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Complex Numbers Model Question Paper 1

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 50
    5 x 1 = 5
  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-2+i|≤2, then the greatest value of |z| is

    (a)

    \(\sqrt { 3 } -2\)

    (b)

    \(\sqrt { 3 } +2\)

    (c)

    \(\sqrt { 5 } -2\)

    (d)

    \(\sqrt { 5 } +2\)

  3. 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

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

  6. 5 x 2 = 10
  7. Find the square roots of 4+3i

  8. Obtain the Cartesian form of the locus of z=x+iy in
    \(\left[ Re\left( iz \right) \right] ^{ 2 }=3\)

  9. If (cosθ + i sinθ)2 = x + iy, then show that x2+y2 =1

  10. Find the argument of -2

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

  12. 5 x 3 = 15
  13. If z1=3,z2=-7i, and z3=5+4i, show that z1(z2+z3)=z1z2+z1z3

  14. If z1=2+5i, z2=-3-4i, and z3=1+i, find the additive and multiplicate inverse of z1,z2 and z3

  15. Find the principal value of -2i.

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

  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. 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

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

  21. Verify that arg(1+i) + arg(1-i) = arg[(1+i) (1-i)]

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

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