Mathematics
Sum of AP — Use S_n = n/2 × (2a + (n-1)d) OR n/2 × (first + last). Remember "N-halves times First-plus-Last" for quick mental calculation.
Divisibility by 11 — Alternating digit sum: (sum of odd-position digits) − (sum of even-position digits) must be 0 or divisible by 11. Quick check for large numbers.
Circle angle theorems — COAT: Circle center angle is twice the circumference angle subtending the same arc. Angle in semicircle always 90°.
Quadratic sum/product — For ax² + bx + c = 0: sum of roots = −b/a, product = c/a. Remember "Minus Bee over Ay, Sea over Ay".
Triangle area shortcuts — Hero's formula: A = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2. For right triangle, just ½ × base × height; no Hero needed.
Coordinate distance — √[(x₂−x₁)² + (y₂−y₁)²]. Remember "X-difference squared plus Y-difference squared, then root"—never mix x and y.
Logarithm laws — PQRD: Product→add, Quotient→subtract, Raise power→multiply, Denominator power→negative. log(ab)=log(a)+log(b); log(a/b)=log(a)−log(b).
Polynomial remainder — By Remainder Theorem: when p(x) divided by (x−a), remainder = p(a). Just substitute a directly into polynomial.
Probability basics — P(event) = favorable/total outcomes. P(not A) = 1 − P(A). For independent events: P(A and B) = P(A) × P(B).
Parallelogram properties — OAAD: Opposite angles equal, Opposite sides equal, Adjacent angles supplementary (sum 180°), Diagonals bisect each other.
Linear equations elimination — Multiply equations to make one variable's coefficients equal (or opposite), then add/subtract. Check: substitute answer back into both originals.
Mensuration cube/cuboid — Cube: TSA=6a², CSA=4a², Volume=a³. Cuboid: TSA=2(lb+bh+hl), Volume=l×b×h. Remember "Total=all 6 faces, Curved=4 sides only".
Logical and Mental Reasoning
Direction sense — NEWS clockwise: North→East→South→West (90° turns). Right turn = clockwise, left = anticlockwise. Draw quick compass mentally.
Blood relations shortcuts — Mother's/Father's brother = Uncle; Sister's/Brother's son = Nephew; Spouse's sibling = Brother/Sister-in-law. Map two generations max.
Syllogism Venn diagrams — Always draw circles for "All A are B" (A inside B), overlapping for "Some", separate for "No". Test each conclusion visually.
Series pattern — Check differences, ratios, squares/cubes first. Alternating series: split into two separate sequences (odd/even positions) and solve each.
Coded inequalities — Create symbol key: @ means >, # means <, etc. Convert to standard inequality, then arrange elements in order on number line.
Mirror/Water image — Mirror image: lateral inversion (left↔right); Water image: vertical inversion (top↔bottom). Letters like A, H unchanged in mirror; b becomes d.