Consider the functions $f_{1}(x) = x$ and $f_{2}(x) = 2 + \ln x$ for $x > 0$. The graphs of these functions intersect:

In the usual notation,the value of $\Delta \nabla$ is equal to

Let $f, g: R \rightarrow R$ be two real-valued functions defined as $f(x)=\begin{cases} -|x+3| & , x < 0 \\ e^{x} & , x \geq 0 \end{cases}$ and $g(x)=\begin{cases} x^{2}+k_{1} x & , x < 0 \\ 4 x+k_{2} & , x \geq 0 \end{cases}$,where $k_{1}$ and $k_{2}$ are real constants. If $(g \circ f)$ is differentiable at $x=0$,then $(g \circ f)(-4)+(g \circ f)(4)$ is equal to

Let $f(x)$ be a quadratic expression which is positive for all real $x$. If $g(x) = f(x) + f'(x) + f''(x)$,then for any real $x$,which one is correct?

For any positive integer $n$,define $f_n:(0, \infty) \rightarrow R$ as $f_n(x)=\sum_{j=1}^n \tan ^{-1}\left(\frac{1}{1+(x+j)(x+j-1)}\right)$ for all $x \in(0, \infty)$. Then,which of the following statement$(s)$ is (are) $TRUE$?
$(A)$ $\sum_{j=1}^5 \tan ^2(f_j(0))=55$
$(B)$ $\sum_{j=1}^{10}(1+f_j'(0)) \sec ^2(f_j(0))=10$
$(C)$ For any fixed positive integer $n$,$\lim _{x \rightarrow \infty} \tan (f_n(x))=\frac{1}{n}$
$(D)$ For any fixed positive integer $n$,$\lim _{x \rightarrow \infty} \sec ^2(f_n(x))=1$

$\lim _{x \rightarrow \frac{\pi}{4}} \frac{\int_2^{\sec ^2 x} f(t) d t}{x^2-\frac{\pi^2}{16}}$ equals