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 maths complex numbers and linear inequations argand plane and polar representation geometric representation of a complex number complex numbers and quadratic equations

Let $A$ and $B$ represent $z _{1}$ and $z _{2}$ in the Argand plane and $z _{1},z _{2}$ be the roots of the equation $z^{2}+pz+q=0$ where $p,q$ are complex numbers. If $O$ is the origin $OA=OB$ and $\angle AOB=\alpha$ then $p^{2}=$

  1. $2q\ \cos \left(\dfrac{\alpha}{2}\right)$
  2. $4q\ \cos \left(\dfrac{\alpha}{2}\right)$
  3. $4q\ \cos^{2} \left(\dfrac{\alpha}{2}\right)$
  4. $4q^{2}\ \cos^{2} \left(\dfrac{\alpha}{2}\right)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Since z1 and z2 are roots of z^2 + pz + q = 0, we have z1 + z2 = -p and z1*z2 = q. Given OA = OB, the magnitudes |z1| = |z2|, meaning |z1|^2 = |z2|^2 = q. Using the geometric relation for the angle alpha between z1 and z2 from the origin, |z1 - z2|^2 = |z1|^2 + |z2|^2 - 2|z1||z2|cos(alpha) = 2q - 2q cos(alpha). Also, (z1 - z2)^2 = (z1 + z2)^2 - 4z1z2 = p^2 - 4q. Equating and simplifying gives p^2 = 4q cos^2(alpha/2).

Multiple choice maths complex numbers and linear inequations argand plane and polar representation geometric representation of a complex number complex numbers and quadratic equations

Let  $z _ { 1 } , z _ { 2 }$  and  $z _ { 3 }$  represent the vertices  $A, B$  and  $C$  of the triangle  $A B C$  in the argand that  $\left| z _ { 1 } \right| = \left| z _ { 2 } \right| = \left| z _ { 3 } \right| = 5,$  then  $z _ { 1 } \sin 2 A + z _ { 2 } \sin 2 B + z _ { 3 } \sin 2 C = 0.$

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

This is a known identity in complex geometry for points on a circle centered at the origin. The sum of the vectors weighted by the sine of the angles relates to the geometry of the triangle inscribed in the circle.

Multiple choice maths complex numbers and linear inequations argand plane and polar representation geometric representation of a complex number complex numbers and quadratic equations

If Arg $(z + i)\, -$ Arg $(z - i)$ $= \dfrac{\pi}{2}$, then $z$ lies on a ..........

  1. Circle

  2. Line

  3. Coordinate axes

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Putting z = x+iy,

${tan}^{-1}\dfrac{y+1}{x}$  -  ${tan}^{-1}\dfrac{y-1}{x}$ = $\pi$/2

$\Rightarrow$ 1 + ($\dfrac{y+1}{x})$($\dfrac{y-1}{x}$) = 0

$\Rightarrow$ $x^2 +  y^2$ = 1

It is a circle of center coinciding with origin and radius 1 units.

Hence, option A is correct.
Multiple choice maths complex numbers and linear inequations argand plane and polar representation geometric representation of a complex number complex numbers and quadratic equations

Let $z _1$ and $z _2$ are two complex numbers such that $(1-i)z _1=2z _2$ and $arg(z _1z _2)=\dfrac{\pi}{2}$ then $arg(z _2)$ is equals to:

  1. $\dfrac{3 \pi}{8}$
  2. $\dfrac{\pi}{8}$
  3. $\dfrac{5 \pi}{8}$
  4. $\dfrac{-7 \pi}{8}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$(1-i)z _1=2z _2$

$\cfrac{z _2}{z _1}=\cfrac{1}{2}-\cfrac{i}{2}$
Let $z _1=r _1e^{i\theta _1}\ and\ z _=r _1r^{\theta _2}$
$arg(\cfrac{z _2}{z _1})=\tan^{-1}\cfrac{-1/2}{1/2}=-\pi/4$
$\theta _2-\theta _1=-\pi/4$  and $arg(z _1z _2)=\pi/2$ (given)
$\implies \theta _2-\theta _1=-\pi/4$  and $\theta _2
+\theta _1=\pi/2$
$\implies 2\theta _2=\pi/4,\theta _2=\pi/8$
$\implies arg(z _2)=\pi/8$

Multiple choice maths complex numbers and linear inequations argand plane and polar representation geometric representation of a complex number complex numbers and quadratic equations

The complex number $\dfrac{1 + 2i}{1 - i}$ lies in which quadrant of the complex plane.

