If $$w=\alpha+\mathrm{i} \beta$$, where $$\beta \neq 0$$ and $$z \neq 1$$, satisfies the condition that $$\left(\frac{w-\bar{w} z}{1-z}\right)$$ is purely real, then the set of values of $$z$$ is:

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