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Complex Numbers One Mark Questions with Answer

12th Standard EM

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Maths

Time : 00:45:00 Hrs
Total Marks : 30
    30 x 1 = 30
  1. The value of (1+i) (1+i2) (1+i3) (1+i4) is

    (a)

    2

    (b)

    0

    (c)

    1

    (d)

    i

  2. 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

  3. If, i2 = -1, then i1 + i2 + i3 + ....+ up to 1000 terms is equal to

    (a)

    1

    (b)

    -1

    (c)

    i

    (d)

    0

  4. If z=cos\(\frac { \pi }{ 4 } \)+i sin\(\frac { \pi }{ 6 } \), then

    (a)

    |z| =1, arg(z) =\(\frac { \pi }{ 4 } \)

    (b)

    |z| =1, arg(z) =\(\frac { \pi }{ 6 } \)

    (c)

    |z|=\(\frac { \sqrt { 3 } }{ 2 } \), arg(z)=\(\frac { 5\pi }{ 24 } \)

    (d)

    |z| =\(\frac { \sqrt { 3 } }{ 2 } \), arg (z) =tan-1\(\left( \frac { 1 }{ \sqrt { 2 } } \right) \)

  5. If a=cosθ + i sinθ, then \(\frac { 1+a }{ 1-a } \) =

    (a)

    cot \(\frac { \theta }{ 2 } \)

    (b)

    cot θ

    (c)

    i cot \(\frac { \theta }{ 2 } \)

    (d)

    i tan\(\frac { \theta }{ 2 } \)

  6. The principal value of the amplitude of (1+i) is

    (a)

    \(\frac { \pi }{ 4 } \)

    (b)

    \(\frac { \pi }{ 12 } \)

    (c)

    \(\frac { 3\pi }{ 4 } \)

    (d)

    \(\pi \)

  7. 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

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

    (a)

    1-i

    (b)

    2i

    (c)

    (1+i)(1-i)

    (d)

    i-1

  9. If z = \(\frac { 1 }{ (2+3i)^{ 2 } } \) then |z| =

    (a)

    \(\frac { 1 }{ 13 } \)

    (b)

    \(\frac { 1 }{ 5} \)

    (c)

    \(\frac { 1 }{ 12 } \)

    (d)

    none of these

  10. If z=1-cosθ + i sinθ, then |z| =

    (a)

    2 sin\(\frac { 1 }{ 3 } \)

    (b)

    2 cos\(\frac { \theta }{ 2 } \)

    (c)

    2|sin\(\frac { \theta }{ 2 } \)|

    (d)

    2|cos\(\frac { \theta }{ 2 } \)|

  11. If z=\(\frac { 1 }{ 1-cos\theta -isin\theta } \), the Re(z) =

    (a)

    0

    (b)

    \(\frac{1}{2}\)

    (c)

    cot\(\frac { \theta }{ 2 } \)

    (d)

    \(\frac{1}{2}\)cot\(\frac { \theta }{ 2 } \)

  12. 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

  13. The amplitude of \(\frac{1}{i}\) is equal to

    (a)

    0

    (b)

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

    (c)

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

    (d)

    \(\pi \)

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

    (a)

    8

    (b)

    4

    (c)

    -8

    (d)

    -4

  15. 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

  16. If z = a + ib lies in quadrant then \(\frac { \bar { z } }{ z } \) also lies in the III quadrant if

    (a)

    a > b > 0

    (b)

    a < b < 0

    (c)

    b < a < 0

    (d)

    b > a > 0

  17. \(\frac { 1+e^{ -i\theta } }{ 1+{ e }^{ i\theta } } \) =

    (a)

    cosθ + i sinθ

    (b)

    cosθ - i sinθ

    (c)

    sinθ - i cosθ

    (d)

    sinθ + icosθ

  18. If zn =\(cos\frac { n\pi }{ 3 } +isin\frac { n\pi }{ 3 } \), then z1, z2 ..... z6 is

    (a)

    1

    (b)

    -1

    (c)

    i

    (d)

    -i

  19. If x =cosθ + i sinθ, then the value of xn+\(\frac { 1 }{ { x }^{ n } } \) is

    (a)

    2 cosθ

    (b)

    2i sin nθ

    (c)

    2i sin nθ

    (d)

    2i cos nθ

  20. 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

  21. The points represented by 3 - 3i, 4 - 2i, 3 - i and 2 - 2i form _____ in the argand plane.

    (a)

    collinear points

    (b)

    Vertices of a parallelogram

    (c)

    Vertices of a rectangle

    (d)

    Vertices of a square

  22. (1+i)3 = ______

    (a)

    3 + 3i

    (b)

    1 + 3i

    (c)

    3 - 3i

    (d)

    2i - 2

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

    (a)

    cos 11θ - isin 11θ

    (b)

    cos 11θ + isin 11θ

    (c)

    cosθ + i sinθ

    (d)

    \(cos\frac { 6\theta }{ 5 } +isin\frac { 6\theta }{ 5 } \)

  24. 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)

  25. The conjugate of \(\frac { 1+2i }{ 1-(1-i)^{ 2 } } \) is _______

    (a)

    \(\frac { 1+2i }{ 1-(1-i)^{ 2 } } \)

    (b)

    \(\frac { 5 }{ 1-(1-i)^{ 2 } } \)

    (c)

    \(\frac { 1-2i }{ 1+(1+i)^{ 2 } } \)

    (d)

    \(\frac { 1+2i }{ 1+(1-i)^{ 2 } } \)

  26. The modular of \(\frac { (-1+i)(1-i) }{ 1+i\sqrt { 3 } } \) is ______

    (a)

    \(\sqrt{2}\)

    (b)

    2

    (c)

    1

    (d)

    \(\frac{1}{2}\)

  27. The value of \(\frac { (cos{ 45 }^{ 0 }+isin{ 45 }^{ 0 })^{ 2 }(cos{ 30 }^{ 0 }-isin{ 30 }^{ 0 }) }{ cos{ 30 }^{ 0 }+isin{ 30 }^{ 0 } } \) is

    (a)

    \(\frac { 1 }{ 2 } +i\frac { \sqrt { 3 } }{ 2 } \)

    (b)

    \(\frac { 1 }{ 2 } -i\frac { \sqrt { 3 } }{ 2 } \)

    (c)

    \(-\frac { \sqrt { 3 } }{ 2 } +\frac { 1 }{ 2 } \)

    (d)

    \(\frac { \sqrt { 3 } }{ 2 } +\frac { 1 }{ 2 } \)

  28. 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

  29. 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

  30. 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

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