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Curated formulas, scientific laws, grammatical concord rules, and syllabus blueprints for JAMB/UTME, WAEC, and NECO revision.
Standard quadratic equation solution and discriminant analysis determining root characteristics.
x = [-b ± √(b² - 4ac)] / 2a Discriminant Δ = b² - 4ac • Δ > 0: Two real, distinct roots • Δ = 0: Two real, equal roots (perfect square) • Δ < 0: No real roots (complex conjugates)
Formulas for nth term, finite sum of progression, and infinite convergent series.
AP nth term: T_n = a + (n - 1)d AP Sum: S_n = n/2 · [2a + (n - 1)d] GP nth term: T_n = a · r^(n - 1) GP Sum: S_n = a(1 - r^n)/(1 - r) (for |r| < 1) Sum to Infinity: S_∞ = a / (1 - r) (only when -1 < r < 1)
Power rule, product rule, quotient rule, and turning points identification.
Power Rule: d/dx(x^n) = n · x^(n - 1) Chain Rule: dy/dx = (dy/du) · (du/dx) Turning Points: Set dy/dx = 0 • Maximum if d²y/dx² < 0 • Minimum if d²y/dx² > 0
Multiplication, division, power, and change of base rules.
log_b(xy) = log_b(x) + log_b(y) log_b(x/y) = log_b(x) - log_b(y) log_b(x^k) = k · log_b(x) Change of base: log_b(a) = log_c(a) / log_c(b) log_b(b) = 1, log_b(1) = 0
Kinematics equations for uniformly accelerated straight-line and vertical motion under gravity.
1) v = u + at 2) s = ut + ½at² 3) v² = u² + 2as 4) s = [(u + v)/2] · t (For vertical motion under gravity, replace a with g = 9.8 or 10 m/s²)
Potential difference, series/parallel circuit resistance, internal resistance, and electrical energy.
V = I · R Resistivity: R = ρL / A Series: R_total = R₁ + R₂ + ... Parallel: 1/R_total = 1/R₁ + 1/R₂ + ... Terminal voltage: V = E - Ir (where E = emf, r = internal resistance) Power: P = IV = I²R = V²/R
Wave velocity, frequency, wavelength, focal length, and optical magnification.
Wave Speed: v = f · λ (v = speed, f = frequency, λ = wavelength) Period: T = 1/f Mirror & Lens Formula: 1/f = 1/u + 1/v Magnification: M = v/u = h_image / h_object Refractive Index: n = sin(i) / sin(r) = c / v
Boyle's, Charles's, General Gas Law, and Avogadro's constant calculations.
General Gas Law: (P₁V₁) / T₁ = (P₂V₂) / T₂ (T in Kelvin! K = °C + 273) Ideal Gas: PV = nRT (R = 8.314 J/(mol·K) or 0.0821 L·atm/(mol·K)) Moles: n = mass / molar_mass = Volume(dm³) / 22.4 dm³ (at s.t.p.) Number of particles = n · N_A (N_A = 6.02 × 10²³)
Quantitative electrolysis, mass deposited, electric charge, and oxidation states.
First Law: m = Z · I · t (m = mass in g, I = current in A, t = time in s) Quantity of Charge: Q = I · t Faraday Equivalent: 1 Faraday = 96,500 Coulombs = 1 mole of electrons m = (Molar_mass · I · t) / (n · 96500) [where n = valence/electrons transferred]
General formulas for hydrocarbons, alcohols, alkanoic acids, esters, and IUPAC nomenclature.
Alkanes: C_n H_(2n+2) (single bonds, substitution) Alkenes: C_n H_(2n) (double bond, addition reactions) Alkynes: C_n H_(2n-2) (triple bond) Alkanols (Alcohols): C_n H_(2n+1)OH Alkanoic Acids: C_n H_(2n+1)COOH Esters: R-COO-R' (sweet fragrance, formed by esterification)
Monohybrid cross, Punnett squares, codominance, sex-linked traits, and ABO blood group alleles.
Monohybrid Phenotypic Ratio: 3:1 (dominant : recessive in F₂) Genotypic Ratio: 1:2:1 (1 RR : 2 Rr : 1 rr) ABO Blood Groups: Alleles I^A, I^B, I^O (I^A and I^B are codominant; I^O is recessive) • Type A: I^A I^A or I^A I^O • Type B: I^B I^B or I^B I^O • Type AB: I^A I^B (universal recipient) • Type O: I^O I^O (universal donor) Rhesus factor: Rh+ is dominant over Rh-
Aerobic vs anaerobic respiration equations, ATP yield, and chloroplast light reactions.
Aerobic Respiration: C₆H₁₂O₆ + 6O₂ → 6CO₂ + 6H₂O + 36-38 ATP Anaerobic in Yeast: C₆H₁₂O₆ → 2C₂H₅OH + 2CO₂ + 2 ATP Anaerobic in Muscles: C₆H₁₂O₆ → 2 Lactic Acid + 2 ATP Photosynthesis: 6CO₂ + 6H₂O + sunlight → C₆H₁₂O₆ + 6O₂ • Light Stage: Occurs in thylakoids (photolysis of water) • Dark Stage (Calvin Cycle): Occurs in stroma (carbon fixation)
Grammatical concord rules that appear on every UTME and WAEC objective paper.
1) Rule of Proximity (Neither... nor / Either... or): Verb agrees with the subject closest to it.
Example: Neither the teacher nor the students WERE present.
2) Accompaniment ('as well as', 'together with', 'in addition to'): Verb agrees with the FIRST subject.
Example: The captain, as well as his crew, WAS rescued.
3) Indefinite Pronouns ('each', 'everyone', 'none', 'neither of'): Takes singular verbs.
Example: Each of the candidates HAS submitted his slip.
4) Plural in form, singular in meaning: 'Mathematics', 'Economics', 'News' take singular verbs.Choose your subjects and test your speed with randomized past questions in Self Practice or the 5-Minute Challenge Arena.