Complex Numbers and Quadratic Equations

Let z1,z2 and z3 be three complex numbers on the circle |z|=1 with (z1)= - 4, (z2)=0 and (z3)= 4. If |z1 z2+z2 z3+z3 z1 |2= + 2, , Z, then the value of 2+ 2 is :

1. 41
2. 29
3. 24
4. 31

Let and be the distinct roots of 2x2+( )x-1=0, (0,2 ). If m and M are the minimum and the maximum values of 4+ 4, then 16(M+m) equals :

1. 27
2. 17
3. 25
4. 24

Let the curve z(1+i)+ z(1-i)=4, z , divide the region |z-3| 1 into two parts of areas and . Then | - | equals :

1. 1+ 3
2. 1+ 6
3. 1+ 2
4. 1+ 4

Let , be the roots of the equation x2 + 2 2x - 1 = 0. The quadratic equation, whose roots are 4 + 4 and 110 ( 6 + 6 ), is:

1. x2 - 180x + 9506 = 0
2. x2 - 195x + 9506 = 0
3. x2 - 190x + 9466 = 0
4. x2 - 195x + 9466 = 0

The sum of the square of the modulus of the elements in the set \z = a + ib : a, b Z, z C, |z - 1| 1, |z - 5| |z - 5i|\ is .

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