Mathematics
Sum of n natural numbers — Use n(n+1)/2. Quick check: sum of first 10 = 10×11/2 = 55. For odd numbers: n², for even: n(n+1).
AP sum without formula — Arithmetic mean of first and last term, multiplied by number of terms: Sum = n × (a + l)/2. Works instantly for sequences like 5,10,15...50.
GP infinite sum condition — Converges only when |r| < 1, sum = a/(1-r). Remember: common ratio must be between -1 and +1 (exclusive).
Binomial middle term trick — For (a+b)ⁿ, if n is even: middle term is (n/2 + 1)th term. If n is odd: two middle terms are (n+1)/2 and (n+3)/2 positions.
Quadratic roots sum/product — Sum of roots = -b/a, Product = c/a from ax²+bx+c=0. Quick verification tool before solving completely.
Binary-to-decimal shortcut — Write powers of 2 backwards (128,64,32,16,8,4,2,1), add wherever bit is 1. Example: 10110101 = 128+32+16+4+1 = 181.
Logarithm product rule — log(xy) = log x + log y. Similarly log(x/y) = log x - log y. Change of base: log_b(a) = log a / log b.
Permutation vs Combination — Arrangement matters? Use nPr = n!/(n-r)!. Selection only? Use nCr = n!/[r!(n-r)!]. Remember: nCr = nC(n-r).
Complex number i-powers cycle — i¹=i, i²=-1, i³=-i, i⁴=1, then repeats. Divide exponent by 4, use remainder: i⁷⁷ = i¹ = i (77÷4 leaves remainder 1).
General Ability Test — English
Idiom recognition — Animals — "Crocodile tears" (fake crying), "let the cat out" (reveal secret), "red herring" (distraction), "wild goose chase" (futile pursuit). Animal idioms dominate NDA English.
Synonym clusters — Similar sounds — Verbose/Wordy/Prolix; Concise/Terse/Brief/Succinct; Frugal/Thrifty/Economical. Group similar-meaning words together for quick recall.
Error spotting — Subject-Verb — Collective nouns (team, committee) take singular verbs in Indian English. "Each/every/either/neither" always singular. Watch for hidden subjects between commas.
One-word substitution — -cide words — Regicide (king killer), Genocide (race killer), Matricide (mother), Patricide (father), Homicide (human), Suicide (self), Infanticide (infant).
Antonym prefix trick — Add un-/in-/im-/ir-/dis-/mis- carefully: improper/proper, irregular/regular, dishonest/honest, misfortune/fortune. Watch false pairs like "inflammable" (still flammable!).
General Ability Test — General Knowledge (Chemistry)
pH scale memory — 7 is neutral — Below 7 = acidic (lemon pH 2, vinegar 3), Above 7 = basic (soap 9-10, bleach 12). Each step is 10× stronger/weaker.
Valency of first 20 elements — H=1, C=4, N=3, O=2 (HONC = 1,2,3,4 backwards). Na,K=1; Mg,Ca=2; Al=3; remember "1-2-3-4" pattern for main groups.
Atom structure — K,L,M,N shells — Maximum electrons: K=2, L=8, M=18, N=32. Formula: 2n² where n=shell number. Valence electrons determine chemical properties.
Acid-base indicator colors — Litmus: red in acid, blue in base. Phenolphthalein: colorless in acid, pink in base. Methyl orange: red in acid, yellow in base.
NPK fertilizer meaning — N=Nitrogen (leaf growth), P=Phosphorus (root/flower), K=Potassium (overall vigor). Urea is N-rich, Superphosphate is P-rich, Muriate of Potash is K-rich.
Chemical combination laws — LAD — Law of Action of Mass (constant composition), Definite proportions, Multiple proportions (Dalton). Elements combine in fixed ratios by mass.
Common compound formulas — Water H₂O, Salt NaCl, Ammonia NH₃, Methane CH₄, Carbon dioxide CO₂. Remember: subscript = number of atoms of that element.