If beginvmatrix a-1- 2 - a- 2 -a-1- 2 lb-1- 2 - b- 2 -b-1- 2c-1- 2 - c-2 -c-1- 2lendvmatrix - k beginvmatrix 1 - a - a - 2 1 - b - b- 2 l1 - c - c- 2lendvmatrix - then the value of k isA- 4B- -4C- -2D- 1

The correct option is B 4 nbsp;C_3→ C_3 C_1 Δ= (a 1)^2 amp; a^2 amp; 4a (b 1)^2 amp; b^2 amp; 4b (c 1)^2 amp; c^2 amp; 4c =4 (a 1)^2 ...

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Byju's Answer

Standard XII

Mathematics

Cofactor

If beginvmatr...

Question

If
$∣ ∣ ∣ ∣ ∣ \begin{matrix} (a - 1)^{2} & a^{2} & (a + 1)^{2} (b - 1)^{2} & b^{2} & (b + 1)^{2} (c - 1)^{2} & c^{2} & (c + 1)^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ = k ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & a & a^{2} 1 & b & b^{2} 1 & c & c^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣,$
then the value of $k$ is

4

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Solution

The correct option is B $- 4$
$C_{3} \to C_{3} - C_{1}$
$Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} (a - 1)^{2} & a^{2} & 4 a (b - 1)^{2} & b^{2} & 4 b (c - 1)^{2} & c^{2} & 4 c \end{matrix} ∣ ∣ ∣ ∣ ∣ = 4 ∣ ∣ ∣ ∣ ∣ \begin{matrix} (a - 1)^{2} & a^{2} & a (b - 1)^{2} & b^{2} & b (c - 1)^{2} & c^{2} & c \end{matrix} ∣ ∣ ∣ ∣ ∣$

$C_{1} \to C_{1} - C_{2} + 2 C_{3}$
$Δ = 4 ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & a^{2} & a 1 & b^{2} & b 1 & c^{2} & c \end{matrix} ∣ ∣ ∣ ∣ ∣ = - 4 ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & a & a^{2} 1 & b & b^{2} 1 & c & c^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣$
$\Rightarrow k = - 4$

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If ∣∣ ∣ ∣∣(a−1)2a2(a+1)2(b−1)2b2(b+1)2(c−1)2c2(c+1)2∣∣ ∣ ∣∣=k∣∣ ∣ ∣∣1aa21bb21cc2∣∣ ∣ ∣∣, then the value of k is