If Delta1 - begin vmatrix 181 - 1 Ma - b - clla- 2 - b - 2 - c- 2 1 - bc - all1 - ca - bll1 - ab - cll

The correct option is B Δ_1+Δ_2=0Δ_1= 1 amp; 1 amp; 1 a amp; b amp; c a^2 amp; b^2 amp; c^2 , amp; Δ_2= 1 amp; bc amp; a 1 ...

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Byju's Answer

Standard XII

Mathematics

Properties of Determinants

If Delta1 = b...

Question

If $Δ_{1} = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 a & b & c a^{2} & b^{2} & c^{2} \end{matrix} ∣ ∣ ∣ ∣, Δ_{2} = ∣ ∣ ∣ ∣ \begin{matrix} 1 & b c & a 1 & c a & b 1 & a b & c \end{matrix} ∣ ∣ ∣ ∣$ , then

Δ_{1} + Δ_{2} = 0

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Δ_{1} + 2 Δ_{2} = 0

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Δ_{1} = Δ_{2}

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Δ_{1} = 2 Δ_{2}

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Solution

The correct option is A $Δ_{1} + Δ_{2} = 0$
$Δ_{1} = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 a & b & c a^{2} & b^{2} & c^{2} \end{matrix} ∣ ∣ ∣ ∣,$ & $Δ_{2} = ∣ ∣ ∣ ∣ \begin{matrix} 1 & b c & a 1 & c a & b 1 & a b & c \end{matrix} ∣ ∣ ∣ ∣$
In $Δ_{2}$ ,Transposing the determinant
$Δ_{2} = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 b c & c a & a b a & a & c \end{matrix} ∣ ∣ ∣ ∣$
$C_{1} \to a C_{1}, C_{2} \to b C_{2}, C_{3} \to C C_{3}$
$Δ_{2} = \frac{1}{a b c} ∣ ∣ ∣ ∣ \begin{matrix} a & b & c a b c & a b c & a b c a^{2} & b^{2} & c^{2} \end{matrix} ∣ ∣ ∣ ∣$
Taking abc common from $R_{2}$
$Δ_{2} = ∣ ∣ ∣ ∣ \begin{matrix} a & b & c 1 & 1 & 1 a^{2} & b^{2} & c^{2} \end{matrix} ∣ ∣ ∣ ∣$
$R_{1} \leftrightarrow R_{2}$
$Δ_{2} = - ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 a & b & c a^{2} & b^{2} & c^{2} \end{matrix} ∣ ∣ ∣ ∣$
$\Rightarrow Δ_{2} = - Δ_{1} \Rightarrow Δ_{2} + Δ_{1} = 0$

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If Δ1=∣∣ ∣∣111abca2b2c2∣∣ ∣∣,Δ2=∣∣ ∣∣1bca1cab1abc∣∣ ∣∣, then

If $Δ_{1} = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 a & b & c a^{2} & b^{2} & c^{2} \end{matrix} ∣ ∣ ∣ ∣, Δ_{2} = ∣ ∣ ∣ ∣ \begin{matrix} 1 & b c & a 1 & c a & b 1 & a b & c \end{matrix} ∣ ∣ ∣ ∣$ , then