Complex Numbers and Quadratic Equations

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

Let A = \ [0, 2 ] : 1 + 10 Re ( 2 + i - 3i ) = 0 \. Then A 2 is equal to

1. 214 2
2. 6 2
3. 274 2
4. 8 2

Let the set of all values of p R, for which both the roots of the equation x2 - (p + 2)x + (2p + 9) = 0 are negative real numbers, be the interval ( , ]. Then - 2 is equal to

1. 5
2. 0
3. 20
4. 9

Among the statements (S1) : The set \z C - \-i\ : |z| = 1 and z - iz + i is purely real \ contains exactly two elements, and (S2) : The set \z C - \-1\ : |z| = 1 and z - 1z + 1 is purely imaginary \ contains infinitely many elements.

1. both are incorrect
2. both are correct
3. only (S2) is correct
4. only (S1) is correct

If the locus of z C, such that Re ( z-12z+i ) + Re ( z-12 z-i ) = 2, is a circle of radius r and center (a, b), then 15abr2 is equal to :

1. 16
2. 24
3. 12
4. 18

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

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

For t > -1, let t and t be the roots of the equation ((t + 2)1/7 - 1 )x2 + ((t + 2)1/6 - 1 )x + ((t + 2)1/21 - 1 ) = 0 If t -1+ t = a and t -1+ t = b, then 72(a + b)2 is equal to .

Consider the equation x2 + 4x - n = 0, where n [20, 100] is a natural number. Then the number of all distinct values of n, for which the given equation has integral roots, is equal to

1. 6
2. 5
3. 8
4. 7

Let A = \z C : |z - 2 - i| = 3\, B = \z C : Re(z - iz) = 2\ and S = A B. Then z S |z|2 is equal to .