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 $b_n>a_n \forall n>n_0$

If $f(x)$ is a twice differentiable function such that $f(A)=0, f(B)=2, f(C)=-1, f(D)=2$, $f(e)=0$, where $a < b < c < d < e$, then the minimum number of zeroes of $g(x)=\left(f^{\prime}(x)\right)^2 +f^{\prime \prime}(x) f(x)$ in the interval $[a, e]$ is :