Complex Numbers Book Back Questions

12th Standard EM

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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: \(\left( -\sqrt { 3 } +3i \right) ^{ 31 }\)

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