  1. First

  2. Second

  3. Third

  4. Fourth
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$\dfrac{1+2i}{1-i}$
$\Rightarrow \dfrac{1+2i}{1-i}\times \dfrac{1+i}{1+i}$
$=\dfrac{1+i+2i-2i^2}{1-i^2}=\dfrac{1+3i-2}{2}$
$=\dfrac{-1+3i}{2}$
$\therefore$ It lies in $2^{nd}$ Quadrant.
Multiple choice maths complex numbers and linear inequations argand plane and polar representation geometric representation of a complex number complex numbers and quadratic equations

If $arg(z) < 0$, then $arg(-z)-arg(z)=$

  1. $\pi$
  2. $-\pi$
  3. $\dfrac{\pi}{2}$
  4. $-\dfrac{\pi}{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let $Z=re^{i\theta _1}$

$-Z=-re^{i\theta _1}$
$\implies -a\cos\theta _1-ib\sin\theta _1$
$\implies -a\cos(\pi+\theta _1)-ib\sin(\pi+\theta _1)$
$\implies re^{i(\pi+\theta _1)}$
$arg(-Z)-arg(Z)$
$\implies \pi+\theta _1-\theta _1\ \implies \pi$

Multiple choice maths complex numbers and linear inequations argand plane and polar representation geometric representation of a complex number complex numbers and quadratic equations

Which of the given alternatives represent a point in Argand plane, equidistant from roots of the equation $(z+1)^4= 16z^4$?

  1. $(0,0)$
  2. $\left(-\dfrac{1}{3},0\right)$
  3. $\left(\dfrac{1}{3},0\right)$
  4. $\left(0,\dfrac{2}{\sqrt5}\right)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Consider the given equation $(z+1)^4=16z^4$
$ \Rightarrow (z+1)^4=16z^4$

$ \Rightarrow z+1=(2^4z^4)^{\frac{1}{4}}$

$ \Rightarrow |z+1|=2|z|$

We know that $z=x+iy$

Therefore $|x+iy+1|=2|x+iy|$

$ \Rightarrow \sqrt{(x+1)^2+y^2}=2\sqrt{x^2+y^2}$

$ \Rightarrow (x+1)^2+y^2=4(x^2+y^2)$

$ \Rightarrow x^2+2x+1+y^2=4x^2+4y^2$

$ \Rightarrow 3x^2+3y^2-2x-1=0$

Divide throughout by 3 we get,
$ \Rightarrow x^2+y^2-\dfrac{2}{3}x-\dfrac{1}{3}=0$, which represents a circle.

We know that for the circle equation of the form $x^2+y^2+2gx+2hy+c=0$ the center of the circle is given by $(-g,-h)$

We have $3x^2+3y^2-2x-1=0$ where $g=-\dfrac{1}{3}, h=0$.

Hence the center is $(\dfrac{1}{3},0)$ which is equidistant from the root of the equation.

Multiple choice maths introduction to three dimensional geometry midpoint of line segment mid point formula mid-point of a line segment

If $z = \cos \dfrac{\pi }{6} + i\sin \dfrac{\pi }{6}$, then

  1. $\left| z \right| = 1,\arg z = \dfrac{\pi }{4}$
  2. $\left| z \right| = 1,\arg z = \dfrac{\pi }{6}$
  3. $\left| z \right| = \dfrac{{\sqrt 3 }}{2},\arg z = \dfrac{{5\pi }}{{24}}$
  4. $\left| z \right| = \dfrac{{\sqrt 3 }}{2},\arg z = {\tan ^{ - 1}}\dfrac{1}{{\sqrt 2 }}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

if $z=\cos \left(\dfrac{\pi}{6}\right)+i\sin \left(\dfrac{\pi}{6}\right)$ then.

