#### 12th Standard Maths English Medium Complex Numbers Reduced Syllabus Important Questions 2021

12th Standard

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Maths

Time : 01:40:00 Hrs
Total Marks : 60

Multiple Choice Questions

15 x 1 = 15
1. in+in+1+in+2+in+3 is

(a)

0

(b)

1

(c)

-1

(d)

i

2. If $z=\cfrac { \left( \sqrt { 3 } +i \right) ^{ 3 }\left( 3i+4 \right) ^{ 2 } }{ \left( 8+6i \right) ^{ 2 } }$ , then |z| is equal to

(a)

0

(b)

1

(c)

2

(d)

3

3. 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

4. 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)

$\sqrt { 5 } +2$

5. 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

6. 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)

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

7. 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

8. 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

9. 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

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

(a)

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

(b)

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

(c)

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

(d)

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

11. 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)

$\cfrac { \pi }{ 2 }$

12. 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

13. The product of all four values of $\left( cos\cfrac { \pi }{ 3 } +isin\cfrac { \pi }{ 3 } \right) ^{ \frac { 3 }{ 4 } }$ is

(a)

-2

(b)

-1

(c)

1

(d)

2

14. 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)

$-\sqrt { 3i }$

15. 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$\frac { \theta }{ 2 }$|

16. 2  Marks

10 x 2 = 20
17. If z1=3-2i and z2=6+4i, find $\cfrac { { z }_{ 1 } }{ z_{ 2 } }$

18. Write the following in the rectangular form:
$\overline { \left( 5+9i \right) +\left( 2-4i \right) }$

19. If z=x+iy, find the following in rectangular form.
$Re\left( \cfrac { 1 }{ z } \right)$

20. Evaluate the following if z=5−2i and w= −1+3i
z w

21. If z1=1-3i,z2=4i, and z3 = 5 ,show that (z1z2)z3=z1(z2z3)

22. Write the following in the rectangular form:
$\overline { 3i } +\cfrac { 1 }{ 2-i }$.

23. Find the modulus of the following complex number $\cfrac { 2-i }{ 1+i } +\cfrac { 1-2i }{ 1-i }$

24. Show that the following equations represent a circle, and, find its centre and radius
$\left| 2z+2-4i \right| =2$

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

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

1. 3 Marks

10 x 3 = 30
27. If $\cfrac { z+3 }{ z-5i } =\cfrac { 1+4i }{ 2 }$, find the complex number z

28. Obtain the Cartesian form of the locus of z=x+iy in
Im[(1−i)z+1]= 0

29. Obtain the Cartesian form of the locus of z=x+iy in
|z+i|=|z-1|

30. If $\omega \neq 1$ is a cube root of unity, show that $\left( 1+\omega \right) \left( 1+{ \omega }^{ 2 } \right) \left( 1+{ \omega }^{ 4 } \right) \left( 1+{ \omega }^{ 8 } \right) ...\left( 1+{ \omega }^{ { 2 }^{ 11 } } \right) =1$.

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

32. Given the complex number z=3+2i, represent the complex numbers z,iz, and z+ iz in one Argand diagram. Show that these complex numbers form the vertices of an isosceles right triangle.

33. Explain the falacy:

34. Show that the complex numbers 3 + 2i, 5i, -3 + 2i and -i form a square.

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

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

1. 5 Marks

7 x 5 = 35
37. If z1,z2, and z3 are three complex numbers such that |z1|=1,|z2|=2|z3|=3 and |z1+z2+z3|=1,show that |9z1z2+4z1z2+z2z3|=6

38. 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

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

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

41. If z=x+iy and arg$\left( \cfrac { z-1 }{ z+1 } \right) =\cfrac { \pi }{ 2 }$ ,then show that x2+y2=1.

42. Show that $\left( \frac { i+\sqrt { 3 } }{ -i+\sqrt { 3 } } \right) ^{ 2\omega }+\left( \frac { i-\sqrt { 3 } }{ i+\sqrt { 3 } } \right) ^{ 2\omega }$=-1

43. Verify that 2 arg(-1) ≠ arg(-1)2