ME – 2007
1. If x y and (x, y) are functions with continuous second derivatives, then x y + i (x, y)
3. An analytic function of a complex variable z = x + iy is expressed as counter clock wise direction. The integral is equal to
8. An analytic function of a complex variable z x + i y is expressed as
9. 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 = √ . If u (x, y) = x – y , then expression for v(x, y) in terms of x, y and a general constant c would be
(A) xy +
CE – 2005
1. Which one of the following is NOT true for complex number and ?
(A) = ̅̅̅̅
| |
(B) | + | ≤ | | + | | (C) | | ≤ | | | | (D) | + | + | |
2| | + 2| |
2. Consider likely applicable of u hy’s integral theorem to evaluate the following integral counter clockwise around the unit circle c.
∮ s z z
z being a complex variable. The value of I will be
(A) I = 0: singularities set = ϕ (B) I = 0: singularities set
= , π n 2 - (C) I π/2: singul riti s s t
{ nπ n 2 } (D) None of the above CE – 2006
3. Using Cauchy’s is integral theorem, the value of the integral (integration being taken in counter clockwise direction)
∮ dz is (A)
4πi (B) πi
(C)
πi (D) 1 CE – 2009
4. The analytic function f(z) =
has singularities at
(A) 1 and 1 (B) 1 and i
(C) 1 and i (D) i and i 5. The value of the integral ∫ dz
(where C is a closed curve given by
|z| = 1) is (A) –πi (B)
(C) (D) πi
CE – 2011
6. For an analytic function,
f(x + iy) = u(x, y)+iv(x, y), u is given by u = 3x 3y . The expression for v considering K to be a constant is
(A) 3y 3x + k (B) 6y – 6x + k
(C) 6x 6y+k (D) 6xy +k
CE – 2014
7. z can be expressed as (A) i
(B) + i
(C) i (D) + i ECE – 2006
1. The value of the contour integral
∮| | z in positive sense is (A)
(B)
(C) (D)
2. For the function of a complex variable W = In Z (where, W = u + jv and
Z = x + jy), the u = constant lines get mapped in Z-plane as
(A) set of radial straight line (B) set of concentric circles (C) set of confocal hyperbolas (D) set of confocal ellipses ECE – 2007
3. If the semi-circular contour D of radius 2 is as shown in the figure, then the value of the integral ∮ is
(A) jπ (B) jπ
(C) π (D) π j2
j2
2
j
ECE – 2008
4. The residue of the function
21
2f z z 2 z 2
at z=2 is
(A) (B)
(C) (D) 5. The equation sin(z)=10 has
(A) no real or complex solution (B) exactly two distinct complex
solutions
(C) a unique solution
(D) an infinite number of complex solutions
ECE – 2009
6. If f(z) = + z , then ∮
z is given by
(A) 2π (B) 2π +
(C) 2πj
(D) 2πj + ECE – 2010
7. The residues of a complex function z at its poles are
(A) and 1 (B) and
(C) and (D) and
ECE – 2011
8. The value of the integral ∮ z where is the circle |z| is given by (A) 0
(B) 1/10
(C) 4/5 (D) 1 ECE\EE\IN – 2012
9. If x = √ then the value of x is (A) ⁄
(B) ⁄
(C) x (D) 1
10. Given f (z)
. If C is a counterclockwise path in the z – plane such that |z+1| =1, the value of
∮ z z is
(A) 2 (B)
(C) (D) 2 ECE – 2014
11. C is a closed path in the z-plane given by
|z|=3. The value of the integral
∮ ( ) z is (A) 4π + j2 (B) 4π j2
(C) 4π + j2 (D) 4π j2 12. The real part of an analytic function z
where z x + jy is given by cos(𝑥).
