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12th Standard Maths English Medium Complex Numbers Reduced Syllabus Important Questions 2021

12th Standard

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Maths

Time : 01:40:00 Hrs
Total Marks : 60

      Multiple Choice Questions


    15 x 1 = 15
  1. in+in+1+in+2+in+3 is

    (a)

    0

    (b)

    1

    (c)

    -1

    (d)

    i

  2. If \(z=\cfrac { \left( \sqrt { 3 } +i \right) ^{ 3 }\left( 3i+4 \right) ^{ 2 } }{ \left( 8+6i \right) ^{ 2 } } \) , then |z| is equal to 
     

    (a)

    0

    (b)

    1

    (c)

    2

    (d)

    3

  3. 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

  4. 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)

    \(\sqrt { 5 } +2\)

  5. 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

  6. 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)

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

  7. If |z1|=1,|z2|=2|z3|=3 and |9z1z2+4z1z3+z2z3|=12, then the value of |z1+z2+z3| is 

    (a)

    1

    (b)

    2

    (c)

    3

    (d)

    4

  8. 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

  9. 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

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

    (a)

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

    (b)

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

    (c)

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

    (d)

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

  11. The principal argument of the complex number \(\cfrac { \left( 1+i\sqrt { 3 } \right) ^{ 2 } }{ 4i\left( 1-i\sqrt { 3 } \right) } \) is

    (a)

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

    (b)

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

    (c)

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

    (d)

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

  12. 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

  13. The product of all four values of \(\left( cos\cfrac { \pi }{ 3 } +isin\cfrac { \pi }{ 3 } \right) ^{ \frac { 3 }{ 4 } }\) is

    (a)

    -2

    (b)

    -1

    (c)

    1

    (d)

    2

  14. 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)

    \(-\sqrt { 3i } \)

  15. 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\(\frac { \theta }{ 2 } \)|

  16. 2  Marks

    10 x 2 = 20
  17. If z1=3-2i and z2=6+4i, find \(\cfrac { { z }_{ 1 } }{ z_{ 2 } } \)

  18. Write the following in the rectangular form:
    \(\overline { \left( 5+9i \right) +\left( 2-4i \right) } \)

  19. If z=x+iy, find the following in rectangular form.
    \(Re\left( \cfrac { 1 }{ z } \right) \)

  20. Evaluate the following if z=5−2i and w= −1+3i
    z w

  21. If z1=1-3i,z2=4i, and z3 = 5 ,show that (z1z2)z3=z1(z2z3)

  22. Write the following in the rectangular form:
     \(\overline { 3i } +\cfrac { 1 }{ 2-i } \).

  23. Find the modulus of the following complex number \(\cfrac { 2-i }{ 1+i } +\cfrac { 1-2i }{ 1-i } \)

  24. Show that the following equations represent a circle, and, find its centre and radius
    \(\left| 2z+2-4i \right| =2\)

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

  26. Find the values of the real number x and y if 3x + (2x - 3y) i = 6 + 3i9.

    1. 3 Marks


    10 x 3 = 30
  27. If \(\cfrac { z+3 }{ z-5i } =\cfrac { 1+4i }{ 2 } \), find the complex number z

  28. Obtain the Cartesian form of the locus of z=x+iy in
     Im[(1−i)z+1]= 0

  29. Obtain the Cartesian form of the locus of z=x+iy in
    |z+i|=|z-1|

  30. If \(\omega \neq 1\) is a cube root of unity, show that \(\left( 1+\omega \right) \left( 1+{ \omega }^{ 2 } \right) \left( 1+{ \omega }^{ 4 } \right) \left( 1+{ \omega }^{ 8 } \right) ...\left( 1+{ \omega }^{ { 2 }^{ 11 } } \right) =1\).

  31. If z=2−2i, find the rotation of z by θ radians in the counter clockwise direction about the origin when\(\theta =\cfrac { 2\pi }{ 3 } \).

  32. Given the complex number z=3+2i, represent the complex numbers z,iz, and z+ iz in one Argand diagram. Show that these complex numbers form the vertices of an isosceles right triangle.

  33. Explain the falacy:

  34. Show that the complex numbers 3 + 2i, 5i, -3 + 2i and -i form a square.

  35. Find the locus of Z if |3z - 5| = 3 |z + 1| where z=x+iy.

  36. If \(\frac { (a+i)^{ 2 } }{ 2a-i } \) =p+iq, show that p2+q2 =\(\frac { ({ a }^{ 2 }+i)^{ 2 } }{ 4a^{ 2 }+1 } \).

    1. 5 Marks


    7 x 5 = 35
  37. If z1,z2, and z3 are three complex numbers such that |z1|=1,|z2|=2|z3|=3 and |z1+z2+z3|=1,show that |9z1z2+4z1z2+z2z3|=6

  38. If z=x+iy and arg \(\left( \cfrac { z-i }{ z+2 } \right) =\cfrac { \pi }{ 4 } \), then show that x2+y2+3x-3y+2=0

  39. Show that \(\left( \cfrac { \sqrt { 3 } }{ 2 } +\cfrac { i }{ 2 } \right) ^{ 5 }+\left( \cfrac { \sqrt { 3 } }{ 2 } -\cfrac { i }{ 2 } \right) ^{ 5 }=-\sqrt { 3 } \)

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

  41. If z=x+iy and arg\(\left( \cfrac { z-1 }{ z+1 } \right) =\cfrac { \pi }{ 2 } \) ,then show that x2+y2=1.

  42. Show that \(\left( \frac { i+\sqrt { 3 } }{ -i+\sqrt { 3 } } \right) ^{ 2\omega }+\left( \frac { i-\sqrt { 3 } }{ i+\sqrt { 3 } } \right) ^{ 2\omega }\)=-1

  43. Verify that 2 arg(-1) ≠ arg(-1)2

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