Statement - I : If $|\begin{matrix} x + 1 & - 3 & 4 \\ - 5 & x + 2 & 2 \\ 4 & 1 & x - 6 \end{matrix}| = 0 t h e n x = 0$
Statement - II : If $|\begin{matrix} 15 - x & 11 & 0 \\ 11 - 3 x & 17 & 16 \\ 7 - x & 14 & 13 \end{matrix}| = 0 t h e n x = 6$ . Which of the above statement(s) is true ?

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Both I and II are true

Only I is true

Only II is true

neither I nor II are true

answer is C.

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Detailed Solution

$|\begin{matrix} x + 1 & - 3 & 4 \\ - 5 & x + 2 & 2 \\ 4 & 1 & x - 6 \end{matrix}| = 0$
$(x + 1) (x^{2} - 4 x - 12 - 2) + 3 (- 5 x + 30 - 8) + 4 (- 5 - 4 x - 8) = 0 x^{3} - 4 x^{2} - 14 x + x^{2} - 4 x - 14 - 15 x + 66 - 52 - 16 x = 0 x^{3} - 3 x^{2} - 49 x = 0 x = 0$
$|\begin{matrix} 15 - x & 11 & 10 \\ 11 - 3 x & 17 & 16 \\ 7 - x & 14 & 13 \end{matrix}| = 0 (15 - x) (- 3) - (11 - 3 x) (3) + (7 - x) (6) = 0 - 45 + 3 x - 33 + 9 x + 42 - 6 x = 0 - 36 + 6 x = 0 \Rightarrow x = 6$