# Let's mix it up

Algebra Level 5

$$\triangle =\begin{vmatrix} x & { x }^{ 2 } & 1+{ x }^{ 3 } \\ y & y^{ 2 } & { 1+y }^{ 3 } \\ z & z^{ 2 } & 1+z^{ 3 } \end{vmatrix}=0$$ 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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