Complex Numbers and Quadratic Equations

The area (in sq. units) of the region S = \z C : |z - 1| 2; (z + z) + i(z - z) 2, Im(z) 0\ is

1. 7 4
2. 3 2
3. 7 3
4. 17 8

Let S = \ 2 2 : ( 4 + 4 )x2 + ( 2 )x + ( 6 + 6 ) = 0 has real roots\. If and be the smallest and largest elements of the set S, respectively, then 3 (( - 2)2 + ( - 1)2 ) equals .

Let S= \z :|z-1|=1 and ( 2-1)(z+ z)-i(z- z)=2 2 \. Let z1, z2 be such that |z1|= z S|z| and |z2|= z S|z|. Then | 2z1-z2 |2 equals :

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

Let S= \x :( 3+ 2)x+( 3- 2)x=10 \. Then the number of elements in S is :

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

Let P=\z C: |z+2-3i| 1\ and Q=\z C: z(1+i)+ z(1-i) -8\. Let in P , |z-3+2i| be maximum and minimum at z1 and z2 respectively. If |z1|2+2|z2|2= + 2, where , are integers, then + equals .

If z is a complex number such that |z| 1, then the minimum value of |z+ 12(3+4 i) | is :

1. 2
2. 52
3. 32
4. 3

Let and be the roots of the equation p x2+q x-r=0, where p 0. If p, q and r be the consecutive terms of a non constant G.P. and 1 + 1 = 34, then the value of ( - )2 is :

1. 8
2. 9
3. 203
4. 809

Let S be the set of positive integral values of a for which ax2+2(a+1)x+9a+4x2-8x+32<0, x R. Then, the number of elements in S is :

1. 0
3. 3
4. 1

For 0<c<b<a, let (a+b-2c)x2+(b+c-2a)x+(c+a-2b)=0 and 1 be one of its roots. Then, among the two statements

1. only (II) is true
2. Both (I) and (II) are true
3. only (I) is true
4. Neither (I) nor (II) is true

Let a be the sum of all coefficients in the expansion of (1-2x+2x2 )2023 (3-4x2+2x3 )2024 and b= x 0 ( 0x (1+t)t2024+1dtx2 ). If the equations cx2+dx+e=0 and 2bx2+ax+4=0 have a common root, where c, d, e R, then d : c : e equals

1. 2 : 1 : 4
2. 1 : 1 : 4
3. 1 : 2 : 4
4. 4 : 1 : 4

If denotes the number of solutions of |1-i|x=2x and = ( |z| (z) ), where z= 4(1+i)4 [ 1- i +i+ -i1+ i ], i= -1, then the distance of the point ( , ) from the line 4x-3y=7 is .

Let z1 and z2 be two complex numbers such that z1+z2=5 and z13+z23=20+15i. Then, |z14+z24 | equals -

1. 15 15
2. 30 3
3. 25 3
4. 75

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

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 .