The sum of all rational terms in the expansion of $$\left(2^{\frac{1}{5}}+5^{\frac{1}{3}}\right)^{15}$$ is equal to :

Let $$a$$ be the sum of all coefficients in the expansion of $$\left(1-2 x+2 x^2\right)^{2023}\left(3-4 x^2+2 x^3\right)^{2024}$$ and $$b=\lim _\limits{x \rightarrow 0}\left(\frac{\int_0^x \frac{\log (1+t)}{t^{2024}+1} d t}{x^2}\right)$$. If the equation $$c x^2+d x+e=0$$ and $$2 b x^2+a x+4=0$$ have a common root, where $$c, d, e \in \mathbb{R}$$, then $$\mathrm{d}: \mathrm{c}:$$ e equals

Suppose $$2-p, p, 2-\alpha, \alpha$$ are the coefficients of four consecutive terms in the expansion of $$(1+x)^n$$. Then the value of $$p^2-\alpha^2+6 \alpha+2 p$$ equals