Complex Numbers Important Questions

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 50
    5 x 1 = 5
  1. The value of \(\sum _{ i=1 }^{ 13 }{ \left( { i }^{ n }+i^{ n-1 } \right) } \) is

    (a)

    1+ i

    (b)

    i

    (c)

    1

    (d)

    0

  2. 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

  3. z1, z2 and z3 are complex number such that z1+z2+z3=0 and |z1|=|z2|=|z3|=1 then z12+z22+z33 is

    (a)

    3

    (b)

    2

    (c)

    1

    (d)

    0

  4. 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

  5. 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

  6. 5 x 1 = 5
  7. Re(z)

  8. (1)

    \(\frac { z+\bar { z } }{ 2 } \)

  9. z is imaginary

  10. (2)

    z=-\(\bar { z } \)

  11. |z1 + z2|

  12. (3)

    ≤ |z1| + |z2|

  13. arg (-i)

  14. (4)

    arg z1 + arg z2

  15. arg (z1z2)

  16. (5)

    \(\frac { \pi }{ 2 } \)

    5 x 2 = 10
  17. Find the values of the real numbers x and y, if the complex numbers (3−i)x−(2−i)y+2i +5 and 2x+(−1+2i)y+3+ 2i are equal.

  18. If z1=1-3i,z2=4i, and z3 = 5 , show that (z1+z2)+z3=z1+(z2+z3)

  19. If z1=3,z2=-7i, and z3=5+4i, show that z1(z2+z3)=z1z2+z1z3

  20. If (cosθ + i sinθ)2 = x + iy, then show that x2+y2 =1

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

  22. 5 x 3 = 15
  23. If \(\cfrac { z+3 }{ z-5i } =\cfrac { 1+4i }{ 2 } \), find the complex number z

  24. If z1=3-2i and z2=6+4i, find \(\cfrac { { z }_{ 1 } }{ z_{ 2 } } \)

  25. Find z−1, if z=(2+3i)(1− i).

  26. Explain the falacy:

  27. Find the circle roots of -27.

  28. 3 x 5 = 15
  29. Find the value of the real numbers x and y, if the complex number (2+i)x+(1−i)y+2i −3
    and x+(−1+2i)y+1+iare equal

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

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

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