Complex Numbers One Mark Question

12th Standard EM

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Maths

Time : 00:45:00 Hrs
Total Marks : 30
    24 x 1 = 24
  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. 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 \(\cfrac { 1+z }{ 1+\overline { z } }\) is

    (a)

    z

    (b)

    \(\bar { z } \)

    (c)

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

    (d)

    1

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

    (a)

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

    (b)

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

    (c)

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

    (d)

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

  6. If z is a complex number such that \(z\varepsilon C/R\quad \)and \(z+\cfrac { 1 }{ z } \epsilon R\) 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 \(\cfrac { 3 }{ -1+i } \)

    (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 \(\left| \begin{matrix} 1 & 1 & 1 \\ 1 & { -\omega }^{ 2 } & { \omega }^{ 2 } \\ 1 & { \omega }^{ 2 } & { \omega }^{ 2 } \end{matrix} \right| \) =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 \(\left| \begin{matrix} z+1 & \omega & { \omega }^{ 2 } \\ \omega & z+{ \omega }^{ 2 } & 1 \\ { \omega }^{ 2 } & 1 & z+\omega \end{matrix} \right| \)

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

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

    (a)

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

    (b)

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

    (c)

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

    (d)

    \(\pi \)

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

  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 =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)

  24. 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

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

  27. (1)

    2\(\sqrt { 2 } \)

  28. Im(z)

  29. (2)

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

  30. z is real

  31. (3)

    z=\(\bar { z } \)

  32. z is imaginary

  33. (4)

    |\(\bar { z } \)|

  34. |z|

  35. (5)

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

  36. |2+2i|

  37. (6)

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

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