Complex Numbers Worksheet

Question 1
True and False
(i) If three complex numbers are in A.P, then they lie on a circle in the complex plane
(ii) The cube root of the unit when represented on the Argand plane form the vertices of an equilateral triangle
(iii)Region represented by $|z + 7| \leq 5$ is On or inside the circle having centre (–7, 0) and radius 5 units
(iv) Region represented by $|z + 7i| \leq 5$ is On or inside the circle having centre (0, -7) and radius 5 units
(v) if the complex numbers $z_1$, $z_2$ and $z_3$ represent the vertices of the equilateral triangle such that $|z_1|=|z_2|=|z_3|$ then $z_1 + z_2 + z_3=0$ (vi) $|z_1 + z_2 + z_3 ...+ z_n| \leq |z_1| + |z_2| + |z_3| +.... + |z_n|$
(vii) If n is a positive integer, then the value of $i^n+ (i)^{n+1} + (i)^{n+2} +(i)^{n+3}$ is 0

Question 2
Fill in the blanks
(i) For any two complex numbers $z_1$ and $z_2$ and real numbers a and b. then $|az_1 -bz_2|^2 + |bz_1 +a z_2|^2$ is ____________
(ii) if arg z < 0, then arg(-z) - arg (z) is _____
(iii) If $z_1$ and $z_2$ are two non-zero complex number such that $|z_1 + z_2| = |z_1| + |z_2|$, then $arg z_1 -arg z_2$ is ____

Question 3
Let A ,B and C be the set of complex number defined as
A={z :||z+2| -|z-2|| =2}
B={z: $arg \frac {z-1}{z} = \frac {\pi}{2}$}
C={z : arg (z-1) =π }
Then find (A ∩ B ∩ C)

Question 4
Let A ,B and C be the set of complex number defined as
A = {z : $Im(z) \geq 1$}
B={z: |z - 2 - i| = 3}
C={z : $Re((1- i)z) =\sqrt 2$ }
Then find n(A ∩ B ∩ C)

Question 5
Find the complex numbers z which simlutanously satisfy the equations
$|\frac {z-12}{z-8i}| = \frac {5}{3}$
$|\frac {z-4}{z-8}| = 1$

Question 6
The complex numbers $z_1$,$z_2$ and the origin form an equilateral triangle only if $z_1^2 + z_2^2 -z_1 z_2=0$

Question 7
Let $A_1,A_2, A_3,...., A_n$ be the vertices of the n-sided polygon such that
$\frac {1}{A_1A_2} = \frac {1}{A_1A_3} + \frac {1}{A_1A_4}$
Find the value of n

Question 8
What is the locus of z, if amplitude of z – 2 – 3i is $\frac {\pi}{4}$ ?

Question 9
Let a complex number $\alpha$, $\alpha \ne 1$, be a root of the equation $z^{p+q} -z^p -z^q +1 =0$ where p and q are distint primes. Show that either
$1 + \alpha + \alpha ^2 +... +\alpha ^{p-1}=0$
$1 + \alpha + \alpha ^2 +... +\alpha ^{q-1}=0$
But not both together

Question 10
Find the locus of the complex number
(i) |z+i|=|z+2|
(ii) $ |\frac {z-2}{z+3}| =1$
(iii)$z^2 + |z| =0$
(iv) $| \frac {z-1}{z+1}| < 1$

Question 11
Prove that. if $\omega= \frac {-1+i \sqrt {3}}{2}$
(i) $a^3+b^3=(a+b)(a \omega+b \omega ^2)(a \omega^2 +b\omega)$
(ii)$(a+b+c)^3+(a+b\omega +c\omega ^2)^3 + (a+b\omega ^2+c\omega)^3=3(a3+b^3 +c^3+6abc)$

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