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Which statement about viruses is NOT true?
Viruses cannot replicate on their own; they require a host cell's machinery. They contain either DNA or RNA, can infect bacteria (bacteriophages), and are subcellular (acellular).
- An oxidizing agent is itself reduced. In option A, the oxidation state of Oxygen in $H_{2}O_{2}$ changes from $-1$ to $-2$ (in $H_{2}O$).
- Since the oxidation state decreases, $H_{2}O_{2}$ is reduced, meaning it acted as an oxidizing agent for the Iodide ions ($I^{-}$ to $I_{2}$).
Who developed the 'Greatest happiness of the greatest number' principle in Utilitarianism?
Jeremy Bentham is the founder of the Utilitarian school of thought.
Let $f:(-1,1)\to\mathbb{R}$ be defined by $f(x)=\max\{-|x|,-\sqrt{1-x^2}\}$. If $K$ is the set of all points where $f$ is not differentiable, find the number of elements in $K$.
The two curves are $y=-|x|$ (a V-shape) and $y=-\sqrt{1-x^2}$ (lower semicircle of unit circle).
$f(x)$ takes whichever is larger (less negative). They intersect where $|x|=\sqrt{1-x^2}\Rightarrow x^2=1-x^2\Rightarrow x=\pm\frac{1}{\sqrt{2}}$.
For $|x|<\frac{1}{\sqrt{2}}$: $-\sqrt{1-x^2}$ is larger; for $|x|>\frac{1}{\sqrt{2}}$: $-|x|$ is larger.
Non-differentiability can only occur at the switching points $x=\pm\frac{1}{\sqrt{2}}$ (where the curves meet) and at $x=0$ (corner of $|x|$, but $f$ uses the semicircle there, so it's smooth).
Checking carefully: $f$ is not differentiable only at $x=\pm\frac{1}{\sqrt{2}}$ — exactly one element... re-examining gives exactly one element in $K$ (by symmetry both switching points are included), so $|K|=1$.
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