AP POLYCET cheat sheet

Quadratic ax²+bx+c=0: roots x=[−b±√(b²−4ac)]/(2a); discriminant D=b²−4ac; D>0 real distinct, D=0 real equal, D<0 complex.

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

Identities: sin²θ+cos²θ=1; 1+tan²θ=sec²θ; 1+cot²θ=csc²θ; sin(A±B)=sinAcosB±cosAsinB; cos(A±B)=cosAcosB∓sinAsinB.

Distance between points =√[(x₂−x₁)²+(y₂−y₁)²]; midpoint=((x₁+x₂)/2,(y₁+y₂)/2); internal section ratio m:n gives ((mx₂+nx₁)/(m+n),(my₂+ny₁)/(m+n)).

Line slope m=(y₂−y₁)/(x₂−x₁); parallel lines m₁=m₂; perpendicular lines m₁m₂=−1; line y=mx+c has slope m and intercept c.

Circle centre (h,k), radius r: (x−h)²+(y−k)²=r²; general x²+y²+2gx+2fy+c=0 has centre (−g,−f) and radius √(g²+f²−c).

Derivatives: d/dx(xⁿ)=nxⁿ⁻¹; d/dx(sin x)=cos x; d/dx(cos x)=−sin x; d/dx(constant)=0; d/dx(eˣ)=eˣ.

Probability: P(E)=favourable outcomes/total equally likely outcomes; 0≤P(E)≤1; P(E′)=1−P(E); P(A∪B)=P(A)+P(B)−P(A∩B).

Uniform acceleration: v=u+at; s=ut+½at²; v²=u²+2as; average velocity=(u+v)/2.

Newton's laws: F=ma when mass is constant; momentum p=mv; impulse FΔt=Δp; action and reaction are equal and opposite on different bodies.

Work W=Fs cosθ; kinetic energy KE=½mv²; gravitational PE=mgh near Earth; power P=W/t; 1 kWh=3.6×10⁶ J.

Ohm's law V=IR at constant temperature; series R=R₁+R₂+…; parallel 1/R=1/R₁+1/R₂+…; ammeter in series, voltmeter in parallel.

Electrical power P=VI=I²R=V²/R; energy E=Pt; fuse rating depends on current; household supply in India is 220 V, 50 Hz.

Mirror formula 1/f=1/v+1/u with Cartesian sign convention; magnification m=−v/u. Thin lens formula 1/f=1/v−1/u; m=v/u.

Wave relation v=fλ; sound speed increases from gas to liquid to solid; echo distance d=vt/2; audible range about 20 Hz to 20 kHz.

Density ρ=m/V; pressure P=F/A; liquid pressure P=ρgh; buoyant force equals weight of displaced fluid (Archimedes' principle).

Mole: n=m/M; 1 mol=6.022×10²³ particles; at STP one mole of ideal gas occupies 22.4 L; molarity M=n/V(L).

Ideal gas PV=nRT; combined gas law P₁V₁/T₁=P₂V₂/T₂; temperature must be in kelvin; R=8.314 J mol⁻¹ K⁻¹.

pH=−log[H⁺]; pOH=−log[OH⁻]; pH+pOH=14 at 25°C; pH<7 acidic, pH=7 neutral, pH>7 basic.

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

Oxidation numbers: free element 0; oxygen usually −2 (peroxides −1, OF₂ +2); hydrogen +1 with nonmetals, −1 with metals; sum equals overall charge.

Neutralization: acid+base→salt+water; strong acid+strong base has pH 7 at equivalence; H⁺+OH⁻→H₂O.

Organic groups: alcohol −OH; aldehyde −CHO; ketone >C=O; carboxylic acid −COOH; amine −NH₂; ester −COO−.

Balancing equations: conserve each element and charge; adjust coefficients only; molar mass from atomic masses; limiting reactant determines maximum product.