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Complex Numbers Model Question Paper

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 45
    6 x 1 = 6
  1. in+in+1+in+2+in+3 is

    (a)

    0

    (b)

    1

    (c)

    -1

    (d)

    i

  2. The value of \(\sum _{ i=1 }^{ 13 }{ \left( { i }^{ n }+i^{ n-1 } \right) } \) is

    (a)

    1+ i

    (b)

    i

    (c)

    1

    (d)

    0

  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 \(\cfrac { z-1 }{ z+1 } \) is purely imaginary, then |z| is

    (a)

    \(\cfrac { 1 }{ 2 } \)

    (b)

    1

    (c)

    2

    (d)

    3

  5. The value of (1+i) (1+i2) (1+i3) (1+i4) is

    (a)

    2

    (b)

    0

    (c)

    1

    (d)

    i

  6. If \(\sqrt { a+ib } \) =x+iy, then possible value of \(\sqrt { a-ib }\) is

    (a)

    x2+y2

    (b)

    \(\sqrt { { x }^{ 2 }+{ y }^{ 2 } } \)

    (c)

    x+iy

    (d)

    x-iy

  7. 2 x 2 = 4
  8. i-1 =
    (i) \(\frac{1}{i}\)
    (ii) i
    (iii) -i
    (4) \(\frac { 1 }{ { i }^{ 2 } } \)

  9. When z =x+iy, then iz is
    (1) x-iy
    (2) i(x+iy)
    (3) -y+ix
    (4) Rotation ofz by 90° in the counter clockwise direction

  10. 3 x 2 = 6
  11. Simplify the following
    i1947+i1950

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

  13. Find Re (z) and im (z) if z = 5i11 + 7i3

  14. 3 x 3 = 9
  15. Simplify the following i7

  16. If \(\cfrac { 1+z }{ 1-z } =cos2\theta +isin2\theta \), show that z=itan\(\theta\)

  17. Show that \(\left| \frac { z-3 }{ z+3 } \right| \) = 2 represent a circle.

  18. 4 x 5 = 20
  19.  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

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

  21. If 1, ω, ω2 are the cube roots of unity then show that (1+5ω24) (1+5ω+ω2) (5+ω+ω5) =64

  22. Find the radius and centre of the circle \(z\bar { z } \)-(2+3i)z-(2-3i)\(\bar { z } \)+9 =0 where z is a complex number.

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