<p>If both the roots of the quadratic equation \(x^2 - mx + 4 = 0\) are real and distinct and they lie in the interval \([1, 5]\) then \(m\) lies in the interval</p>
Step-by-Step Solution
Key Concept: For both roots of a quadratic to lie in an interval [a,b], check that discriminant is positive, the function is positive at both endpoints, and the vertex lies within the interval.
<p><strong>Solution:</strong> According to the given information, both roots lie in $[1, 5]$.</p><p>For a quadratic $f(x) = x^2 - mx + 4$, with both roots in $[1, 5]$, we need:</p><p>(a) $\Delta \geq 0$ (roots are real and distinct): $m^2 - 16 > 0$ ⟹ $m > 4$ or $m < -4$</p><p>(b) $af(1) > 0$: $f(1) = 1 - m + 4 = 5 - m > 0$ ⟹ $m < 5$</p><p>(c) $af(5) > 0$: $f(5) = 25 - 5m + 4 = 29 - 5m > 0$ ⟹ $m < 5.8$</p><p>(d) $1 < -\frac{b}{2a} < 5$: $1 < \frac{m}{2} < 5$ ⟹ $2 < m < 10$</p><p>Combining all conditions: $4 < m < 5$</p><p>∴ Answer is (a) $(4, 5)$</p>
Correct Answer: a