Complex Numbers and Quadratic Equations

Let integers a, b [-3,3] be such that a+b 0. Then the number of all possible ordered pairs (a, b), for which | z-az+b |=1 and vmatrixz+1& & 2\ &z+ 2&1\ 2&1&z+ vmatrix=1, z C, where and 2 are the roots of x2+x+1=0, is equal to .

The sum, of the squares of all the roots of the equation x2 + |2x - 3| - 4 = 0, is

1. 6(2 - 2)
2. 3(3 - 2)
3. 3(2 - 2)
4. 6(3 - 2)

Let O be the origin, the point A be z1 = 3 + 2 2i, the point B(z2) be such that 3|z2| = |z1| and (z2) = (z1) + 6. Then

1. area of triangle ABO is 114
2. area of triangle ABO is 11 3
3. ABO is a scalene triangle
4. ABO is an obtuse angled isosceles triangle

If + i and + i are the roots of x2 - (3 - 2i)x - (2i - 2) = 0, i = -1, then + is equal to:

1. 2
2. -6
3. 6
4. -2

The product of all the rational roots of the equation (x2-9x+11)2-(x-4)(x-5)=3, is equal to

1. 7
2. 21
3. 28
4. 14

If and are the roots of the equation 2z2-3z-2i=0, where i= -1, then 16 ( 19+ 19+ 11+ 11 15+ 15 ) ( 19+ 19+ 11+ 11 15+ 15 ) is equal to

1. 441
2. 312
3. 409
4. 398

Let | z-i2 z+i |= 13, z C, be the equation of a circle with center at C. If the area of the triangle, whose vertices are at the points (0,0), C and ( ,0) is 11 square units, then 2 equals:

1. 12125
2. 100
3. 8125
4. 50

If the equation a(b-c)x2+b(c-a)x+c(a-b)=0 has equal roots, where a+c=15 and b= 365, then a2+c2 is equal to .

The number of complex numbers z, satisfying |z|=1 and | z z+ zz |=1, is :

1. 8
2. 10
3. 4
4. 6

Let , be the roots of the equation x2-ax-b=0 with Im( )<Im( ). Let Pn= n- n. If P3=-5 7 i, P4=-3 7 i, P5=11 7 i and P6=45 7 i, then | 4+ 4 | is equal to .