Compound Tricks
4 principle pairs that spike HARD
When two principles appear together, the question is reliably hard — these compounds run 40–67% HARD vs the 22.5% bank average. Most students lose marks here not because they don't know either principle — but because they didn't see them chained. Drill the recipe, not the silos.
- compound recipes
- 4
- compound questions
- 41
- HARD rate (vs 22.5% bank avg)
- 40–67%
- harder than average
- 1.8–3×
Across the 487 HARD questions in the 2,160-q bank, four principle pairings recur with markedly elevated HARD rates. Each compound below runs 1.8× to 3× the bank-average HARD rate (22.5%), and the questions that match almost always require both tricks in sequence. The chain is the skill — drill them as compounds.
AM-GM + GP three-term
20 compound questions·35% HARD·1.6× the bank average
When a question asks for a minimum or maximum given a GP constraint, direct calculus rarely works — you need AM ≥ GM with equality at x = y = z. The GP relation b² = ac then closes the system. The chapter label might say Sequence & Series or Logarithms, but the technique is the same compound.
What it looks like
- If a, b, c are in GP with abc = 8, find the minimum of a + b + c.
- If log₁₀x, log₁₀y, log₁₀z are in AP, what is the relation between x, y, z?
Best single drill — the interrelating-progressions subtopic is where GP meets the means.
AP + GP
10 compound questions·40% HARD·1.8× the bank average
Two sequences interleaved. Common pattern: 'if a, b, c are in AP and b, c, d are in GP, find d'. The trick is to spot the chain — applying 2b = a+c and c² = bd gives a system of two equations in three unknowns, which closes via ratios.
What it looks like
- If a, b, c are in AP and a², b², c² are in GP, find the common ratio.
- If p, q, r in AP and 1/p, 1/q, 1/r in HP, prove the sequence is constant.
ω + Vieta
6 compound questions·67% HARD·3.0× the bank average
Cube roots of unity treated as polynomial roots. The classic shape: 'if α, β are roots of x² + x + 1 = 0, find α^n + β^n'. The roots are ω, ω² — and ω satisfies 1 + ω + ω² = 0 with ω³ = 1. Powers cycle every 3, so the answer is a small case match. DB-tagged intersection: 6 q · 67% HARD (4 of 6) — the most HARD-concentrated compound on this page.
What it looks like
- If α, β are roots of x² − x + 1 = 0, find (α − 1/α)² + (β − 1/β²)² + (α − 1/β⁴)².
- If 1, ω, ω² are cube roots of unity, evaluate |a + bω + cω²|² given a + b + c = 0.
The Cube Roots of Unity subtopic (18 q) is where most ω + Vieta lives.
Extrema + Logarithms
5 compound questions·40% HARD·1.8× the bank average
Finding the minimum or maximum of a logarithmic expression. Standard mistake: differentiate, set f'(x) = 0, lose to bad algebra. The shorter path is AM-GM on the arguments inside the log, then apply log monotonicity. log(AM) ≥ log(GM) gives the bound for free.
What it looks like
- Find the minimum value of log_{10}(x² + 2x + 11) for real x.
- If x, y > 0 and x + y = 1, find the maximum of log x + log y.
Why compounds matter more than the individual tricks
The bank’s difficulty isn’t evenly distributed across the 120 questions of any paper. Hard questions cluster — typically 25–30 questions per paper sit at the top of the difficulty curve, and a disproportionate share of those involve one of the four compound recipes above. A student who knows AM-GM and GP separately but hasn’t practiced them chained will solve the easy versions and lose the hard ones. The chain is the skill.