Let a function $$f: \mathbb{R} \rightarrow \mathbb{R}$$ be defined as :

$$f(x)= \begin{cases}\int\limits_{0}^{x}(5-|t-3|) d t, & x>4 \\ x^{2}+b x & , x \leq 4\end{cases}$$

where $$\mathrm{b} \in \mathbb{R}$$. If $$f$$ is continuous at $$x=4$$, then which of the following statements is NOT true?

For $$k \in \mathbb{R}$$, let the solutions of the equation $$\cos \left(\sin ^{-1}\left(x \cot \left(\tan ^{-1}\left(\cos \left(\sin ^{-1} x\right)\right)\right)\right)\right)=k, 0<|x|<\frac{1}{\sqrt{2}}$$ be $$\alpha$$ and $$\beta$$, where the inverse trigonometric functions take only principal values. If the solutions of the equation $$x^{2}-b x-5=0$$ are $$\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}$$ and $$\frac{\alpha}{\beta}$$, then $$\frac{b}{k^{2}}$$ is equal to ____________.

The mean and variance of 10 observations were calculated as 15 and 15 respectively by a student who took by mistake 25 instead of 15 for one observation. Then, the correct standard deviation is _____________.

An ellipse $$E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$ passes through the vertices of the hyperbola $$H: \frac{x^{2}}{49}-\frac{y^{2}}{64}=-1$$. Let the major and minor axes of the ellipse $$E$$ coincide with the transverse and conjugate axes of the hyperbola $$H$$, respectively. Let the product of the eccentricities of $$E$$ and $$H$$ be $$\frac{1}{2}$$. If $$l$$ is the length of the latus rectum of the ellipse $$E$$, then the value of $$113 l$$ is equal to _____________.