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

If $(\cos  \theta  + i  \sin  \theta)(\cos  2 \theta 
+ i  \sin  2  \theta) ... (\cos  n  \theta + i  \sin  n  \theta) = 1$, then the value of $\theta$ is , $m\in N$  

  1. $4m\pi$
  2. $\displaystyle \frac{2m\pi}{n(n+1)}$
  3. $\displaystyle \frac{4m\pi}{n(n+1)}$
  4. $\displaystyle \frac{m\pi}{n(n+1)}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Changing the above expression to Eular's form, we get
$e^{i\theta}e^{2i\theta}e^{3i\theta}...e^{in\theta})=1$
$e^{i(\theta+2\theta+3\theta+...n\theta}=1$
$e^{i\cfrac{n(n+1)}{2}\theta}=e^{2m\pi}$
Therefore, simplifying we get
$\dfrac{n(n+1)}{2}\theta=2m\pi$
$\theta=\dfrac{4m\pi}{n(n+1)}$

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

If $z _1$ and $z _2$ are the complex roots of the equation $(x-3)^3+1 = 0$, then $z _1 + z _2$ equals to 

  1. 1

  2. 3

  3. 5

  4. 7
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$cis\left( \theta  \right) =\cos { \theta  } +i\sin { \theta  } $
De Moivre's Theorem for fractional power:
${ \left( cis\theta  \right)  }^{ \frac { 1 }{ n }  }=cis\left( \frac { 2k\Pi +\theta  }{ n }  \right) $

${ \left( x-3 \right)  }^{ 3 }+1=0$
$\Longrightarrow x=3+{ \left( cis\left( \Pi  \right)  \right)  }^{ \frac { 1 }{ 3 }  }$
$x=3+{ \left( cis\left( \frac { 2k\Pi +\Pi  }{ 3 }  \right)  \right)  }$      ...{De Moivre's Theorem}
Where, $k=0,1,2$
for  $k=0$,
$x _{ 1 }=3+cis\left( \frac { \Pi  }{ 3 }  \right)$ 

for $k=1$,
$x _{ 2 }=3+cis\left( \Pi  \right) $

for $k=2,$
$x _{ 3 }=3+cis\left( \frac { 5\Pi  }{ 3 }  \right) $

$\Longrightarrow { x } _{ 1 }+{ x } _{ 3 }=6+cis\left( \frac { \Pi  }{ 3 }  \right) +cis\left( \frac { 5\Pi  }{ 3 }  \right) \ \Longrightarrow { x } _{ 1 }+{ x } _{ 3 }=7$
 
Ans: D

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

Given $z$ is a complex number with modulus $1$. Then the equation $\dfrac{(1+ia)}{(1-ia)}$ = $z$ has

  1. all roots real and distinct

  2. two real and one imaginary

  3. three roots real and one imaginary

  4. one root real and three imaginary
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\displaystyle { \left( \frac { 1+ia }{ 1-ia }  \right)  }^{ 4 }=z\quad $         ...(1)
$\displaystyle & \quad \left| z \right| =1$
$\displaystyle z=cisA=\cos { A } +i\sin { A } $
Substitute $z$  in equation (1)
$\displaystyle { \left( \frac { 1+ia }{ 1-ia }  \right)  }={ cisA }^{ \frac { 1 }{ 4 }  }=cis\frac { 2k\pi +A }{ 4 } $       ...{De Moivre's Theorem}
where $ k=0,1,2,3$

Let $\displaystyle B=\frac { 2k\pi +A }{ 4 } $

$\displaystyle \Longrightarrow ia=\frac { -1+cisB }{ 1+cisB } =\frac { \sin { \frac { B }{ 2 } \left( i\cos { \frac { B }{ 2 }  } -\sin { \frac { B }{ 2 }  }  \right)  }  }{ \cos { \frac { B }{ 2 } \left( \cos { \frac { B }{ 2 }  } +i\sin { \frac { B }{ 2 }  }  \right)  }  } $

$\displaystyle \Longrightarrow ia=\frac { i\sin { \frac { B }{ 2 } \left( \cos { \frac { B }{ 2 }  } +i\sin { \frac { B }{ 2 }  }  \right)  }  }{ \cos { \frac { B }{ 2 } \left( \cos { \frac { B }{ 2 }  } +i\sin { \frac { B }{ 2 }  }  \right)  }  } $

$\displaystyle \Longrightarrow a=\tan { \frac { B }{ 2 }  } $

Therefore roots are real and distinct.

Ans: A

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

If ${ x }^{ 6 }={ \left( 4-3i \right)  }^{ 5 }$, then the product of all of its roots is (where $\displaystyle \theta =-\tan ^{ -1 }{ \frac { 3 }{ 4 }  } $)

  1. ${ 5 }^{ 5 }\left( \cos { 5\theta } +i\sin { 5\theta } \right) $
  2. $-{ 5 }^{ 5 }\left( \cos { 5\theta } +i\sin { 5\theta } \right) $
  3. ${ 5 }^{ 5 }\left( \cos { 5\theta } -i\sin { 5\theta } \right) $
  4. $-{ 5 }^{ 5 }\left( \cos { 5\theta } -i\sin { 5\theta } \right) $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\displaystyle { x }^{ 6 }={ \left( 4-3i \right)  }^{ 5 }\Rightarrow { x }^{ 6 }={ 5 }^{ 6 }\left( \frac { 4 }{ 5 } -\frac { 3i }{ 5 }  \right) ={ 5 }^{ 5 }{ \left( \cos { \theta  } +i\sin { \theta  }  \right)  }^{ 5 }$

where $\displaystyle \theta =-\tan ^{ -1 }{ \frac { 3 }{ 4 }  } ={ 5 }^{ 5 }\left( \cos { 5\theta  } +i\sin { 5\theta  }  \right) $
$\displaystyle x={ 5 }^{ 5/6 }{ \left( \cos { 5\theta  } +i\sin { 5\theta  }  \right)  }^{ 1/6 }={ 5 }^{ 5/6 }\left[ \cos { \left( \frac { 2k\pi +5\theta  }{ 6 }  \right)  } +i\sin { \left( \frac { 2k\pi +5\theta  }{ 6 }  \right)  }  \right] $
${ x } _{ 1 }{ x } _{ 2 }{ x } _{ 3 }...{ x } _{ 6 }={ 5 }^{ 5 }\left( \cos { \left( 5\pi +5\theta  \right)  } +i\sin { \left( 5\pi +5\theta  \right)  }  \right) \ ={ 5 }^{ 5 }\left( -\cos { 5\theta  } -i\sin { 5\theta  }  \right) =-{ 5 }^{ 5 }\left( \cos { 5\theta  } +i\sin { 5\theta  }  \right) $

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 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 value of the expression (2 + 3i)(4 - 5i)?

  1. 22 - 11i

  2. 22 + 11i

  3. 14 - 22i

  4. 14 + 22i
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Using the distributive property, we can expand the expression as follows: (2 + 3i)(4 - 5i) = 2(4 - 5i) + 3i(4 - 5i) = 8 - 10i + 12i - 15i^2 = 8 + 2i - 15(-1) = 22 + 11i.

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}}).