Complex Numbers and Quadratic Equations

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 = -1 + i 32 and = -1 - i 32, i = -1. If (7 - 7 + 9 )20 + (9 + 7 - 7 )20 + (-7 + 9 + 7 )20 + (14 + 7 + 7 )20 = m10, then m is

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

If x2 + x + 1 = 0, then the value of (x + 1x )4 + (x2 + 1x2 )4 + (x3 + 1x3 )4 + + (x25 + 1x25 )4 is:

1. 162
2. 145
3. 128
4. 175

The sum of all the roots of the equation (x - 1)2 - 5|x - 1| + 6 = 0, is :

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

Let and be the roots of the equation x2 + 2ax + (3a + 10) = 0 such that < 1 < . Then the set of all possible values of a is :

1. (- , -115 ) (5, )
2. (- , -115 )
3. (- , -3)
4. (- , -2) (5, )

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

Let be a solution of x2 + x + 1 = 0, and for some a and b in R, bmatrix 4 & a & b bmatrix bmatrix 1 & 16 & 13 \ -1 & -1 & 2 \ -2 & -14 & -8 bmatrix = bmatrix 0 & 0 & 0 bmatrix. If 4 4 + m a + n b = 3, then m + n is equal to

1. 11
2. 3
3. 8
4. 7

The sum of the squares of the roots of |x-2|2 + |x-2| - 2 = 0 and the squares of the roots of x2 - 2|x-3| - 5 = 0, is

1. 24
2. 26
3. 36
4. 30