$\mathrm{A}=\left[\begin{array}{lll}1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1\end{array}\right]$, if $\mathrm{U}_1, \mathrm{U}_2$ and $\mathrm{U}_3$ are columns matrices satisfying. $\mathrm{AU}_1=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right], \mathrm{AU}_2=\left[\begin{array}{l}2 \\ 3 \\ 0\end{array}\right], \mathrm{AU}_3=\left[\begin{array}{l}2 \\ 3 \\ 1\end{array}\right]$ and U is $3 \times 3$ matrix whose columns are $\mathrm{U}_1, \mathrm{U}_2, \mathrm{U}_3$ then answer the following questions

The value of $\left[\begin{array}{lll}3 & 2 & 0\end{array}\right] U\left[\begin{array}{l}3 \\ 2 \\ 0\end{array}\right]$ is :

If roots of the equation $x^2-10 c x-11 d=0$ are $a, b$ and those of $x^2-10 a x-11 b=0$ are $c, d$, then the value of $a+b+c+d$ is $(a, b, c$ and $d$ are distinct numbers)

$$ \text { The value of } 5050 \frac{\int_0^1\left(1-x^{50}\right)^{100} d x}{\int_0^{\frac{1}{1}}\left(1-x^{50}\right)^{101} d x} \text { is : } $$

If $a_n=\frac{3}{4}-\left(\frac{3}{4}\right)^2+\left(\frac{3}{4}\right)^3+\cdots \cdots(-1)^{n-1}\left(\frac{3}{4}\right)^n$ and $b_n=1-a_n$, then find the minimum natural number $n_0$ such that $b_n>a_n \forall n>n_0$