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#### Complex Numbers Important Questions

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 50
5 x 1 = 5
1. If |z-2+i|≤2, then the greatest value of |z| is

(a)

$\sqrt { 3 } -2$

(b)

$\sqrt { 3 } +2$

(c)

$\sqrt { 5 } -2$

(d)

$\sqrt { 5 } +2$

2. If $\left| z-\cfrac { 3 }{ z } \right| =2$ then the least value |z| is

(a)

1

(b)

2

(c)

3

(d)

5

3. The principal argument of the complex number $\cfrac { \left( 1+i\sqrt { 3 } \right) ^{ 2 } }{ 4i\left( 1-i\sqrt { 3 } \right) }$ is

(a)

$\cfrac { 2\pi }{ 3 }$

(b)

$\cfrac { \pi }{ 6 }$

(c)

$\cfrac { 5\pi }{ 6 }$

(d)

$\cfrac { \pi }{ 2 }$

4. If z=cos$\frac { \pi }{ 4 }$+i sin$\frac { \pi }{ 6 }$, then

(a)

|z| =1, arg(z) =$\frac { \pi }{ 4 }$

(b)

|z| =1, arg(z) =$\frac { \pi }{ 6 }$

(c)

|z|=$\frac { \sqrt { 3 } }{ 2 }$, arg(z)=$\frac { 5\pi }{ 24 }$

(d)

|z| =$\frac { \sqrt { 3 } }{ 2 }$, arg (z) =tan-1$\left( \frac { 1 }{ \sqrt { 2 } } \right)$

5. If x+iy =$\frac { 3+5i }{ 7-6i }$, they y =

(a)

$\frac { 9 }{ 85 }$

(b)

-$\frac { 9 }{ 85 }$

(c)

$\frac { 53 }{ 85 }$

(d)

none of these

6. 5 x 2 = 10
7. Simplify $\left( \cfrac { 1+i }{ 1-i } \right) ^{ 3 }-\left( \cfrac { 1-i }{ 1+i } \right) ^{ 3 }$

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

9. Find the modulus of the following complex numbers
2i(3−4i)(4−3i).

10. If z=$\left( \frac { \sqrt { 3 } }{ 2 } +\frac { i }{ 2 } \right) ^{ 107 }+\left( \frac { \sqrt { 3 } }{ 2 } -\frac { i }{ 2 } \right) ^{ 107 }$ , then show that Im (z) =0

11. Find the modules of (1+ 3i)3

12. 5 x 3 = 15
13. Find the values of the real numbers x and y, if the complex numbers (3−i)x−(2−i)y+2i +5 and 2x+(−1+2i)y+3+ 2i are equal.

14. If z1=3,z2=-7i, and z3=5+4i, show that z1(z2+z3)=z1z2+z1z3

15. If z=2−2i, find the rotation of z by θ radians in the counter clockwise direction about the origin when $\theta =\cfrac { \pi }{ 3 }$.

16. Find the principal value of -2i.

17. If $\frac { (a+i)^{ 2 } }{ 2a-i }$ =p+iq, show that p2+q2 =$\frac { ({ a }^{ 2 }+i)^{ 2 } }{ 4a^{ 2 }+1 }$.

18. 4 x 5 = 20
19. Show that $\left( \cfrac { \sqrt { 3 } }{ 2 } +\cfrac { i }{ 2 } \right) ^{ 5 }+\left( \cfrac { \sqrt { 3 } }{ 2 } -\cfrac { i }{ 2 } \right) ^{ 5 }=-\sqrt { 3 }$

20. Solve the equation z3+27=0 .

21. Find all the roots $(2-2i)^{ \frac { 1 }{ 3 } }$ and also find the product of its roots.

22. Find the radius and centre of the circle $z\bar { z }$-(2+3i)z-(2-3i)$\bar { z }$+9 =0 where z is a complex number.