Question

Find the values of m for which the roots of quadratic equation $x^2$ - (m - 3) x + m = 0 lies between 1 and 2.

• A. m $ϵ$ (1, 2)
• B. m $ϵ$ (5, 7)
• C. m $ϵ$ (-$\infty$, 1]$\cup$[9, $\infty$)
• D. m $ϵ$ $\varphi$

Answer 1

The correct option is D

m $ϵ$ $\varphi$

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

Visualizing using graph the given conditions are represented as,

Observe that i) f(1) > 0, f(2) > 0

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

iii) Also D ≥ 0 for roots to be real

i) f(1) =1 - (m - 3) + m = 4 > 0 (Always true)

f(2) = 4 - 2 (m - 3) + m = 4 - 2m + 6 + m = 10 - m > 0

m < 10 ------------- (1)

ii) 1 < [(m-3)/2] < 2 ⇒ 2 < m - 3 < 4

5 < m < 7 ------------ (2)

iii) D ≥ 0 $\Rightarrow$ ${(m-3)}^2$ - 4 m ≥ 0 $\Rightarrow$ $m^2$ + 9 - 6m - 4m ≥ 0

$m^2$ - 10m + 9 ≥ 0

$m^2$ - 9m - m + 9 ≥ 0

m(m - 9) -1(m - 9) ≥ 0

m ≤ 1 (or) m ≥ 9 ----------- (3)

Intersection of (1), (2) & (3) have no values of m in common.

So, m $ϵ$ $\varphi$