Let $$a, b, c$$ be the sides of a triangle. No two of them are equal and $$\lambda \in R$$. If the roots of the equation $$x^{2}+2(a+b+c) x+3 \lambda(a b+b c+c a)=0$$ are real, then,

If $$f''(x)=-f(x)$$ and $$g(x)=f'(x)$$ and $$\mathrm{F}(x)=\left(f\left(\frac{x}{2}\right)\right)^{2}+\left(g\left(\frac{x}{2}\right)\right)^{2}$$ and given that $$\mathrm{F}(5)=5$$, then $$\mathrm{F}(10)$$ is equal to :

Let $$\theta \in\left(0, \frac{\pi}{4}\right)$$ and $$t_{1}=(\tan \theta)^{\tan \theta}, t_{2}=(\tan \theta)^{\cot \theta}, t_{3}=(\cot \theta)^{\tan \theta}$$ and $$t_{4}=(\cot \theta)^{\cot \theta}$$, then

There are $$n$$ urns each containing $$n+1$$ balls such that the $$i^{\text {th }}$$ urn contains $$i$$ white balls and $$(n+1-i)$$ red balls. Let $$u_{i}$$ be the event of selecting $$i^{\text {th }}$$ urn, $$i =1,2,3 \ldots, n$$ and $$w$$ denotes the event of getting a white ball.

If $$\mathrm{P}\left(u_{i}\right) \propto i$$, where $$i=1,2,3, \ldots n$$, then $$\lim_\limits{n \rightarrow \infty} \mathrm{P}(w)$$ is equal to: