Let z1 , z2 and z3 be non-zero complex numbers satisfyi

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 Multiple Choice QuestionsMultiple Choice Questions

31.

If α and β are the roots of the equation 1 + x + x2 = 0, then the matrix product 1βαααβ1β = ?

  • 1112

  •  - 1 - 1 - 12

  • 1 - 1 - 12

  •  - 1 - 1 - 1 - 2


32.

The sum of all real roots of the equation

x - 32 + x - 3 - 2 = 0 is :

  • 2

  • 3

  • 4

  • 6


33.

It is given that the roots of the equation x2 - 4x - log3(P) = 0 are real. For this, the minimum value of P is

  • 127

  • 164

  • 181

  • 1


34.

The smallest positive integer n for which = 1 + i1 - in = 1 is :

  • 1

  • 4

  • 8

  • 16


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35.

If α and β are the roots of the equation 3x2 + 2x + 1 = 0, then the equation whose roots are α + β - 1 is :

  • 3x2 + 8x + 16 = 0

  • 3x2 - 8x - 16 = 0

  • 3x2 + 8x - 16 = 0

  • x2 + 8x + 16 = 0


36.

If x is any real number, then x21 + x4 belongs to which one of the following intervals ?

  • 0, 1

  • 0, 12

  • 0, 12

  • [0, 1]


37.

Suppose ω is a cube root unity with ω  1. Suppose P and Q are the points on the complex plane defined by ω and ω2. If O is the origin,then what is the angle between OP and OQ ?

  • 60°

  • 90°

  • 120°

  • 150°


38.

If x2 - px + 4 > 0 for all real values of x, then which one of the following is correct ?

  • p < 4

  • p  4

  • p > 4

  • p  4


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39.

If z = x + iy = 12 - i2 - 25, where i =  - 1, then what is the fundamental amplitude of z - 2z - i2 ?

  • π

  • π2

  • π3

  • π4


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40.

Let z, z2 and z3 be non-zero complex numbers satisfying

z2 = iz, where i =  - 1.

What is z1 + z2 + z3 = ?

  • i

  •  - i

  • 0

  • 1


C.

0

Denote conjugate of z by z*

We are given z2= i z*.

Taking modulus on both sides we see that |z2| = |z|

since z is non-zero, we have |z| = 1

i.e. zz* = 1 or z* = 1/z.

Given equation is  z2= iz*

or z2 = i/z (since z* = 1/z) 

or z- i = 0.

This is a cubic with 3 roots, and the sum of roots is 0.

i.e. z+ z+ z3 = 0


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