/(2πi)) ∮_C f(z) dz is

Accepted Answer

1. The contour |z+1|=1 encloses only z=-1. The residue of 1/(z+1) there is 1, and the term z^2/(z+3) has its pole at z=-3 outside the contour. So (1/(2*pi*i))*integral = 1, not -1.

Question 22

∮ (z² - 4)/(z² - 4) dz evaluated anticlockwise around the circle |z - i| = 2, where i = √(-1), is

Accepted Answer

0. The integrand simplifies to 1 everywhere except at the poles, which are at z = ±2. Since both poles lie outside the contour defined by |z - i| = 2, the integral evaluates to zero due to the absence of singularities within the contour.

Question 23

∮ (z² - 4)/(z² + 4) dz evaluated anticlockwise around the circle |z| = 2, where i = √(-1), is

Accepted Answer

0. The integral evaluates to zero because the integrand has no poles inside the contour |z| = 2. The poles at z = 2i and z = -2i are outside this contour, leading to a result of zero by the residue theorem.

Question 24

Square roots of i, where i = √−1, are

Accepted Answer

cos(π/4) + i sin(π/4), cos(3π/4) + i sin(3π/4). The correct option represents the square roots of i in polar form, where i can be expressed as e^(iπ/2). The square roots correspond to angles π/4 and 3π/4, leading to the expressions cos(π/4) + i sin(π/4) and cos(3π/4) + i sin(3π/4).

Question 25

∫ (z² - 4)/(z² - 4) dz evaluated anticlockwise around the circle |z| = 2, where i = √(-1), is

Accepted Answer

0. The integrand simplifies to 1 for all points except where the denominator is zero, which occurs at z = ±2. Since the contour |z| = 2 does not enclose any singularities, the integral evaluates to zero by the residue theorem.

Question 26

∫ (z² − 4)/(z² + 4) dz evaluated anticlockwise around the circle |z| = 2, where i = √−1, is

Accepted Answer

0. The integral evaluates to zero because the integrand has no poles inside the contour |z| = 2, as the only singularities occur at z = ±2i, which lie outside the circle. By the residue theorem, since there are no residues to account for within the contour, the integral results in zero.

Question 27

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) = z z* where z* denotes the complex conjugate of z. The f(z) maps S to which one of the following in the complex plane

Accepted Answer

the point (1, 0). The function f(z) = z z* simplifies to |z|², which equals 1 for any point z on the unit circle. Since |z|² is always 1, f(z) evaluates to 1 for all z in S, mapping every point on the unit circle to the single point (1, 0) in the complex plane.

Question 28

All the values of the multi-valued complex function 1ⁱ, where i = √−1, are

Accepted Answer

real and non-negative.. Writing 1 = e^(i 2 pi k), 1^i = e^(i * i 2 pi k) = e^(-2 pi k) for integer k. These are all positive real numbers, so the values are real and non-negative; only the k=0 value lies on the unit circle, so 'on the unit circle' is not correct.

Question 29

Integration of the complex function f(z) = z²/(z² - 1), in the counterclockwise direction, around |z| = 1, is

Accepted Answer

0. The function f(z) = z²/(z² - 1) has poles at z = 1 and z = -1, which lie outside the contour |z| = 1. Since there are no singularities within the contour, the integral evaluates to zero by the residue theorem.

Question 30

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 ?

Accepted Answer

f(z) is continuous but not analytic. The function f(z) = z + z* is continuous because it is composed of continuous functions (the identity and conjugate functions). However, it is not analytic since it does not satisfy the Cauchy-Riemann equations, which are necessary for a function to be analytic in the complex plane.

Question 31

The value of the integral ∮_C (z+1)/(z²-4) dz in counter clockwise direction around a circle C of radius 1 with center at the point z = -2 is

Accepted Answer

πi/2. The integral evaluates the residue of the function at the poles inside the contour. The only pole within the circle of radius 1 centered at z = -2 is at z = -1, which contributes a residue of 1/2, leading to the result of πi/2 when multiplied by 2πi.

Question 32

Which one of the following functions is analytic in the region |z| ≤ 1?

Accepted Answer

(z²−1)/(z+2). The function (z²−1)/(z+2) is analytic in the region |z| ≤ 1 because it is a rational function with no singularities or poles within that region, as the denominator z+2 does not equal zero for any z such that |z| ≤ 1.

Question 33

Which of the following complex functions is/are analytic on the complex plane?

