Complex Numbers and Quadratic Equations

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

If satisfies the equation x2+x+1=0 and (1+ )7=A+B +C 2, A, B, C 0, then 5(3A-2B-C) is equal to .

If , are the roots of the equation, x2-x-1=0 and Sn=2023 n+2024 n, then :

1. 2S12=S11+S10
2. S12=S11+S10
3. S11=S10+S12
4. 2S11=S12+S10

Let the complex numbers and 1 lie on the circles |z-z0 |2=4 and |z-z0 |2=16 respectively, where z0=1+i. Then, the value of 100| |2 is .