Statement $$-$$ I : If $$x+1-34-5x+2241x-6$$ $$=$$ $$0$$ then $$x=0Statement$$ $$-$$ II : If $$15-x11011-3x17167-x1413$$ $$=$$ $$0$$ then $$x=$$ $$6$$. Which of the above $$statement(s)$$ is true ?

$$x+1-34-5x+2241x-6$$ $$=$$ $$0$$ $$(x+1)(x2-4x-12-2)+3(-5x+30-8)+4(-5-4x-8)=0$$ $$x3-4x2-14x+x2-4x-14-15x+66-52-16x=0$$ $$x3-3x2-49x=0$$ $$x=015-x111011-3x17167-x1413$$ $$=$$ $$0$$ $$(15-x)(-3)-(11-3x)(3)+(7-x)(6)=0$$ $$-45+3x-33+9x+42-6x=0$$ $$-36+6x=0$$⇒$$x=6$$

Questions

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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Only I is true

Only II is true

Both I and II are true

neither I nor II are true

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detailed solution

Correct option is C

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