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#### Complex Numbers One Mark Question

12th Standard EM

Reg.No. :
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Maths

Time : 00:45:00 Hrs
Total Marks : 30
24 x 1 = 24
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. 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)

$\sqrt { 5 } +2$

4. If |z|=1, then the value of $\cfrac { 1+z }{ 1+\overline { z } }$ is

(a)

z

(b)

$\bar { z }$

(c)

$\cfrac { 1 }{ z }$

(d)

1

5. The solution of the equation |z|-z=1+2i is

(a)

$\cfrac { 3 }{ 2 } -2i$

(b)

$-\cfrac { 3 }{ 2 } +2i$

(c)

$2-\cfrac { 3 }{ 2 } i$

(d)

$2+\cfrac { 3 }{ 2 } i$

6. If z is a complex number such that $z\varepsilon C/R\quad$and $z+\cfrac { 1 }{ z } \epsilon R$ then|z| is

(a)

0

(b)

1

(c)

2

(d)

3

7. If z=x+iy is a complex number such that |z+2|=|z−2|, then the locus of z is

(a)

real axis

(b)

imaginary axis

(c)

ellipse

(d)

circle

8. The principal argument of $\cfrac { 3 }{ -1+i }$

(a)

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

(b)

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

(c)

$\cfrac { -3\pi }{ 4 }$

(d)

$\cfrac { -\pi }{ 2 }$

9. If (1+i)(1+2i)(1+3i)...(1+ni)=x+iy, then $2\cdot 5\cdot 10...\left( 1+{ n }^{ 2 } \right)$ is

(a)

1

(b)

i

(c)

x2+y2

(d)

1+n2

10. If $\alpha$ and $\beta$ are the roots of x2+x+1=0, then ${ \alpha }^{ 2020 }+{ \beta }^{ 2020 }$ is

(a)

-2

(b)

-1

(c)

1

(d)

2

11. If $\omega \neq 1$ is a cubic root of unity and $\left| \begin{matrix} 1 & 1 & 1 \\ 1 & { -\omega }^{ 2 } & { \omega }^{ 2 } \\ 1 & { \omega }^{ 2 } & { \omega }^{ 2 } \end{matrix} \right|$ =3k, then k is equal to

(a)

1

(b)

-1

(c)

$\sqrt { 3i }$

(d)

$-\sqrt { 3i }$

12. If $\omega =cis\cfrac { 2\pi }{ 3 }$, then the number of distinct roots of $\left| \begin{matrix} z+1 & \omega & { \omega }^{ 2 } \\ \omega & z+{ \omega }^{ 2 } & 1 \\ { \omega }^{ 2 } & 1 & z+\omega \end{matrix} \right|$

(a)

1

(b)

2

(c)

3

(d)

4

13. The value of (1+i) (1+i2) (1+i3) (1+i4) is

(a)

2

(b)

0

(c)

1

(d)

i

14. If $\sqrt { a+ib }$ =x+iy, then possible value of $\sqrt { a-ib }$ is

(a)

x2+y2

(b)

$\sqrt { { x }^{ 2 }+{ y }^{ 2 } }$

(c)

x+iy

(d)

x-iy

15. If a=cosθ + i sinθ, then $\frac { 1+a }{ 1-a }$ =

(a)

cot $\frac { \theta }{ 2 }$

(b)

cot θ

(c)

i cot $\frac { \theta }{ 2 }$

(d)

i tan$\frac { \theta }{ 2 }$

16. The principal value of the amplitude of (1+i) is

(a)

$\frac { \pi }{ 4 }$

(b)

$\frac { \pi }{ 12 }$

(c)

$\frac { 3\pi }{ 4 }$

(d)

$\pi$

17. The least positive integer n such that $\left( \frac { 2i }{ 1+i } \right) ^{ n }$  is a positive integer is

(a)

16

(b)

8

(c)

4

(d)

2

18. If z = $\frac { 1 }{ (2+3i)^{ 2 } }$ then |z| =

(a)

$\frac { 1 }{ 13 }$

(b)

$\frac { 1 }{ 5}$

(c)

$\frac { 1 }{ 12 }$

(d)

none of these

19. If z=1-cosθ + i sinθ, then |z| =

(a)

2 sin$\frac { 1 }{ 3 }$

(b)

2 cos$\frac { \theta }{ 2 }$

(c)

2|sin$\frac { \theta }{ 2 }$|

(d)

2|cos$\frac { \theta }{ 2 }$|

20. If z=$\frac { 1 }{ 1-cos\theta -isin\theta }$, the Re(z) =

(a)

0

(b)

$\frac{1}{2}$

(c)

cot$\frac { \theta }{ 2 }$

(d)

$\frac{1}{2}$cot$\frac { \theta }{ 2 }$

21. The complex number z which satisfies the condition $\left| \frac { 1+z }{ 1-z } \right|$ =1 lies on

(a)

circle x2+y2 =1

(b)

x-axis

(c)

y-axis

(d)

the lines x+y=1

22. If zn =$cos\frac { n\pi }{ 3 } +isin\frac { n\pi }{ 3 }$, then z1, z2 ..... z6 is

(a)

1

(b)

-1

(c)

i

(d)

-i

23. If a =cosα + i sinα, b= -cosβ + i sinβ then $\left( ab-\frac { 1 }{ ab } \right)$ is _________

(a)

-2i sin(α - β)

(b)

2i sin(α - β)

(c)

2 cos(α - β)

(d)

-2 cos(α - β)

24. If x=cosθ + i sinθ, then xn+$\frac { 1 }{ { x }^{ n } }$ is ______

(a)

2 cos nθ

(b)

2 i sin nθ

(c)

2n cosθ

(d)

2n i sinθ

25. 6 x 1 = 6
26. Re(z)

27. (1)

|$\bar { z }$|

28. Im(z)

29. (2)

z=$\bar { z }$

30. z is real

31. (3)

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

32. z is imaginary

33. (4)

z=-$\bar { z }$

34. |z|

35. (5)

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

36. |2+2i|

37. (6)

2$\sqrt { 2 }$