Complex Numbers and Quadratic Equations

The number of values of z C, satisfying the equations |z-(4+8i)|= 10 and |z-(3+5i)|+|z-(5+11i)|=4 5, is :

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

Let z be a complex number such that |z+2|=|z-2| and ( z+3z-i )= 4. Then |z|2 is equal to:

1. 9
2. 4
3. 5
4. 1

Let the circles C1 : |z|=r and C2 : |z-3-4i|=5, z C, be such that C2 lies within C1. If z1 moves on C1, z2 moves on C2 and |z1-z2|=2, then |z1-z2| is equal to :

1. 12
2. 17
3. 22
4. 24

Let z be a complex number such that |z − 6| = 5 and |z + 2 − 6i| = 5. Then the value of z3 + 3z2 - 15z + 141 is equal to :

1. 61
2. 37
3. 42
4. 50

Let A = \z C : |z - 2| 4\ and B = \z C : |z - 2| + |z + 2| = 5\. Then the \|z1 - z2| : z1 A and z2 B\ is :

1. 172
2. 8
3. 9
4. 152

Let S = \z C : | z-6iz-2i | = 1 and | z-8+2iz+2i | = 35 \. Then z S |z|2 is equal to :

1. 413
2. 398
3. 385
4. 423

Let S = \z : 3 |2z - 3(1 + i)| 7 \ be a set of complex numbers. Then Minz S | (z + 12(5 + 3i) ) | is equal to :

1. 12
2. 52
3. 2
4. 32

Let z be the complex number satisfying |z - 5| 3 and having maximum positive principal argument. Then 34 | 5z - 125iz + 16 |2 is equal to:

1. 20
2. 26
3. 12
4. 16

Among the statements (S1) : The set \z C - \-i\ : |z| = 1 and z - iz + i is purely real \ contains exactly two elements, and (S2) : The set \z C - \-1\ : |z| = 1 and z - 1z + 1 is purely imaginary \ contains infinitely many elements.

1. both are incorrect
2. both are correct
3. only (S2) is correct
4. only (S1) is correct

If the locus of z C, such that Re ( z-12z+i ) + Re ( z-12 z-i ) = 2, is a circle of radius r and center (a, b), then 15abr2 is equal to :

1. 16
2. 24
3. 12
4. 18

If z1, z2, z3 C are the vertices of an equilateral triangle whose centroid is z0, then k=13 (zk - z0)2 is equal to

1. 0
2. 1
3. i
4. -i

Let |z1 - 8 - 2i| 1 and |z2 - 2 + 6i| 2, z1, z2 C. Then the minimum value of |z1 - z2| is :

1. 3
2. 10
3. 7
4. 13

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

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

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 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 S1 = \z C : |z| 5\, S2 = \z C : Im ( z + 1 - 3 i1 - 3 i ) 0 \ and S3 = \z C : Re(z) 0\. Then the area of the region S1 S2 S3 is :

1. 125 24
2. 125 6
3. 125 12
4. 125 4

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= \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

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

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

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