Among the statements : I: If vmatrix 1 1 1 vmatrix=vmatrix 0 0 0 vmatrix, then ^2 + ^2 + ^…

Question

Among the statements :

I: If $\begin{vmatrix} 1 \quad  \cos \alpha  \quad  \cos \beta \ \cos \alpha  \quad  1 \quad  \cos \gamma \ \cos \beta  \quad  \cos \gamma  \quad  1 \end{vmatrix}=\begin{vmatrix} 0 \quad  \cos \alpha  \quad  \cos \beta \ \cos \alpha  \quad  0 \quad  \cos \gamma \ \cos \beta  \quad  \cos \gamma  \quad  0 \end{vmatrix}$, then ${\cos }^{2}\alpha +{\cos }^{2}\beta +{\cos }^{2}\gamma =\frac{3}{2}$, and

II: If $\begin{vmatrix} {x}^{2}+x \quad  x+1 \quad  x-2 \ 2{x}^{2}+3x-1 \quad  3x \quad  3x-3 \ {x}^{2}+2x+3 \quad  2x-1 \quad  2x-1 \end{vmatrix}=px+q$, then ${p}^{2}=196{q}^{2}$,

Accepted Answer

$\begin{gathered} |\begin{array}{ccc}1 \quad  \cos \alpha  \quad  \cos \beta \ \cos \alpha  \quad  1 \quad  \cos \gamma \ \cos \beta  \quad  \cos \gamma  \quad  1\end{array}|=1(\begin{gathered} 1-{\cos }^{2}\gamma )-\cos \alpha (\cos \alpha -\cos \beta \cos \gamma )+\cos \beta (\cos \alpha \cos \gamma -\cos \beta ) \ =1-{\cos }^{2}\gamma -{\cos }^{2}\alpha -{\cos }^{2}\beta +2\cos \beta +2+\cos \alpha \cos \beta \cos \gamma \ |\begin{array}{ccc}0 \quad \cos \alpha \quad \cos \beta \ \cos \alpha \quad 0 \quad \cos \gamma \ \cos \beta \quad \cos \gamma \quad 0\end{array}|=-\cos \alpha (-\cos \beta \cos \gamma )+\cos \beta (\cos \alpha \cos \gamma )=2\cos \alpha \cos \beta -\cos \gamma \ \Rightarrow {\cos }^{2}\alpha +{\cos }^{2}\beta +{\cos }^{2}\gamma =1 \end{gathered}$

$\begin{gathered} \quad \because q=|\begin{array}{ccc}0 \quad 1 \quad -2 \ -1 \quad 0 \quad -3 \ 3 \quad -1 \quad -1\end{array}|=-10-2=-12 \ -p+q=|\begin{array}{ccc}0 \quad 0 \quad -3 \ -2 \quad -3 \quad -6 \ 2 \quad -3 \quad -3\end{array}|=-36 \ \Rightarrow p=q+36=24 \ \Rightarrow {p}^{2}
e 196{q}^{2} \end{gathered} \end{gathered}$

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