Question

Let x > 0, y > 0, z > 0 are respectively the $2^{nd},3^{rd},4^{th},$ terms of a G.P and $\Delta =\left|\begin{array}{ccc}{x}^k& x^{k+1}& x^{k+2}\\ y^k& y^{k+1}& y^{k+2}\\ z^k& z^{k+1}& z^{k+2}\end{array}\right|=(r-1{)}^2(1-\frac{1}{r^2})$ (where r is the common ratio) then

• A. k=-1
• B. k=1
• C. k=0
• D. None of these

Answer 1

The correct option is A k=-1

$x^k{y}^k{z}^k\left|\begin{array}{ccc}1& ar& a^2{r}^2\\ 1& a{r}^2& a^2{r}^4\\ 1& a{r}^3& a^2{r}^6\end{array}\right|a^{3(k+1)}.r^{3(2k+1)}\left[\right(r-1\left)\right(r^4-1)-(r^2-1{)}^2]\Rightarrow k=-1$