Complex Analysis MCQ

Test your knowledge with important Complex Analysis MCQ and their applications. These MCQs are beneficial for competitive exams too. Explore 30 + more Complex Analysis MCQs at Bissoy

• Let f(z)=z4+a1z3+a2z2+a3z+a4=0; a1, a2, a3, a4 being real and non-zero. If f has a purely imaginary root, then what is the value of the expression a3/(a1a2)+ a1a4/(a2a3) ?
  ক

  0

  খ

  1

  গ

  -2

  ঘ

  2
• For a∈R, let f(z)=z5-5z+a. Select the correct statement for α satisfying f(α)=0.
  ক

  α has exactly three possible real values for a>4

  খ

  α has exactly one possible real value for a>4

  গ

  α has exactly three possible real values for a<-4

  ঘ

  α has exactly one possible real value for -4<a<4
• Let f(z)=&vert;1–z&vert;, if zk=cos(2kπ/10)+isin(2kπ/10), then find the value of f(z1)×f(z2)×…×f(z9).
  ক

  10

  খ

  15

  গ

  20

  ঘ

  30
• Consider a function f(z) of degree two, having real coefficients. If z1 and z2 satisfying f(z1)=f(z2)=0 are such that Re z1=Re z2=0 and if z3 satisfies f(f(z3))=0, then select the correct statement.
  ক

  Re z3=0

  খ

  Im z3=0

  গ

  Re z3×Imz3≠0

  ঘ

  Re z3=0 and Im z3=0
• Let f(z)=&vert;z&vert;2+Re z(2(z+z̅)+3(z-z̅)/2i, the find the maximum value of &vert;z&vert;2/f(z).
  ক

  1

  খ

  2

  গ

  3

  ঘ

  4
• Find the range of the function defined by f(z)=Re. &Union; &Union;
  ক

  . ] (−∞, 0) &Union; (0, ∞)

  খ

  2iz/(1-z2)

  গ

  2iz/(1-z2)

  ঘ

  2iz/(1-z2)
• Define f(z)=z2+bz−1=0 and g(z)=z2+z+b=0. If there exists α satisfying f(α)=g(α)=0, which of the following cannot be a value of b?
  ক

  √3i

  খ

  -√3i

  গ

  0

  ঘ

  √3i/2
• Let f(z)=arg 1/(1 – z), then find the range of f(z) for &vert;z&vert;=1, z≠1.
  ক

  (-∞, π/2)

  খ

  (-π/2, π/2)

  গ

  (-∞, ∞)

  ঘ

  [0, π/2)
• Let f(z)=(z2–z–1)7. If α2+α+1=0 and Im(α)>0, then find f(α).
  ক

  128α

  খ

  -128α

  গ

  128α2

  ঘ

  -128α2
• For the function f(z)=zi, what is the value of &vert;f(ω)&vert;+Arg f(ω), ω being the cube root of unity with Im(ω)>0?
  ক

  e-2π/3

  খ

  e2π/3

  গ

  e-2π/3+2π/3

  ঘ

  e-2π/3-2π/3
• Let f(z)=z+1/z. What will be the definition of this function in polar form?
  ক

  (r+1/r)cosθ+i(r-1/r)sinθ

  খ

  (r-1/r)cosθ+i(r+1/r)sinθ

  গ

  (r+1/r)sinθ+i(r-1/r)cosθ

  ঘ

  (r+1/r)sinθ+i(r-1/r)cosθ
• Find the domain of the function defined by f(z)=z/(z+z̅).
  ক

  Im(z)≠0

  খ

  Re(z)≠0

  গ

  Im(z)=0

  ঘ

  Re(z)=0
• The area of the region enclosed by the curve zz̅+a(z̅+z)+a=0 is 2π. If a2–7a+10=0, find the area of the region enclosed by the curve zz̅+2a(z̅+z)+a=0.
  ক

