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Assertion :If $$ bc+qr=ca+rp=ab+pq=-1$$, then $$\begin{vmatrix} ap & a & p \\ bq & b & q \\ cr & c & r \end{vmatrix}=0\quad (abc,pqr\neq 0)$$ Reason: If system of equations $${ a }_{ 1 }x+{ b }_{ 1 }y+{ c }_{ 1 }=0,\quad { a }_{ 2 }x+{ b }_{ 2 }y+{ c }_{ 2 }=0,{ \quad a }_{ 3 }x+{ b }_{ 3 }y+{ c }_{ 3 }=0$$ has non-trivial solutions, $$\begin{vmatrix} { a }_{ 1 } & { b }_{ 1 } & { c }_{ 1 } \\ { a }_{ 2 } & { b }_{ 2 } & { c }_{ 2 } \\ { a }_{ 3 } & { b }_{ 3 } & { c }_{ 3 } \end{vmatrix}=0$$


A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Assertion is incorrect but Reason is correct
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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
Reason is true
Assertion
Given equations can be rewritten as
$$bc+qr+1=0$$   ...(1)
$$ca+rp+1=0$$   ...(2)
$$ab+pq+1=0$$  ...(3)
Multiplying (1),(2) and (3) by ap, bq,cr respectively, we get
$$\left( abc \right) p+\left( pqr \right) a+ap=0\\ \left( abc \right) q+\left( pqr \right) b+bq=0\\ \left( abc \right) r+\left( pqr \right) c+cr=0$$
These equation are consistent,
Hence $$\begin{vmatrix} p\quad  & a\quad  & ap \\ q\quad  & b\quad  & bq \\ r\quad  & c\quad  & cr \end{vmatrix}=0\Rightarrow \begin{vmatrix} p & q & r \\ a & b & c \\ ap & bq & cr \end{vmatrix}=0$$ ( interchanging rows into columns)  
$$\Rightarrow \left( -1 \right) \begin{vmatrix} ap\quad  & bq\quad  & cr \\ a & b & c \\ p & q & r \end{vmatrix}=0\quad \left( { R }_{ 1 }{ \leftrightarrow R }_{ 2 } \right) \\ \Rightarrow \begin{vmatrix} ap\quad  & bq\quad  & cr \\ a & b & c \\ p & q & r \end{vmatrix}=0$$

Mathematics

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