If the vectors $\overline{A B}=3 \hat{i}+4 \hat{k}$ and $\overline{A C}=5 \hat{i}-2 \hat{j}+4 \hat{k}$ are the sides of the triangle $A B C$, then the length of the median through $A$ is

If $\mathrm{f}(x)=x^3+b x^2+c x+d$ and $0< b^2< c$, then in $(-\infty, \infty)$

If $\mathrm{f}\left(\frac{x-4}{x-2}\right)=2 x+1, x \in \mathbb{R}-\{1,-2\}$, then $\int \mathrm{f}(x) \mathrm{d} x$ is equal to

If $\sin (\theta-\alpha), \sin \theta$ and $\sin (\theta+\alpha)$ are in H.P., then the value of $\cos ^2 \theta$ is