Matrices and Determinants 2 Question 17
19. The parameter on which the value of the determinant
$\begin{vmatrix}1 & a & a^{2} \\ \cos (p-d) x & \cos p x & \cos (p+d) x \\ \sin (p-d) x & \sin p x & \sin (p+d) x \end{vmatrix}$
does not depend upon, is
(1997, 2M)
(a) $a$
(b) $p$
(c) $d$
(d) $x$
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Answer:
Correct Answer: 19. (b)
Solution:
- Let $\Delta=$ $\begin{vmatrix}1 & a & a^{2} \\ \cos (p-d) x & \cos p x & \cos (p+d) x \\ \sin (p-d) x & \sin p x & \sin (p+d) x \end{vmatrix}$
Applying $C_{1} \rightarrow C_{1}+C_{3}$
$ \begin{aligned} & \Rightarrow \Delta=\begin{vmatrix} 1+a^{2} & a & a^{2} \\ \cos (p-d) x+\cos (p+d) x & \cos p x & \cos (p+d) x \\ \sin (p-d) x+\sin (p+d) x & \sin p x & \sin (p+d) x \end{vmatrix} \end{aligned} $
$ \begin{aligned} & \Rightarrow \Delta=\begin{vmatrix} 1+a^{2} & a & a^{2} \\ 2 \cos p x \cos d x & \cos p x & \cos (p+d) x \\ 2 \sin p x \cos d x & \sin p x & \sin (p+d) x \end{vmatrix} \end{aligned} $
Applying $C_{1} \rightarrow C_{1}-2 \cos d x C_{2}$
$ \Rightarrow \Delta=\begin{vmatrix} 1+a^{2}-2 a \cos d x & a & a^{2} \\ 0 & \cos p x & \cos (p+d) x \\ 0 & \sin p x & \sin (p+d) x \end{vmatrix} $
$\Rightarrow \Delta=\left(1+a^{2}-2 a \cos d x\right)[\sin (p+d) x \cos p x$$ -\sin p x \cos (p+d) x] $
$ \Rightarrow \quad \Delta=\left(1+a^{2}-2 a \cos d x\right) \sin d x $
which is independent of $p$.
