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Complex Numbers Worksheet





Question 1
Find the Multiplicative inverse of the following
(i) $3 - \sqrt {7} i$
(ii) 2 + 3i
(iii) $(1 + \sqrt {3} i)^2$
(iv) $( 3 + 4i)$


Question 2
Find the conjugate of the following
(i) $1 - \sqrt {2} i$
(ii) 5 - i
(iii) $\frac {1}{3 - 4i}$
(iv) $\frac {1+i}{1-i}$
(v)$\frac { \sqrt {3 + 12 i} + \sqrt {3 - 12 i}}{\sqrt {3 + 12 i} - \sqrt {3 - 12 i}}$


Question 3
Express the following complex number in the standard form
(i) $(1 + 2i)( 4 + 3i)$
(ii) $(1 + i)^6 + (1 – i)^3$
(iii) $(2 +i)^4$
(iv)$\frac {(3+4i)(4 + 3i)}{1+i}$


Question 4
Find the polar form of the following complex numbers
(i) (1 + i)
(ii) $\frac {1 + i}{1-i}$
(iii) 5
(iv) $2\frac { i-1}{1 + i \sqrt 3}$


Question 5
Find the square roots of the following complex numbers
(a) 5 -2i
(b) -24 -18i


Question 6
True and False
(i) Conjugate of 1+i lies in IV Quadrant
(ii) Polar form of the complex number $(i^{25})^3$ is $cos \frac {\pi}{2} -i sin \frac {\pi}{2}$
(iii) Principle argument for $z=-10$ is $-\pi$
(iv) The order relation is defined on the set of complex numbers.
(v)5 is a complex number
(vi) If $|z_1| =|z_2|$ then $z_1=z_2$
(vii) if |z|=5, then locus of the complex number is a circle
(viii) $|z_1 + z_2 + z_3 ...+ z_n| \leq |z_1| + |z_2| + |z_3| +.... + |z_n|$


Question 7
if $(x+ iy)^{1/3} =a + ib$, then prove that
$\frac {x}{a} + \frac {y}{b}= 4(a^2 -b^2)$


Question 8
If $z_1$ and $z_2$ both satisfy $z + \bar {z} = 2|z-1|$ and $arg (z_1 -z_2) = \frac {\pi}{4}$
then find $Im (z_1 + z_2)$



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