S1 : Let a,b,c ∈ C and ax2 + bx + c = 0 be a quadratic equation. Then b2 -4ac = 0 ⇒ roots are real and equal.

Question

Consider the following statements :

S₁ : Let a,b,c ∈ C and ax² + bx + c = 0 be a quadratic equation. Then b² -4ac = 0 ⇒ roots are real and equal.

S₂ : Let b,c ∈ I and b² - 4c be a perfect square. Then roots of the equation x² + bx + c = 0 may not be integers.

S₃ : _{If the quadratic equations} a₁x² + b₁x + c₁ = 0 and a₂x²+ b₂x + c₂ = 0 have common root, then a₁/a₂ = b₁/b₂ = c₁/c₂.

S₄ : f(x) = a₁x² + b₁x + c₁ /a₂x²+ b₂x + c₂ , g (x) = lx + m/ax + b

where a1 , a2 , l , a are non zero real and other coefficients are also real. Then range of f(x) ≠ range of g(x). State, in order, whether S₁ , S₂ , S₃ , S₄ are true or false

(A) TTFT

(B)  TTTF

(C)  FFTT

(D)  FFFF 

Answer 1

Correct option (D) FFFF

Explanation:

S₁ : D = 0 ⇒ roots are real and equal if a, b, c ∈ R. 

S₂ : Roots are integers

S₃ : a₁/a₂ = b₁/b₂ = c₁/c₂ is true iff roots are both common

S₄ : If Numerator and Denominator have common factor then Rf can be equal to R_g .