Number of integral values of k for which the equation 4 ${\cos }^{-1}\left(-\left|x\right|\right)=k$ has exactly two solutions is

$\begin{array}{l}4{\cos }^{-1}\left(-\left|x\right|\right)=k\\ \Rightarrow {\cos }^{-1}\left(-\left|x\right|\right)=\frac{k}{4} \left(1\right)\end{array}$

The graph of $y={\cos }^{-1}\left(-\left|x\right|\right)$ is as shown in the following figure

Clearly from the figure equation (1) has two solutions if line 

$\begin{array}{l} y=\frac{k}{4} \mathrm{intersect} \mathrm{the} \mathrm{graph} \mathrm{between} \frac{\pi }{2} \mathrm{and} \pi .\\ \therefore \frac{\pi }{2}<\frac{k}{4}\le \pi \\ \therefore 2\pi <k\le 4\pi \end{array}$

Therefore, integral values of k are 7,8, 9, 10, I1,12.