The imaginary part of z is (A) os x
(B) sin x
(C) sin x (D) sin x EE – 2007
1. The value of ∮ where C is the contour |z-i/2| = 1 is
(A) 2πi (B) π
(C) t n z (D) πi t n z EE – 2011
2. A point z has been plotted in the complex plane, as shown in figure below.
lmlm nit ir l
y
lm nit ir l
y
lm nit ir l
y
lm nit ir l
y
nit ir l
z
EE – 2013
3. ∮ z evaluated anticlockwise around the circle |z i| 2 where i √ , is (A) 4π
(B)
(C) 2 + π (D) 2 +2i 4. Square roots of – i, where i = √ , are
(A) i, i
(B) os ( ) + i sin ( ) os ( ) + i sin ( ) (C) os ( ) + i sin ( )
os ( ) + i sin ( ) (D) os ( ) + i sin ( )
os ( ) + i sin ( )
EE – 2014
5. Let S be the set of points in the complex plane corresponding to the unit circle.
(That is, {z: |z| } . 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
(A) unit circle
(B) horizontal axis line segment from origin to (1, 0)
(C) the point (1, 0)
(D) the entire horizontal axis
6. All the values of the multi-valued complex function , where i √ are
(A) purely imaginary.
(B) real and non-negative.
(C) on the unit circle.
(D) equal in real and imaginary parts.
7. Integration of the complex function z
, in the counter clockwise direction, around |z 1| = 1, is
(A) πi (B)
(C) πi (D) 2πi
IN – 2005
1. Consider the circle | | 2 in the complex plane (x, y) with z = x + iy. The minimum distant form the origin to the circle is
(A) √2 2 (B) √ 4
(C) √ 4 (D) √2
2. Let ̅, where z is a complex number not equal to zero. The z is a solution of (A) z
(B) z
(C) z (D) z IN – 2006
3. The value of the integral of the complex function
f(s) 3s 4
(s 1)(s 2)
Along the path s 3is (A) 2j
(B) 4j
(C) 6j (D) 8j
IN – 2007
4. For the function of a complex variable z, the point z=0 is
(A) a pole of order 3 (B) a pole of order 2 (C) a pole of order 1 (D) not a singularity
5. Let j = √ .Then one value of is (A) √j
(B) 1
(C) (D) IN – 2008
6. A complex variable x + j has its real part x varying in the range to + . Which one of the following is the locus (shown in thick lines) of 1/Z in the complex plane?
IN – 2009
7. The value of ∮ where the contour of integration is a simple closed curve around the origin, is
(A) 0 (B) 2πj
(C) (D)
8. If z = x+jy, where x and y are real. The value of | | is
(A) 1 (B) √
(C) (D)
9. One of the roots of the equation 𝑥 =j, where j is positive square root of 1, is (A) j
(B) √ + j
(C) √ j (D) √ j
IN – 2010
10. The contour C in the adjoining figure is described by x + y . The value of
∮ z is.
(Note: √ )
(A) 2πj (B) 2πj
(C) 4πj (D) 4πj IN – 2011
11. The contour integral ∮ / with C as the counter-clockwise unit circle in the z-plane is equal to
(A) 0 (B) 2π
(C) 2π√
(D) y
x pl n
m gin ry xis
l xis j
l xis j
m gin ry xis
l xis j
l xis j
m gin ry xis m gin ry xis
Answer Keys and Explanations
ME
1. [Ans. B]
By definition C-R equation holds 2. [Ans. A]
By Milne Thomson method Let w = u + iv
v y x y x + onst nt 9. [Ans. C]
iv n u x y v v
x x + v y y v
y u x
v x
u y v u
y x + u x y 2y x + 2x y
rm ont ing y t rms only llow v 2 xy +
10. [Ans. B]
∫ z
z ln z|
ln i ln ln + ln i ln ln ln + ln i
+ i (
z os + i sin z i i sin π/2
z / ln i ln z ln /
iπ
2 i ) CE
1. [Ans. C]
