Determinants Matrices Question 21
Question: The determinant $ \begin{vmatrix} x & \sin \theta & \cos \theta \\ -\sin \theta & -x & 1 \\ \cos \theta & 1 & x \\ \end{vmatrix} $ is independent of
Options:
A) x only
B) $ \theta $ only
C) x and $ \theta $ both
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
- [b] $ \begin{vmatrix} x & \sin \theta & \cos \theta \\ -\sin \theta & -x & 1 \\ \cos \theta & 1 & x \\ \end{vmatrix} $
$ =x(x^{2}-1)-\sin \theta (-x\sin \theta -\cos \theta ) $
$ +\cos \theta (-\sin \theta +x\cos \theta ) $
$ =-x^{3}-x+x{{\sin }^{2}}\theta +\sin \theta \cos \theta $
$ -\cos \theta \sin \theta +x{{\cos }^{2}}\theta ) $
$ =x^{3}-x+x=x^{3} $
BETA
