Question

The number of positive integral solutions of the equation det$\left[\begin{array}{ccc}{y}^3+1& y^{2}z& y^{2}x\\ y{z}^2& z^3+1& z^2\\ y{x}^2& x^z& x^3+1\end{array}\right]=11is$

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The correct option is C 3

Multiply by y,z and x in rows 1,2 and 3 respectively and then take common y, z and x from column 1,2 and 3 respectively, then
det$\left[\begin{array}{ccc}{y}^3+1& y^3& y^3\\ z^3& z^3+1& z^3\\ x^3& x^3& x^3+1\end{array}\right]=11$
$\Rightarrow det\left[\begin{array}{ccc}1& 0& y^3\\ -1& 1& z^3\\ 0& -1& x^3+1\end{array}\right]=11(C_1\to C_1-C_{2}and{C}_2\to C_2-C_3)\Rightarrow (x^3+1+z^3)+y^3\left(1\right)=11\Rightarrow x^3+y^3+z^3=10Sosolutionare(1,1,2),(1,2,1)or(2,1,1)$