Post-UTME cheat sheet

Concord rule: a singular subject takes a singular verb regardless of intervening phrases - 'The list of items is long'; 'Each of the boys has a book' takes singular verbs after each/every/either/neither.

Collective nouns (team, committee, government) take a singular verb when acting as one unit and a plural verb when members act individually: 'The jury has reached a verdict' vs 'The jury are divided'.

Tense sequence: after a past-tense main verb, use past or past perfect in the subordinate clause ('He said he was ill'), except for universal truths ('He said the sun rises in the east').

Subjunctive after wish/as if/if only and after verbs of command (suggest, demand, insist, recommend): 'I wish I were taller'; 'The chairman insisted that he resign'.

Distinguish homophones tested in objective questions: affect (verb) / effect (noun); principal (main/head) / principle (rule); stationary (not moving) / stationery (writing materials); complement (complete) / compliment (praise).

Idiomatic preposition pairs: 'married to', 'comprise' takes no 'of', 'differ from' (things) vs 'differ with' (persons), 'superior to', 'prefer to', 'consist of', 'insist on', 'capable of'.

Punctuation: semicolon joins two independent clauses; colon introduces a list or explanation; comma separates non-defining relative clauses ('My brother, who lives in Lagos, is a doctor').

Vocabulary in context: antonym/synonym items reward reading the whole sentence - 'candid' = frank, 'frugal' = thrifty, 'obstinate' = stubborn, 'transient' = temporary, 'ambiguous' = unclear.

Question tags mirror the auxiliary and reverse polarity: 'She doesn't sing, does she?'; 'They left early, didn't they?' - 'I am' takes 'aren't I?'.

Quadratic formula: x = [-b ± √(b² − 4ac)] / 2a; discriminant b² − 4ac > 0 gives two real roots, = 0 gives equal roots, < 0 gives no real roots; sum of roots = −b/a, product = c/a.

Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1; a⁻ⁿ = 1/aⁿ; a^(1/n) = ⁿ√a.

Logarithms: log(ab) = log a + log b; log(a/b) = log a − log b; log aⁿ = n log a; logₐ a = 1; logₐ 1 = 0; change of base logₐ b = log b / log a.

Arithmetic progression: nth term a + (n−1)d; sum Sₙ = n/2[2a + (n−1)d]. Geometric progression: nth term arⁿ⁻¹; sum Sₙ = a(rⁿ − 1)/(r − 1), r ≠ 1; S∞ = a/(1 − r) for |r| < 1.

Mensuration: circle area πr², circumference 2πr; sphere volume (4/3)πr³, surface 4πr²; cone volume (1/3)πr²h, curved surface πrl; cylinder volume πr²h.

Trigonometry: sin²θ + cos²θ = 1; tan θ = sin θ / cos θ; sine rule a/sin A = b/sin B = c/sin C; cosine rule a² = b² + c² − 2bc cos A; area of triangle = ½ab sin C.

Probability: P(E) = favourable outcomes / total outcomes; P(not E) = 1 − P(E); P(A or B) = P(A) + P(B) − P(A and B); independent events P(A and B) = P(A) × P(B).

Permutation and combination: ⁿPᵣ = n!/(n−r)!; ⁿCᵣ = n!/[r!(n−r)!]; 0! = 1; arrangements of n objects with repeats = n!/(p!q!...).

Coordinate geometry: distance √[(x₂−x₁)² + (y₂−y₁)²]; midpoint ((x₁+x₂)/2, (y₁+y₂)/2); gradient (y₂−y₁)/(x₂−x₁); parallel lines m₁ = m₂; perpendicular m₁m₂ = −1.

Nigeria's federal structure: 36 states plus the Federal Capital Territory (Abuja); the three arms are Executive (President), Legislature (National Assembly: Senate + House of Representatives) and Judiciary (Supreme Court at apex).

Nigeria's independence was 1 October 1960; it became a republic in 1963; the current constitution dates from 1999 (as amended). The National Assembly has 109 senators and 360 House members.

International organisations: UN founded 1945, headquarters New York, 193 member states; AU headquarters Addis Ababa; ECOWAS founded 1975 (Treaty of Lagos), headquarters Abuja; OPEC founded 1960, headquarters Vienna; WHO headquarters Geneva.

