MHT-CET cheat sheet

Projectile motion on level ground: T = 2u sinθ/g, H = u² sin²θ/(2g), R = u² sin 2θ/g; R is maximum at θ = 45°.

Uniform circular motion: speed is constant but velocity changes; a_c = v²/r = ω²r directed toward the centre, and centripetal force = mv²/r.

Work-energy theorem: W_net = ΔK = ½mv_f² − ½mv_i²; for a conservative force, F_x = −dU/dx.

Rolling without slipping: v_cm = ωR at the contact; K_total = ½mv_cm²(1 + I_cm/(mR²)).

SHM: T = 2π√(m/k) for a mass-spring system and T = 2π√(L/g) for a small-amplitude simple pendulum; v = ±ω√(A² − x²).

Waves: v = fλ; string fixed at both ends f_n = nv/(2L); open pipe f_n = nv/(2L); closed pipe f_n = (2n − 1)v/(4L).

First law with gas work: Q = ΔU + W_by gas; Carnot efficiency η = 1 − T_c/T_h, with temperatures in kelvin.

Optics: mirror 1/f = 1/v + 1/u; lens 1/f = 1/v − 1/u using the Cartesian sign convention; Snell n₁ sin i = n₂ sin r; critical angle sin C = n₂/n₁ for n₁ > n₂.

Current electricity: R = ρL/A; series resistors carry the same current and add, parallel resistors share the same voltage and add reciprocals; Wheatstone balance P/Q = R/S.

Mole relations: n = m/M = N/N_A; ideal gas PV = nRT; at 273 K and 1 atm, molar volume is about 22.4 L/mol, while at 1 bar it is about 22.7 L/mol.

Equilibrium: K_p = K_c(RT)^Δn; Q < K favours the forward reaction, Q > K favours the reverse; catalysts change rates but not K.

Acid-base: pH = −log[H⁺]; at 25 °C K_w = 1.0×10⁻¹⁴; acidic buffer pH = pKa + log([A⁻]/[HA]).

Thermodynamics: ΔG = ΔH − TΔS; ΔG < 0 is spontaneous at constant T and P; ΔG° = −RT ln K.

Kinetics: first-order t₁/₂ = 0.693/k; Arrhenius k = A e^(−Ea/RT); rate law must be determined experimentally, not from stoichiometry.

Organic substitution: SN1 proceeds via carbocation, favoured by tertiary substrates and polar protic solvents; SN2 is concerted with backside attack and inversion, favoured by primary substrates and polar aprotic solvents.

Alkene addition: HX adds Markovnikov (H to carbon with more H); HBr with peroxide gives anti-Markovnikov bromination by a radical pathway.

Periodic trends: across a period atomic radius decreases and ionization energy generally increases; down a group radius increases and ionization energy decreases; F has the highest electronegativity.

Electrochemistry: E°cell = E°cathode − E°anode; Nernst at 298 K E = E° − (0.059/n) log Q; 1 F ≈ 96,500 C/mol e⁻.

Quadratic ax² + bx + c = 0: sum of roots = −b/a, product = c/a; D = b² − 4ac; D > 0 real distinct, D = 0 real equal, D < 0 complex conjugates.

Derivatives: d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(tan x) = sec²x, d/dx(ln x) = 1/x, d/dx(e^x) = e^x; use chain, product, quotient rules.

Integrals: ∫x^n dx = x^(n+1)/(n+1) + C for n ≠ −1; ∫(1/x) dx = ln|x| + C; ∫u dv = uv − ∫v du.

Limits: lim x→0 sin x/x = 1, lim x→0 (1 − cos x)/x² = 1/2, lim x→0 (e^x − 1)/x = 1, lim x→0 ln(1+x)/x = 1.

Probability: P(A∪B) = P(A) + P(B) − P(A∩B); P(A|B) = P(A∩B)/P(B); Bayes theorem reverses conditional probabilities using prior and likelihood.

Vectors: a·b = |a||b|cosθ = a_xb_x + a_yb_y + a_zb_z; |a×b| = |a||b|sinθ; scalar triple product gives parallelepiped volume and is zero for coplanar vectors.

Circle x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c); parabola y² = 4ax has focus (a, 0) and directrix x = −a.

Matrices: det(AB) = det(A)det(B); A⁻¹ = adj(A)/det(A) exists only when det(A) ≠ 0; a square matrix with det 0 is singular.

Triangle rules: sine rule a/sin A = b/sin B = c/sin C; cosine rule a² = b² + c² − 2bc cos A; area = ½ab sin C.

Cell cycle: DNA replicates in S phase; mitosis has prophase, metaphase, anaphase, telophase; meiosis I includes crossing over in pachytene and independent assortment at metaphase I, producing haploid cells.

Photosynthesis: light reactions occur in thylakoid membranes and produce ATP and NADPH; the Calvin cycle occurs in stroma, where RuBisCO fixes CO₂.

C4 plants use PEP carboxylase in mesophyll and RuBisCO in bundle-sheath cells, reducing photorespiration; CAM plants fix CO₂ at night.

Cellular respiration: glycolysis in cytoplasm gives net 2 ATP and 2 NADH per glucose; Krebs cycle in mitochondrial matrix; electron transport chain and oxidative phosphorylation in inner mitochondrial membrane.

Mendelian genetics: law of segregation separates alleles during gamete formation; law of independent assortment applies to genes on different chromosomes; test cross reveals genotype by crossing with homozygous recessive.

Molecular biology: DNA replication is semiconservative; transcription uses RNA polymerase; translation occurs on ribosomes; AUG is start codon and the genetic code is triplet and degenerate.

Human physiology: SA node is pacemaker; action potential depolarization involves Na⁺ influx and repolarization involves K⁺ efflux; nephron glomerular filtration is followed by tubular reabsorption and secretion.

Ecology: energy transfer between trophic levels is about 10%; pyramids of energy are always upright; exponential growth dN/dt = rN, logistic growth dN/dt = rN(1 − N/K).

Enzymes: lower activation energy; competitive inhibitors bind active site and raise apparent Km; noncompetitive/allosteric inhibitors reduce activity; high temperature or extreme pH can denature.