Complex Numbers and Quadratic Equations

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