The integer $k$ for which the inequality $x^2-2(4k-1)x+15k^2-2k-7>0$ is valid for any $x$, is

The given inequality is valid for any ‘$x$’, if and only if $\Delta <0$ as $a=1>0$

Discriminant is negative. 

i.e if  $4{(4K-1)}^2-4(15K^2-2K-7)<0$

i.e if $K^2-6K+8<0$

$$\begin{gathered} \Rightarrow (K-4)(K-2)<0 \\ \Rightarrow 2<K<4 \\ \Rightarrow K=3 \end{gathered}$$