  4π sq. units

  খ

  10π sq. units

  গ

  14π sq. units

  ঘ

  22π sq. units
• Describe the region given by &vert;z-i&vert;z&vert;&vert;-&vert;z+i&vert;z&vert;&vert;=0.
  ক

  real axis

  খ

  imaginary axis

  গ

  circle centered at origin

  ঘ

  quadrant 2
• Find the locus of z/(1-z2), where z lies on the circle of radius 1 centered at origin and z≠±1.
  ক

  line not passing through origin

  খ

  &vert;z&vert;=√2

  গ

  real axis

  ঘ

  imaginary axis
• Find the area of the region bounded by arg&vert;z&vert;≤π/4 and &vert;z-1&vert;<&vert;z-3&vert;.
  ক

  1 sq. units

  খ

  2 sq. units

  গ

  3 sq. units

  ঘ

  4 sq. units
• Given a vertex of the square circumscribing the circle &vert;z-1&vert;=√2 as 2+√3i, which of the following is not a vertex of this square ?
  ক

  (1-√3)+i

  খ

  –i√3

  গ

  (√3+i)-i

  ঘ

  i√3
• Find the area enclosed by the curve formed by iz3+z2–z+i=0.
  ক

  π/2

  খ

  π

  গ

  3π/4

  ঘ

  2π
• Consider the shape formed by the set of points z=ω-1/ω, where &vert;ω&vert;=2. Which of the following is incorrect?
  ক

  eccentricity=4/5

  খ

  &vert;z&vert;≤3

  গ

  shape is an ellipse

  ঘ

  major axis is of length=5/2
• On the arg and plane, the complex numbers z1, z2, z3, z4 are the vertices of a parallelogram. Evaluate (z4–z1+z2)/z3.
  ক

  1

  খ

  2

  গ

  3

  ঘ

  4
• Find the area of the region given by 11≤&vert;z&vert; ≤ 19.
  ক

  120π sq. units

  খ

  180π sq. units

  গ

  240π sq. units

  ঘ

  320π sq. units
• What is the shape of the region formed by the set of complex numbers z satisfying &vert;z-ω&vert;≤ α?
  ক

  circle of radius ω

  খ

  circle with center ω

  গ

  disk of radius α

  ঘ

  disk with center α
• Which of the following is not equal to (-1)1/3?
  ক

  -1

  খ

  (-√3+i)/(2i)

  গ

  (√3+i)/(2i)

  ঘ

  (√3–i)/(2i)
• Find ∑r=1(ar+b) ωr-1 if ω is a complex nth root of unity.
  ক

  n(n+1)a/2

  খ

  ωr-1 if ω is a complex nth root of unity. a) n(n+1)a/2 [ nb/(1-n)

  গ

  na/(ω-1)

  ঘ

  n(n+1)a/(ω-1)
• If ω is the complex cube root of unity, then which among the following is a factor of the polynomial x6+ 4x5+3x4+2x3+x+1?
  ক

  x+ω

  খ

  x+ω2

  গ

  (x+ω)(x+ω2)

  ঘ

  (x–ω)(x–ω2)
• Find the cube root of 8i lying in the first quadrant of the complex plane.
  ক

  i-√3

  খ

  2i+√3

  গ

  i+2√3

  ঘ

  i+√3
• Let ω and ω2 be the non-real cube roots of unity and 1/(a+ω)+1/(b+ω)+1/(c+ω)=2ω2 and 1/(a+ω2)+1/(b+ω2)+1/(c+ω2)=2ω, then calculate 1/(a+1)+1/(b+1)+1/(c+1).
  ক

  1

  খ

  2

  গ

  3

  ঘ

  4
• Find the possible value(s) of Re(i1/2)+&vert;Im(i1/2)&vert;.
  ক

  -1, 1

  খ

  0, √2

  গ

  0, 1

  ঘ

  1, √2
• If α,β,ȣ are the roots of equation x3–3x2+3x+7=0 and ω is cube root of unity, then evaluate (α-1)/(β-1)+(β-1)/(ȣ-1)+(ȣ-1)/(α-1).
  ক

  ω

  খ

  ω2

  গ

  3ω

  ঘ

  3ω2
• Find the value of (1+ω)(1+ω2)(1+ω4)(1+ω8)…to 2n factors.
  ক

  1

  খ

  2

  গ

  3

  ঘ

  4