If x + iy = 32 + cosθ + isin&theta

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 Multiple Choice QuestionsMultiple Choice Questions

121.

The complex numbers sin(x) + i cos(2x) and cos(x) - i sin(2x) are conjugate to each other for

  • x = 

  • x = n + 12π

  • x = 0

  • No value of x


122.

The locus of the points z which satisfy the condition argz - 1z + 1 = π3

  • a straight line

  • a circle

  • a parabola

  • None of the above


123.

If α is an nth root of unity, then 1 + 2α + 3α2 + ... + n - 1equals

  • - n1 - α

  • - n1 + α2

  • n1 - α

  • None of these


124.

If z  3, then the least value of z + 14

  • 112

  • 114

  • 3

  • 14


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125.

If the complex number z lies on a circle with centre at the origin and radius = 14, then the 4 complex number - 1 + 8z lies on a circle with radius

  • 4

  • 1

  • 3

  • 2


126.

If x2 + x + 1 = 0, then the value of n = 16xn + 1xn2 is

  • 13

  • 12

  • 9

  • 14


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127.

If x + iy = 32 + cosθ + isinθ,  then x2 + y2 is equal to

  • 3x - 4

  • 4x - 3

  • 4x + 3

  • None of these


B.

4x - 3

x + iy = 32 + cosθ + isinθ          = 32 + cosθ - isinθ2 + cosθ2 + sin2θ          = 6 + 3cosθ - 3isinθ4 + cos2θ + 4cosθ + sin2θ         = 6 + 3cosθ - 3isinθ5 + 4cosθ         = 6 + 3cosθ5 + 4cosθ + i- 3isinθ5 + 4cosθ

On equating real and imaginary parts, we get              x = 32 + cosθ5 + 4cosθand         y = - 3sinθ5 + 4cosθ x2 + y2 = 95 + 4cosθ2                4 + cos2θ + 4cosθ + sin2θ                 = 95 + 4cosθ2 = 46 + 5cosθ5 + 4cosθ - 3                 = 4x - 3


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128.

If z + 4  3 then the greatest and the least value of z + 1 are

  • - 1, 6

  • 6, 0

  • 6, 3

  • None of these


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129.

If the area of the triangle on the complex plane formed by the points z, z + iz and iz is 200, then the value of 3lz I must be equal to

  • 20

  • 40

  • 60

  • 80


130.

A cmplex number z is such that argz - 2z + 2 = z3. The points representing this complex number will lie on

  • an ellipse

  • a parabola

  • a circle

  • a straight line


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