The points representing the complex number z for which arg z 

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 Multiple Choice QuestionsMultiple Choice Questions

81.

If α + β and α - β are the roots of the equation x2 + px + q = 0, where α, β, p and q are real, then the roots of the equation (p2 - 4q)(p2x2 + 4px) - 16q = 0 are

  • 1α + 1β and 1α - 1β

  • 1α + 1β and 1α - 1β

  • 1α + 1β and 1α - 1β

  • α + β and α - β


82.

The number of solutions of the equation log2x2 + 2x - 1 = 1 is

  • 0

  • 1

  • 2

  • 3


83.

Let R be the set of real numbers and the functions f : R ➔ R and g : R ➔ R be defined by f(x) = x2 + 2x - 3 and g(x) = x + 1. Then, the value of x for which f(g(x)) = g(f(x)) is

  • - 1

  • 0

  • 1

  • 2


84.

The maximum value of z, when the complex number z satisfies the condition z + 2z = 2 is

  • 3

  • 3 + 2

  • 3 + 1

  • 3 - 1


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85.

If 32 + i3250 = 325x +iy, where x and y are real, then the ordered pair (x, y) is

  • - 3, 0

  • 0, 3

  • 0, - 3

  • 12, 32


86.

If z - 1z + 1 is pure imaginary, then

  • z = 12

  • z = 1

  • z = 2

  • z = 3


87.

Let f(x) = ax2 + bx + c, g(x) = px2 + qx + r such that f(1) = g(1), f(2) = g(2) and f(3) - g(3) = 2. Then, f(4) - g(4) is

  • 4

  • 5

  • 6

  • 7


88.

The equations x2 + x + a= 0 and x2 + ax + 1 = 0 have a common real root

  • for no value of a

  • for exactly one value of a

  • for exactly two value of a

  • for exactly three value of a


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89.

The points representing the complex number z for which arg z - 2z + 2 = π3 lie on

  • a circle

  • a straight line

  • an ellipse

  • a parabola


A.

a circle

Let z = x + iy

 z - 2z + 2 = x +iy - 2x +iy + 2= x - 2 + iyx + 2 + iy × x + 2 - iyx + 2 - iy= x - 2x + 2 + iyx + 2 - iyx - 2 - i2y2x + 22 - iy2= x2 + y2 - 4 + 4iyx + 22 + y2 = x2 + y2 - 4x2 + 4 + 4x + y2 + 4iyx2 + 4 + 4x + y2 argz - 2z + 2 = tan-14yx2 + 4 + 4x + y2 × x2 + 4 + 4x + y2x2 + y2 - 4 = π3            given      tanπ3 = 4yx2 + y2 - 4               3 = 4yx2 + y2 - 4x2 + y2 - 4 - 43y = 0, which represents a circle.


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90.

The quadratic equation 2x2 - (a3 + 8a - 1)x + a2 - 4a = 0 posses roots of opposite sign. Then,

  •  0

  • 0 < a < 4

  • 4  a < 8

  • a  8


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