Multiple choice

Let the equation $x^2$ + 2(a - 1)x + a + 5 = 0, where 'a' is a parameter, match the real value of 'a' so that the given equation has :- Column - I Column - II (A) Imaginary roots (P) $\displaystyle{\left(-\infty, -\frac{8}{7} \right)}$ (B) One root less than 3 other root is greater than 3 (Q) (-1, 4) (C) One root less than 1 & other root is greater than 3 (R) $\displaystyle{\left(-\infty, -\frac{4}{3} \right)}$

  1. (A) $\rightarrow$ (Q), (B) $\rightarrow$ (P), (C) $\rightarrow$ (R)
  2. (A) $\rightarrow$ (P), (B) $\rightarrow$ (Q), (C) $\rightarrow$ (R)
  3. (A) $\rightarrow$ (Q), (B) $\rightarrow$ (R), (C) $\rightarrow$ (P)
  4. (A) $\rightarrow$ (R), (B) $\rightarrow$ (P), (C) $\rightarrow$ (Q)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For imaginary roots, discriminant D < 0: 4(a-1)^2 - 4(a+5) < 0 => a^2 - 2a + 1 - a - 5 < 0 => a^2 - 3a - 4 < 0 => (a-4)(a+1) < 0 => -1 < a < 4. This matches (Q). For (B), one root < 3 and one > 3, f(3) < 0: 9 + 6(a-1) + a + 5 < 0 => 9 + 6a - 6 + a + 5 < 0 => 7a + 8 < 0 => a < -8/7. This matches (P). For (C), one root < 1 and one > 3, f(1) < 0 and f(3) < 0: f(1) = 1 + 2a - 2 + a + 5 = 3a + 4 < 0 => a < -4/3. This matches (R).

AI explanation

For imaginary roots in condition A, the discriminant of the quadratic must be less than zero: [2(a-1)]^2 - 4(1)(a+5) < 0, which simplifies to 4a^2 - 12a - 16 < 0, or (a-4)(a+1) < 0. This yields the interval (-1, 4) for A, matching A with Q. For condition B, one root less than 3 and one root greater than 3 requires the quadratic evaluated at 3 to be negative: 3^2 + 2(a-1)(3) + a + 5 < 0. Simplifying this gives 7a + 8 < 0, meaning a < -8/7 or a in the interval (-infinity, -8/7), so B matches P. For condition C, one root less than 1 and one root greater than 3 requires the quadratic evaluated at 1 to be negative (1 + 2(a-1) + a + 5 < 0), giving 3a + 4 < 0 and a < -4/3. Since a < -4/3 is a subset of a < -8/7, this matches the remaining option R.

Keep practicing — related questions