Mathematics - Quadratic Equation Question with Solution | TestHub

Let $3^x=t\Rightarrow t^2-ta-a+3\le 0:t>0$

Let $f(t)=t^2-at+3-a$

Discriminate of $f(t)=0$ is $a^2-4(3-a)a$

i.e., $a^2+4a-12$.

$D\ge 0\Rightarrow a\le -6$ or $a\ge 2$.

Similarly $x\in (1,2)$ then $t\in (3,9)$

Similar to the above the question

$$\begin{array}{rlr}& f(\theta )=\left(a^2+b^2+c^2\right){\cos }^2\theta & f^1(\theta )={\left(a^2+b^2+c^2\right)}^3\times (-\sin 2\theta )\end{array}$$