Complex Numbers and Quadratic Equations

If the set R = \(a, b) : a + 5b = 42, a, b N\ has m elements and n=1m (1 - in! ) = x + iy, where i = -1, then the value of m + x + y is

1. 12
2. 4
3. 8
4. 5

The sum of all possible values of [- , 2 ], for which 1 + i 1 - 2i is purely imaginary, is equal to :

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

The number of distinct real roots of the equation |x + 1||x + 3| - 4|x + 2| + 5 = 0, is

Let , be the distinct roots of the equation x2 - (t2 - 5t + 6 )x + 1 = 0, t R and an = n + n. Then the minimum value of a2023 + a2025a2024 is

1. -1/2
2. -1/4
3. 1/4
4. 1/2

Let x1, x2, x3, x4 be the solution of the equation 4x4 + 8x3 - 17x2 - 12x + 9 = 0 and (4 + x12 ) (4 + x22 ) (4 + x32 ) (4 + x42 ) = 12516m. Then the value of m is .

If z1, z2 are two distinct complex numbers such that | z1 - 2z2 12 - z1 z2 | = 2, then

1. either z1 lies on a circle of radius 12 or z2 lies on a circle of radius 1.
2. z1 lies on a circle of radius 12 and z2 lies on a circle of radius 1.
3. either z1 lies on a circle of radius 1 or z2 lies on a circle of radius 12.
4. both z1 and z2 lie on the same circle.

Let , be roots of x2 + 2x - 8 = 0. If Un = n + n, then U10 + 2 U92U8 is equal to .

Consider the following two statements : Between the above two statements,

1. both Statement I and Statement II are incorrect.
2. Statement I is correct but Statement II is incorrect.
3. Statement I is incorrect but Statement II is correct.
4. both Statement I and Statement II are correct.

The number of distinct real roots of the equation |x||x + 2| - 5|x + 1| - 1 = 0 is .

Let S1 = \z C : |z| 5\, S2 = \z C : Im ( z + 1 - 3 i1 - 3 i ) 0 \ and S3 = \z C : Re(z) 0\. Then the area of the region S1 S2 S3 is :

1. 125 24
2. 125 6
3. 125 12
4. 125 4