New ! Maths MCQ Practise Tests



Complex Numbers One Mark Question

12th Standard

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Maths

Time : 00:45:00 Hrs
Total Marks : 30
    24 x 1 = 24
  1.  The value of \(\sum_{n=1}^{13}\left(i^{n}+i^{n-1}\right)\) 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. 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)

    \(\sqrt { 5 } +2\)

  4. If |z| = 1, then the value of \(\frac { 1+z }{ 1+\overline { z } }\) is

    (a)

    z

    (b)

    \(\bar { z } \)

    (c)

    \(\cfrac { 1 }{ z } \)

    (d)

    1

  5. The solution of the equation |z| - z = 1 + 2i is

    (a)

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

    (b)

    \(-\frac { 3 }{ 2 } +2i\)

    (c)

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

    (d)

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

  6. If z is a complex number such that \(z \in \mathbb{C} \backslash \mathbb{R}\) and \(z+\frac { 1 }{ z } \epsilon R\), then |z| is

    (a)

    0

    (b)

    1

    (c)

    2

    (d)

    3

  7. If z = x + iy is a complex number such that |z+2| = |z−2|, then the locus of z is

    (a)

    real axis

    (b)

    imaginary axis

    (c)

    ellipse

    (d)

    circle

  8. The principal argument of \(\cfrac { 3 }{ -1+i } \) is

    (a)

    \(\cfrac { -5\pi }{ 6 } \)

    (b)

    \(\cfrac { -2\pi }{ 3 } \)

    (c)

    \(\cfrac { -3\pi }{ 4 } \)

    (d)

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

  9. If (1+i)(1+2i)(1+3i)...(1+ni) = x + iy, then \(2\cdot 5\cdot 10...\left( 1+{ n }^{ 2 } \right) \) is

    (a)

    1

    (b)

    i

    (c)

    x2+y2

    (d)

    1+n2

  10. If \(\alpha \) and \(\beta \) are the roots of x2+x+1 = 0, then \({ \alpha }^{ 2020 }+{ \beta }^{ 2020 }\) is

    (a)

    -2

    (b)

    -1

    (c)

    1

    (d)

    2

  11. If \(\omega \neq 1\) is a cubic root of unity and \(\left| \begin{matrix} 1 & 1 & 1 \\ 1 & { -\omega }^{ 2 }-1 & { \omega }^{ 2 } \\ 1 & { \omega }^{ 2 } & { \omega }^{ 7 } \end{matrix} \right| \) = 3k, then k is equal to 

    (a)

    1

    (b)

    -1

    (c)

    \(\sqrt { 3i } \)

    (d)

    \(-\sqrt { 3i } \)

  12. If \(\omega =cis\cfrac { 2\pi }{ 3 } \), then the number of distinct roots of \(\left| \begin{matrix} z+1 & \omega & { \omega }^{ 2 } \\ \omega & z+{ \omega }^{ 2 } & 1 \\ { \omega }^{ 2 } & 1 & z+\omega \end{matrix} \right| \)=0

    (a)

    1

    (b)

    2

    (c)

    3

    (d)

    4

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

    (a)

    2

    (b)

    0

    (c)

    1

    (d)

    i

  14. 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

  15. If a = cos θ + i sin θ, then \(\frac { 1+a }{ 1-a } \) = ___________

    (a)

    cot \(\frac { \theta }{ 2 } \)

    (b)

    cot θ

    (c)

    i cot \(\frac { \theta }{ 2 } \)

    (d)

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

  16. .If a = 3+i and z = 2-3i, then the points on the Argand diagram representing az, 3az and - az are _________

    (a)

    Vertices of a right angled triangle

    (b)

    Vertices of an equilateral triangle

    (c)

    Vertices of an isosceles

    (d)

    Collinear

  17. 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

  18. 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

  19. If z = 1-cos θ + i sin θ, then |z| = _____________

    (a)

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

    (b)

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

    (c)

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

    (d)

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

  20. 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\(\frac { \theta }{ 2 } \)

  21. 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

  22. 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

  23. If a = cos α + i sin α, b = -cos β + i sin β 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)

  24. If x = cos θ + i sin θ, then x\(\frac { 1 }{ { x }^{ n } } \) is ______

    (a)

    2 cos nθ

    (b)

    2 i sin nθ

    (c)

    2n cosθ

    (d)

    2n i sinθ

  25. 6 x 1 = 6
  26. Re(z)

  27. (1)

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

  28. Im(z)

  29. (2)

    2\(\sqrt { 2 } \)

  30. z is real

  31. (3)

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

  32. z is imaginary

  33. (4)

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

  34. |z|

  35. (5)

    z =\(\bar { z } \)

  36. |2+2i|

  37. (6)

    |\(\bar { z } \)|

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