Mathematics - Determinant Question with Solution | TestHub

The given equation is:

$|\begin{array}{ccc}{y}^3+1& y^{2}z& y^{2}x\\ y{z}^2& z^3+1& z^{2}x\\ y{x}^2& x^{2}z& x^3+1\end{array}|=11$

By applying determinant properties (factoring out $x,y,z$ from columns, multiplying rows, and column operations) or direct expansion, the equation simplifies to the Diophantine equation:

$x^3+y^3+z^3=10$

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Step 2: Find Positive Integral Solutions

We need to find positive integral solutions for

$x^3+y^3+z^3=10$

where $x,y,z\ge 1$. Since $3^3=27$, which is greater than 10, the only possible positive integer values for $x,y,z$ are 1 and 2.

* If $x=1,y=1,z=1$, then $1^3+1^3+1^3=3\ne 10$.

* If any one variable is 2 and the other two are 1, the sum of cubes is $2^3+1^3+1^3=8+1+1=10$.

The possible ordered triplets are permutations of (1, 1, 2):

* (1, 1, 2)

* (1, 2, 1)

* (2, 1, 1)

Any other combination involving values $\ge 2$ will exceed 10.

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Answer: The number of positive integral solutions is **3**.