Question

${\Delta }_1=\left|\begin{array}{ccc}{y}^5{z}^6(z^3-y^3)& x^4{z}^6(x^3-z^3)& x^4{y}^5(y^3-x^3)\\ y^2{z}^3(y^6-z^6)& x{z}^3(z^6-x^6)& x{y}^2(x^6-y^6)\\ y^2{z}^3(z^3-y^3)& x{z}^3(x^3-z^3)& x{y}^2(y^3-x^3)\end{array}\right|$ and ${\Delta }_2=\left|\begin{array}{ccc}x& y^2& z^3\\ x^4& y^5& z^6\\ x^7& y^8& z^9\end{array}\right|$. then ${\Delta }_1$ is equal to

• A. ${{\Delta }_2}^3$
• B. ${{\Delta }_2}^2$
• C. ${{\Delta }_2}^4$
• D. none of these

Answer 1

The correct option is A ${{\Delta }_2}^3$

On solving
${\Delta }_2=\left|\begin{array}{ccc}x& y^2& z^3\\ x^4& y^5& z^6\\ x^7& y^8& z^9\end{array}\right|=\left(x{y}^2{z}^3\right)\left((z^3-y^3)(z^3-x^3)(y^3-x^3)\right)$
Now
${\Delta }_1=\left|\begin{array}{ccc}{y}^5{z}^6(z^3-y^3)& x^4{z}^6(x^3-z^3)& x^4{y}^5(y^3-x^3)\\ y^2{z}^3(y^6-z^6)& x{z}^3(z^6-x^6)& x{y}^2(x^6-y^6)\\ y^2{z}^3(z^3-y^3)& x{z}^3(x^3-z^3)& x{y}^2(y^3-x^3)\end{array}\right|=\left(y^2{z}^3(z^3-y^3)\right)\left(x{z}^3(z^3-x^3)\right)\left(x{y}^2(y^3-x^3)\right)\left|\begin{array}{ccc}{y}^3{z}^3& {-x}^3{z}^3& x^3{y}^3\\ -(y^3+z^3)& (z^3+x^3)& -(x^3+y^3)\\ 1& -1& 1\end{array}\right|$
Applying $C_2\to C_2+C_1,C_3\to C_3-C_1$
$=x^2{y}^4{z}^6(z^3-y^3)(z^3-x^3)(y^3-x^3)\left|\begin{array}{ccc}{y}^3{z}^3& z^3(y^3-x^3)& y^3(x^3-z^3)\\ -(y^3+z^3)& x^3-y^3& z^3-x^3\\ 1& 0& 0\end{array}\right|={\left(x{y}^2{z}^3\right)}^2{\left((z^3-y^3)(z^3-x^3)(y^3-x^3)\right)}^2={{\Delta }_2}^3$
Therefore, $\Delta ={{\Delta }_2}^3$
Hence, option 'A' is correct.