Complex Numbers and Quadratic Equations

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

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 .

Let the product of 1 = (8 + i) + (7 + 4i) and 2 = (1 + 8i) + (4 + 7i) be + i , i = -1. Let p and q be the maximum and the minimum values of + respectively. Then p + q is equal to :

1. 130
2. 150
3. 160
4. 140

If is a root of the equation x2+x+1=0 and k=1n ( k+ 1 k )2=20, then n is equal to .

Let and be the roots of x2 + 3x - 16 = 0, and and be the roots of x2 + 3x - 1 = 0. If Pn = n + n and Qn = n + n, then P25 + 3P242P23 + Q25 - Q23Q24 is equal to

1. 4
2. 3
3. 5
4. 7

Let z C be such that z2+3iz-2+i = 2+3i. Then the sum of all possible values of z2 is :

1. -19+2i
2. -19-2i
3. 19-2i
4. 19+2i

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)