Let $f(x)=a x^2+b x+c$ and the $GCD$ of $a, b, c$ be $1$. If $\frac{-7+\sqrt{11} i}{6}$ is a root of $f(x)=0$ and $f\left(\frac{x}{k}\right)-L=(x+4)(3 x-5)$,then $k$ and $L$ are respectively:

If $x = \frac{1}{2} \left( \sqrt{7} + \frac{1}{\sqrt{7}} \right)$,then the value of $\frac{\sqrt{x^2 - 1}}{x - \sqrt{x^2 - 1}}$ is equal to

If a polynomial $P(x)$ of degree $4$ is given by $P(x) = 2x^4 + ax^3 + bx^2 + cx + d$ such that $P(1) = 4, P(2) = 7, P(3) = 12$,and $P(4) = 19$,then find the value of $P(5)$.

If $\cos^4 \theta + \alpha$ and $\sin^4 \theta + \alpha$ are the roots of the equation $x^2 + 2bx + b = 0$,and $\cos^2 \theta + \beta$ and $\sin^2 \theta + \beta$ are the roots of the equation $x^2 + 4x + 2 = 0$,then find the value of $b$.

If $\frac{1}{2} \leq \frac{x^2+x+a}{x^2-x+a} \leq 2$ for all $x \in R$,then $a=$

If $\sqrt{2} \sin^2 x + (3\sqrt{2} + 1) \sin x + 3 > 0$ and $x^2 - 7x + 10 < 0$,then $x$ lies in the interval