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3 and 4 .Determinants and Matrices
normal
The values of $\theta, \lambda$ for which the following equations $\sin \theta x - cos\theta y + (\lambda +1)z = 0$; $\cos\theta x + \sin\theta\, y - \lambda z = 0$;$ \lambda x +(\lambda + 1)y + \cos\theta z = 0$ have non trivial solution, is
A
$\theta = n\pi , \lambda \in R - {0}$
B
$\theta = 2n\pi , \lambda $ is any rational number
C
$\theta = (2n + 1)\pi , \lambda \in R+, n \in I$
D
$\theta = (2n + 1),\frac{\pi }{2} \lambda \in R, n \in I$
Solution
for non trivial solution $\left| {\,\begin{array}{*{20}{c}}{\sin \theta }&{ – \cos \theta }&{\lambda + 1}\\{\cos \theta }&{\sin \theta }&{ – \lambda }\\\lambda &{\lambda + 1}&{\cos \theta }\end{array}\,} \right|$ $= 0$ ;
this gives $2$ $\cos\theta (\lambda^2 + \lambda + 1) = 0$
Standard 12
Mathematics
