Let p, q and r be three distinct positive real numbers. If D=| array lll p & q & r q & r & p r & p & q array | , then which one of the following is correct?

Correct answer: A. D < 0 aligned D & =| array lll p & q & r q & r & p r & p & q array | & =p(r q-p^2)+q(p r-q^2)+r(p q-r^2) & =3 p q r-p^3-q^3-r^3 & =-(p^3+q^3+r^3-3 p q r) aligned For real and distinct positive real value of p, q and r . aligned & p^3+q^3+r^3-3 p q r 0 & D 0 . aligned

NDA 2020

Exam: NDA
Year: 2020
Subject: Mathematics
Chapter: Determinants

NDA 2020 · Mathematics · Determinants

Let p, q and r be three distinct positive real numbers. If D=|...

Let

p, q

and

r

be three distinct positive real numbers. If

D=\left|

\begin{array}{lll}p & q & r \\ q & r & p \\ r & p & q\end{array}

\right|

, then which one of the following is correct?

Options

A $D < 0$
B $D \leq 0$
C $D > 0$
D $D \geq 0$

Correct answer

A. $D < 0$

Step-by-step solution

\begin{aligned} D & =\left|\begin{array}{lll} p & q & r \\ q & r & p \\ r & p & q \end{array}\right| \\ & =p\left(r q-p^2\right)+q\left(p r-q^2\right)+r\left(p q-r^2\right) \\ & =3 p q r-p^3-q^3-r^3 \\ & =-\left(p^3+q^3+r^3-3 p q r\right) \end{aligned}

For real and distinct positive real value of

p, q and r

\begin{aligned} & p^3+q^3+r^3-3 p q r \gt 0 \\ & \therefore D \lt 0 . \end{aligned}

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