Questions

1. Question 1

   Plot the following points:

   (i)

   $(2, 75^\circ)$

   (ii)

   $(-3, 120^\circ)$

   (iii)

   $(2, \dfrac{\pi}{6})$

   (iv)

   $(5, \dfrac{5\pi}{6})$

   (v)

   $(-\dfrac{5}{2}, \dfrac{\pi}{3})$

   (vi)

   $(-3, -\dfrac{2\pi}{3})$

   (vii)

   $(\dfrac{9}{2}, \dfrac{19\pi}{12})$

   (viii)

   $(-\dfrac{5}{2}, \dfrac{5\pi}{12})$

   SolutionTheory
2. Question 2

   Express the following complex numbers in polar form:

   (i)

   $4+3i$

   (ii)

   $1+i$

   (iii)

   $\dfrac{1}{2}+\dfrac{\sqrt{3}}{2}i$

   (iv)

   $-\dfrac{5}{2}-\dfrac{5\sqrt{3}}{2}i$

   (v)

   $\dfrac{1-i}{1+i}$

   (vi)

   $\dfrac{\sqrt{3}+i}{1+\sqrt{3}i}$

   (vii)

   $\dfrac{3+4i}{4+3i}$

   (viii)

   $\dfrac{3}{2}+\dfrac{\sqrt{3}}{2}i$

   SolutionTheory
3. Question 3

   Convert each of the complex numbers $z$ in the rectangular form $x+iy$:

   (i)

   $4\left(\cos\dfrac{5\pi}{3}+i\sin\dfrac{5\pi}{3}\right)$

   (ii)

   $\dfrac{3}{2}\left(\cos\dfrac{7\pi}{6}+i\sin\dfrac{7\pi}{6}\right)$

   (iii)

   $|z|=7,\arg(z)=\dfrac{23\pi}{12}$

   (iv)

   $|z|=11,\arg(z)=-\dfrac{11\pi}{12}$

   (v)

   $|z|=\dfrac{10}{3},\arg(z)=-\dfrac{17\pi}{12}$

   (vi)

   $2\cos(-33^\circ)+i\,2\sin(-33^\circ)$

   SolutionTheory
4. Question 4

   If $z_1=9\left(\cos\dfrac{5\pi}{4}+i\sin\dfrac{5\pi}{4}\right)$ and $z_2=5\left(\cos\dfrac{\pi}{3}+i\sin\dfrac{\pi}{3}\right)$ then find:

   (i)

   $z_1+z_2$

   (ii)

   $z_1-z_2$

   (iii)

   $z_1\cdot z_2$

   (iv)

   $\dfrac{z_1}{z_2}$

   SolutionTheory
5. Question 5

   If $z_1=7\left(\cos\dfrac{23\pi}{12}+i\sin\dfrac{23\pi}{12}\right)$ and $z_2=11\left(\cos\dfrac{11\pi}{12}+i\sin\dfrac{11\pi}{12}\right)$ then find the following and express the result into $x+iy$ form:

   (i)

   $z_1+z_2$

   (ii)

   $z_1-z_2$

   (iii)

   $z_1\cdot z_2$

   (iv)

   $\dfrac{z_1}{z_2}$

   SolutionTheory
6. Question 6

   If $z_1$ and $z_2$ are two complex numbers, show that:

   (i)

   $\operatorname{Arg}(z_1z_2)=\operatorname{Arg}(z_1)+\operatorname{Arg}(z_2)$

   (ii)

   $\operatorname{Arg}\left(\dfrac{z_1}{z_2}\right)=\operatorname{Arg}(z_1)-\operatorname{Arg}(z_2)$

   SolutionTheory
7. Question 7

   Divide $z_1=6(\cos 150^\circ+i\sin 150^\circ)$ by $z_2=3(\cos 30^\circ+i\sin 30^\circ)$ and express in $x+iy$ form.

   SolutionTheory
8. Question 8

   Multiply $z_1=2(\cos 60^\circ+i\sin 60^\circ)$ and $z_2=5(\cos 90^\circ+i\sin 90^\circ)$ and express in $x+iy$ form.

   SolutionTheory
9. Question 9

   Find the modulus and argument of $z=-2-2i$.

   SolutionTheory
10. Question 10

    Write the equation $\operatorname{Arg}\left(\dfrac{z-2+i}{-2-2i}\right)=\dfrac{2\pi}{3}$ in cartesian form, if $z=x+iy$.

    SolutionTheory
11. Question 11

    If $z=x+iy$ and $\arg\left(\dfrac{z-1+2i}{z+1-2i}\right)=\dfrac{9\pi}{4}$, show that $x^2+y^2+4x+2y-5=0$.

    SolutionTheory
12. Question 12

    If $z=x+iy$ and $\arg(z-2-3i)-\arg(z+2+3i)=2\pi$, show that $2y=3x$.

    SolutionTheory
13. Question 13

    Solve the equation $|z-2|=|z+2|$ for $z=x+iy$.

    SolutionTheory
14. Question 14

    For $z=x+iy$, solve the equation $|5z+4+i|=|5z-3+2i|$.

    SolutionTheory
15. Question 15

    Determine the set of points $z=x+iy$ that satisfy $|3z-2+i|=|3z+i|$.

    SolutionTheory
16. Question 16

    If $z=x+iy$ and $w=\dfrac{1-iz}{z-i}$, show that $|w|=1 \Rightarrow z$ is real.

    SolutionTheory
17. Question 17

    If $z_1$ and $z_2$ are different complex numbers with $|z_2|=1$, find $\left|\dfrac{z_2-z_1}{1-z_1z_2}\right|$.

    SolutionTheory
18. Question 18

    An AC source supplies a voltage $V=120\left(\cos\dfrac{\pi}{4}+i\sin\dfrac{\pi}{4}\right)$ volts to a circuit with impedance $Z=\dfrac{1+i\sqrt{3}}{2}$ ohms. Calculate the current in polar form.

    SolutionTheory
19. Question 19

    An AC circuit has an impedance $Z=3-6i$ ohms and is connected to a voltage source $V=90+30i$ volts. Find the current in both rectangular and polar form.

    SolutionTheory
20. Question 20

    Encrypt the word "CODE" by multiplying the complex encryption key $k=2-i$. Then decrypt it back to the original word.

    SolutionTheory
21. Question 21

    Consider the complex encryption key $k=3-3i$. Encrypt the word "QUIZ", and then recover the original word using the inverse of the key.

    SolutionTheory
22. Question 22

    Encrypt the word "CLASS" by adding the complex encryption key $k=-3+4i$. Then decrypt it back to the original word.

    SolutionTheory