Question

Let $x^2$ - (m - 3) x + m = 0, m ∈ R be a quadratic equation. The values of m for which both roots lie in between 1 and 2 is given by

• A. m (5, 7)
• B. m < 10
• C. m [1, 9]
• D. m

Answer 1

The correct option is D

m

For the given equation, a > 0, when compared to a $x^2$ + bx + c = 0

So, the graph is upward parabolas shown.

For roots α, β to be in between 1 & 2, f(1) > 0 , f(2) > 0 and 1 < < 2

Firstly, for roots to be real

D ≥ 0

$b^2$ - 4ac ≥ 0

$\Rightarrow$ m ≤ 1 or m ≥ 9 ---------------- (1)

f(x) = $x^2$ - (m - 3) x + m (given)

Now, f(1) = 1 - (m - 3) + m

= 4 > 0 [Always]

f(2) = 4 - (m - 3)2 + m

= 4 - 2m + 6 + m

= 10 - m

10 - m > 0

m < 10 ------------------- (2)

1 < $\frac{(-b)}{2a}$ < 2

1 < (m-3)/2 < 2

2 < (m - 3) < 4

5 < m < 7 -------------------------- (3)

From (1), (2) & (3)

No intersection

Hence m belongs to null set