Accepted Answer

f(z) = z² - z. The function f(z) = z² - z is a polynomial, and all polynomials are entire functions, meaning they are analytic everywhere in the complex plane. In contrast, the other options involve operations that do not satisfy the Cauchy-Riemann equations or are not defined for all complex numbers.

Question 34

F(z) is a function of the complex variable z = x + iy given by F(z) = iz + k Re(z) + i Im(z). For what value of k will F(z) satisfy the Cauchy-Riemann equations?

Accepted Answer

1. F=iz+k Re(z)+i Im(z)=(kx-y)+i(x+y), so u=kx-y, v=x+y. Cauchy-Riemann requires u_x=v_y => k=1 (and u_y=-v_x gives -1=-1, satisfied). Hence k=1.

Question 35

Let C represent the unit circle in the complex variable, z = x + iy. If f = cosh 3z, then ∮_C f(z) dz = ?

Accepted Answer

0. The integral of a holomorphic function over a closed contour is zero if the function has no singularities inside the contour. Since cosh(3z) is an entire function (holomorphic everywhere), the integral over the unit circle is zero.

Question 36

An analytic function f(z) of complex variable z = x + iy may be written as f(z) = u(x,y) + iv(x,y). Then, u(x,y) and v(x,y) must satisfy

Accepted Answer

∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x. The correct option is right because it reflects the Cauchy-Riemann equations, which are necessary conditions for a function to be analytic. These equations establish a relationship between the partial derivatives of the real part u and the imaginary part v of the function, ensuring that the function is differentiable in the complex sense.

Question 37

If φ(x,y) and ψ(x,y) are functions with continuous second derivatives, then φ(x,y) + iψ(x,y) can be expressed as an analytic function of x + iy (i = √−1), when

Accepted Answer

∂φ/∂y = −∂ψ/∂x; ∂φ/∂x = ∂ψ/∂y. The correct option states the Cauchy-Riemann equations, which are necessary conditions for a function to be analytic. These equations ensure that the real and imaginary parts of the function are related in a way that allows for differentiability in the complex sense.

Question 38

The modulus of the complex number |(3 + 4i)/(1 − 2i)| is

Accepted Answer

√5. |3+4i|=5 and |1-2i|=sqrt(5). The modulus of the quotient is 5/sqrt(5)=sqrt(5). The stored 1/sqrt(5) inverts the ratio.

Question 39

The product of two complex numbers 1 + i and 2 − 5i is

Accepted Answer

7 − 3i. To find the product of the complex numbers 1 + i and 2 - 5i, we use the distributive property (FOIL method), which results in 1*2 + 1*(-5i) + i*2 + i*(-5i). This simplifies to 2 - 5i + 2i + 5, combining like terms gives 7 - 3i.

Question 40

The argument of the complex number (1 + i)/(1 - i), where i = √(-1), is

Accepted Answer

π/2. To find the argument of the complex number (1 + i)/(1 - i), we can simplify it by multiplying the numerator and denominator by the conjugate of the denominator, resulting in a complex number whose argument can be determined using the arctangent function. The final result shows that the angle corresponding to this complex number is π/2, indicating it lies on the positive imaginary axis.

Question 41

An analytic function of a complex variable z = x + i y is expressed as f(z) = u(x, y) + i v(x, y), where i = √−1. If u(x, y) = x² − y², then expression for v(x, y) in terms of x, y and a general constant c would be

Accepted Answer

2xy + c. With u = x^2 - y^2, u_x = 2x = v_y gives v = 2xy + g(x), and u_y = -2y = -v_x forces g'(x)=0. Hence v = 2xy + c (index 2), not xy + c.

Question 42

The value of the integral

∫ from −∞ to ∞ of [sin x / (x² + 2x + 2)] dx

evaluated using contour integration and the residue theorem is

Accepted Answer

−π sin(1)/e. The integral evaluates to −π sin(1)/e because the integrand has poles that contribute to the residue calculation, and the sine function in the numerator leads to a negative sign in the final result when applying the residue theorem.

Question 43

Let f(z) be an analytic function, where z = x + iy. If the real part of f(z) is cosh x cos y, and the imaginary part of f(z) is zero for y = 0, then f(z) is

Accepted Answer

cosh z. The function f(z) is determined to be cosh z because it is analytic and its real part matches the expression for cosh x cos y, while the imaginary part being zero for y = 0 indicates that the function does not have any imaginary component at that line, consistent with the properties of the hyperbolic cosine function.

GATE Engineering Mathematics: Complex Variables questions with solutions

Questions