Complex Numbers and Quadratic Equations · Question 39

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