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Polar Coordinates and Complex Numbers Worksheet

Polar Coordinates and Complex Numbers Worksheet
  • Page 1
 1.  
If argument of x + iy = π 6, then choose the relation between x and y.
a.
x = 3y
b.
x = - 3y
c.
y = - 3x
d.
y = 3x


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


Correct answer : (1)
 2.  
If z = x + iy and w = 1- izz-i , |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.


Correct answer : (4)
 3.  
If the rectangular coordinates of a point P with polar coordinates (r, 3π4) are (- 22, 22), then find the value of r.
a.
8
b.
2
c.
1
d.
4


Answer: (d)


Correct answer : (4)
 4.  
Express in standard form (cos 60° + i sin 60°).
a.
32 + (1 2)i
b.
1 2 + (1 2)i
c.
32 + (32)i
d.
1 2 + (32)i


Answer: (d)


Correct answer : (4)
 5.  
Express in polar form 1 2 + (32)i.
a.
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°)


Answer: (c)


Correct answer : (3)
 6.  
Express the complex number - 3 + 2i in polar form.
a.
7(cos 49° + i sin 49°)
b.
7(cos 131° + i sin 131°)
c.
(cos 131° + i sin 131°)
d.
(cos 49° + i sin 49°)


Answer: (b)


Correct answer : (2)
 7.  
The polar coordinates of point P are (2, π4). Which of the following are the rectangular coordinates for point P?
a.
(- 1, - 1)
b.
(1, 1)
c.
(- 1, 1)
d.
(1, - 1)


Answer: (b)


Correct answer : (2)
 8.  
Using De Moivre's theorem, express (1 - i)4 in the form of a + ib.
a.
4
b.
- 4 + 4i
c.
4 - 4i
d.
- 4


Answer: (d)


Correct answer : (4)
 9.  
Simplify 11(cos 4θ - i sin 4θ)15.
a.
1115(cos 44θ - i sin 44θ)
b.
11(cos15 4θ - i sin15 4θ)
c.
11(cos 60θ - i sin 60θ)
d.
1115(cos 60θ - i sin 60θ)


Answer: (c)


Correct answer : (3)
 10.  
Simplify [12(cos 20° - i sin 20°)]6.
a.
864 - 8643i
b.
864 + 8643i
c.
- 864 - 8643i
d.
- 864 + 8643i


Answer: (c)


Correct answer : (3)

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