Disproportionation occurs when an element in an intermediate oxidation state simultaneously undergoes oxidation and reduction.

• $\text{BrO}^-$ (Br = +1): can disproportionate to Br$^-$ ($-1$) and BrO$_3^-$ ($+5$) ✓
• $\text{BrO}_2^-$ (Br = +3): can disproportionate to $\text{BrO}^-$ and $\text{BrO}_3^-$ ✓
• $\text{BrO}_3^-$ (Br = +5): can disproportionate under certain conditions ✓
• $\text{BrO}_4^-$ (Br = +7): Br is in its highest oxidation state for bromine — it cannot be further oxidised, so disproportionation is not possible.

$\text{BrO}_4^-$ does not undergo disproportionation.

Influenza virus is an enveloped virus (lipid bilayer derived from host cell). Enterovirus, Hepatitis A, and Polio are non-enveloped (naked) picornaviruses.

Split into right and left limits.

Right limit ($x\to0^+$): $[x]=0$, so $\sin(x[x])=0$, $|x|=x$.

$\dfrac{\tan(\pi\sin^2x)+x^2}{x^2}$. Using $\tan(\pi\sin^2x)\approx\pi\sin^2x\approx\pi x^2$:

$=\pi+1$

Left limit ($x\to0^-$): $[x]=-1$, $\sin(x[x])=\sin(-x)=-\sin x$, $|x|=-x$.

$=\dfrac{\tan(\pi\sin^2x)+(-x-(-\sin x))^2}{x^2}=\dfrac{\pi x^2+(\sin x-x)^2}{x^2}\to\pi+0=\pi$

Left $\neq$ Right, so the limit does not exist.

John Locke's work argued against absolute monarchy and for consent of the governed.

Using the Principle of Calorimetry: Heat lost by metal = Heat gained by (water + calorimeter).

$m_m s_m (100 - 21.5) = (m_w s_w + m_c s_c) (21.5 - 8.4)$

$0.192 \times s_m \times 78.5 = (0.24 \times 4184 + 0.128 \times 394) \times 13.1$

$15.072 s_m = (1004.16 + 50.432) \times 13.1 = 1054.592 \times 13.1 = 13815.15$

$s_m \approx 916.6 \text{ J kg}^{-1}\text{K}^{-1}$

If $1$ USD $= 0.93$ peso, then $1$ peso $= \dfrac{1}{0.93} \approx 1.075$ USD.

If $1$ USD $= 32.08$ shillings, then $1$ shilling $= \dfrac{1}{32.08} \approx 0.031$ USD.

Quantity A $\approx 1.075$ USD > Quantity B $\approx 0.031$ USD.

Quantity A is greater.