Statement-1: If a , b , c , d ∈ R such that a < b < c < d, then t

1.	Statement-1: If a , b , c , d ∈ R such that a < b < c < d, then the equation (x – a) (x – c) + 2(x – b) (x – d) = 0 are real and distinct.Statement-2: If f(x) = 0 is a polynomial equation and a, b are two real numbers such that f(a) f(b) < 0 has at least one real root.(A) Statement – 1 is True, Statement – 2 is True; Statement – 2 is a correct explanation for Statement – 1. (B) Statement – 1 is True, Statement – 2 is True; Statement – 2 is NOT a correct explanation for Statement – 1.(C) Statement – 1 is True, Statement – 2 is False. (D) Statement – 1 is False, Statement – 2 is True
Answer» Correct option (A) Statement – 1 is True, Statement – 2 is True; Statement – 2 is a correct explanation for Statement – 1. Explanation: x² + x + 1 > 0 ∀x ∈R a = 1 > 0 b² – 4ac = 1 – 4 = -3 < 0 x² + 2x + 5 > 0 ∀x ∈R a = 1 > 0 b² – 4ac = 4 – 20 = -16 < 0 So x² + x + 1/x² + 2x + 5 > 0 ∀x∈R ‘a’ is correct

Statement-1: If a , b , c , d ∈ R such that a < b < c < d, then the equation (x – a) (x – c) + 2(x – b) (x – d) = 0 are real and distinct.Statement-2: If f(x) = 0 is a polynomial equation and a, b are two real numbers such that f(a) f(b) < 0 has at least one real root.(A) Statement – 1 is True, Statement – 2 is True; Statement – 2 is a correct explanation for Statement – 1. (B) Statement – 1 is True, Statement – 2 is True; Statement – 2 is NOT a correct explanation for Statement – 1.(C) Statement – 1 is True, Statement – 2 is False. (D) Statement – 1 is False, Statement – 2 is True

Answer»

Correct option (A) Statement – 1 is True, Statement – 2 is True; Statement – 2 is a correct explanation for Statement – 1.

Explanation:

x² + x + 1 > 0 ∀x ∈R

a = 1 > 0

b² – 4ac = 1 – 4 = -3 < 0

x² + 2x + 5 > 0 ∀x ∈R

a = 1 > 0

b² – 4ac = 4 – 20 = -16 < 0

So x² + x + 1/x² + 2x + 5 > 0 ∀x∈R ‘a’ is correct