1. Find the modulus or the absolute value of

.

2. Find the modulus and argument of the following complex numbers:-
(1)

(2)

(3)

.

3. Prove that the complex numbers

are the vertices of an equilateral triangle in the complex plane.

4. Prove that the points representing the complex numbers 2i, 1+i, 4+4i, and 3+5i on the Argand plane are the vertices of a rectangle.

5. Find the square root of (-7+24i).

6. If (1+i)(1+2i)(1+3i)......(1+ni) = x+iy, show that
2.5.10.......(1+n²) = x² + y²

7. Express the following complex numbers in polar form.
(1) 2 + 2√3
(2) -1 + i√3

8. Prove that |z| =1 if

.

9. P represents the variable complex number z, find the locus of P if
(i)

(ii)

.

10. If one of the roots is 2+√3 i, solve the equation

.