If a a-1 a-1 -b b-1 b-1 c c-1 c-1 - a-1 b-1 c-1 a-1 b-1 c-1 -1 n-2 a -1 n-2 b -1 n c -0- Then the value of n is

The correct option is C any integer a+1 amp;b+1 amp;c 1 a 1 amp;b 1 amp;c+1 ( 1)^n+2a amp;( 1)^n+2b amp;( 1)^nc=2 amp;2 amp; 2 ...

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Properties of Determinants

If a a+1 ...

Question

If $∣ ∣ ∣ ∣ \begin{matrix} a & a + 1 & a - 1 - b & b + 1 & b - 1 c & c - 1 & c + 1 \end{matrix} ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ \begin{matrix} a + 1 & b + 1 & c - 1 a - 1 & b - 1 & c + 1 {(- 1)}^{n + 2} a & {(- 1)}^{n + 2} b & {(- 1)}^{n} c \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0$ . Then the value of $n$ is

zero

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any even integer

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any odd integer

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any integer

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Solution

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\neq

∣ ∣ ∣ ∣ \begin{matrix} a & a + 1 & a - 1 - b & b + 1 & b - 1 c & c - 1 & c + 1 \end{matrix} ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ \begin{matrix} a + 1 & b + 1 & c - 1 a - 1 & b - 1 & c + 1 (- 1)^{n + 2} a & (- 1)^{n - 1} b & (- 1)^{n} c \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0

a, b, c

b (a + c) \neq 0

∣ ∣ ∣ ∣ \begin{matrix} a & a + 1 & a - 1 - b & b + 1 & b - 1 c & c - 1 & c + 1 \end{matrix} ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ \begin{matrix} a + 1 & b + 1 & c - 1 a - 1 & b - 1 & c + 1 (- 1)^{n + 2} a & (- 1)^{n + 1} b & (- 1)^{n} c \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0

n

Let a, b, c be such that (b+c) ≠0. If

Then, the value of 'n' is

∣ ∣ ∣ ∣ ∣ \begin{matrix} (- 1)^{n} a & (- 1)^{n + 1} b & (- 1)^{n + 1} c a + 1 & 1 - b & 1 + c a - 1 & b + 1 & 1 - c \end{matrix} ∣ ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ \begin{matrix} (- 1)^{n + 1} a & a + 1 & a - 1 (- 1)^{n} b & 1 - b & b + 1 (- 1)^{n + 2} c & 1 + c & 1 - c \end{matrix} ∣ ∣ ∣ ∣ ∣

$t a n^{- 1} n + c o t^{- 1} (n + 1)$ is equal to

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