Let $$f(x)=2^x-x^2, x \in \mathbb{R}$$. If $$m$$ and $$n$$ are respectively the number of points at which the curves $$y=f(x)$$ and $$y=f^{\prime}(x)$$ intersect the $$x$$-axis, then the value of $$\mathrm{m}+\mathrm{n}$$ is ___________.

If the solution curve $$y=y(x)$$ of the differential equation $$\left(1+y^2\right)\left(1+\log _{\mathrm{e}} x\right) d x+x d y=0, x > 0$$ passes through the point $$(1,1)$$ and $$y(e)=\frac{\alpha-\tan \left(\frac{3}{2}\right)}{\beta+\tan \left(\frac{3}{2}\right)}$$, then $$\alpha+2 \beta$$ is _________.

The resistance $$R=\frac{V}{I}$$ where $$\mathrm{V}=(200 \pm 5) \mathrm{V}$$ and $$I=(20 \pm 0.2) \mathrm{A}$$, the percentage error in the measurement of $$\mathrm{R}$$ is :

The de-Broglie wavelength of an electron is the same as that of a photon. If velocity of electron is $$25 \%$$ of the velocity of light, then the ratio of K.E. of electron and K.E. of photon will be: