IIT JAM MA cheat sheet

Restriction endonucleases recognize palindromic DNA sequences; Type II enzymes (e.g., EcoRI: GAATTC) cleave within the recognition site and are used for cloning.

The central dogma flow is DNA → RNA → protein; transcription uses RNA polymerase, translation uses ribosomes, tRNA, and mRNA codons read 5′→3′.

PCR cycle steps: denaturation (~94 °C), primer annealing (~50–65 °C), extension (~72 °C, Taq polymerase); n cycles give ~2^n-fold amplification.

Plasmid cloning requires an origin of replication, selectable marker (e.g., ampR), and multiple cloning site; insert ligation uses compatible sticky ends and DNA ligase.

Sanger sequencing uses dideoxynucleotides (ddNTPs) to terminate chain elongation; each ddNTP is labeled for detection by capillary electrophoresis.

Enzyme kinetics: Michaelis–Menten v = Vmax[S]/(Km+[S]); Km is the substrate concentration at half Vmax and reflects enzyme–substrate affinity inversely.

Western blot detects proteins via antibody binding after SDS-PAGE; Southern blot detects DNA, Northern blot detects RNA.

CRISPR-Cas9 uses a guide RNA complementary to a target sequence and Cas9 endonuclease to create a double-strand break; repair occurs by NHEJ or HDR.

Ideal gas law PV = nRT, with R = 8.314 J mol⁻¹ K⁻¹; at STP (273.15 K, 1 bar) molar volume ≈ 22.71 L.

First law of thermodynamics: ΔU = q + w; for reversible isothermal expansion of an ideal gas, ΔU = 0 and w = −nRT ln(V2/V1).

Gibbs free energy ΔG = ΔH − TΔS; a reaction is spontaneous at constant T, P when ΔG < 0.

Arrhenius equation k = A e^(−Ea/RT); a plot of ln k vs 1/T is linear with slope −Ea/R.

For a weak acid, Ka·Kb = Kw = 1.0×10⁻¹⁴ at 25 °C; pH = pKa + log([A⁻]/[HA]) (Henderson–Hasselbalch).

Nernst equation: E = E° − (RT/nF) ln Q; at 298 K, E = E° − (0.0592/n) log Q.

Rate law for a first-order reaction: [A] = [A]₀e^(−kt), t₁/₂ = ln2/k ≈ 0.693/k.

Crystal field theory: in an octahedral field, d orbitals split into t2g (lower) and eg (upper) sets; Δo depends on ligand field strength (spectrochemical series).

Price elasticity of demand: Ed = (%ΔQd)/(%ΔP); |Ed| > 1 elastic, |Ed| < 1 inelastic, |Ed| = 1 unit elastic.

Consumer equilibrium under utility maximization: MUx/Px = MUy/Py, subject to budget constraint PxX + PyY = M.

Perfect competition: firm maximizes profit where P = MC; short-run shutdown if P < AVC; long-run equilibrium at P = minimum ATC.

Monopoly profit maximization: MR = MC; markup over marginal cost follows Lerner index (P − MC)/P = 1/|Ed|.

GDP (expenditure approach) = C + I + G + (X − M); real GDP uses constant base-year prices to remove inflation.

Quantity theory of money: MV = PY; in classical long-run view, money growth raises prices proportionally if V and Y are constant.

Phillips curve: short-run trade-off between inflation and unemployment; long-run curve is vertical at the natural rate of unemployment.

Comparative advantage: a country gains from trade by specializing where its opportunity cost of production is lower, even if it has absolute disadvantage in both goods.

Mohs hardness scale: talc (1), gypsum (2), calcite (3), fluorite (4), apatite (5), orthoclase (6), quartz (7), topaz (8), corundum (9), diamond (10).

Bowen's reaction series: discontinuous branch olivine → pyroxene → amphibole → biotite; continuous branch Ca-plagioclase → Na-plagioclase → K-feldspar → muscovite → quartz.

Seismic wave speeds: P waves (compressional) travel faster than S waves (shear); S waves cannot propagate through liquids, defining the outer core as liquid.

