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Solution


Q.

a, b, c, d are real numbers. If D₁ and D₂ are discriminants of the quadratic equations x² + ax + b = 0 and x² + cx + d = 0 respectively, and ac = 2(b + d), then which of the following statements is true:

(i) At least one D₁ or D₂ is greater than or equal to zero.
(ii) Both D₁ and D₂ are greater than zero.
(iii) At least one of the two equations has real roots.
(iv) Both the equations have real roots.

 

Choose the correct option from the following:
1 (i) and (iii)    
2 (i) and (iv)    
3 (ii) and (iii)   
4 (ii) and (iv)  

Ans.

(1)

D₁ = a² − 4b and D₂ = c² − 4d

Now, D₁ + D₂ = a² + c² − 4(b + d) = a² + c² − 2ac [Since 2(b + d) = ac]

⇒ D₁ + D₂ = (a − c)² ≥ 0

So, at least one of D₁ and D₂ is greater than or equal to zero. Thus, at least one of the given two equations has real roots.