The number of values of \[\theta \] \[ \in \] (0, \[\pi \]) for which the system of linear equations
x + 3y + 7z = 0
- x + 4y + 7z = 0
(sin3\[\theta \])x + (cos2\[\theta \])y + 2z = 0.
has a non-trival solution, is -
Solutions · 1
Official solution
\[\left| {1−1sin3θ34cos2θ772 } \right| = 0\]
(8 - 7 cos 2\[\theta \]) - 3(-2 - 7 sin 3\[\theta \])
+7 (- cos 2\[\theta \] - 4 sin 3\[\theta \]) = 0
14 - 7 cos 2\[\theta \] + 21 sin 3\[\theta \] - 7 cos 2\[\theta \]
- 28 sin 3\[\theta \] = 0
14 - 7 sin 3\[\theta \] - 14 cos 2\[\theta \] = 0
14 - 7 (3 sin \[\theta \] - 4 sin3\[\theta \] ) - 14 (1 - 2 sin2 \[\theta \]) = 0
- 21 sin \[\theta \] + 28 sin3 \[\theta \] + 28 sin2 \[\theta \] = 0
7 sin \[\theta \] [- 3 + 4 sin2 \[\theta \] + 4 sin \[\theta \]] = 0 sin\[\theta \],
4 sin2 \[\theta \] + 6 sin \[\theta \] - 2 sin \[\theta \] - 3 = 0
2 sin \[\theta (2 sin \[\theta \] + 3) - 1 (2 sin \[\theta \] + 3) = 0
sin \[\theta \] = \[{{ - 3} \over 2}\]; sin\[\theta \] = \[{1 \over 2}\]
Hence, 2 solutions in (0, \[\pi \])