Earthquake magnitude: moment magnitude Mw = (2/3) log₁₀ M₀ − 6.07, where M₀ is seismic moment in N·m; each unit increase ≈ 31.6× energy.

Plate boundaries: divergent (rifting, mid-ocean ridges), convergent (subduction, collision), transform (lateral slip, e.g., San Andreas).

Rock cycle: igneous (crystallization from melt), sedimentary (weathering, transport, deposition, lithification), metamorphic (heat and pressure without melting).

Isostasy: buoyancy of lithosphere on asthenosphere; Airy model assumes uniform density with varying root thickness; Pratt model assumes varying density at constant depth.

Geologic time scale eras: Paleozoic (541–252 Ma), Mesozoic (252–66 Ma), Cenozoic (66 Ma–present); Precambrian covers ~88% of Earth history.

Limit definition of derivative: f′(x) = lim_{h→0} [f(x+h) − f(x)]/h; differentiability implies continuity, not conversely.

Rolle's theorem: if f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then ∃ c ∈ (a,b) with f′(c)=0.

Mean value theorem: under continuity on [a,b] and differentiability on (a,b), ∃ c with f′(c) = [f(b)−f(a)]/(b−a).

Taylor series about x=a: f(x) = Σ_{n=0}^∞ f^(n)(a)(x−a)^n/n! within radius of convergence.

For a matrix A, eigenvalues λ satisfy det(A − λI) = 0; eigenvectors solve (A − λI)v = 0.

Gradient ∇f is perpendicular to level curves/surfaces; directional derivative along unit vector u is ∇f·u.

Double integral in polar coordinates: ∬_D f(x,y) dA = ∬ f(r cosθ, r sinθ) r dr dθ; Jacobian factor is r.

Cauchy–Riemann equations for f(z)=u+iv: u_x = v_y and u_y = −v_x; satisfied plus continuity of partials implies analyticity.

Probability axioms: P(∅)=0, P(S)=1, and for disjoint events P(∪A_i)=ΣP(A_i); inclusion–exclusion for two events: P(A∪B)=P(A)+P(B)−P(A∩B).

Bayes' theorem: P(A|B) = P(B|A)P(A)/P(B), with P(B) = Σ P(B|A_i)P(A_i) over a partition.

Binomial(n,p): mean np, variance np(1−p); Poisson(λ): mean λ, variance λ; Normal(μ,σ²): mean μ, variance σ².

Central limit theorem: for i.i.d. variables with mean μ and variance σ², (X̄ − μ)/(σ/√n) → N(0,1) in distribution as n → ∞.

Unbiased estimator: E(θ̂)=θ; sample variance S² = Σ(X_i − X̄)²/(n−1) is unbiased for σ².

Maximum likelihood estimator maximizes the likelihood L(θ); often found by solving ∂ ln L/∂θ = 0.

Confidence interval for normal mean with known σ: X̄ ± z_{α/2}·σ/√n; use t_{n−1} when σ is unknown.

Hypothesis test: reject H₀ when p-value < α; Type I error probability = α, Type II error probability = β, power = 1 − β.

Newton's second law: F = dp/dt = ma for constant mass; work–energy theorem: W_net = ΔK.

Conservation of mechanical energy: K + U = constant when only conservative forces act; U_gravity = mgh, U_spring = ½kx².

Simple harmonic motion: x(t) = A cos(ωt + φ); ω = √(k/m) for a spring, ω = √(g/L) for a pendulum; T = 2π/ω.

Wave equation: v = fλ; for a string, v = √(T/μ); for sound in air, v ≈ 331 + 0.6T(°C) m/s.

Ideal gas: PV = nRT; kinetic theory gives average translational kinetic energy (3/2)k_BT per molecule.

Coulomb's law: F = k q₁q₂/r² with k = 1/(4πε₀) ≈ 8.99×10⁹ N·m²/C²; electric field of a point charge E = kq/r².

Ohm's law V = IR; power P = VI = I²R = V²/R; resistors in series add, in parallel reciprocals add.

Lorentz force on a charge: F = q(E + v × B); magnetic force on a current-carrying wire: F = I L × B.

Photoelectric effect: maximum kinetic energy K_max = hf − φ, where φ is the work function; threshold frequency f₀ = φ/h.