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 e1 and e2 be two distinct roots of the equation x2-ax+2=0. Let the sets \a R : e1 and e2 are the eccentricities of hyperbolas\=( , ), and \a R : e1 and e2 are the eccentricities of an ellipse and a hyperbola, respectively\=( , ). Then 2+ 2+ 2 is equal to :

1. 18
2. 22
3. 26
4. 34

Let the set of all values of k R such that the equation z ( z+2+i )+k(2+3i)=0, z C, has at least one solution, be the interval [ , ]. Then 9( + ) is equal to :

1. -10
2. -8
3. 10 13
4. 8 13

Let S= \z C : z2+ 6 iz-3=0 \. Then z S z8 is equal to :

1. 162
2. 184
3. 262
4. 324

Let a, b C. Let , be the roots of the equation x2+ax+b=0. If - = 11 and 2- 2=3i 11, then ( 3- 3 )2 is equal to:

1. 160
2. 176
3. 194
4. 187

Let , be the roots of the equation x2-x+p=0 and , be the roots of the equation x2-4x+q=0; p, q Z. If , , , are in G.P., then |p+q| equals :

1. 16
2. 32
3. 34
4. 38

Let z1, z2 C be the distinct solutions of the equation z2+4z-(1+12i)=0. Then |z1|2+|z2|2 is equal to :

1. 18
2. 22
3. 29
4. 34

If the set of all solutions of |x2+x-9|=|x|+|x2-9| is [ , ] [ , ), then ( 2+ 2+ 2 ) is equal to:

1. 9
2. 18
3. 36
4. 72

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 S=\z C : z2+4z+16=0\. Then z S |z+ 3 i |2 is equal to :

1. 42
2. 23
3. 27
4. 38

If the quadratic equation ( +2)x2-3 x+4 =0, -2, has two positive roots, then the number of possible integral values of is :

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

Let , +2, Z, be the roots of the quadratic equation x(x+2)+(x+1)(x+3)+(x+2)(x+4)+ +(x+n-1)(x+n+1)=4n for some n N. Then n+ is equal to :

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

Let x and y be real numbers such that 50 ( 2x1+3i- y1-2i )=31+17i, i= -1. Then the value of 10(x-3y) is :

1. 20
2. 31
3. 35
4. 75

Let , be the roots of the equation x2-3x+r=0, and 2, 2 be the roots of the equation x2+3x+r=0. If the roots of the equation x2+6x=m are 2 + +2r and -2 - r2, then m is equal to :

1. -135
2. -567
3. 135
4. 567

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

If α, β, where α < β, are the roots of the equation x2 - ( + 3)x + 3 = 0 such that 1 - 1 = 13, then the sum of all possible values of λ is

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

Let S = \x3 + ax2 + bx + c : a, b, c N and a, b, c 20\ be a set of polynomials. Then the number of polynomials in S, which are divisible by x2 + 2, is

1. 6
2. 120
3. 20
4. 10

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 the arithmetic mean of 1a and 1b be 516, a > 2. If is such that a, 4, , b are in A.P., then the equation x2 - ax + 2( - 2b) = 0 has :

1. one root in (1, 4) and another in (-2, 0)
2. one root in (0, 2) and another in (-4, -2)
3. both roots in the interval (-2, 0)
4. complex roots of magnitude less than 2

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

The number of the real solutions of the equation: x|x + 3| + |x - 1| - 2 = 0 is

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

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

The smallest positive integral value of a, for which all the roots of x4 - ax2 + 9 = 0 are real and distinct, is equal to

1. 7
2. 3
3. 4
4. 9

If α and β (α < β) are the roots of the equation (-2 + 3 ) ( | x - 3 | ) + (x - 6 x ) + (9 - 2 3 ) = 0, x 0, then + is equal to :

1. 8
2. 10
3. 9
4. 11

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

A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in :

1. 36 days
2. 24 days
3. 30 days
4. 42 days

If z = 32 + i2, i = -1, then (z201 - i )8 is equal to

1. 1
2. 0
3. -1
4. 256

The number of distinct real solutions of the equation x|x+4| + 3|x+2| + 10 = 0 is

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

Let , be the roots of the quadratic equation 12x2 - 20x + 3 = 0, Z. If 12 | - | 32, then the sum of all possible values of is :

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

Let S = \z C : 4z2 + z = 0\. Then z S |z|2 is equal to:

1. 564
2. 116
3. 764
4. 316