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Complex Numbers Book Back Questions

12th Standard EM

Reg.No. :
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Maths

Time : 00:45:00 Hrs
Total Marks : 30
5 x 1 = 5
1. 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

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 |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 |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

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

(a)

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

(b)

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

(c)

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

(d)

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

6. 3 x 2 = 6
7. Evaluate the following if z=5−2i and w= −1+3i
z+w

8. Find the following $\left| \cfrac { 2+i }{ -1+2i } \right|$

9. If $\omega \neq 1$ is a cube root of unity, then the show that $\cfrac { a+b\omega +c{ \omega }^{ 2 } }{ b+c\omega +{ a\omega }^{ 2 } } +\cfrac { a+b\omega +{ c\omega }^{ 2 } }{ c+a\omega +b{ \omega }^{ 2 } } =1$

10. 3 x 3 = 9
11. 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.

12. If $\left| z-\cfrac { 2 }{ z } \right| =2$ show that the greatest and least value of |z| are $\sqrt { 3 } +1$ and $\sqrt { 3 } -1$ respectively.

13. 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

14. 2 x 5 = 10
15. 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)
${ x }^{ m }{ y }^{ n }+\cfrac { 1 }{ { x }^{ m }{ y }^{ n } } =2cos(m\alpha +n\beta )$

16. Simplify: $\left( -\sqrt { 3 } +3i \right) ^{ 31 }$