Let α,  β be real and z be a complex number. If z² + αz + β = 0 has two distinct roots on the line Re z = 1, then it is necessary that

• β ∈(0, 1)

• β ∈(-1, 0)

• |β| = 1

• |β| = 1

If ω(≠1) is a cube root of unity, and (1 + ω)⁷ = A + Bω.Then (A, B) equals

• (0,1)

• (1,1)

• (1,0)

• (1,0)

The Number of complex numbers z such that |z– 1| = |z + 1| = |z – i| equals 

• 0

• 1

• 2

• 2

If α and β are the roots of the equation x² – x +1 =0, then α²⁰⁰⁹ + β²⁰⁰⁹ =

• -2

• -1

• 1

• 1

If, for a positive integer n, the quadratic equation,
x(x + 1) + (x + 1) (x + 2) + .....
+ (x + n -1 ) (x + n) = 10n
has two consecutive integral solutions, then n is equal to :

• 11

• 12

• 9

• 9

If the roots of the equation bx²+ cx + a = 0 be imaginary, then for all real values of x, the expression 3b²x² + 6bcx + 2c² is

• greater than 4ab

• less than 4ab

• greater than -4ab

• greater than -4ab

The conjugate of a complex number is . Then the complex number is

The quadratic equations x² – 6x + a = 0 and x² – cx + 6 = 0 have one root in common. The other roots of the first and second equations are integers in the ratio 4 : 3. Then the common root is

• 1

• 4

• 3

• 3

A body weighing 13 kg is suspended by two strings 5 m and 12 m long, their other ends being fastened to the extremities of a rod 13 m long. If the rod be so held that the body hangs immediately below the middle point. The tensions in the strings are 

• 12 kg and 13 kg

• 5 kg and 5 kg

• 5 kg and 12 kg

• 5 kg and 12 kg

If the difference between the roots of the equation x² + ax + 1 = 0 is less than 5 , then the set of possible values of a is

•  (−3, 3)

• (−3, ∞)

• (3, ∞)

• (3, ∞)