Live practice

Proceed from the given expression and keep every transformation explicit.

The last line satisfies the required domain and establishes the result.

The modulus is defined for every real \(x\), so the domain is \(\mathbb R\). Its value is non-negative, reaching 0 at \(x=1\), and every \(y\ge0\) occurs at \(x=1\pm y\).

1. \(A-B=\{3,6,9,15,18,21\}\).
2. \(A-C=\{3,9,15,18,21\}\).
3. \(A-D=\{3,6,9,12,18,21\}\).
4. \(B-A=\{4,8,16,20\}\).
5. \(C-A=\{2,4,8,10,14,16\}\).
6. \(D-A=\{5,10,20\}\).
7. \(B-C=\{20\}\).
8. \(B-D=\{4,8,12,16\}\).
9. \(C-B=\{2,6,10,14\}\).
10. \(D-B=\{5,10,15\}\).
11. \(C-D=\{2,4,6,8,12,14,16\}\).
12. \(D-C=\{5,15,20\}\).

Radius \(r=2.5\times10^{-4}\) m; \[\begin{aligned}A &= \pi r^2\\&= 1.963\times10^{-7}\end{aligned}\] m².

Doubling diameter quadruples area, so resistance becomes \(\frac{10}{4}=2.50\) Ω, a decrease of 7.50 Ω.

\(u=-27\) cm, \(f=-18\) cm.

Place the screen \(54.0\,\mathrm{cm}\) in front. The image is real, inverted and enlarged.

Aerobic respiration uses oxygen, completely oxidises glucose to carbon dioxide and water in mitochondria, and releases much ATP. Anaerobic respiration occurs without oxygen in cytoplasm, incompletely breaks down glucose and releases much less ATP. Yeast produces ethanol and carbon dioxide; some bacteria respire anaerobically; human muscle temporarily forms lactic acid during vigorous activity.

• Metals lose electrons to form cations; non-metals gain or share electrons.
• Metal oxides are generally basic or amphoteric; non-metal oxides are generally acidic or neutral.
• Reactive metals displace hydrogen from dilute acids; non-metals do not.
• A more reactive metal displaces a less reactive metal from its salt solution; non-metals do not show this metal-displacement pattern.
• Metals commonly form ionic chlorides and hydrides; non-metals commonly form covalent compounds.

Hydrochloric acid is a strong acid and ionises almost completely in water. Ethanoic acid is weak and ionises only partially. Thus the HCl solution has a higher concentration of available hydronium ions and reacts faster with magnesium.

Hydrogen bubbles are therefore produced more vigorously in the HCl tube.

Define the sample space and count each favourable outcome once.

The final value lies in [0,1]; complement pairs also sum to one, providing an independent check.

Simplify the expression before substitution and preserve one-sided information.

Substitution into the original expression verifies the stated domain and final value.