$∣ ∣ ∣ ∣ \begin{matrix} x + 1 & x + 2 & x + 4 x + 3 & x + 5 & x + 8 x + 7 & x + 10 & x + 14 \end{matrix} ∣ ∣ ∣ ∣ =$

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-2

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x^{2} - 2

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None of these

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Solution

The correct option is B -2
$Δ = ∣ ∣ ∣ ∣ \begin{matrix} - 1 & - 2 & x + 4 - 2 & - 3 & x + 8 - 3 & - 4 & x + 14 \end{matrix} ∣ ∣ ∣ ∣,$
by $C_{1} \to C_{1} - C_{2} C_{2} \to C_{2} - C_{3} = ⎡ ⎢ ⎣ \begin{matrix} - 1 & - 1 & x - 2 & - 1 & x - 3 & - 1 & x + 2 \end{matrix} ⎤ ⎥ ⎦,$
by $C_{2} \to C_{2} - C_{1} C_{3} \to C_{3} - 4 C_{1} = - (- x - 2 + x) + 1. (- 2 x - 4 + 3 x) + x (2 - 3) = 2 + x - 4 - x = - 2$ .
Trick : Put x = 1. Then $∣ ∣ ∣ ∣ \begin{matrix} 2 & 3 & 5 4 & 6 & 9 8 & 11 & 15 \end{matrix} ∣ ∣ ∣ ∣ = - 2$ .

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