A value of `theta in (0, pi//3),` for which
`|{:(1+"cos"^(2)theta, "sin"^(2)theta, 4"cos" 6 theta), ("cos"^(2)theta, 1+"sin"^(2)theta, 4"cos"6theta), ("cos"^(2)theta, "sin"^(2)theta, 1+4"cos"6theta):}| = 0,` is

Answer

9

Explanation

`|(1+cos^(2) theta, sin^(2) theta, 4 cos 6 theta),(cos^(2)theta,1+sin^(2)theta, 4 cos 6 theta),(cos^(2) theta, sin^(2)theta, 1+4 cos 6 theta)|=0`
`C_(1)toC_(1)toC_(2) " "R_(1)toR_(1)-R_(2)`
`|(2, sin^(2) theta, 4 cos 6 theta),(2, 1+sin^(2) theta, 4 cos 6 theta),(1 sin^(2) theta, 1 +4, cos 6 theta)|=0implies|(0,-1,0),(2,1+sin^(2)theta, 4 cos 5 theta),(1, sin^(2) theta, 1+4 cos 6 theta)|=0`
`2+4 cos 6 theta=-1/2`
`6 theta=(2pi)/3,(4pi)/3, theta=(pi)/9,(2pi)/9`