The idea behind the pure premium is to fund only the likelihood-weighted cost of the death benefit. It’s the base amount needed to cover the expected mortality costs, computed from mortality probabilities and the benefit amount (often using present-value calculations to reflect time value). It does not include any extra charges for expenses, commissions, risk, or profit. That is why the pure premium is described as the base actuarial cost to cover only the expected mortality—no loading. The full premium charged to a policyholder includes those additional loadings, which is why it’s larger than the pure premium. And pricing isn’t something the policyholder simply chooses; actuarial pricing determines the premium by building in expenses, risk loads, and profit.

The main idea is that groundwater flow follows gradients in hydraulic head, but moves from high head to low head, so the flow is opposite the gradient. Darcy's law expresses this as q = -K ∇h, where q is the Darcy flux (per unit area), K is the hydraulic conductivity (a positive property of the medium), and h is the hydraulic head (h = pressure head plus elevation head). The negative sign ensures flow goes toward lower head, with its magnitude set by how strong the head gradient is and by how permeable the material is (K). The other statements don’t fit because flow does depend on hydraulic conductivity, it isn’t proportional to the square of the gradient, and it isn’t directed along the gradient but opposite to it.

The key idea is how discharge responds to channel roughness in Manning’s equation when cross-sectional area, hydraulic radius, and slope are fixed. Manning’s equation shows Q = (1/n) A R^{2/3} S^{1/2}. If A, R, and S are constant, Q changes directly with 1/n, so increasing the roughness coefficient n reduces Q in proportion to 1/n. Physically, a rougher channel slows the flow due to greater friction, lowering discharge even though the channel’s size and grade don’t change. That’s why discharge decreases as n increases, matching the statement that Q is inversely proportional to n.

The main idea is that soil shear strength is controlled by the stress carried by the solid grain skeleton, not by the water in the pores. That stress is the effective stress, defined as the total stress in the soil minus the pore water pressure: sigma' = sigma - u. Why this matters: when water fills the pores, some of the load is carried by the water instead of the grains, so the grains press on each other with less force. Since shear strength comes from those intergranular contacts and friction, lower effective stress means lower shear strength. In drained soils, shear strength is often expressed as c' + sigma' tan(phi'), showing how sigma' directly sets the available resistance. In undrained or rapid loading, pore pressure can rise (u increases), which lowers sigma' and can lead to failure even without a large increase in total stress. The other forms would misrepresent how pore water affects the skeleton: they either add pressures together or subtract in the wrong way, implying water pressure increases the grain contact forces or multiplies stresses, which contradicts observed soil behaviour.

Liquefaction is when saturated soils temporarily lose their ability to carry shear stress because the pore water pressure builds up under dynamic loading, so the effective stress that keeps the soil grains interlocked drops and the soil behaves more like a liquid. In saturated sands, cyclic shaking—such as during an earthquake—causes repeated compression and dilation of the pore space, and the water pressure increases faster than it can drain away. With the effective stress nearly gone, the soil’s shear strength collapses, leading to sudden ground flow, settlements, or lateral spreading. This mechanism is specific to the combination of saturation, cyclic loading, and rapid pore pressure rise. Other possibilities described—strength loss simply from saturation, chemical degradation, or consolidation under static loads—do not capture the dynamic pore-pressure-driven loss of strength that characterizes liquefaction.