﻿ Polar Coordinates and Complex Numbers Worksheet | Problems & Solutions

Polar Coordinates and Complex Numbers Worksheet

Polar Coordinates and Complex Numbers Worksheet
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1.
If argument of $x$ + $\mathrm{iy}$ = $\frac{\pi }{6}$, then choose the relation between $x$ and $y$.
 a. $x$ = $\sqrt{3}$$y$ b. $x$ = - $\sqrt{3}$$y$ c. $y$ = - $\sqrt{3}$$x$ d. $y$ = $\sqrt{3}$$x$

Solution:

Let θ be the argument and r be the modulus of the complex number x + iy

So, r = x²+y², sin θ = yr and cos θ = xr

sin π / 6= yx²+y²
[Use θ = π / 6, r = x²+y².]

1 / 2 = yx²+y²
[Use sin π / 6= 1 / 2.]

x²+y² = 2y
[Simplify.]

x² + y² = 4y²
[Square on both the sides.]

x² = 3y²
[Subtract y² on both the sides.]

x = 3y
[Take square root on both the sides.]

Therefore, the relation between x and y is x = 3y

2.
If $z$ = $x$ + $i$$y$ and w = , |w| = 1, then where does $z$ lie in the complex plane?
 a. a line parallel to $x$-axis b. lies on the unit circle c. the imaginary axis d. the real axis

Solution:

|w| = 1

|1-izz-i| = 1
[Substitute the value of w.]

|1-i(x+iy)x+iy-i| = 1
[Substitute the value of z.]

|1-ix+yx+iy-i| = 1

|(1+y)-ixx+i(y-1)| = 1

(1+y)2+x²x²+(y-1)2 = 1
[Formula |x + iy| = x²+y².]

(1+y)2+x² =(y-1)2+x² = 1

(1+y)2 + x2 = x2 + (y-1)2
[Square on both the sides.]

(1+y)2 + x2 - x2 + (y-1)2 = 0
[Use (a + b)² = a² + b² + 2ab and (a - b)² = a² + b² - 2ab.]

1 + y² + 2y + x² -x² - y² - 1 + 2y = 0
[Simplify.]

4y = 0 y = 0
[Solve for y.]

As y = 0, z lies on the real axis.

3.
If the rectangular coordinates of a point P with polar coordinates ($r$, $\frac{3\pi }{4}$) are (- 2$\sqrt{2}$, 2$\sqrt{2}$), then find the value of $r$.
 a. 8 b. 2 c. 1 d. 4

4.
Express in standard form (cos 60° + $i$ sin 60°).
 a. $\frac{\sqrt{3}}{2}$ + ($\frac{1}{2}$)$i$ b. $\frac{1}{2}$ + ($\frac{1}{2}$)$i$ c. $\frac{\sqrt{3}}{2}$ + ($\frac{\sqrt{3}}{2}$)$i$ d. $\frac{1}{2}$ + ($\frac{\sqrt{3}}{2}$)$i$

5.
Express in polar form $\frac{1}{2}$ + ($\frac{\sqrt{3}}{2}$)$i$.
 a. $\sqrt{2}$(cos 30° + $i$ sin 30°) b. 2(cos 60° + $i$ sin 60°) c. (cos 60° + $i$ sin 60°) d. (cos 300° + $i$ sin 300°)

6.
Express the complex number - $\sqrt{3}$ + 2$i$ in polar form.
 a. $\sqrt{7}$(cos 49° + $i$ sin 49°) b. $\sqrt{7}$(cos 131° + $i$ sin 131°) c. (cos 131° + $i$ sin 131°) d. (cos 49° + $i$ sin 49°)

7.
The polar coordinates of point P are ($\sqrt{2}$, $\frac{\pi }{4}$). Which of the following are the rectangular coordinates for point P?
 a. (- 1, - 1) b. (1, 1) c. (- 1, 1) d. (1, - 1)

8.
Using De Moivre's theorem, express (1 - $i$)4 in the form of $a$ + $i$$b$.
 a. 4 b. - 4 + 4$i$ c. 4 - 4$i$ d. - 4

9.
Simplify 11(cos 4$\theta$ - $i$ sin 4$\theta$)15.
 a. 1115(cos 44$\theta$ - $i$ sin 44$\theta$) b. 11(cos15 4$\theta$ - $i$ sin15 4$\theta$) c. 11(cos 60$\theta$ - $i$ sin 60$\theta$) d. 1115(cos 60$\theta$ - $i$ sin 60$\theta$)

10.
Simplify [$\sqrt{12}$(cos 20° - $i$ sin 20°)]6.
 a. 864 - 864$\sqrt{3}i$ b. 864 + 864$\sqrt{3}i$ c. - 864 - 864$\sqrt{3}i$ d. - 864 + 864$\sqrt{3}i$