Equations and I...

E₁ : a + b + c = 0 , if 1 is root of aχ² + bχ + c = 0, E₂ : b² - a² = 2ac, if sin θ and cos θ are the roots of aχ² + bχ + c = 0, which of the following is correct ?

If a, b and c are distinct positive real numbers in AP, then the roots of the equation aχ² + 2bχ + c = 0 are

If a = cos(2π/7) + i sin(2π/7), then the quadratic equation whose roots are α = a + a² + a⁴ and β = a³ + a⁵ + a⁶, is

If the roots of the equation ⋋² +8⋋ + μ² + 6μ = 0 are real, then μ lies between

Let p and q be real numbers. If α is the root of χ² + 3p² χ + 5q² = 0, β is the root of χ² + 9p²χ + 15q² = 0 and 0 < α< β, then the equation χ² + 6p²χ + 10q² = 0 has root γ that always satisfies

If the equations x² + 2χ + 3 = 0 and aχ² + bχ + c = 0; a, b, c ∈ R, have a common root, then a : b : c is equal to

Let a, b and c be real numbers, a ≠ 0, if α ≠0. If α is a root of a²x² + bx + c = 0, β is a root of a²χ² - bχ - c = 0 and 0 < α< β. Then, the equation a²χ² + 2bχ - c = 0 has a root γ that always satisfies

A value of b for which the equations χ² + bχ - 1 = 0, χ² + χ + b = 0 have one root in common, is

If α and β are the roots of the equation aχ² + bχ + c = 0, then the equation whose roots are k/α and k/β is

If both the roots of the equation χ² - 6aχ + 2 – 2a + 9a² = 0 exceed 3, then