Question

If n is an integer and if $\left|\begin{array}{ccc}{x}^n& x^{n+2}& x^{n+3}\\ y^n& y^{n+2}& y^{n+3}\\ z^n& z^{n+2}& z^{n+3}\end{array}\right|=(x-y(y-z)(z-x)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)$ then n equals

• A. 1
• B. -1
• C. 2
• D. None of these.

Answer 1

Answer: (B)

$\left|\begin{array}{ccc}{x}^n& x^{n+2}& x^{n+3}\\ y^n& y^{n+2}& y^{n+3}\\ z^n& z^{n+2}& z^{n+3}\end{array}\right|=x^n{y}^n{z}^n\left|\begin{array}{ccc}1& x^2& x^3\\ 1& y^2& y^3\\ 1& z^2& z^3\end{array}\right|$
$x^n{y}^n{z}^n\left|\begin{array}{ccc}1& x^2& x^3\\ 1& y^2-x^3& y^3-x^3\\ 0& z^2-x^2& z^3-x^3\end{array}\right|$
(By operating $R_3-R_1{R}_2-R_1$)
= $x^n{y}^n{z}^n(y-x)(z-x)\left|\begin{array}{ccc}1& x^2& x^3\\ 0& x+y& x^2+y^2+xy\\ 0& x+z& x^2+z^2+xz\end{array}\right|$
= $x^n{y}^n{z}^n(x-y)(x-z)(x^3+x{z}^2+x^{2}z+y{z}^2+y{z}^2+xyz-x^3-x{y}^2-x^{2}y-z{x}^2-z{y}^2-xyz)$
= $x^n{y}^n{z}^n(x-y)(x-z)[x{z}^2+y{z}^2-x{y}^2-2y^2]$
= $x^n{y}^n{z}^n(x-y)(x-z)[x(z^2-y^2)+yz(z-y)]$
= $x^n{y}^n{z}^n(x-y)(x-z)(z-y)(xy+yz+zx)$
= $(x^n{y}^n{z}^n)(x-y)(x-z)(z-y)(xy+yz+zx)$
= $x^{n+1}{y}^{n+1}{z}^{n+1}(x-y)(y-Z)(z-X)$
$\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)$
Comparing with given value of determinant $n$ + 1
= 0 $\Rightarrow$ n = - 1.