Complex Numbers and Quadratic Equations

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

The number of real solutions of the equation x|x + 5| + 2|x + 7| - 2 = 0 is .

If 2 and 6 are the roots of the equation ax2+bx+1=0, then the quadratic equation, whose roots are 12a+b and 16a+b, is :

1. x2+8x+12=0
2. 2x2+11x+12=0
3. 4x2+14x+12=0
4. x2+10x+16=0

Let and be the sum and the product of all the non-zero solutions of the equation ( z)2+|z|=0, z . Then 4( 2+ 2) is equal to :

1. 4
2. 2
3. 6
4. 8