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

12th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 100

      Multiple Choice Questions


    15 x 1 = 15
  1. 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

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

    (a)

    -2

    (b)

    -1

    (c)

    1

    (d)

    2

  4. 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

  5. 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

  6. If a = 1+i, then a2 equals

    (a)

    1-i

    (b)

    2i

    (c)

    (1+i)(1-i)

    (d)

    i-1

  7. 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

  8. The value of (1+i)4 + (1-i)4 is

    (a)

    8

    (b)

    4

    (c)

    -8

    (d)

    -4

  9. 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

  10. If ω is the cube root of unity, then the value of (1-ω) (1-ω2) (1-ω4) (1-ω8) is

    (a)

    9

    (b)

    -9

    (c)

    16

    (d)

    32

  11. \(\frac { (cos\theta +isin\theta )^{ 6 } }{ (cos\theta -isin\theta )^{ 5 } } \) = ________

    (a)

    cos 11θ - isin 11θ

    (b)

    cos 11θ + isin 11θ

    (c)

    cosθ + i sinθ

    (d)

    \(cos\frac { 6\theta }{ 5 } +isin\frac { 6\theta }{ 5 } \)

  12. 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)

  13. 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

  14. If z1, z2, z3 are the vertices of a parallelogram, then the fourth vertex z4 opposite to z2 is _____

    (a)

    z1 + z2 - z2

    (b)

    z1 + z2 - z3

    (c)

    z1 + z2 - z3

    (d)

    z1 - z2 - z3

  15. If xr=\(cos\left( \frac { \pi }{ 2^{ r } } \right) +isin\left( \frac { \pi }{ 2^{ r } } \right) \) then x1, x2 ... x is

    (a)

    -∞

    (b)

    -2

    (c)

    -1

    (d)

    0

    1. 2 Marks


    10 x 2 = 20
  16. Find z−1, if z=(2+3i)(1− i).

  17. Obtain the Cartesian equation for the locus of z=x+iy in
    |z-4|=16

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

  19. Show that the following equations represent a circle, and, find its centre and radius
    |3z-6+12i|=8

  20. Simplify the following:
     \(\sum _{ n=1 }^{ 102 }{ { i }^{ n } } \)

  21. Find the following \(\left| \overline { (1+i) } (2+3i)(4i-3 \right| \)

  22. Find the modulus and principal argument of the following complex numbers.
    \(-\sqrt { 3 } +i\)

  23. It z1 and z2 are two complex numbers, such that |z1| = Iz2|, then is it necessary that z1 = z2?

  24. If 1, ω, ω2 are the cube roots of unity show that (1+ω2)3 - (1+ω)3 =0

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

    1. 3 Marks


    10 x 3 = 30
  26. The complex numbers u,v, and w are related by \(\cfrac { 1 }{ u } =\cfrac { 1 }{ v } +\cfrac { 1 }{ w } \) If v=3−4i and w=4+3i, find u in rectangular form.

  27. If z1=3+4i,z2=5-12i, and z3 =6+8 , find |z1|,|z2|,|z3|,|z1+z2|,|z2-z3|,and|z1+z3|

  28. Which one of the points10 − 8i , 11+ 6i is closest to1+ i .

  29. Show that the equation \({ z }^{ 3 }+2\bar { z } =0\) has five solutions

  30. If z=x+iy is a complex number such that \(\left| \cfrac { z-4i }{ z+4i } \right| =1\) show that the locus of z is real axis.

  31. Find the product \(\cfrac { 3 }{ 2 } \left( cos\cfrac { \pi }{ 3 } +isin\cfrac { \pi }{ 3 } \right) .6\left( cos\cfrac { 5\pi }{ 6 } +isin\cfrac { 5\pi }{ 6 } \right) \)in rectangular from

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

  33. Find the circle roots of -27.

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

  35. Find the locus of z if Re\(\\ \left( \frac { \bar { z } +1 }{ \bar { z } -i } \right) \) =0.

    1. 5 Marks


    7 x 5 = 35
  36. Show that \(\left( 2+i\sqrt { 3 } \right) ^{ 10 }+\left( 2-i\sqrt { 3 } \right) ^{ 10 }\) is real ii)  \(\left( \cfrac { 19+9i }{ 5-3i } \right) ^{ 15 }-\left( \cfrac { 8+i }{ I+2i } \right) ^{ 15 }\)  is purely imaginary.

  37.  If z=x+iy is a complex number such that Im \(\left( \cfrac { 2z+1 }{ iz+1 } \right) =0\) show that the locus of z is 2x2+2y2+x-2y=0

  38. If \(2cosa=x+\cfrac { 1 }{ x } \) and \(2cos\beta =y+\cfrac { 1 }{ y } \), show that 
    i) \(\cfrac { x }{ y } +\cfrac { y }{ x } =2cos\left( \alpha -\beta \right) \).
    ii) \(xy-\cfrac { 1 }{ xy } =2isin\left( \alpha +\beta \right) \)
    iii)
    \(\cfrac { { x }^{ m } }{ { y }^{ n } } -\cfrac { { y }^{ n } }{ { x }^{ m } } =2isin\left( m\alpha -n\beta \right) \)
    iv)
    \({ x }^{ m }{ y }^{ n }+\cfrac { 1 }{ { x }^{ m }{ y }^{ n } } =2cos(m\alpha +n\beta )\)

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

  40. 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

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

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

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