The integer ‘k’ for which the inequality ${\mathrm{x}}^2-2(3\mathrm{k}-1)\mathrm{x}+8{\mathrm{k}}^2-7>0$ is valid for every x in R, is

${\mathrm{x}}^2-2(3\mathrm{k}-1)\mathrm{x}+8{\mathrm{k}}^2-7>0\Rightarrow$The roots are imaginary or(real and equal)

$\Rightarrow$Discriminant $\le 0\Rightarrow 4(3\mathrm{k}-1{)}^2-4\times 1\times \left(8{\mathrm{k}}^2-7\right)\le 0\Rightarrow 9{\mathrm{k}}^2-6\mathrm{k}+1-8{\mathrm{k}}^2+7\le 0$

$\Rightarrow {\mathrm{k}}^2-6\mathrm{k}+8\le 0\Rightarrow (\mathrm{k}-4)(\mathrm{k}-2)\le 0\Rightarrow \mathrm{k}\in [2,4]$

The number of integer values of k is 3.