Mathematics

Complex Variables and Numbers

206 Questions

Complex numbers and variables form a crucial part of advanced mathematics syllabi. This topic covers roots of unity, Argand plane geometry, and Z-transforms. These concepts are frequently tested in engineering entrance exams and UPSC mathematics optional papers.

Roots of unityArgand plane geometryZ-transform sequencesExponential complex formsComplex number quadrantsRiemann hypothesis

Complex Variables and Numbers Questions

Multiple choice

What is the real part of the complex number (z = 3 + 4i)?

  1. 3

  2. 4

  3. 7

  4. 12
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The real part of a complex number is the part without the imaginary unit 'i'. Therefore, the real part of (z = 3 + 4i) is 3.

Multiple choice

What is the imaginary part of the complex number (z = 3 + 4i)?

  1. 3

  2. 4

  3. 7

  4. 12
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The imaginary part of a complex number is the part that contains the imaginary unit 'i'. Therefore, the imaginary part of (z = 3 + 4i) is 4.

Multiple choice

What is the complex conjugate of the complex number (z = 3 + 4i)?

  1. 3 - 4i

  2. 3 + 4i

  3. 6 + 8i

  4. 6 - 8i
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The complex conjugate of a complex number (z = a + bi) is (\bar{z} = a - bi). Therefore, the complex conjugate of (z = 3 + 4i) is (3 - 4i).

Multiple choice

What is the modulus (absolute value) of the complex number (z = 3 + 4i)?

  1. 5

  2. 7

  3. 12

  4. 16
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The modulus (absolute value) of a complex number (z = a + bi) is (|z| = \sqrt{a^2 + b^2}). Therefore, the modulus of (z = 3 + 4i) is (|z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5).

Multiple choice

What is the argument (phase) of the complex number (z = 3 + 4i)?

  1. \(\arctan(4/3)\)
  2. \(\arctan(3/4)\)
  3. \(\pi/4\)
  4. \(\pi/2\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The argument (phase) of a complex number (z = a + bi) is (\theta = \arctan(b/a)). Therefore, the argument of (z = 3 + 4i) is (\theta = \arctan(4/3)).

Multiple choice

What is the polar form of the complex number (z = 3 + 4i)?

  1. \(5(\cos(\arctan(4/3)) + i\sin(\arctan(4/3))\)\)
  2. \(5(\cos(\pi/4) + i\sin(\pi/4))\)
  3. \(5(\cos(\pi/2) + i\sin(\pi/2))\)
  4. \(5(\cos(\pi) + i\sin(\pi))\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The polar form of a complex number (z = a + bi) is (z = r(\cos(\theta) + i\sin(\theta))), where (r = |z|) and (\theta = \arg(z)). Therefore, the polar form of (z = 3 + 4i) is (5(\cos(\arctan(4/3)) + i\sin(\arctan(4/3)))).

Multiple choice

What is the exponential form of the complex number (z = 3 + 4i)?

  1. \(5e^{i\arctan(4/3)}\)
  2. \(5e^{i\pi/4}\)
  3. \(5e^{i\pi/2}\)
  4. \(5e^{i\pi}\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The exponential form of a complex number (z = a + bi) is (z = re^{i\theta}), where (r = |z|) and (\theta = \arg(z)). Therefore, the exponential form of (z = 3 + 4i) is (5e^{i\arctan(4/3)}).

Multiple choice

What is the value of the integral (\int_C \frac{1}{z} dz), where (C) is the unit circle centered at the origin?

  1. 0

  2. 1

  3. 2\pi i
  4. $\infty$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The value of the integral (\int_C \frac{1}{z} dz) is given by (2\pi i), where (C) is a positively oriented simple closed curve around the origin. This is known as Cauchy's integral theorem.

Multiple choice

What is the residue theorem?

  1. A theorem that gives a formula for evaluating integrals of complex functions around closed curves

  2. A theorem that gives a formula for finding the derivative of a complex function

  3. A theorem that gives a formula for finding the Taylor series expansion of a complex function

  4. A theorem that gives a formula for finding the zeros of a complex function
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The residue theorem is a powerful tool for evaluating integrals of complex functions around closed curves. It states that the value of the integral is equal to (2\pi i) times the sum of the residues of the function at its poles inside the curve.

Multiple choice

Which of the following is a fundamental theorem in complex analysis?

  1. Cauchy's Integral Formula

  2. Residue Theorem

  3. Maximum Modulus Principle

  4. Open Mapping Theorem
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Cauchy's Integral Formula provides a powerful tool for evaluating integrals of complex functions.

Multiple choice

What is the Riemann hypothesis?

  1. A conjecture that states that the Riemann zeta function has its zeros only at negative even integers and complex numbers with real part equal to 1/2.

  2. A conjecture that states that the Riemann zeta function has its zeros only at negative even integers and complex numbers with real part equal to 0.

  3. A conjecture that states that the Riemann zeta function has its zeros only at negative even integers and complex numbers with real part equal to -1/2.

  4. A conjecture that states that the Riemann zeta function has its zeros only at negative even integers and complex numbers with real part equal to -1.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Riemann hypothesis is a famous unsolved problem in number theory that states that the Riemann zeta function has its zeros only at negative even integers and complex numbers with real part equal to 1/2.