Mathematics

Complex Variables and Numbers

157 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 Z-Transform of the sequence (x[n] = n \cos(\omega_0 n))?

  1. \(X(z) = \frac{z(z - \cos(\omega_0))}{(z - \cos(\omega_0))^2 + \sin^2(\omega_0)}\)
  2. \(X(z) = \frac{z(z + \cos(\omega_0))}{(z + \cos(\omega_0))^2 + \sin^2(\omega_0)}\)
  3. \(X(z) = \frac{z}{(z - \cos(\omega_0))^2 + \sin^2(\omega_0)}\)
  4. \(X(z) = \frac{1}{(z - \cos(\omega_0))^2 + \sin^2(\omega_0)}\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Z-Transform of (x[n] = n \cos(\omega_0 n)) is (X(z) = \sum_{n=0}^\infty n \cos(\omega_0 n) z^{-n} = \frac{z(z - \cos(\omega_0))}{(z - \cos(\omega_0))^2 + \sin^2(\omega_0)}).

Multiple choice

What is the Z-Transform of the sequence (x[n] = \left{\begin{array}{ll} 1, & n = 0\ -1, & n \text{ is odd}\ 0, & n \text{ is even and } n \ne 0 \end{array}\right.)?

  1. \(X(z) = \frac{1 - z^{-1}}{1 + z^{-1}}\)
  2. \(X(z) = \frac{1 + z^{-1}}{1 - z^{-1}}\)
  3. \(X(z) = \frac{1}{1 + z^{-1}}\)
  4. \(X(z) = \frac{1}{1 - z^{-1}}\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Z-Transform of (x[n]) is (X(z) = \sum_{n=0}^\infty x[n] z^{-n} = 1 - z^{-1} - z^{-3} - z^{-5} + \cdots = \frac{1 - z^{-1}}{1 + z^{-1}}).

Multiple choice

What is the Z-Transform of the sequence (x[n] = \left{\begin{array}{ll} 1, & n = 0\ 2, & n = 1\ 3, & n = 2\ 4, & n = 3\ 0, & \text{otherwise} \end{array}\right.)?

  1. \(X(z) = \frac{1 + 2z^{-1} + 3z^{-2} + 4z^{-3}}{1 - z^{-1}}\)
  2. \(X(z) = \frac{1 + 2z^{-1} + 3z^{-2} + 4z^{-3}}{1 + z^{-1}}\)
  3. \(X(z) = \frac{1 - 2z^{-1} + 3z^{-2} - 4z^{-3}}{1 - z^{-1}}\)
  4. \(X(z) = \frac{1 - 2z^{-1} + 3z^{-2} - 4z^{-3}}{1 + z^{-1}}\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Z-Transform of (x[n]) is (X(z) = \sum_{n=0}^\infty x[n] z^{-n} = 1 + 2z^{-1} + 3z^{-2} + 4z^{-3} = \frac{1 + 2z^{-1} + 3z^{-2} + 4z^{-3}}{1 - z^{-1}}).

Multiple choice

What is the Riemann hypothesis?

  1. The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at negative even integers and complex numbers with real part 1/2.

  2. The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at negative odd integers and complex numbers with real part 1/2.

  3. The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at negative even integers and complex numbers with real part 3/4.

  4. The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at negative odd integers and complex numbers with real part 3/4.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Riemann hypothesis is one of the most famous unsolved problems in mathematics. It states that the Riemann zeta function has its zeros only at negative even integers and complex numbers with real part 1/2.

Multiple choice

What is the Riemann hypothesis?

  1. The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at negative even integers and complex numbers with real part 1/2.

  2. The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at negative odd integers and complex numbers with real part 1/2.

  3. The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at positive even integers and complex numbers with real part 1/2.

  4. The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at positive odd integers and complex numbers with real part 1/2.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at negative even integers and complex numbers with real part 1/2. The Riemann zeta function is a function that is defined for all complex numbers except for 1, and it is related to the distribution of prime numbers.

Multiple choice

The equation (\zeta(2) = \frac{\pi^2}{6}) is known as:

  1. Riemann zeta function

  2. Goldbach's conjecture

  3. Catalan's conjecture

  4. Pólya's conjecture
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation (\zeta(2) = \frac{\pi^2}{6}) is a special case of the Riemann zeta function, which is a function that is defined for complex numbers.

Multiple choice

What is the unity element of the ring (ℤ/6ℤ, +, ×)?

  1. 0

  2. 1

  3. 2

  4. 3
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The unity element of a ring is the element that acts as the multiplicative identity. In the ring (ℤ/6ℤ, +, ×), the unity element is 1, since 1 × a = a × 1 = a for any element a in the ring.

Multiple choice

Which of the following is the imaginary unit?

  1. $i$
  2. $\pi$
  3. $\sqrt{-1}$
  4. $\infty$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The imaginary unit, denoted by $i$, is defined as the square root of -1, i.e., $i = \sqrt{-1}$. It is used to represent imaginary numbers, which are numbers that have a real part of 0.

Multiple choice

What is the Cauchy-Riemann equation in complex analysis?

  1. $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
  2. $\frac{\partial u}{\partial x} = -\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = \frac{\partial v}{\partial x}$
  3. $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = \frac{\partial v}{\partial x}$
  4. $\frac{\partial u}{\partial x} = -\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Cauchy-Riemann equation is a system of two partial differential equations that are necessary and sufficient conditions for a complex function to be differentiable at a point.

Multiple choice

What is the Laurent expansion of a complex function around a point?

  1. An infinite series representation of the function in terms of powers of $z - a$
  2. An infinite series representation of the function in terms of powers of $z$
  3. An infinite series representation of the function in terms of powers of $1/z$
  4. An infinite series representation of the function in terms of powers of $1/(z - a)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Laurent expansion of a complex function around a point is an infinite series representation of the function in terms of powers of $z - a$.

Multiple choice

What is the imaginary unit in complex numbers?

  1. i

  2. j

  3. k

  4. 1
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The imaginary unit in complex numbers is denoted by 'i' and is defined as the square root of -1, i.e., (i = \sqrt{-1}).

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).