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#### Complex Numbers - Two Marks Study Materials

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 30
15 x 2 = 30
1. Simplify $\left( \cfrac { 1+i }{ 1-i } \right) ^{ 3 }-\left( \cfrac { 1-i }{ 1+i } \right) ^{ 3 }$

2. If z1=3-2i and z2=6+4i, find $\cfrac { { z }_{ 1 } }{ z_{ 2 } }$

3. Find the modulus of the following complex numbers
$\cfrac { 2i }{ 3+4i }$

4. Find the square roots of 4+3i

5. Show that the following equations represent a circle, and, find its centre and radius|
$\left| z-2-i \right| =3$

6. Find the principal argument Arg z , when z = $\cfrac { -2 }{ 1+i\sqrt { 3 } }$

7. Evaluate the following if z=5−2i and w= −1+3i
z−iw

8. Write the following in the rectangular form:
$\overline { 3i } +\cfrac { 1 }{ 2-i }$.

9. Find the modulus of the following complex numbers
(1-i)10

10. Show that the following equations represent a circle, and, find its centre and radius
|3z-6+12i|=8

11. Simplify the following:
$\sum _{ n=1 }^{ 102 }{ { i }^{ n } }$

12. Represent the complex numbe $1+i\sqrt { 3 }$ in polar form.

13. If (cosθ + i sinθ)2 = x + iy, then show that x2+y2 =1

14. Find the modulus of the complex number i25.

15. Find the values of the real number x and y if 3x + (2x - 3y) i = 6 + 3i9.