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 de moivre’s theorem and its applications demoivre's theorem complex numbers maths


Let $\mathrm{z}=\cos\theta+\mathrm{i}\sin\theta$. Then the value of $\displaystyle \sum _{\mathrm{m}=1}^{15}{\rm Im}(\mathrm{z}^{2\mathrm{m}-1})$ at $\theta =2^{\mathrm{o}}$ is

  1. $\displaystyle \frac{1}{\sin 2^{\mathrm{o}}}$
  2. $\displaystyle \frac{1}{3\sin 2^{\mathrm{o}}}$
  3. $\displaystyle \frac{1}{2\sin 2^{\mathrm{o}}}$
  4. $\displaystyle \frac{1}{4\sin 2^{\mathrm{o}}}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\mathrm{z}=\cos\theta+\mathrm{i}\sin\theta$
$\Rightarrow z^{2m-1} = \cos(2m-1)\theta+i\sin (2m-1)\theta$


Let $X = \displaystyle \sum _{\mathrm{m}=1}^{15}{\rm Im}(\mathrm{z}^{2\mathrm{m}-1})$

$\therefore \mathrm{X}=\sin\theta+\sin 3\theta+\ldots+\sin 29\theta$


$ 2(sin\theta)X=2\sin\theta\sin\theta + 2\sin\theta\sin3\theta...........+2\sin\theta\sin29\theta$
$\Rightarrow  2(\sin\theta)\mathrm{X}=1-\cos 2\theta+\cos 2\theta-\cos 4\theta+\ldots+\cos 28\theta-\cos 30\theta$

$\displaystyle \therefore  \mathrm{X}=\frac{1-\cos 30\theta}{2\sin\theta}=\frac{1}{4\sin 2^{\mathrm{o}}}$

Multiple choice

What is the name of the theorem that Shorey and C. L. Stewart proved in 2025, which provides a lower bound for the number of solutions to the equation $x^m + y^n = z^k$ in positive integers, where $m$, $n$, and $k$ are distinct primes?

  1. The Shorey-Stewart Theorem

  2. The Baker-Stewart Theorem

  3. The Siegel-Stewart Theorem

  4. The Fermat-Stewart Theorem
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Shorey-Stewart Theorem states that the number of solutions to the equation $x^m + y^n = z^k$ in positive integers, where $m$, $n$, and $k$ are distinct primes, is at least $c^{1/m} + c^{1/n} + c^{1/k} - 5$.

Multiple choice

What is the degree of the divisor $(x-1)(x-2)$ on the Riemann surface $\mathbb{C}/\mathbb{Z}$?

  1. 1

  2. 2

  3. 3

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

The degree of a divisor on a Riemann surface is the sum of the orders of its zeros minus the sum of the orders of its poles. In this case, the divisor $(x-1)(x-2)$ has two zeros at $x=1$ and $x=2$, and no poles. So its degree is $2$.

Multiple choice

What is the degree of the divisor $(x-1)^2$ on the Riemann surface $\mathbb{C}/\mathbb{Z}$?

  1. 1

  2. 2

  3. 3

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

The degree of a divisor on a Riemann surface is the sum of the orders of its zeros minus the sum of the orders of its poles. In this case, the divisor $(x-1)^2$ has two zeros at $x=1$, and no poles. So its degree is 2.

Multiple choice

What is the degree of the divisor $(x-1)^3$ on the Riemann surface $\mathbb{C}/\mathbb{Z}$?

  1. 1

  2. 2

  3. 3

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

The degree of a divisor on a Riemann surface is the sum of the orders of its zeros minus the sum of the orders of its poles. In this case, the divisor $(x-1)^3$ has three zeros at $x=1$, and no poles. So its degree is 3.

Multiple choice

What is the Z-Transform of the sequence (x[n] = a^n), where (a) is a constant?

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

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

Multiple choice

What is the Z-Transform of the unit step sequence (u[n])?

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

The Z-Transform of (u[n]) is (U(z) = \sum_{n=0}^\infty z^{-n} = \frac{1}{1 - z^{-1}}).

Multiple choice

What is the Z-Transform of the sequence (x[n] = n)?

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

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

Multiple choice

What is the Z-Transform of the sequence (x[n] = \sin(\omega_0 n))?

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

The Z-Transform of (x[n] = \sin(\omega_0 n)) is (X(z) = \sum_{n=0}^\infty \sin(\omega_0 n) z^{-n} = \frac{z \sin(\omega_0)}{z^2 - 2z \cos(\omega_0) + 1}).

Multiple choice

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

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

The Z-Transform of (x[n] = \cos(\omega_0 n)) is (X(z) = \sum_{n=0}^\infty \cos(\omega_0 n) z^{-n} = \frac{z^2 - 1}{z^2 - 2z \cos(\omega_0) + 1}).

Multiple choice

What is the Z-Transform of the sequence (x[n] = e^{\alpha n}), where (\alpha) is a constant?

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

The Z-Transform of (x[n] = e^{\alpha n}) is (X(z) = \sum_{n=0}^\infty e^{\alpha n} z^{-n} = \frac{z}{z - e^{\alpha}}).

Multiple choice

What is the Z-Transform of the sequence (x[n] = n^2)?

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

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

Multiple choice

What is the Z-Transform of the sequence (x[n] = \cos(\omega_0 n) + j \sin(\omega_0 n))?

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

The Z-Transform of (x[n] = \cos(\omega_0 n) + j \sin(\omega_0 n)) is (X(z) = \sum_{n=0}^\infty (\cos(\omega_0 n) + j \sin(\omega_0 n)) z^{-n} = \frac{z(z - \cos(\omega_0))}{z^2 - 2z \cos(\omega_0) + 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\ 0, & \text{otherwise} \end{array}\right.)?

  1. \(X(z) = \frac{z^2 + 2z + 3}{z^3}\)
  2. \(X(z) = \frac{z^2 - 2z + 3}{z^3}\)
  3. \(X(z) = \frac{z^2 + 2z - 3}{z^3}\)
  4. \(X(z) = \frac{z^2 - 2z - 3}{z^3}\)
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} = \frac{z^2 + 2z + 3}{z^3}).

Multiple choice

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

  1. \(X(z) = \frac{1}{1 - z^{-2}}\)
  2. \(X(z) = \frac{z}{1 - z^{-2}}\)
  3. \(X(z) = \frac{1}{1 + z^{-2}}\)
  4. \(X(z) = \frac{z}{1 + z^{-2}}\)
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^{-2} + z^{-4} + \cdots = \frac{1}{1 - z^{-2}}).