Question

Assertion :If $bc+qr=ca+rp=ab+pq=-1$, then $\left|\begin{array}{ccc}ap& a& p\\ bq& b& q\\ cr& c& r\end{array}\right|=0(abc,pqr\ne 0)$ Reason: If system of equations $a_{1}x+b_{1}y+c_1=0,a_{2}x+b_{2}y+c_2=0,a_{3}x+b_{3}y+c_3=0$ has non-trivial solutions, $\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|=0$

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

Answer 1

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 $\left|\begin{array}{ccc}p& a& ap\\ q& b& bq\\ r& c& cr\end{array}\right|=0\Rightarrow \left|\begin{array}{ccc}p& q& r\\ a& b& c\\ ap& bq& cr\end{array}\right|=0$ ( interchanging rows into columns)
$\Rightarrow (-1)\left|\begin{array}{ccc}ap& bq& cr\\ a& b& c\\ p& q& r\end{array}\right|=0\left(R_1{\leftrightarrow R}_2\right)\Rightarrow \left|\begin{array}{ccc}ap& bq& cr\\ a& b& c\\ p& q& r\end{array}\right|=0$