Iwate Entrance Science cheat sheet

Quadratic formula: x = (-b ± √(b²-4ac)) / 2a; the discriminant b²-4ac gives two, one, or zero real roots when positive, zero, or negative.

Sine and cosine rules for any triangle: a/sin A = b/sin B = c/sin C, and c² = a² + b² - 2ab cos C.

Derivative rules: d/dx(xⁿ) = nxⁿ⁻¹, product rule (uv)' = u'v + uv', quotient rule (u/v)' = (u'v - uv')/v², chain rule dy/dx = dy/du · du/dx.

Definite integral as limit of Riemann sums; ∫ₐᵇ f(x)dx = F(b) - F(a) when F' = f (Fundamental Theorem of Calculus).

Arithmetic sequence: aₙ = a₁ + (n-1)d, sum Sₙ = n(a₁+aₙ)/2. Geometric: aₙ = a₁rⁿ⁻¹, sum Sₙ = a₁(1-rⁿ)/(1-r) for r ≠ 1.

Infinite geometric series converges to a₁/(1-r) only when |r| < 1; diverges otherwise.

Circle: area πr², circumference 2πr. Sphere: volume (4/3)πr³, surface area 4πr².

Logarithm laws: log(xy) = log x + log y, log(x/y) = log x - log y, log(xⁿ) = n log x; logₐ b = ln b / ln a.

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

Kinematics with constant acceleration: v = u + at, s = ut + ½at², v² = u² + 2as.

Conservation of mechanical energy: ½mv² + mgh = constant when only gravity does work; friction converts energy to heat.

Wave relation v = fλ; for a string fixed at both ends, resonant wavelengths λ = 2L/n.

Snell's law: n₁ sin θ₁ = n₂ sin θ₂; total internal reflection occurs when the angle exceeds the critical angle sin θc = n₂/n₁.

Ohm's law V = IR; resistors in series add, in parallel 1/R = 1/R₁ + 1/R₂; electrical power P = VI = I²R = V²/R.

Ideal gas law PV = nRT; absolute temperature must be used in kelvin.

First law of thermodynamics: ΔU = Q - W, where W is work done by the gas; adiabatic means Q = 0.

Ionic bonding transfers electrons between metal and nonmetal; covalent bonding shares electron pairs, usually between nonmetals.

Octet rule: atoms gain, lose, or share electrons to reach eight valence electrons; exceptions include H (2) and expanded octets in period 3+.

Mole concept: n = m/M; one mole contains 6.02 × 10²³ particles; molar volume of an ideal gas is about 22.4 L at STP.

Balanced equations conserve atoms and charge; limiting reactant determines the maximum product yield.

Acid-base neutralization: acid + base → salt + water; pH = -log[H⁺] for dilute aqueous solutions.

Le Chatelier's principle: a system at equilibrium shifts to oppose changes in concentration, pressure, or temperature.

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

Organic functional groups: alcohols -OH, carboxylic acids -COOH, aldehydes -CHO, ketones C=O between carbons, esters -COO-.

Prokaryotic cells lack a nucleus and membrane-bound organelles; eukaryotic cells have both.

Mitosis produces two genetically identical diploid cells; meiosis produces four haploid gametes with genetic recombination.

Mendel's law of segregation: two alleles separate during gamete formation; law of independent assortment applies to unlinked genes.

DNA base pairing: adenine pairs with thymine, guanine with cytosine; replication is semi-conservative.

Transcription makes mRNA from DNA; translation at ribosomes uses codons (three bases) to specify amino acids.

Natural selection requires heritable variation, differential reproductive success, and change across generations.

Photosynthesis: 6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂, driven by light energy in chloroplasts; respiration reverses this overall equation.

Ecosystem energy flows from producers to consumers, with roughly 10% transferred between trophic levels; nutrients cycle.

Correlation does not establish causation; a controlled experiment or a plausible mechanism is needed for causal claims.

Independent variable is manipulated; dependent variable is measured; controlled variables are held constant.

Precision describes repeatability of measurements; accuracy describes closeness to the true value.

Significant figures: results of multiplication and division carry the fewest significant figures of the inputs; addition and subtraction follow decimal places.

A graph's slope equals the rate of change of y with respect to x; the intercept often represents a baseline or systematic offset.

Outliers should be investigated, not automatically deleted; report whether they are included and why.

Percent error = |measured - accepted| / accepted × 100%; percent difference compares two measurements.

Sample size affects reliability: larger samples reduce random error, but do not remove systematic bias.

A hypothesis must be testable and falsifiable; predictions state expected outcomes under specified conditions.

Include a control group and, where possible, random assignment or blocking to reduce confounding.

Use appropriate glassware: graduated cylinders for volume delivery, volumetric flasks for precise fixed volumes, burettes for titration.

Read the bottom of the meniscus at eye level for transparent liquids; record the smallest graduation plus one estimated digit.

Wear goggles and gloves; handle concentrated acids by adding acid to water, never water to acid.

Titration endpoint is signaled by an indicator color change; use a white tile and swirl constantly.

Calibrate balances and meters before use; record instrument model, uncertainty, and ambient conditions.

Repeat measurements at least three times and report mean and standard deviation to characterize variability.