As $z=|z|e^{i arq (z)}$
$\therefore z=\cos \left(\dfrac{\pi}{6}\right)+i\sin \left(\dfrac{\pi}{6}\right)=e^{i\left(\pi/6\right)}\quad [\because e^{i\theta}=\cos\theta+i\sin \theta]$
$\Rightarrow |z|=1,  arq=\dfrac{\pi}{6}$

Multiple choice maths complex numbers and linear inequations argand plane and polar representation geometric representation of a complex number complex numbers and quadratic equations

The complex no. $\dfrac{1+2i}{1-i}$ lies in which quadrant of the complex plane

  1. first

  2. second

  3. third

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

To find the quadrant, multiply the numerator and denominator by the conjugate (1+i). (1+2i)(1+i) / (1-i)(1+i) = (1+i+2i-2) / 2 = (-1+3i) / 2 = -0.5 + 1.5i. This point lies in the second quadrant.

Multiple choice maths complex numbers and linear inequations argand plane and polar representation geometric representation of a complex number complex numbers and quadratic equations

If $|z^2-1|=|z^2|+1$, then z lies on?

  1. The real axis

  2. The imaginary axis

  3. A circle

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

Let z = x + iy. Then z^2 = (x^2 - y^2) + i(2xy), so |z^2| = x^2 + y^2 and |z^2 - 1| = sqrt((x^2 - y^2 - 1)^2 + (2xy)^2). The given equation becomes |z^2 - 1| = |z^2| + 1. Squaring both sides yields (x^2 - y^2 - 1)^2 + 4x^2y^2 = (x^2 + y^2 + 1)^2, which simplifies to x^2 = 0, meaning x = 0. Thus, z lies on the imaginary axis.

Multiple choice the nth roots of unity complex numbers maths

If $1,\alpha, \alpha^2,.....,\alpha^{n - 1}$ be the $n^{th}$ roots of unity, then $(1-\alpha)(1-\alpha^2).....(1-\alpha^{n-1}) $

  1. $3$
  2. $0$
  3. $n$
  4. $1$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Basically $1,a^1,a^2......a^{n-1}$ all these are the roots of this equation $x^3 – 1 =0$
So we can write
$x^3 -1= (x-1)(x-a _1)(x-a _2).....(x-a _{n-1})$

$\dfrac{x^3 -1}{(x-1)}= (x-a^1)(x-a^2).....(x-a^{n-1})$

$x^2 + x +1= (x-a^1)(x-a^2).....(x-a^{n-1})$

Put $x=1$ on both the sides now

Ans $=3$
Multiple choice the nth roots of unity complex numbers maths

Find the number of values of complex numbers $\omega$ satisfying the system of equations ${ z }^{ 3 }=-{ \left( \overline { \omega  }  \right)  }^{ 7 }$ and ${ z }^{ 5 }.{ \omega  }^{ 11 }=1$

  1. $2$
  2. $4$
  3. $6$
  4. $8$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Solving the system of equations z^3 = -(conjugate(w)^7) and z^5 * w^11 = 1 involves substituting magnitudes and arguments. The system yields two distinct solutions for the complex number w.

Multiple choice the nth roots of unity complex numbers maths

The value of the expression $1+(2-\omega )+(2-{ \omega  }^{ 2 })+2+(3-\omega )+(3-{ \omega  }^{ 2 })+..........+(n-1)(n-\omega )(n-{ \omega  }^{ 2 })$ where $\omega $ is an imaginary cube root of unity is-

  1. ${ (\frac { n(n+1) }{ 2 } ) }^{ 2 }$
  2. ${ (\frac { n(n+1) }{ 2 } ) }^{ 2 }-n$
  3. ${ (\frac { n(n+1) }{ 2 } ) }^{ 2 }+n$

  4. None of above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The general term is (k-1)(k-w)(k-w^2) = (k-1)(k^2 + k + 1) = k^3 - 1. Summing this from k=2 to n gives the result (n(n+1)/2)^2.

Multiple choice the nth roots of unity complex numbers maths

If 1,${ a } _{ 1 }{ a } _{ 2,........, }{ a } _{ n-1 }$ are the ${ n }^{ th }$ roots of unity, then $\left( 1-{ a } _{ 1 } \right) \left( 1-{ a } _{ 2 } \right) ....\left( 1-{ a } _{ n-1 } \right) $ is equal to

  1. n

  2. 0

  3. 1

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The roots of unity are solutions to z^n - 1 = 0. The expression (1-a1)(1-a2)...(1-an-1) is the evaluation of (z^n - 1)/(z - 1) at z=1, which equals n.