67 The least integral value of λ for which x2 4x + λ 0, ∀x∈R - Maths - Complex Numbers and Quadratic Equations

Question

67 The least integral value of λ for which x
2 – 4x + λ > 0,
∀x∈R is
(1) 4 (2) 5 (3) 6 (4) 7

68 If b > a then the equation (x – a)
(x – b) – 1 = 0 has
(1) Both roots in [a, b]
(2) Both roots in (–∞, a)
(3) Both roots in (b, ∞)
(4) One roots in (–∞, a) and other in
(b, ∞)

Shivam Mishra · Accepted Answer

Dear Student,
Please find below the solution to the asked query:

67 . Point to Remember : Any quadratic inequality {ax}^{2} + bx + c > 0 holds for all x \in R if and only if a > 0 and Discriminant < 0 Any quadratic inequality {ax}^{2} + bx + c < 0 holds for all x \in R if and only if a < 0 and Discriminant < 0 Now x^{2} - 4 x + λ > 0 for x \in R Now a = 1 > 0 Hence Discriminant < 0 \Rightarrow {(- 4)}^{2} - 4 λ < 0 \Rightarrow 4 λ > 16 \Rightarrow λ > 4 Hence least integral value is 5 . . . . Option (2) Please ask one question in one thread .

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67. The least integral value of λ for which x 2 – 4x + λ > 0, ∀x∈R is (1) 4 (2) 5 (3) 6 (4) 7 68. If b > a then the equation (x – a) (x – b) – 1 = 0 has (1) Both roots in [a, b] (2) Both roots in (–∞, a) (3) Both roots in (b, ∞) (4) One roots in (–∞, a) and other in (b, ∞)