If the roots of the equation are real and less than 3, then
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: A
Solution
For roots to be real: discriminant ≥ 0 → 4a² - 4(a² + a - 3) ≥ 0 → -4a + 12 ≥ 0 → a ≤ 3.
For both roots less than 3: since the parabola opens upward, we need f(3) > 0 and vertex x-coordinate < 3.
f(3) = 9 - 6a + a² + a - 3 = a² - 5a + 6 = (a-2)(a-3) > 0 → a < 2 or a > 3.
Vertex condition: 2a/2 = a < 3 → a < 3.
Combining all three: a ≤ 3 AND (a < 2 or a > 3) AND a < 3 gives a < 2.
For both roots less than 3: since the parabola opens upward, we need f(3) > 0 and vertex x-coordinate < 3.
f(3) = 9 - 6a + a² + a - 3 = a² - 5a + 6 = (a-2)(a-3) > 0 → a < 2 or a > 3.
Vertex condition: 2a/2 = a < 3 → a < 3.
Combining all three: a ≤ 3 AND (a < 2 or a > 3) AND a < 3 gives a < 2.
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