Complex Numbers and Quadratic Equations

Let z be a complex number such that the real part of z - 2iz + 2i is zero. Then, the maximum value of |z - (6 + 8i)| is equal to

1. 8
2. 12
3. 10

Let , ; > , be the roots of the equation x2 - 2x - 3 = 0. Let Pn = n - n, n N. Then (11 3 - 10 2)P10 + (11 2 + 10)P11 - 11P12 is equal to

1. 10 3P9
2. 11 3P9
3. 11 2P9
4. 10 2P9

Let z be a complex number such that |z + 2| = 1 and Im ( z + 1z + 2 ) = 15. Then the value of |Re ( z + 2 ) | is

1. 2 65
2. 245
3. 65
4. 1 + 65

The sum of all the solutions of the equation (8)2x - 16 (8)x + 48 = 0 is :

1. 1 + 8(6)
2. 1 + 6(8)
3. 8(6)
4. 8(4)

If the set R = \(a, b) : a + 5b = 42, a, b N\ has m elements and n=1m (1 - in! ) = x + iy, where i = -1, then the value of m + x + y is

1. 12
2. 4
3. 8
4. 5

The sum of all possible values of [- , 2 ], for which 1 + i 1 - 2i is purely imaginary, is equal to :

1. 4
2. 3
3. 2
4. 5

The number of distinct real roots of the equation |x + 1||x + 3| - 4|x + 2| + 5 = 0, is

Let , be the distinct roots of the equation x2 - (t2 - 5t + 6 )x + 1 = 0, t R and an = n + n. Then the minimum value of a2023 + a2025a2024 is

1. -1/2
2. -1/4
3. 1/4
4. 1/2

Let x1, x2, x3, x4 be the solution of the equation 4x4 + 8x3 - 17x2 - 12x + 9 = 0 and (4 + x12 ) (4 + x22 ) (4 + x32 ) (4 + x42 ) = 12516m. Then the value of m is .

If z1, z2 are two distinct complex numbers such that | z1 - 2z2 12 - z1 z2 | = 2, then

1. either z1 lies on a circle of radius 12 or z2 lies on a circle of radius 1.
2. z1 lies on a circle of radius 12 and z2 lies on a circle of radius 1.
3. either z1 lies on a circle of radius 1 or z2 lies on a circle of radius 12.
4. both z1 and z2 lie on the same circle.