$$\text { If }\left|\begin{array}{lll} \alpha & 3 & 4 \\ 1 & 2 & 1 \\ 1 & 4 & 1 \end{array}\right|=0 \text {, then the value of } \alpha \text { is: }$$

The derivative of $$x^{2 x}$$ w.r.t. $$x$$ is:

The function $$f(x)=[x]$$, where $$[x]$$ denotes the greatest integer less than or equal to $$x$$, is continuous at:

If $$x=A \cos 4 t+B \sin 4 t$$, then $$\frac{d^2 x}{d t^2}$$ is equal to: