Short-Cuts

Use when: any ×11

Use when: squaring anything ending in 5

Use when: both factors cluster near 10/100/1000

Use when: the two numbers straddle a round number

Use when: one side is an ugly %, the other friendly

Use when: multiplying by these — halving beats long multiplication

Use when: estimating a surd to pick the nearest option

Use when: asked for a sum/product/symmetric combo of roots without the roots themselves — never solve

Use when: asked for α²+β², 1/α+1/β etc. — combine with Vieta, never find a single root

Use when: hunting a cubic factor fast — test x=1 and x=−1 first, they're free

Use when: factorising with no obvious root — only test ±(factors of constant)/(factors of leading)

Use when: checking a factor or getting a remainder without doing the division

Use when: factoring, or simplifying a fraction before it explodes

Use when: a question about the nature/number of roots, or 'for what k are roots real' — solve Δ ≥ 0, never the quadratic

Use when: max/min of a quadratic without calculus

Use when: asked for one specific coefficient — compute that term only

Use when: the unknown is in an exponent, or logs of products/powers to collapse

Use when: summing an arithmetic progression

Use when: summing a geometric progression or to infinity

Use when: summing consecutive integers (the only power sum on the spec)

Use when: a right triangle has two of these sides — write the third instantly

Use when: any exact-value triangle or exact trig value

Use when: no-calculator trig — recall, don't derive

Use when: θ is small (pendulums, diffraction, 'for small angles…')

Use when: area without splitting into base/height

Use when: circle geometry — recall the set

Use when: coordinate geometry

Use when: halve the work on symmetric even integrands (even powers, cos, |x|)

Use when: add the integral to its reflection so the integrand simplifies — rare at ESAT, try odd/even & area first

Use when: differentiate/integrate polynomials

Use when: max/min/inflection — but if it's a quadratic, complete the square instead

Use when: where a function rises/falls, or matching a graph to its derivative

Use when: a symmetry / graph-matching question about f and f'

Use when: 'at least one' appears

Use when: r is large — swap to the smaller one

Use when: expected totals — sum the piece-wise expectations

Use when: counting arrangements

Use when: swapping between sin² and cos² to simplify or solve a trig equation

Use when: 'how many solutions between a and b' — the spec only asks for solutions in a given interval

Use when: matching or sketching a transformed graph without plotting

Use when: finding or sketching an inverse

Use when: nested functions — work inside-out

Use when: sketching or matching rational functions

Use when: reading a polynomial's shape straight from its factors

Use when: simplifying a fraction or factorising a large number fast

Use when: given one of HCF/LCM, get the other instantly

Use when: converting a recurring decimal

Use when: nPr or factorial fractions — cancel, never expand the whole n!

Use when: simplifying powers or surds before they get messy

Use when: condition-reasoning questions — mixing these up is the classic trap

Use when: deciding whether a restated implication says the same thing

Use when: 'which statement is false' — hunt a single counterexample

Use when: a claim hard to prove directly (irrationality, infinitude of primes)

Use when: negating or interpreting a logical statement

Use when: length or direction of a vector

Use when: the angle between two vectors, or testing perpendicularity

Use when: three points collinear ⇔ AB parallel to AC

Use when: solving forwards is messy but checking is quick

Use when: an identity/inequality must hold for all x — one counterexample eliminates

Use when: options are far apart

Use when: symmetric problems