Nigerian currency is the naira, issued by the Central Bank of Nigeria; the capital moved from Lagos to Abuja in 1991; the Niger and Benue rivers meet at Lokoja.

Continents and oceans: seven continents (Asia largest, Australia smallest); five oceans (Pacific largest and deepest, Arctic smallest); the Sahara is the largest hot desert, the Nile is usually cited as the world's longest river.

Key global bodies and roles: IMF (balance-of-payments support), World Bank (development lending), WTO (trade rules), UNESCO (education/culture), UNICEF (children), ILO (labour standards).

Nigerian geography facts: highest point is Chappal Waddi (Taraba); largest state by area is Niger; smallest is Lagos; the country has 774 local government areas.

Cell theory: all living things are made of cells, cells arise from pre-existing cells; prokaryotes lack a membrane-bound nucleus and organelles, eukaryotes have them.

Organelles and functions: mitochondrion = aerobic respiration/ATP; ribosome = protein synthesis; chloroplast = photosynthesis; lysosome = intracellular digestion; nucleus = genetic control; Golgi body = packaging and secretion.

Photosynthesis equation: 6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂ in the presence of light and chlorophyll; light-dependent reactions occur in the thylakoid membranes, the Calvin cycle in the stroma.

Respiration: aerobic yields about 38 ATP per glucose and produces CO₂ and water; anaerobic in muscle gives lactic acid (about 2 ATP), in yeast gives ethanol and CO₂ (fermentation).

Blood: red cells carry haemoglobin and oxygen, white cells defend, platelets clot; humans have 46 chromosomes (23 pairs); ABO and Rhesus factors determine transfusion compatibility.

Genetics: dominant and recessive alleles; genotype vs phenotype; a cross of Aa × Aa gives 3:1 phenotype and 1:2:1 genotype ratios; sex is determined by X and Y chromosomes.

Taxonomy hierarchy: Kingdom, Phylum, Class, Order, Family, Genus, Species; binomial nomenclature uses Genus + species, e.g. Homo sapiens.

Ecology: food chain producers → consumers; decomposers recycle nutrients; symbiosis types are mutualism, commensalism and parasitism; energy decreases at each trophic level (roughly 10% transfer).

Mole concept: 1 mole = 6.02 × 10²³ particles; molar mass in g/mol; at STP one mole of gas occupies 22.4 dm³; concentration = moles / volume in dm³.

Acid–base: pH = −log[H⁺]; pH < 7 acidic, 7 neutral, > 7 alkaline; strong acids fully ionise; neutralisation H⁺ + OH⁻ → H₂O; titration uses n = CV.

Periodic trends: atomic radius decreases across a period and increases down a group; electronegativity and ionisation energy increase across a period and decrease down a group; metals are on the left, non-metals on the right.

Bonding: ionic bonds transfer electrons between metals and non-metals; covalent bonds share electrons between non-metals; metallic bonds involve delocalised electrons; ionic compounds conduct when molten or dissolved, not solid.

Gas laws: Boyle's pV = constant at constant T; Charles's V/T = constant at constant p; combined p₁V₁/T₁ = p₂V₂/T₂; ideal gas pV = nRT with R ≈ 8.314 J/mol·K.

Oxidation and reduction: oxidation is loss of electrons, reduction is gain (OIL RIG); oxidising agents are reduced; assign oxidation numbers to balance redox equations.

Reaction rates increase with higher temperature, higher concentration, greater surface area, catalysts, and higher pressure for gases; catalysts lower activation energy without being consumed.

Organic chemistry: alkanes CₙH₂ₙ₊₂ (single bonds), alkenes CₙH₂ₙ (double bonds, decolourise bromine water), alkynes CₙH₂ₙ₋₂; alkanols contain –OH; alkanoic acids contain –COOH.

Newton's laws: a body stays at rest or uniform motion unless acted on by a net force (F = ma); every action has an equal and opposite reaction; momentum p = mv is conserved in collisions.

Equations of motion: v = u + at; s = ut + ½at²; v² = u² + 2as; weight W = mg with g ≈ 9.8 m/s²; kinetic energy ½mv²; potential energy mgh.

