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

12th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 100

Multiple Choice Questions

15 x 1 = 15
1. 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

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

(a)

-2

(b)

-1

(c)

1

(d)

2

4. 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

5. 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

6. If a = 1+i, then a2 equals

(a)

1-i

(b)

2i

(c)

(1+i)(1-i)

(d)

i-1

7. If x+iy =$\frac { 3+5i }{ 7-6i }$, they y =

(a)

$\frac { 9 }{ 85 }$

(b)

-$\frac { 9 }{ 85 }$

(c)

$\frac { 53 }{ 85 }$

(d)

none of these

8. The value of (1+i)4 + (1-i)4 is

(a)

8

(b)

4

(c)

-8

(d)

-4

9. 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

10. If ω is the cube root of unity, then the value of (1-ω) (1-ω2) (1-ω4) (1-ω8) is

(a)

9

(b)

-9

(c)

16

(d)

32

11. $\frac { (cos\theta +isin\theta )^{ 6 } }{ (cos\theta -isin\theta )^{ 5 } }$ = ________

(a)

cos 11θ - isin 11θ

(b)

cos 11θ + isin 11θ

(c)

cosθ + i sinθ

(d)

$cos\frac { 6\theta }{ 5 } +isin\frac { 6\theta }{ 5 }$

12. 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)

13. 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

14. 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

15. 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

1. 2 Marks

10 x 2 = 20
16. Find z−1, if z=(2+3i)(1− i).

17. Obtain the Cartesian equation for the locus of z=x+iy in
|z-4|=16

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

19. Show that the following equations represent a circle, and, find its centre and radius
|3z-6+12i|=8

20. Simplify the following:
$\sum _{ n=1 }^{ 102 }{ { i }^{ n } }$

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

22. Find the modulus and principal argument of the following complex numbers.
$-\sqrt { 3 } +i$

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

24. If 1, ω, ω2 are the cube roots of unity show that (1+ω2)3 - (1+ω)3 =0

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

1. 3 Marks

10 x 3 = 30
26. The complex numbers u,v, and w are related by $\cfrac { 1 }{ u } =\cfrac { 1 }{ v } +\cfrac { 1 }{ w }$ If v=3−4i and w=4+3i, find u in rectangular form.

27. If z1=3+4i,z2=5-12i, and z3 =6+8 , find |z1|,|z2|,|z3|,|z1+z2|,|z2-z3|,and|z1+z3|

28. Which one of the points10 − 8i , 11+ 6i is closest to1+ i .

29. Show that the equation ${ z }^{ 3 }+2\bar { z } =0$ has five solutions

30. If z=x+iy is a complex number such that $\left| \cfrac { z-4i }{ z+4i } \right| =1$ show that the locus of z is real axis.

31. Find the product $\cfrac { 3 }{ 2 } \left( cos\cfrac { \pi }{ 3 } +isin\cfrac { \pi }{ 3 } \right) .6\left( cos\cfrac { 5\pi }{ 6 } +isin\cfrac { 5\pi }{ 6 } \right)$in rectangular from

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

33. Find the circle roots of -27.

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

35. Find the locus of z if Re$\\ \left( \frac { \bar { z } +1 }{ \bar { z } -i } \right)$ =0.

1. 5 Marks

7 x 5 = 35
36. Show that $\left( 2+i\sqrt { 3 } \right) ^{ 10 }+\left( 2-i\sqrt { 3 } \right) ^{ 10 }$ is real ii)  $\left( \cfrac { 19+9i }{ 5-3i } \right) ^{ 15 }-\left( \cfrac { 8+i }{ I+2i } \right) ^{ 15 }$  is purely imaginary.

37.  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

38. If $2cosa=x+\cfrac { 1 }{ x }$ and $2cos\beta =y+\cfrac { 1 }{ y }$, show that
i) $\cfrac { x }{ y } +\cfrac { y }{ x } =2cos\left( \alpha -\beta \right)$.
ii) $xy-\cfrac { 1 }{ xy } =2isin\left( \alpha +\beta \right)$
iii)
$\cfrac { { x }^{ m } }{ { y }^{ n } } -\cfrac { { y }^{ n } }{ { x }^{ m } } =2isin\left( m\alpha -n\beta \right)$
iv)
${ x }^{ m }{ y }^{ n }+\cfrac { 1 }{ { x }^{ m }{ y }^{ n } } =2cos(m\alpha +n\beta )$

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

40. 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

41. Verify that 2 arg(-1) ≠ arg(-1)2

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