$$\begin{gathered} \mathrm{If} \mathrm{f}\left(\mathrm{x}\right)={\cos }^2\mathrm{x}+{\cos }^{2}2\mathrm{x}+{\cos }^{2}3\mathrm{x}, \mathrm{then} \mathrm{the} \\ \mathrm{number} \mathrm{of} \mathrm{values} \mathrm{of} \mathrm{x}\in \left[0,2\mathrm{\pi }\right] \mathrm{for} \mathrm{which} \\ \mathrm{f}\left(\mathrm{x}\right)=1 \mathrm{is} \end{gathered}$$

$$\begin{gathered} \begin{array}{l}f\left(x\right)=1\\ {\cos }^{2}3x+{\cos }^{2}2x+{\cos }^{2}x=1\\ {\left(4{\cos }^{3}x-3\cos x\right)}^2+{\left(2{\cos }^{2}x-1\right)}^2+{\cos }^{2}x=1\\ \begin{array}{l}16{\cos }^{6}x-20{\cos }^{4}x+6\cos x=0\\ {\cos }^{2}x\left(8{\cos }^{4}x-10{\cos }^{2}x+3\right)=0\\ {\cos }^{2}x=0\text{ or }\left(8{\cos }^{4}x-10{\cos }^{2}x+3\right)=0\\ {\cos }^{2}x=\frac{10\pm \sqrt{100-96}}{16} \\ {\cos }^{2}x=0,\frac{1}{2},\frac{3}{4} \\ \text{ no. of solutions are }\frac{\pi }{6},\frac{5\pi }{6},\frac{\pi }{2},\frac{\pi }{3},\frac{3\pi }{2},\frac{2\pi }{3},\frac{4\pi }{3},\frac{5\pi }{3},\frac{7\pi }{6},\frac{11\pi }{6}\end{array}\end{array} \end{gathered}$$