AP EAMCET flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Quadratic roots sum and product

For ax²+bx+c=0 (a≠0), the sum of roots is −b/a and the product is c/a; the discriminant b²−4ac determines real or equal roots.

Binomial general term

In (a+b)^n, the (r+1)th term is C(n,r)a^(n−r)b^r, where C(n,r)=n!/[r!(n−r)!] and 0≤r≤n.

sin(x)/x limit

As x approaches 0, sin x divided by x tends to 1; also (1−cos x)/x² tends to 1/2, and (e^x−1)/x tends to 1.

Derivative of x^n

For real n, d/dx(x^n)=n x^(n−1). At a local extremum, f′(x)=0; the second derivative sign classifies maximum or minimum when nonzero.

Definite integral area

The definite integral of a nonnegative function f from a to b equals the area between its graph, the x-axis, and the vertical lines x=a and x=b.

Straight line slope form

A line through (x₁,y₁) with slope m is y−y₁=m(x−x₁). Two lines are perpendicular when m₁m₂=−1 and parallel when m₁=m₂.

Circle center and radius

For x²+y²+2gx+2fy+c=0, the center is (−g,−f) and radius is √(g²+f²−c); the radius is real only when g²+f²−c>0.

Parabola y²=4ax

The parabola y²=4ax has focus (a,0), directrix x=−a, vertex (0,0), and latus rectum length 4a; a point on it satisfies distance to focus equals distance to directrix.

Ellipse eccentricity

For x²/a²+y²/b²=1 with a>b, eccentricity e=√(1−b²/a²); foci are (±ae,0), and the sum of focal distances is 2a.

Vector dot and cross

Dot product a·b=|a||b|cosθ is scalar; cross product a×b has magnitude |a||b|sinθ and direction perpendicular to both by the right-hand rule.

Conditional probability formula

P(A|B)=P(A∩B)/P(B) when P(B)>0. Events A and B are independent exactly when P(A∩B)=P(A)P(B).

Bayes theorem statement

For a partition of outcomes, P(A_i|B)=P(B|A_i)P(A_i)/Σ_j P(B|A_j)P(A_j), provided each conditioning event has positive probability.

Projectile range

For projectile motion on level ground with initial speed u and angle θ, horizontal range is R=u² sin(2θ)/g; maximum range u²/g occurs at θ=45° ignoring air resistance.

Centripetal acceleration

An object moving in a circle of radius r at speed v has centripetal acceleration v²/r directed toward the center; its centripetal force is mv²/r.

SHM displacement and acceleration

Simple harmonic motion has x=A sin(ωt+φ), velocity v=±ω√(A²−x²), and acceleration a=−ω²x; acceleration is always directed opposite displacement.

Spring oscillator period

A mass m attached to an ideal spring of force constant k oscillates with period T=2π√(m/k), independent of amplitude for small oscillations.

Wave speed frequency relation

For a wave, speed v equals frequency f times wavelength λ, so v=fλ; frequency is set by the source while speed depends on the medium.

First law thermodynamics

The first law states ΔU=Q−W, where ΔU is internal energy change, Q is heat added to the system, and W is work done by the system.

Capacitor stored energy

A capacitor of capacitance C with voltage V stores energy U=½CV²; it also equals Q²/(2C) and ½QV, where Q is the stored charge.

Ohm law and power

For an ohmic conductor at constant temperature, V=IR; electrical power dissipated is P=VI=I²R=V²/R, with SI units volts, amperes, ohms, and watts.

Photoelectric equation

Einstein's photoelectric equation is hf=φ+K_max, where hf is incident photon energy, φ is work function, and K_max is maximum kinetic energy of emitted electrons.

de Broglie wavelength

A particle with momentum p has de Broglie wavelength λ=h/p; for an electron accelerated through potential difference V, λ≈1.23 nm/√(V in volts).

Mole and Avogadro number

One mole contains 6.022×10²³ particles. Number of moles n=mass/molar mass, and number of particles N=n×6.022×10²³.

Molarity definition

Molarity is moles of solute per litre of solution: M=n/V(L). It changes with temperature because solution volume expands or contracts.

Ideal gas equation

For an ideal gas, PV=nRT, where P is pressure, V volume, n moles, R=8.314 J mol⁻¹ K⁻¹, and T absolute temperature in kelvin.

Gibbs free energy criterion

ΔG=ΔH−TΔS. A process is spontaneous at constant temperature and pressure when ΔG<0; equilibrium corresponds to ΔG=0.

pH definition

pH=−log₁₀[H⁺] for dilute aqueous solutions. At 25°C, pH+pOH=14, and neutral water has pH 7.

Nernst equation

At 298 K, E=E°−(0.05916/n)log Q for a cell reaction, where n is electrons transferred and Q is the reaction quotient.

First order half-life

For a first-order reaction, half-life t₁/₂=0.693/k and is independent of initial concentration; concentration follows [A]=[A]₀e^(−kt).

Periodic ionization enthalpy trend

Ionization enthalpy generally increases across a period and decreases down a group; exceptions occur at stable half-filled or fully filled subshells and from poor shielding.

Coordination number and geometry

Coordination number counts donor atoms bonded to the central metal. Common geometries: 2 linear, 4 tetrahedral or square planar, and 6 octahedral.

SN1 versus SN2

SN1 proceeds via a carbocation intermediate with first-order kinetics and racemization; SN2 is one-step, second-order, and gives inversion at the reacting carbon.