If $$x+\frac{1}{x}=1$$ and $$p=x^{4000}+\frac{1}{x^{4000}}$$ and $$q$$ is the digit at unit place in the number $$2^{2 n}+1$$, then the value of $$(p+q)$$ is equal to

Let $$z_k=\cos \left(\frac{2 k \pi}{10}\right)+i \sin \left(\frac{2 k \pi}{10}\right) ; k=1,2, \ldots \ldots \ldots$$ 9, then $$\frac{1}{10}\left\{\left|1-z_1\right|\left|1-z_2\right| \ldots .\left|1-z_a\right|\right\}$$ equals to

If $$m$$ and $$n$$ are order and degree of the question $$\left(\frac{d^2 y}{d x^2}\right)^4+8 \frac{\left(d^2 y / d x^2\right)^3}{\left(d^4 y / d x^4\right)^5}+\left(\frac{d^4 y}{d x^4}\right)=x^2+4$$, then $$m-n$$ is equal to

If $$\left(\sin ^{-1} x\right)^2-\left(\cos ^{-1} x\right)^2=a \pi^2$$, then the range of $$a$$ is