Electricity: Ohm's law V = IR; power P = IV = I²R = V²/R; resistors in series add, in parallel 1/R = 1/R₁ + 1/R₂; energy consumed = power × time.

Waves: v = fλ; frequency is set by the source, speed by the medium; refraction bends waves toward the normal when entering a denser medium; sound needs a medium, light does not.

Optics: angle of incidence equals angle of reflection; real images are inverted and can be projected, virtual images are upright; Snell's law n₁ sin θ₁ = n₂ sin θ₂; power of a lens in dioptres = 1/f (metres).

Thermal physics: heat flows from hot to cold; specific heat capacity Q = mcΔθ; latent heat Q = mL; absolute zero is 0 K = −273 °C; the three modes are conduction, convection and radiation.

Work, energy and power: work = force × distance in the direction of force; power = work / time; efficiency = useful output / total input × 100%.

Pressure and fluids: pressure = force / area; liquid pressure p = ρgh; Archimedes' principle states upthrust equals the weight of fluid displaced; a floating body displaces its own weight.

Basic problem is scarcity of resources versus unlimited wants; three questions are what, how and for whom to produce; opportunity cost is the value of the next best alternative forgone.

Demand and supply: demand curve slopes downward, supply upward; equilibrium is where they intersect; a price above equilibrium creates a surplus, below creates a shortage; price controls cause black markets.

Elasticity: price elasticity of demand = % change in quantity / % change in price; demand is elastic when |E| > 1, inelastic when |E| < 1; necessities are inelastic, luxuries elastic.

Types of markets: perfect competition (many sellers, identical products), monopoly (one seller), oligopoly (few sellers), monopolistic competition (differentiated products).

National income: GDP measures output within borders, GNP adds net factor income from abroad; national income = C + I + G + (X − M); real values are adjusted for inflation.

Money and banking: functions of money are medium of exchange, store of value, unit of account and standard of deferred payment; the central bank controls money supply and is lender of last resort.

Inflation is a sustained rise in the general price level, measured by CPI; demand-pull and cost-push are the main types; unemployment types include frictional, structural, cyclical and seasonal.

Public finance and international trade: fiscal policy uses taxation and government spending; monetary policy uses interest rates and money supply; a balance-of-payments deficit means outflows exceed inflows.

Forms of government: democracy (rule by the people via elections), monarchy (hereditary), oligarchy (rule by a few), dictatorship (absolute power); sovereignty is the supreme power of the state.

Separation of powers (Montesquieu): legislature makes laws, executive implements them, judiciary interprets them; checks and balances prevent abuse; the rule of law means all are subject to law.

Federalism divides power between central and regional governments; a unitary system concentrates power; a confederation gives maximal power to member states.

Constitution types: written vs unwritten, rigid vs flexible; a constitution defines powers and rights; Nigeria's 1999 Constitution is written and rigid.

Civic duties include obeying laws, paying taxes, voting, jury/community service and respecting others' rights; human rights include life, liberty, fair hearing, freedom of expression and association.

Electoral concepts: franchise is the right to vote; universal adult suffrage; constituency is a voting area; a free and fair election requires an independent electoral body, secret ballot and an impartial judiciary.

Political parties aggregate interests and contest elections; pressure groups (e.g. trade unions) influence policy without seeking office; the civil service is a permanent administrative body.

Genres: drama is meant for performance with dialogue and stage directions; prose fiction (novel, novella, short story) is narrative; poetry uses compressed, patterned language and often rhythm and rhyme.

Literary devices: simile uses 'like/as', metaphor states one thing is another, personification gives human traits to non-human things, hyperbole is exaggeration, oxymoron joins contradictory terms.

Sound devices: alliteration is repetition of initial consonants, assonance of vowel sounds, onomatopoeia imitates sounds; rhyme and rhythm control pace.

Drama terms: protagonist is the central character, antagonist opposes them; soliloquy is spoken alone on stage, aside is heard by the audience only; tragedy ends in the protagonist's downfall.

Structure: plot is the sequence of events (exposition, rising action, climax, falling action, resolution); setting is time and place; theme is the central idea; conflict drives the narrative.

Narrative perspective: first-person uses 'I', third-person omniscient knows all characters' thoughts, third-person limited stays with one character; unreliable narrators distort events.

