Question

If $x^2$ - ( a - 3)x + a = 0 has atleast one positive root then :

• A. a (- , 0 ) (7, 9).
• B. a (- , 0 ) (7, ).
• C. a (- , 0 ) (9, ).
• D. None of these

Answer 1

The correct option is C

a (- , 0 ) (9, ).

$x^2$- (a-3)x+a=0 ......(1)

(1) has at least one positive root.

$\therefore$ roots are real $\Rightarrow$ $(a-3{)}^2-4a\ge 0$

$\Rightarrow$ a $\le 1ora\ge 9$ ....(2)

let both roots be negative or zero. Then

f(0) = a $\ge$ 0 and $\alpha$ + $\beta$ = a - 3 $\le$ 0 i.e., a $\le$ 3 .

$\therefore$ At least one root is positive for

(- $\infty$,1 ] $\cup$ [9 , $\infty$ ) - [0,3]

i.e., when a $\in$ (- $\infty$ , 0 ) $\cup$ (9, $\infty$).