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#### Complex Numbers Important Questions

12th Standard EM

Reg.No. :
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Maths

Time : 01:00:00 Hrs
Total Marks : 50
5 x 1 = 5
1. The value of $\sum _{ i=1 }^{ 13 }{ \left( { i }^{ n }+i^{ n-1 } \right) }$ is

(a)

1+ i

(b)

i

(c)

1

(d)

0

2. If z is a non zero complex number, such that 2iz2=$\bar { z }$ then |z| is

(a)

$\cfrac { 1 }{ 2 }$

(b)

1

(c)

2

(d)

3

3. z1, z2 and z3 are complex number such that z1+z2+z3=0 and |z1|=|z2|=|z3|=1 then z12+z22+z33 is

(a)

3

(b)

2

(c)

1

(d)

0

4. If z1, z2, z3 are the vertices of a parallelogram, then the fourth vertex z4 opposite to z2 is _____

(a)

z1 + z2 - z2

(b)

z1 + z2 - z3

(c)

z1 + z2 - z3

(d)

z1 - z2 - z3

5. If xr=$cos\left( \frac { \pi }{ 2^{ r } } \right) +isin\left( \frac { \pi }{ 2^{ r } } \right)$ then x1, x2 ... x is

(a)

-∞

(b)

-2

(c)

-1

(d)

0

6. 5 x 1 = 5
7. Re(z)

8. (1)

z=-$\bar { z }$

9. z is imaginary

10. (2)

$\frac { z+\bar { z } }{ 2 }$

11. |z1 + z2|

12. (3)

arg z1 + arg z2

13. arg (-i)

14. (4)

$\frac { \pi }{ 2 }$

15. arg (z1z2)

16. (5)

≤ |z1| + |z2|

5 x 2 = 10
17. 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.

18. If z1=1-3i,z2=4i, and z3 = 5 , show that (z1+z2)+z3=z1+(z2+z3)

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

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

21. If z1 and z2 are 1-i, -2+4i then find Im$\left( \frac { { z }_{ 1 }{ z }_{ 2 } }{ \bar { { z }_{ 1 } } } \right)$.

22. 5 x 3 = 15
23. If $\cfrac { z+3 }{ z-5i } =\cfrac { 1+4i }{ 2 }$, find the complex number z

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

25. Find z−1, if z=(2+3i)(1− i).

26. Explain the falacy:

27. Find the circle roots of -27.

28. 3 x 5 = 15
29. Find the value of the real numbers x and y, if the complex number (2+i)x+(1−i)y+2i −3
and x+(−1+2i)y+1+iare equal

30. Find the following $\left| \overline { (1+i) } (2+3i)(4i-3 \right|$

31. If 1, ω, ω2 are the cube roots of unity then show that (1+5ω24) (1+5ω+ω2) (5+ω+ω5) =64