Complex Numbers and Quadratic Equations

Match each entry in List-I to the correct entry in List-II and choose the correct option.

1. (P) → (1), (Q) → (2), (R) → (5), (S) → (4)
2. (P) → (3), (Q) → (1), (R) → (4), (S) → (5)
3. (P) → (1), (Q) → (2), (R) → (4), (S) → (5)
4. (P) → (2), (Q) → (3), (R) → (5), (S) → (4)

Let a, b, c be positive integers in arithmetic progression such that the equation ax2+bx+c=0 has only integer solutions. Then which of the following statements is (are) TRUE ?

1. c-b is an integer multiple of a
2. Both the roots of the equation ax2+bx+c=0 are odd integers
3. If c=15, then ab=8
4. If b=8, then x=3 is a root of the equation ax2+bx+c=0

Let R denote the set of all real numbers. Let ai, bi R for i \1, 2, 3\. Define the functions f: R R, g: R R, and h: R R by f(x) = a1 + 10x + a2x2 + a3x3 + x4, g(x) = b1 + 3x + b2x2 + b3x3 + x4, h(x) = f(x+1) - g(x+2). If f(x) g(x) for every x R, then the coefficient of x3 in h(x) is

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

Let R denote the set of all real numbers. Let z1 = 1 + 2i and z2 = 3i be two complex numbers, where i = -1. Let S = \(x, y) R R : |x + iy - z1| = 2|x + iy - z2|\. Then which of the following statements is (are) TRUE?

1. S is a circle with centre (- 13, 103 )
2. S is a circle with centre ( 13, 83 )
3. S is a circle with radius 23
4. S is a circle with radius 2 23

For a non-zero complex number z, let (z) denote the principal argument of z, with - < (z) . Let be the cube root of unity for which 0< ( )< . Let = ( n=12025(- )n ). Then the value of 3 is .

Let f(x)=x4+ax3+bx2+c be a polynomial with real coefficients such that f(1)=-9. Suppose that i 3 is a root of the equation 4x3+3ax2+2bx=0, where i= -1. If 1, 2, 3, and 4 are all the roots of the equation f(x)=0, then | 1|2+| 2|2+| 3|2+| 4|2 is equal to .