Multiple choice

$x^2-(m-3)x+m=0::(m \in R)$ be a quadratic equation. Find the value of $m$ for which, a t least one root lie in the interval $(1,2)$.

  1. $(0,10)$
  2. $(10,\infty)$
  3. $(-\infty,10)$

  4. None of the above.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let f(x) = x^2 - (m - 3)x + m. Since f(1) = 4 and f(2) = 10 - m, a root lies strictly between 1 and 2 when these values have opposite signs. Therefore, 10 - m < 0, giving m > 10.

AI explanation

Let f(x) = x^2 - (m-3)x + m. For at least one root to lie in the interval (1, 2), the necessary condition is f(1)f(2) < 0. Evaluating the function gives f(1) = 4 and f(2) = 10 - m, so their product is 4(10 - m). This product is negative when m > 10, meaning the values of m lie in the interval (10, infinity).

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