Quadratic formula: x = (-b ± √(b²-4ac)) / 2a; the discriminant b²-4ac gives two, one, or zero real roots when positive, zero, or negative.

Sine and cosine rules for any triangle: a/sin A = b/sin B = c/sin C, and c² = a² + b² - 2ab cos C.

Derivative rules: d/dx(xⁿ) = nxⁿ⁻¹, product rule (uv)' = u'v + uv', quotient rule (u/v)' = (u'v - uv')/v², chain rule dy/dx = dy/du · du/dx.

Definite integral as limit of Riemann sums; ∫ₐᵇ f(x)dx = F(b) - F(a) when F' = f (Fundamental Theorem of Calculus).

Arithmetic sequence: aₙ = a₁ + (n-1)d, sum Sₙ = n(a₁+aₙ)/2. Geometric: aₙ = a₁rⁿ⁻¹, sum Sₙ = a₁(1-rⁿ)/(1-r) for r ≠ 1.

Infinite geometric series converges to a₁/(1-r) only when |r| < 1; diverges otherwise.

Circle: area πr², circumference 2πr. Sphere: volume (4/3)πr³, surface area 4πr².

Logarithm laws: log(xy) = log x + log y, log(x/y) = log x - log y, log(xⁿ) = n log x; logₐ b = ln b / ln a.

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

Kinematics with constant acceleration: v = u + at, s = ut + ½at², v² = u² + 2as.

Conservation of mechanical energy: ½mv² + mgh = constant when only gravity does work; friction converts energy to heat.

Wave relation v = fλ; for a string fixed at both ends, resonant wavelengths λ = 2L/n.

Snell's law: n₁ sin θ₁ = n₂ sin θ₂; total internal reflection occurs when the angle exceeds the critical angle sin θc = n₂/n₁.

Ohm's law V = IR; resistors in series add, in parallel 1/R = 1/R₁ + 1/R₂; electrical power P = VI = I²R = V²/R.

Ideal gas law PV = nRT; absolute temperature must be used in kelvin.

First law of thermodynamics: ΔU = Q - W, where W is work done by the gas; adiabatic means Q = 0.

Ionic bonding transfers electrons between metal and nonmetal; covalent bonding shares electron pairs, usually between nonmetals.

Octet rule: atoms gain, lose, or share electrons to reach eight valence electrons; exceptions include H (2) and expanded octets in period 3+.

Mole concept: n = m/M; one mole contains 6.02 × 10²³ particles; molar volume of an ideal gas is about 22.4 L at STP.

Balanced equations conserve atoms and charge; limiting reactant determines the maximum product yield.

Acid-base neutralization: acid + base → salt + water; pH = -log[H⁺] for dilute aqueous solutions.

Le Chatelier's principle: a system at equilibrium shifts to oppose changes in concentration, pressure, or temperature.

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

Organic functional groups: alcohols -OH, carboxylic acids -COOH, aldehydes -CHO, ketones C=O between carbons, esters -COO-.

Prokaryotic cells lack a nucleus and membrane-bound organelles; eukaryotic cells have both.

Mitosis produces two genetically identical diploid cells; meiosis produces four haploid gametes with genetic recombination.

Mendel's law of segregation: two alleles separate during gamete formation; law of independent assortment applies to unlinked genes.

DNA base pairing: adenine pairs with thymine, guanine with cytosine; replication is semi-conservative.

Transcription makes mRNA from DNA; translation at ribosomes uses codons (three bases) to specify amino acids.

Natural selection requires heritable variation, differential reproductive success, and change across generations.

Photosynthesis: 6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂, driven by light energy in chloroplasts; respiration reverses this overall equation.

Ecosystem energy flows from producers to consumers, with roughly 10% transferred between trophic levels; nutrients cycle.

Correlation does not establish causation; a controlled experiment or a plausible mechanism is needed for causal claims.

Independent variable is manipulated; dependent variable is measured; controlled variables are held constant.

Precision describes repeatability of measurements; accuracy describes closeness to the true value.

Significant figures: results of multiplication and division carry the fewest significant figures of the inputs; addition and subtraction follow decimal places.

A graph's slope equals the rate of change of y with respect to x; the intercept often represents a baseline or systematic offset.

Outliers should be investigated, not automatically deleted; report whether they are included and why.

Percent error = |measured - accepted| / accepted × 100%; percent difference compares two measurements.

Sample size affects reliability: larger samples reduce random error, but do not remove systematic bias.

A hypothesis must be testable and falsifiable; predictions state expected outcomes under specified conditions.

Include a control group and, where possible, random assignment or blocking to reduce confounding.

Use appropriate glassware: graduated cylinders for volume delivery, volumetric flasks for precise fixed volumes, burettes for titration.

Read the bottom of the meniscus at eye level for transparent liquids; record the smallest graduation plus one estimated digit.

Wear goggles and gloves; handle concentrated acids by adding acid to water, never water to acid.

Titration endpoint is signaled by an indicator color change; use a white tile and swirl constantly.

Calibrate balances and meters before use; record instrument model, uncertainty, and ambient conditions.

Repeat measurements at least three times and report mean and standard deviation to characterize variability.