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

Consider the system of linear equations $$x+y+z=5, x+2 y+\lambda^2 z=9, x+3 y+\lambda z=\mu$$, where $$\lambda, \mu \in \mathbb{R}$$. Then, which of the following statement is NOT correct?

Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$|\vec{b}|=1$$ and $$|\vec{b} \times \vec{a}|=2$$. Then $$|(\vec{b} \times \vec{a})-\vec{b}|^2$$ is equal to

If the domain of the function $$f(x)=\log _e\left(\frac{2 x+3}{4 x^2+x-3}\right)+\cos ^{-1}\left(\frac{2 x-1}{x+2}\right)$$ is $$(\alpha, \beta]$$, then the value of $$5 \beta-4 \alpha$$ is equal to