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#### Complex Numbers Model Question Paper 1

12th Standard EM

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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-2+i|≤2, then the greatest value of |z| is

(a)

$\sqrt { 3 } -2$

(b)

$\sqrt { 3 } +2$

(c)

$\sqrt { 5 } -2$

(d)

$\sqrt { 5 } +2$

3. If |z1|=1,|z2|=2|z3|=3 and |9z1z2+4z1z3+z2z3|=12, then the value of |z1+z2+z3| is

(a)

1

(b)

2

(c)

3

(d)

4

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 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$\frac { \theta }{ 2 }$

6. 5 x 2 = 10
7. Find the square roots of 4+3i

8. Obtain the Cartesian form of the locus of z=x+iy in
$\left[ Re\left( iz \right) \right] ^{ 2 }=3$

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

10. Find the argument of -2

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

12. 5 x 3 = 15
13. If z1=3,z2=-7i, and z3=5+4i, show that z1(z2+z3)=z1z2+z1z3

14. If z1=2+5i, z2=-3-4i, and z3=1+i, find the additive and multiplicate inverse of z1,z2 and z3

15. Find the principal value of -2i.

16. Find the locus of Z if |3z - 5| = 3 |z + 1| where z=x+iy.

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. If z=x+iy and arg $\left( \cfrac { z-i }{ z+2 } \right) =\cfrac { \pi }{ 4 }$, then show that x2+y2+3x-3y+2=0

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

21. Verify that arg(1+i) + arg(1-i) = arg[(1+i) (1-i)]

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