Complex Numbers

Let a,b be real and z be a complex number

Let a,b be real and z be a complex number. If z² + az + b = 0 has two distinct roots on the line Re(z) = 1, then it is necessary that

|b| = 1
b ∈ (0,1)
b ∈ (1,∞)
b ∈ (-1,0)

Let z1 and z2 be two roots of the equation z^2 + az + b = 0

Let z₁ and z₂ be two roots of the equation z² + az + b = 0, z being complex further, assume that the origin, z₁ and z₂ form an equilateral triangle, then

a² = 4b
a² = 3b
a² = 2b
a² = b

If z^2 + z + 1 = 0, where z is a complex number

If z² + z + 1 = 0, where z is a complex number, then the value of

(z + 1/z)² + (z² + 1/z²)² +... + (z⁶ + 1/z⁶)²

The locus of the centre of a circle which touches the circle

The locus of the centre of a circle which touches the circle |z - z₁| = a and |z - z₂| = b externaly (z, z₁ & z₂ are complex numbers) will be

a hyperbola
an ellipse
a circle
a straight line

If |z - 4/z| = 2, then the maximum value of |Z| is equal to

√3 + 1
√5 + 1
2 + √2
2

If w (≠1) is a cube root of unity and (1 + w)^7 = A + Bw

If w (≠1) is a cube root of unity and (1 + w)⁷ = A + Bw. Then (A, B) equals to

(1, 1)
(1, 0)
(0, 1)
(-1, 1)

The Number of complex numbers z such that |z – 1| = |z + 1| = |z – i| equals

0
1
2
infinity

If |z - 4| less than |z - 2|, its solution is given by

If |z - 4| < |z - 2|, its solution is given by

Re(z) > 3
Re(z) > 0
Re(z) < 0
Re(z) > 2

If ((1 + i)/(1 - i))^x = 1, then

x = 4n, where n is any positive integer.
x = 2n, where n is any positive integer.
x = 4n + 1, where n is any positive integer.
x = 2n + 1, where n is any positive integer.

If 2+3i is one of the roots of the equation 2x^3 – 9x^2 + kx – 13 = 0

If 2+3i is one of the roots of the equation 2x³ – 9x² + kx – 13 = 0, k ∈ R, then the real root of this equation:

does not exist
exists and is equal to 1/2
exists and is equal to -1/2
exists and is equal to 1