Q51 / 160 · AP EAMCET 2024 (22 May Shift 1) · Mathematics
If both the roots of the equation $x^2-6 a x+2-2 a+9 a^2=0$ exceeds 3 , then
- A$a \lt \frac{3}{2}$
- B$a\gt\frac{3}{2}$
- C$a \lt \frac{5}{2}$
- Correct answerD$a\gt\frac{11}{9}$
Solution
$\begin{aligned}
& x^2-6 a x+2-2 a+9 a^2=0 \\
& \because \quad \mathrm{D}\gt0 \Rightarrow 36 a^2-4\left(2-2 a+9 a^2\right)\gt0 \\
& \Rightarrow 8(a-1)\gt0 \Rightarrow a\gt1
\end{aligned}$
$f(3)\gt0 \Rightarrow 9-18 a-2 a+9 a^2+2\gt0$
$\begin{aligned}
& \Rightarrow 9 a^2-20 a+11\gt0 \\
& \Rightarrow(9 a-11)(a-1)\gt0 \Rightarrow a\gt\frac{11}{9} \\
& f^{\prime}(3) \lt 0 \Rightarrow 6-6 a \lt 0 \Rightarrow a\gt1
\end{aligned}$
Condition for roots to be greater than 3 is $a\gt\frac{11}{9}$.
Condition for roots to be greater than 3 is $a\gt\frac{11}{9}$.
Hence, the correct answer is option: 'D'.