Complex Numbers and Quadratic Equations · Question 16

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

Let z = (1 + i)(1 + 2i)(1 + 3i) (1 + ni), where i = -1. If |z|2 = 44200, then n is equal to .

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