Complex Numbers and Quadratic Equations

Let a, b, c be the lengths of three sides of a triangle satisfying the condition (a2+b2 )x2-2b(a+c)x+ (b2+c2 )=0. If the set of all possible values of x is the interval ( , ), then 12 ( 2+ 2 ) is equal to .

If z=x+iy, xy 0, satisfies the equation z2+i z=0, then |z2 | is equal to :

1. 9
2. 14
3. 4
4. 1

Let , N be roots of the equation x2-70x+ =0, where 2, 3 N. If assumes the minimum possible value, then ( -1+ -1)( +35)| - | is equal to :

If z is a complex number, then the number of common roots of the equations z1985+z100+1=0 and z3+2z2+2z+1=0, is equal to

1. 0
2. 2
3. 1
4. 3

The number of real solutions of the equation x (x2+3|x|+5|x-1|+6|x-2| )=0 is .

If z= 12-2i is such that |z+1|= z+ (1+i), i= -1 and , R, then + is equal to

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

Let , be the roots of the equation x2-x+2=0 with Im( )>Im( ). Then 6+ 4+ 4-5 2 is equal to .

Let r and respectively be the modulus and amplitude of the complex number z=2-i (2 5 8 ), then (r, ) is equal to

1. (2 11 8, 11 8 )
2. (2 3 8, 3 8 )
3. (2 5 8, 3 8 )
4. (2 3 8, 5 8 )

Let , be the roots of the equation x2- 6x+3=0 such that Im( )>Im( ). Let a, b be integers not divisible by 3 and n be a natural number such that 99 + 98=3n(a+ib), i= -1. Then n+a+b is equal to .

If S=\z :|z-i|=|z+i|=|z-1|\, then, n(S) is :

1. 1
2. 2
3. 3
4. 0