Question

The number of positive integral triplets (x, y, z) satisfying the equation $\left|\begin{array}{ccc}{y}^3+1& y^{2}z& y^{2}x\\ y{z}^2& z^3+1& z^{2}x\\ y{x}^2& x^{2}z& x^3+1\end{array}\right|=11$ is

• 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 $\left|\begin{array}{ccc}{y}^3+1& y^3& y^3\\ z^3& z^3+1& z^3\\ x^3& x^3& x^3+1\end{array}\right|=1$
$\Rightarrow \left|\begin{array}{ccc}1& 0& y^3\\ -1& 1& z^3\\ 0& -1& x^3+1\end{array}\right|=(C_1\to C_1-C_{2}and{C}_2\to C_2-C_3)$
$\Rightarrow 1(x^3+1+z^3)+y^3\left(1\right)=11\Rightarrow x^3+y^3+z^3=10$