Mie Entrance Science cheat sheet

Quadratic formula: x = [-b ± √(b² - 4ac)] / (2a); discriminant D = b² - 4ac: D > 0 two real roots, D = 0 one repeated root, D < 0 no real roots.

Laws of logarithms: log_a(xy) = log_a x + log_a y; log_a(x/y) = log_a x - log_a y; log_a(x^n) = n log_a x; log_a b = (ln b)/(ln a).

Sum of arithmetic series: S_n = n/2 [2a + (n-1)d]; sum of geometric series (r ≠ 1): S_n = a(1 - r^n)/(1 - r); infinite sum |r| < 1: S = a/(1 - r).

Circle equation: (x - h)² + (y - k)² = r² has center (h, k) and radius r; general form x² + y² + Dx + Ey + F = 0 gives center (-D/2, -E/2) and r = √[(D² + E²)/4 - F].

Derivative rules: d/dx x^n = n x^(n-1); product rule (uv)' = u'v + uv'; quotient rule (u/v)' = (u'v - uv')/v²; chain rule dy/dx = dy/du · du/dx.

Integration: ∫ x^n dx = x^(n+1)/(n+1) + C for n ≠ -1; definite integral ∫_a^b f(x) dx = F(b) - F(a) where F' = f.

Trigonometric identities: sin²θ + cos²θ = 1; tanθ = sinθ/cosθ; sin(A ± B) = sinA cosB ± cosA sinB; cos(A ± B) = cosA cosB ∓ sinA sinB.

Probability: P(A ∪ B) = P(A) + P(B) - P(A ∩ B); conditional P(A|B) = P(A ∩ B)/P(B); independent events satisfy P(A ∩ B) = P(A)P(B).

Vector dot product: a·b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃; perpendicular vectors have dot product zero.

Newton's second law: F_net = ma; weight W = mg with g ≈ 9.8 m/s² near Earth's surface.

Kinematics: v = u + at; s = ut + ½at²; v² = u² + 2as (constant acceleration only).

Work and energy: W = Fd cosθ; kinetic energy KE = ½mv²; gravitational potential energy ΔPE = mgΔh; conservation of mechanical energy when only conservative forces act.

Ohm's law: V = IR; electrical power P = VI = I²R = V²/R; resistors in series add (R_eq = R₁ + R₂ + …); in parallel 1/R_eq = 1/R₁ + 1/R₂ + …

Wave equation: v = fλ; for a wave on a string v = √(T/μ) where T is tension and μ is linear mass density.

Snell's law: n₁ sinθ₁ = n₂ sinθ₂; refractive index n = c/v; total internal reflection occurs when θ₁ exceeds critical angle θ_c = sin⁻¹(n₂/n₁) for n₁ > n₂.

Ideal gas law: PV = nRT; pressure P = F/A; absolute temperature must be in kelvin (K = °C + 273.15).

Magnetic force on a moving charge: F = qvB sinθ; on a current-carrying wire: F = BIL sinθ; direction given by Fleming's left-hand rule.

Half-life: N = N₀(1/2)^(t/T½); activity A = λN with decay constant λ = ln2 / T½.

Mole concept: n = m/M; number of particles N = n × 6.022 × 10²³ mol⁻¹; molar volume of ideal gas at STP (0 °C, 1 atm) ≈ 22.4 L/mol.

Stoichiometry: balanced equation coefficients give mole ratios; limiting reactant is the one that produces the least product.

Concentration: molarity M = moles solute / liters solution; dilution M₁V₁ = M₂V₂.

pH = -log[H₃O⁺]; pOH = -log[OH⁻]; pH + pOH = 14 at 25 °C; strong acid/base dissociates completely, weak partially (K_a or K_b).

Ideal gas equation PV = nRT with R = 8.314 J mol⁻¹ K⁻¹ or 0.0821 L atm mol⁻¹ K⁻¹; combined gas law P₁V₁/T₁ = P₂V₂/T₂.

Le Chatelier's principle: increasing reactant concentration or pressure (for gas reactions with fewer moles on product side) shifts equilibrium toward products; increasing temperature favors the endothermic direction.

Oxidation is loss of electrons (OIL), reduction is gain (RIG); oxidizing agent is reduced, reducing agent is oxidized.

Periodic trends: atomic radius decreases across a period and increases down a group; electronegativity increases across a period and decreases down a group.

Bond polarity: ΔEN > 1.7 generally ionic, 0.4–1.7 polar covalent, < 0.4 nonpolar covalent (Pauling scale guideline).

Cell theory: all living organisms are composed of one or more cells; the cell is the basic unit of life; new cells arise from pre-existing cells.

Photosynthesis equation: 6CO₂ + 6H₂O + light energy → C₆H₁₂O₆ + 6O₂; occurs in chloroplasts, light reactions in thylakoid membranes, Calvin cycle in stroma.

Cellular respiration: C₆H₁₂O₆ + 6O₂ → 6CO₂ + 6H₂O + ATP; glycolysis in cytoplasm, Krebs cycle in mitochondrial matrix, electron transport chain on inner mitochondrial membrane.

DNA structure: double helix with antiparallel strands; adenine pairs with thymine (2 H-bonds), guanine with cytosine (3 H-bonds); RNA replaces thymine with uracil.

Mendelian genetics: law of segregation (alleles separate during meiosis) and law of independent assortment (genes on different chromosomes assort independently); Punnett square predicts genotype ratios.

Enzyme kinetics: enzymes are biological catalysts that lower activation energy; activity affected by temperature, pH, substrate concentration; denaturation at high temperature or extreme pH.

Mitosis produces two genetically identical diploid daughter cells; meiosis produces four genetically distinct haploid gametes with crossing over in prophase I.

Homeostasis maintains stable internal conditions via negative feedback loops: sensor detects change, control center compares to set point, effector reverses the change.

Ecological pyramids: energy pyramid always upright (≈10% transfer between trophic levels); numbers and biomass pyramids can be inverted in some ecosystems.