\(\triangle =\begin{vmatrix} x & { x }^{ 2 } & 1+{ x }^{ 3 } \\ y & y^{ 2 } & { 1+y }^{ 3 } \\ z & z^{ 2 } & 1+z^{ 3 } \end{vmatrix}=0\) where \(x\neq y\neq z\).

Consider a determinant above.

Now find the value of \(x\times y\times z\) and let it be equal to \(A\), where \(A\) is an integer.

Now consider a binomial \(({ \sqrt [ 4 ]{ 6 } +\sqrt [ 6 ]{ 4 } ) }^{ 50 }\)

Let the number of rational terms in the expansion of above binomial is \(B\) and the number of irrational terms be \(C\).

Now if, it is given that \(\frac { C-\lambda B - 1}{ 2 } =-A\) where \(\lambda \) is a constant. Then find the value of \(B+C+\lambda - 1 - A\).

Hint: Use properties of determinant to solve the determinant.

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