Complex Numbers and Quadratic Equations

Let O be the origin, the point A be z1 = 3 + 2 2i, the point B(z2) be such that 3|z2| = |z1| and (z2) = (z1) + 6. Then

1. area of triangle ABO is 114
2. area of triangle ABO is 11 3
3. ABO is a scalene triangle
4. ABO is an obtuse angled isosceles triangle

If + i and + i are the roots of x2 - (3 - 2i)x - (2i - 2) = 0, i = -1, then + is equal to:

1. 2
2. -6
3. 6
4. -2

The product of all the rational roots of the equation (x2-9x+11)2-(x-4)(x-5)=3, is equal to

1. 7
2. 21
3. 28
4. 14

If and are the roots of the equation 2z2-3z-2i=0, where i= -1, then 16 ( 19+ 19+ 11+ 11 15+ 15 ) ( 19+ 19+ 11+ 11 15+ 15 ) is equal to

1. 441
2. 312
3. 409
4. 398

Let | z-i2 z+i |= 13, z C, be the equation of a circle with center at C. If the area of the triangle, whose vertices are at the points (0,0), C and ( ,0) is 11 square units, then 2 equals:

1. 12125
2. 100
3. 8125
4. 50

If the equation a(b-c)x2+b(c-a)x+c(a-b)=0 has equal roots, where a+c=15 and b= 365, then a2+c2 is equal to .

The number of complex numbers z, satisfying |z|=1 and | z z+ zz |=1, is :

1. 8
2. 10
3. 4
4. 6

Let , be the roots of the equation x2-ax-b=0 with Im( )<Im( ). Let Pn= n- n. If P3=-5 7 i, P4=-3 7 i, P5=11 7 i and P6=45 7 i, then | 4+ 4 | is equal to .

Let z1,z2 and z3 be three complex numbers on the circle |z|=1 with (z1)= - 4, (z2)=0 and (z3)= 4. If |z1 z2+z2 z3+z3 z1 |2= + 2, , Z, then the value of 2+ 2 is :

1. 41
2. 29
3. 24
4. 31

Let and be the distinct roots of 2x2+( )x-1=0, (0,2 ). If m and M are the minimum and the maximum values of 4+ 4, then 16(M+m) equals :

1. 27
2. 17
3. 25
4. 24

Let the curve z(1+i)+ z(1-i)=4, z , divide the region |z-3| 1 into two parts of areas and . Then | - | equals :

1. 1+ 3
2. 1+ 6
3. 1+ 2
4. 1+ 4

Let , be the roots of the equation x2 + 2 2x - 1 = 0. The quadratic equation, whose roots are 4 + 4 and 110 ( 6 + 6 ), is:

1. x2 - 180x + 9506 = 0
2. x2 - 195x + 9506 = 0
3. x2 - 190x + 9466 = 0
4. x2 - 195x + 9466 = 0

The sum of the square of the modulus of the elements in the set \z = a + ib : a, b Z, z C, |z - 1| 1, |z - 5| |z - 5i|\ is .

Let z be a complex number such that the real part of z - 2iz + 2i is zero. Then, the maximum value of |z - (6 + 8i)| is equal to

1. 8
2. 12
3. 10

Let , ; > , be the roots of the equation x2 - 2x - 3 = 0. Let Pn = n - n, n N. Then (11 3 - 10 2)P10 + (11 2 + 10)P11 - 11P12 is equal to

1. 10 3P9
2. 11 3P9
3. 11 2P9
4. 10 2P9

Let z be a complex number such that |z + 2| = 1 and Im ( z + 1z + 2 ) = 15. Then the value of |Re ( z + 2 ) | is

1. 2 65
2. 245
3. 65
4. 1 + 65

The sum of all the solutions of the equation (8)2x - 16 (8)x + 48 = 0 is :

1. 1 + 8(6)
2. 1 + 6(8)
3. 8(6)
4. 8(4)

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

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