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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. 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)

    \(\sqrt { 5 } +2\)

  2. If \(\left| z-\cfrac { 3 }{ z } \right| =2\) then the least value |z| is

    (a)

    1

    (b)

    2

    (c)

    3

    (d)

    5

  3. 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)

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

  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 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

  6. 5 x 2 = 10
  7. Simplify \(\left( \cfrac { 1+i }{ 1-i } \right) ^{ 3 }-\left( \cfrac { 1-i }{ 1+i } \right) ^{ 3 }\)

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

  9. Find the modulus of the following complex numbers
     2i(3−4i)(4−3i).

  10. If z=\(\left( \frac { \sqrt { 3 } }{ 2 } +\frac { i }{ 2 } \right) ^{ 107 }+\left( \frac { \sqrt { 3 } }{ 2 } -\frac { i }{ 2 } \right) ^{ 107 }\) , then show that Im (z) =0

  11. Find the modules of (1+ 3i)3

  12. 5 x 3 = 15
  13. 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.

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

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

  16. Find the principal value of -2i.

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

  18. 4 x 5 = 20
  19. Show that \(\left( \cfrac { \sqrt { 3 } }{ 2 } +\cfrac { i }{ 2 } \right) ^{ 5 }+\left( \cfrac { \sqrt { 3 } }{ 2 } -\cfrac { i }{ 2 } \right) ^{ 5 }=-\sqrt { 3 } \)

  20. Solve the equation z3+27=0 .

  21. Find all the roots \((2-2i)^{ \frac { 1 }{ 3 } }\) and also find the product of its roots.

  22. Find the radius and centre of the circle \(z\bar { z } \)-(2+3i)z-(2-3i)\(\bar { z } \)+9 =0 where z is a complex number.

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