Complex Numbers and Quadratic Equations

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 the equation x(x + 2)(12 - k) = 2 have equal roots. Then the distance of the point (k, k2 ) from the line 3x + 4y + 5 = 0 is

1. 15
2. 12
3. 5 3
4. 15 5

Let z be a complex number such that |z| = 1. If 2 + k2 zk + z = kz, k R, then the maximum distance of k + ik2 from the circle |z - (1 + 2i)| = 1 is :

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

Let Pn = n + n, n N. If P10 = 123, P9 = 76, P8 = 47 and P1 = 1, then the quadratic equation having roots 1 and 1 is :

1. x2 + x - 1 = 0
2. x2 - x + 1 = 0
3. x2 + x + 1 = 0
4. x2 - x - 1 = 0

If the set of all a R - \1\, for which the roots of the equation (1 - a)x2 + 2(a - 3)x + 9 = 0 are positive is (- , - ] [ , ), then 2 + + is equal to .

The number of solutions of the equation ( 9x - 9 x + 2 ) ( 2x - 7 x + 3 ) = 0 is :

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

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

If the set of all a R, for which the equation 2x2 + (a - 5)x + 15 = 3a has no real root, is the interval ( , ), and X = \x Z; < x < \, then x X x2 is equal to:

1. 2139
2. 2119
3. 2109
4. 2129

Let integers a, b [-3,3] be such that a+b 0. Then the number of all possible ordered pairs (a, b), for which | z-az+b |=1 and vmatrixz+1& & 2\ &z+ 2&1\ 2&1&z+ vmatrix=1, z C, where and 2 are the roots of x2+x+1=0, is equal to .

The sum, of the squares of all the roots of the equation x2 + |2x - 3| - 4 = 0, is

1. 6(2 - 2)
2. 3(3 - 2)
3. 3(2 - 2)
4. 6(3 - 2)