(A) is true since = ̅̅̅̅̅̅̅̅ = ̅̅̅̅
| |
(B) is true by triangle inequality of complex number
(C) is not true since | | ≥ | | | | (D) is true since
| + |2 = ( + ) ̅̅̅̅̅̅̅̅̅̅̅̅ +
= ( + ) (z̅ + z̅ )
= z̅ + z̅ + z̅ + z̅ i And | |2 = ( + ) ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
= ( + ) (z̅ z̅ )
= z̅ + z̅ z̅ + z̅ ii Adding (i) and (ii) we get
| + |2 + | |2 = 2 z̅ + 2 z̅ = 2| | + 2| |
2. [Ans. B]
∫ s z z ∫
os z z
The poles are at z = n + /2 π
= π/2 π/2 + π/2
None of these poles lie inside the unit circle |z| =1
Hence, sum of residues at poles = 0 Singularities set = ϕ and
2πi [sum o r si u s o t z t th poles]
2 πi 3. [Ans. A]
u hy’s int gr l th or m is f(a) = ∮ z
i.e. ∮ z 2πi Now, ∮ z = ∮
( )
pplying u hy’s int gr l th or m, using z z
.2πi (i )/
.2πi 0(i
) 1/
.2πi 0(i
) 1/
2π
i 4πi 2π
4πi 4. [Ans. D]
z z z +
z
z z z i z + i The singularities are at z = i and –i 5. [Ans. C]
r ∫ os 2πz 2z z
2∫
* +
*z +
in z is point with in |z|=1(the los urv w n us u hy’s integral theorem and say that
2[2πi (
For f to be analysis, we have Cauchy-Riemann conditions,
Now applying equation (iii) we get u Substitute in equation (iii) v= 3x2 + 6yx – 3x2 + K Since, u is constant, therefore
2 2
Only s= +1 lies inside the given contour Residues at s= +1 =lim s 1 f sS 1
4. [Ans. A] infinite number of complex solutions sin z has infinite no. of complex
12. [Ans. B]
Suppose that z u x y + iv x y is analytic then, u and v satisfy the Cauchy Riemann equation
u x
v
y n u
y v x r u x y os x
u
x sin x v y u
y os x v x v sin x
EE
1. [Ans. B]
Pole (z=i) lies inside the circle. |z-i/2|=1.
Hence
∮ z + z ∮
z + i z i z
∮ z 2 πi i , wh r z - 2 πi
2i π 2. [Ans. D]
Let + i
Since Z is shown inside the unit circle in I quadrant, a and B are both +ve and √ +
ow + i i
+ + + i
Since
√ +
+
o in qu r nt
| | √(
+ ) + (
+ )
√ + √ +
in √ +
√ +
o / is outside the unit circle is IV quadrant
3. [Ans. A]
∮z 4 z + 4
|z i| 2 z + 4 z 4 z 2i For z 2i Residue at z +2i
4 4 z + 2i
+4i +2i t z 2i li insi t z 2i li outsi o ∮z 4
z + 4 2πi sum o r si u 2πi 2i
4π 4. [Ans. B]
Let + i √ i
Squaring both sides we get + 2 i i
Equating real and imaginary parts
2
wh n 2 2 i
√2 wh n i
√2 i
√2 + i i
√2+ i ( i
√2)
√2+ i
√2 wh n i
√2 i
√2 + i i
√2+ i ( i
√2) i
√2+
√2
√2+ ( i
√2) os (π
4) + i sin ( π 4 )
or singularity in contour c
r |z | n
For distance to be min. The point P will be on the line passes through origin and centre of the circle.
Slope of line OP = Slope of line OC
Expand by Laurent series
3
5. [Ans. D]
t x j ( ⁄ )
⁄
log x log ( ⁄ )
⁄
log ⁄ log ( ⁄ ) log jπ
2 ⁄ log jπ
2 j x / 6. [Ans. B]
x + j x + j
x j x + x
x + j x +
|
j
j
|
lim
{
x +
j x + } j
ption s tis y th ov on itions 7. [Ans. A]
u hy’s int gr l ormul is
∫
Here a = 0, then f(0) = sin 0 = 0 8. [Ans. D]
z x + iy
p | | = | |
= | | = | | = 9. [Ans. B]
Given x3 = j = e+jπ/2 x ⁄
x os π
+ j sin π √
2 + j 2
10. [Ans. D]
∮ z = ∮ z Pole z j ⟹ z j
Residue at z j ⟹ 2[ j + 2
∮ z 2πj [sum o r si o pol 2πj 2 4πj
11. [Ans. C]
z ∮ ⁄ z ∮ ( +
z+ 2z +
z + ) z
The only pole of z is at z , which lies within |z|
∫ z z 2πi (residue)
Note: Residue of z at z is coefficient of z⁄ i.e. 1, here.