If both the roots of the quadratic equation are real and distinct and they lie in the interval then lies in the interval
- A.(4, 5)
- B.(5, 6)
- C.
- D.(3, 4)
Show answer and solution
Correct option: A
Solution
For both roots to be real, distinct, and lying in [1,5], let f(x) = x² - mx + 4.
Conditions:
1. Discriminant > 0: m² - 16 > 0 → m > 4 or m < -4
2. f(1) ≥ 0: 1 - m + 4 ≥ 0 → m ≤ 5
3. f(5) ≥ 0: 25 - 5m + 4 ≥ 0 → m ≤ 29/5
4. Vertex x-coordinate in [1,5]: 1 ≤ m/2 ≤ 5 → 2 ≤ m ≤ 10
Combining all conditions (taking positive m): m > 4 AND m ≤ 5 AND m ≤ 29/5 AND 2 ≤ m ≤ 10 gives m ∈ (4, 5).
Conditions:
1. Discriminant > 0: m² - 16 > 0 → m > 4 or m < -4
2. f(1) ≥ 0: 1 - m + 4 ≥ 0 → m ≤ 5
3. f(5) ≥ 0: 25 - 5m + 4 ≥ 0 → m ≤ 29/5
4. Vertex x-coordinate in [1,5]: 1 ≤ m/2 ≤ 5 → 2 ≤ m ≤ 10
Combining all conditions (taking positive m): m > 4 AND m ≤ 5 AND m ≤ 29/5 AND 2 ≤ m ≤ 10 gives m ∈ (4, 5).
[JEE Main 2019 · 9 Jan · Shift II]