Let 4x2−4(α−2)x+α−2=0,(α∈R)) be a quadratic equation. The set of values of α of which

A) 4(α−2)4>12+12⇒α>3⇒∴α∈(3∝)Δ=16(α−2)2−16(α−2)=16(α−2)(α−2−1)=16(α−2)(α−3)≥0; α≤2 or α≥3 ) B) ∴f(0) 0⇒1−4(α−2)12+α−2>0⇒α<3∴α∈(−∞,2] C) α−24<0⇒α<2 ∴α∈(−∞,2)

Let 4x2−4(α−2)x+α−2=0,(α∈R)) be a quadratic equation. The set of values of α of which

A) 4(α−2)4>12+12⇒α>3⇒∴α∈(3∝)Δ=16(α−2)2−16(α−2)=16(α−2)(α−2−1)=16(α−2)(α−3)≥0; α≤2 or α≥3 ) B) ∴f(0) 0⇒1−4(α−2)12+α−2>0⇒α<3∴α∈(−∞,2] C) α−24<0⇒α<2 ∴α∈(−∞,2)

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Let $4 x^{2} - 4 (α - 2) x + α - 2 = 0, (α \in R)$ ) be a quadratic equation. The set of values of α of which

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Detailed Solution

$\begin{array}{l} A) \frac{4 (α - 2)}{4} > \frac{1}{2} + \frac{1}{2} \Rightarrow α > 3 \Rightarrow∴ α \in (3 \propto) \\ (Δ = 16 (α - 2)^{2} - 16 (α - 2) \\ = 16 (α - 2) (α - 2 - 1) = 16 (α - 2) (α - 3) \geq 0; \\ α \leq 2 or α \geq 3) \\ B) ∴ f (0) < 0, f (\frac{1}{2}) > 0 \Rightarrow 1 - 4 (α - 2) \frac{1}{2} + α - 2 > 0 \\ \Rightarrow α < 3 ∴ α \in (- \infty, 2] \\ C) \frac{α - 2}{4} < 0 \Rightarrow α < 2 ∴ α \in (- \infty, 2) \end{array}$