Live practice

A metal oxide is converted into metal by reduction, that is, removal of oxygen. For a moderately reactive metal, carbon or carbon monoxide may act as the reducing agent:

Highly reactive metals require electrolytic reduction instead.

As the concentration of hydrogen ions increases, pH decreases and acidic character becomes stronger. When hydrogen-ion concentration is smaller than hydroxide-ion concentration, the solution is basic. In neutral water their concentrations are equal and pH is 7.

Principle and reasoning. Work from the governing physical relation and apply it to every stated part.

(a) separates into a fixed spatial factor and one time factor, so it is a standing wave. (b) is a function only of x−vt and translates in +x, so it is a travelling profile.

In (c) , both terms have the same phase argument and combine into a single sinusoid, 5sin(5x−0.5t+δ), so it is travelling. Expression (d) superposes two standing modes with different spatial and temporal frequencies; it is neither one travelling wave nor one normal-mode standing wave.

Principle and reasoning. Work from the governing physical relation and apply it to every stated part.

Static spring extension depends on its tension. In both arrangements the spring tension is F, so its extension is \(\frac{F}{k}\).

For one mass, m x¨=−kx, hence T₁=2π√(\(\frac{m}{k}\)). For two masses moving oppositely by x, spring extension changes by 2x. Each mass obeys m x¨=−2kx, so ω²=2k/m and T₂=2π√(m/(2k)).

Principle. Convert mass flow to volume flow, apply Poiseuille's law, and evaluate Reynolds number.

Principle. For a small compression, Δρ/ρ≈ΔP/B because mass is fixed and density rises as volume falls.

Principle. Gravity does mgh on either equal vertical drop. Net work equals change in kinetic energy, so air-drag work is Delta-K minus gravity's work.

An x–t graph plots one coordinate against time; it is not the geometrical trajectory in space.

The terms subtracted inside the bracket must be dimensionless, so v² must be divided by a quantity with dimensions of speed squared.

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.