Search any question & find its solution
Question:
Answered & Verified by Expert
The value of $\left|\begin{array}{lll}x & p & q \\ p & x & q \\ p & q & x\end{array}\right|$ is
Options:
Solution:
1059 Upvotes
Verified Answer
The correct answer is:
$(x-p)(x-q)(x+p+q)$
Given
$$
\begin{aligned}
&\left|\begin{array}{lll}
x & p & q \\
p & x & q \\
p & q & x
\end{array}\right| \\
=& x\left[x^{2}-q^{2}\right]-p[p x-p q]+q[p q-p x] \\
=& x^{3}-x q^{2}-p^{2} x+p^{2} q+p q^{2}-p q x \\
=& x(x-q)(x+q)-p^{2}(x-q)-p q(x-q) \\
=&(x-q)\left[x^{2}+x q-p^{2}-p q\right] \\
=&(x-q)(x-p)(x+p+q)
\end{aligned}
$$
$$
\begin{aligned}
&\left|\begin{array}{lll}
x & p & q \\
p & x & q \\
p & q & x
\end{array}\right| \\
=& x\left[x^{2}-q^{2}\right]-p[p x-p q]+q[p q-p x] \\
=& x^{3}-x q^{2}-p^{2} x+p^{2} q+p q^{2}-p q x \\
=& x(x-q)(x+q)-p^{2}(x-q)-p q(x-q) \\
=&(x-q)\left[x^{2}+x q-p^{2}-p q\right] \\
=&(x-q)(x-p)(x+p+q)
\end{aligned}
$$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.