If   then show that |z| = 1.

Let z and ω be two complex numbers such that |z| ≤ 1 and |w| ≤ 1 then show that .

Find all non zero complex numbers satisfying .

Let and  be the roots of the equation  where the coefficients p and q may be complex numbers. Let A and B represent  in the complex plane. If  and OB = OA where O is the origin, prove that .

Let and  are two complex numbers such that  then prove that .

Prove that there exists no complex number z such that  and .

True/False
If the complex numbers  represent the vertices of an equilateral triangle with  then .

a) True

b) False

If three complex numbers are in arithmetic progression then they lie on a circle in the complex plane.

a) True

b) False

The cube roots of unity when represented on argand diagram form the vertices of an equilateral triangle.

a) True

b) False

For any two complex numbers  and any real numbers  is equal to .  .  .  .

a)  

b)  

c)  

d)  

If a and b are real numbers between 0 and 1 such that the points  form an equilateral triangle then a is equal to .  .  .  .

a)  

b)  

c)  

d)  

Solve for x
6.2^2x – 13.6^x + 6.3^2x = 0

a) x = Φ

b) x = 1

c) x = −1

d) x = ± 1

Solve for x
 

a) x = 0, 1

b) x = 1, 2

c) x = 2, 3

d) x = 0, 1, 2, 3

Solve for x
5^{2x + 1} + 7^{x + 1} – 175^x – 35 = 0

a) x = Φ

b)

c)

d) , 7

Solve for x
 

a) x = Φ

b) x = 1

c) x = 3

d) all of the above

Solve for x
 

a) x = Φ

b) x = 1

c) x = - 3

d) x = 1, - 3

Solve for x
 

a) x = Φ

b) x = 3

c) x = 6

d) x = 9

Solve for x
 

a) x = Φ

b)

c)

d) , 1

Solve for x
4^{2x – 3} = 7^{x – 1.5}

a) x = 0

b)

c)

d)

Solve for x
9^{x + 2} = (13)^{2x – 1}

a) x = 1

b) x = ln9 + ln13

c)

d)

Solve for x
27^{x – 2/3} – 9^{x – 1}  = 2.3^{2x – 1} – 2.3^{3x – 1}

a) x = 0

b) x = 1

c) x = 2

d) x = 3

Solve for x
4^x + 6^x – 9^x = 0

a) x = 0

b) x = 1

c)

d)

Solve for x
3^{2x + 1} + 8^{x + 1} – (72)^x – 24 = 0

a) x = Φ

b)

c)

d) ,