Question

If the quadratic equation $ax ^2- bx +12=0$, where $a$ and $b$ are positive integers not exceding 10 , has roots both greater than 2 , then the number of possible ordered pair $(a, b)$ is

• A. 1
• B. 2
• C. 4
• D. 8

Answer 1

Answer: (A)

$\Theta $ Both roots are greater than 2
$\therefore$
(i) a f (2) >0
$\Rightarrow a(4 a-2 b+12)>0 \Rightarrow 2 a-b+6>0$
(ii) $D \geq 0 \Rightarrow b ^2-48 a \geq 0 \Rightarrow b \geq 4 \sqrt{3} a$
(iii) $\frac{-(- b )}{2 a }>2 \Rightarrow b >4 a$
$\therefore b \geq 4 \sqrt{3} a$
If $a=1, b \geq 4 \sqrt{3}, b \geq 7$
$\Theta 2 a - b +6>0$
$\therefore a=1, b=7$ satisfies it
$\therefore $ Number of ordered pairs $(a, b)=1$