Let $$A(1,1), B(-4,3), C(-2,-5)$$ be vertices of a triangle $$A B C, P$$ be a point on side $$B C$$, and $$\Delta_{1}$$ and $$\Delta_{2}$$ be the areas of triangles $$A P B$$ and $$A B C$$, respectively. If $$\Delta_{1}: \Delta_{2}=4: 7$$, then the area enclosed by the lines $$A P, A C$$ and the $$x$$-axis is :

If the circle $$x^{2}+y^{2}-2 g x+6 y-19 c=0, g, c \in \mathbb{R}$$ passes through the point $$(6,1)$$ and its centre lies on the line $$x-2 c y=8$$, then the length of intercept made by the circle on $$x$$-axis is :

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 ____________.