Question

For the equation $x^2$ - (a - 3) x + a = 0 (a ∈ R), find the values of 'a' such that exactly one root lies in between 1 and 2.

• A. a (10, )
• B. a (-, -10)
• C. a (-, -1)
• D. None of these

Answer 1

The correct option is A

a (10, )

Consider the graph of f(x) = $x^2$ - (a - 3) x + a

For roots to be real

D ≥ 0

$b^2$ - 4ac $\ge$ 0

$\Rightarrow$ a less than or equal to 1 (or) a ≥ 9 ----------- (1)

For exactly 1 root to lie in between 1 and 2, f(1) and f(2) should be of opposite signs.

f(1) f(2) < 0

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

f(1) = 1 - (a - 3) + a = 4

f(2) =4 - 2(a - 3) + a = 4 - 2a + 6 + a

= 10 - a

f(1) (2) < 0

4 (10 - a) < 0

a $ϵ$ (10, $\infty$) ----------------- (2)

From (1) & (2), a $ϵ$ (10, $\infty$)