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#### Complex Numbers Model Question Paper

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 45
6 x 1 = 6
1. in+in+1+in+2+in+3 is

(a)

0

(b)

1

(c)

-1

(d)

i

2. The value of $\sum _{ i=1 }^{ 13 }{ \left( { i }^{ n }+i^{ n-1 } \right) }$ is

(a)

1+ i

(b)

i

(c)

1

(d)

0

3. 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

4. If $\cfrac { z-1 }{ z+1 }$ is purely imaginary, then |z| is

(a)

$\cfrac { 1 }{ 2 }$

(b)

1

(c)

2

(d)

3

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

(a)

2

(b)

0

(c)

1

(d)

i

6. 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

7. 2 x 2 = 4
8. i-1 =
(i) $\frac{1}{i}$
(ii) i
(iii) -i
(4) $\frac { 1 }{ { i }^{ 2 } }$

9. When z =x+iy, then iz is
(1) x-iy
(2) i(x+iy)
(3) -y+ix
(4) Rotation ofz by 90° in the counter clockwise direction

10. 3 x 2 = 6
11. Simplify the following
i1947+i1950

12. It z1 and z2 are two complex numbers, such that |z1| = Iz2|, then is it necessary that z1 = z2?

13. Find Re (z) and im (z) if z = 5i11 + 7i3

14. 3 x 3 = 9
15. Simplify the following i7

16. If $\cfrac { 1+z }{ 1-z } =cos2\theta +isin2\theta$, show that z=itan$\theta$

17. Show that $\left| \frac { z-3 }{ z+3 } \right|$ = 2 represent a circle.

18. 4 x 5 = 20
19.  If z=x+iy is a complex number such that Im $\left( \cfrac { 2z+1 }{ iz+1 } \right) =0$ show that the locus of z is 2x2+2y2+x-2y=0

20. Prove that the values of $\sqrt [ 4 ]{ -1 } arr\quad \pm \cfrac { 1 }{ \sqrt { 2 } } \left( 1\pm i \right)$.Let z=(-1)

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

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.