Poetry forms: sonnet has 14 lines, ode addresses a subject, elegy mourns, ballad narrates; a stanza is a grouped set of lines; free verse lacks regular metre and rhyme.

Christian Bible structure: Old Testament (39 books) and New Testament (27 books); the Gospels are Matthew, Mark, Luke and John; the Acts of the Apostles records the early church.

Christian doctrines: Trinity (Father, Son, Holy Spirit), Incarnation, Atonement, Resurrection; the Ten Commandments guide moral conduct; the Lord's Prayer is the model prayer in Matthew 6.

Christian ethics: the two great commandments are love of God and love of neighbour; the Beatitudes (Matthew 5) describe kingdom values; parables teach moral lessons, e.g. the Good Samaritan on neighbourliness.

Qur'an structure: 114 surahs revealed to Prophet Muhammad over about 23 years; the longest is Al-Baqarah and the shortest Al-Kawthar; the Qur'an is written in Arabic and recited in salah.

Five Pillars of Islam: Shahadah (declaration of faith), Salah (five daily prayers), Zakah (almsgiving), Sawm (fasting in Ramadan), Hajj (pilgrimage to Makkah once if able).

Islamic ethics and sources: Shari'ah derives from the Qur'an, Sunnah (Prophet's practice), Ijma (consensus) and Qiyas (analogy); hadith are recorded sayings of the Prophet; halal and haram govern conduct.

Shared moral themes: honesty, justice, charity, patience and respect for parents; both traditions teach accountability to God and reward/punishment in the hereafter.

Soil science: soil texture depends on proportions of sand, silt and clay; loam is best for most crops; pH around 6.0–7.0 suits many crops; liming raises soil pH, gypsum treats sodic soils.

Plant nutrients: macronutrients N, P, K, Ca, Mg, S; micronutrients include Fe, Zn, Cu, Mn, B; nitrogen promotes leaf growth, phosphorus roots and fruiting, potassium overall vigour and disease resistance.

Fertiliser: NPK 15-15-15 means 15% each of nitrogen, phosphorus and potassium; organic manure improves structure and water-holding capacity; apply fertiliser at the right stage to avoid leaching.

Crop production: land preparation, planting, weeding, thinning, pest control and harvesting; crop rotation reduces pests and restores nutrients; intercropping grows two crops together.

Animal husbandry: ruminants (cattle, sheep, goats) digest cellulose via rumen microbes; poultry need protein-rich feed; signs of disease include loss of appetite, dull coat and abnormal droppings.

Pests and diseases: pests include insects, rodents and birds; control methods are cultural, biological, chemical and integrated pest management; quarantine prevents introduction of new pests.

Farm economics and tools: fixed costs (land, buildings) vs variable costs (seed, labour); gross margin = revenue − variable costs; a hoe is for tilling, a cutlass for clearing, a dibber for making planting holes.

Accounting equation: Assets = Capital + Liabilities; the double-entry rule is debit the receiver, credit the giver; every transaction affects at least two accounts and the equation stays balanced.

Books of account: journal is the book of original entry; ledger classifies accounts; trial balance tests arithmetic accuracy (debits = credits) but does not detect all errors.

Financial statements: trading account gives gross profit = sales − cost of goods sold; profit and loss account gives net profit = gross profit + other income − expenses; balance sheet shows position at a date.

Cost of goods sold = opening stock + purchases + carriage inwards − returns outwards − closing stock; depreciation (e.g. straight-line) allocates cost over useful life.

Key ratios: gross profit margin = gross profit / sales × 100; net profit margin = net profit / sales × 100; current ratio = current assets / current liabilities (about 2:1 is often cited as healthy).

Commerce concepts: trade is home or foreign; aids to trade include banking, insurance, transport, warehousing, advertising and communication; a contract requires offer, acceptance, consideration and capacity.

Business documents: invoice requests payment and shows goods supplied; credit note corrects an overcharge; debit note corrects an undercharge; receipt acknowledges payment; statement of account summarises transactions.

Types of business organisation: sole proprietorship (owned by one), partnership (2–20 partners generally), limited liability company (shareholders protected), cooperative (owned by members).