AP POLYCET flashcards
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- Quadratic formula
- For ax²+bx+c=0 with a≠0, x equals negative b plus or minus the square root of b²−4ac, all divided by 2a.
- Discriminant
- The discriminant of ax²+bx+c=0 is D=b²−4ac. D>0 gives two real roots, D=0 gives equal real roots, and D<0 gives complex roots.
- AP nth term
- For an arithmetic progression with first term a and common difference d, the nth term is aₙ=a+(n−1)d; consecutive terms differ by d.
- GP sum
- For a geometric progression with first term a and ratio r≠1, the sum of n terms is Sₙ=a(rⁿ−1)/(r−1); each term is r times the previous term.
- Trigonometric Pythagorean identity
- The Pythagorean identity states sin²θ+cos²θ=1 for every angle θ. Dividing by cos²θ gives 1+tan²θ=sec²θ; dividing by sin²θ gives 1+cot²θ=csc²θ.
- Distance formula
- The distance between points (x₁,y₁) and (x₂,y₂) in a plane is √[(x₂−x₁)²+(y₂−y₁)²], derived from the Pythagorean theorem.
- Section formula
- If point P divides the segment joining A(x₁,y₁) and B(x₂,y₂) in ratio m:n internally, then P is ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)).
- Slope of line
- For a nonvertical line through (x₁,y₁) and (x₂,y₂), slope m=(y₂−y₁)/(x₂−x₁). Parallel lines have equal slopes; perpendicular lines have product −1.
- Circle equation
- A circle with centre (h,k) and radius r has equation (x−h)²+(y−k)²=r². Expanded form x²+y²+2gx+2fy+c=0 has centre (−g,−f).
- Derivative power rule
- The power rule states that d/dx(xⁿ)=nxⁿ⁻¹ for constant n. For example, d/dx(x³)=3x² and d/dx(x)=1.
- Probability complement
- For an event E with probability P(E), the complement E′ satisfies P(E′)=1−P(E), and probabilities always lie between 0 and 1 inclusive.
- First equation of motion
- For uniform acceleration a, velocity after time t is v=u+at, where u is initial velocity and v is final velocity.
- Newton's second law
- Newton's second law states that net force equals mass times acceleration, F=ma. The acceleration is in the same direction as the net force.
- Work done by force
- Work done by a constant force is W=Fs cosθ, where F is force, s is displacement, and θ is the angle between force and displacement.
- Kinetic energy
- The kinetic energy of a body of mass m moving with speed v is KE=½mv². It is a scalar quantity measured in joules.
- Ohm's law
- Ohm's law states that current I through a conductor is directly proportional to potential difference V at constant temperature: V=IR, where R is resistance.
- Resistors in series
- In a series circuit, the same current flows through each resistor and the equivalent resistance is R=R₁+R₂+R₃+…; voltage divides across resistors.
- Resistors in parallel
- In a parallel circuit, the voltage across each resistor is the same and the reciprocal equivalent resistance is 1/R=1/R₁+1/R₂+…; current divides.
- Electrical power
- Electrical power consumed by a component is P=VI=I²R=V²/R, where V is voltage, I is current, and R is resistance.
- Thin lens formula
- Using the Cartesian sign convention, the thin lens formula is 1/f=1/v−1/u, where f is focal length, v image distance, and u object distance.
- Speed of sound wave
- For a wave, speed equals frequency times wavelength: v=fλ. Sound travels faster in solids than in gases at the same temperature.
- Density formula
- Density is mass per unit volume, ρ=m/V. It is measured in kg/m³ in SI units and helps predict whether an object floats.
- Mole formula
- The number of moles n in a sample equals its mass m divided by molar mass M: n=m/M. One mole contains Avogadro's number of particles.
- Avogadro's number
- Avogadro's number is approximately 6.022×10²³ particles per mole. It counts atoms, molecules, or ions in one mole of a substance.
- Molar volume at STP
- At standard temperature and pressure, one mole of an ideal gas occupies about 22.4 litres. This volume is called the molar volume.
- Ideal gas law
- The ideal gas law is PV=nRT, where P is pressure, V volume, n moles, R the gas constant, and T absolute temperature.
- pH definition
- pH is defined as −log₁₀[H⁺], where [H⁺] is hydrogen ion concentration in mol/L. A pH below 7 is acidic at 25°C.
- Atomic radius trend
- Atomic radius generally decreases across a period from left to right and increases down a group, because added shells outweigh increased nuclear charge.
- Oxygen oxidation number
- Oxygen usually has oxidation number −2 in compounds, except in peroxides where it is −1 and in OF₂ where it is +2.
- Neutralization reaction
- An acid reacts with a base to form salt and water: acid+base→salt+water. For example, HCl+NaOH→NaCl+H₂O.
- Carboxylic acid group
- The carboxylic acid functional group is −COOH. It contains a carbonyl group bonded to a hydroxyl group and is acidic in water.
- Balancing chemical equations
- A balanced chemical equation has the same number of each type of atom on both sides. Coefficients are adjusted, never subscripts, to conserve mass.