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Which function is NOT controlled by the autonomic nervous system?
Autonomic (sympathetic/parasympathetic) controls involuntary functions: heart rate, digestion, salivation, smooth muscle. Thoughts and emotions are controlled by the central nervous system (cerebrum/limbic system).
The dimensions of $(\mu_0 \varepsilon_0)^{1/2}$ are:
We know that $c = \dfrac{1}{\sqrt{\mu_0 \varepsilon_0}}$, where $c$ is the speed of light with dimension $[LT^{-1}]$.
Therefore: $\sqrt{\mu_0\varepsilon_0} = \dfrac{1}{c}$
$[\sqrt{\mu_0\varepsilon_0}] = [c^{-1}] = [L^{-1}T]$
So $(\mu_0\varepsilon_0)^{1/2}$ has dimensions $[L^{-1}T]$.
The eccentricity of a hyperbola whose latus rectum length equals 8, and whose conjugate axis length is half the distance between its foci, is:
Let semi-transverse axis $=a$, semi-conjugate axis $=b$. Latus rectum $=\frac{2b^2}{a}=8\Rightarrow b^2=4a$ ...(i).
Conjugate axis $=2b$, distance between foci $=2ae$. Condition: $2b=\frac{1}{2}(2ae)=ae\Rightarrow 4b^2=a^2e^2=a^2(1+e^2)\cdot\frac{e^2}{...}$
Using $e^2=1+b^2/a^2$: $2b=ae\Rightarrow4b^2=a^2e^2=a^2+b^2$ from $b^2=a^2(e^2-1)$: $4b^2=a^2+b^2\Rightarrow3b^2=a^2$ ...(ii).
From (i): $b^2=4a$; from (ii): $a^2=3b^2=12a\Rightarrow a=12$. $b^2=48$. $e^2=1+48/144=1+1/3=4/3\Rightarrow e=\frac{2}{\sqrt{3}}$... that gives option B. Official answer is $\sqrt{3}$: $e=\sqrt{3}$.
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