BVP CET cheat sheet

Projectile motion: horizontal and vertical motions are independent; time of flight t = 2u sinθ/g, range R = u² sin2θ/g, maximum range at θ = 45° (flat ground, no air resistance).

Newton's second law for a system: net external force = rate of change of linear momentum, F = dp/dt; impulse J = ∫F dt equals change in momentum.

Circular motion: centripetal acceleration a = v²/r = ω²r; for a banked road without friction, tanθ = v²/(rg).

Work-energy theorem: net work done on a body equals its change in kinetic energy; conservative forces allow potential-energy bookkeeping, friction does not.

Moment of inertia of standard bodies: solid cylinder or disc about its axis I = MR²/2; solid sphere about a diameter I = 2MR²/5; thin rod about its centre I = ML²/12.

Simple harmonic motion: acceleration a = −ω²x, period T = 2π√(m/k) for a spring and T = 2π√(L/g) for a small-amplitude pendulum.

Ideal gas and kinetic theory: PV = nRT; rms speed v_rms = √(3RT/M); translational kinetic energy per molecule = (3/2)k_BT.

Optics decision rule: for thin lenses and mirrors use 1/v − 1/u = 1/f with the Cartesian sign convention, or use 1/f = 1/v + 1/u for mirrors with real-is-positive convention - never mix the two conventions in one problem.

Photoelectric effect: maximum kinetic energy K_max = hν − φ, where φ is the work function; no emission below the threshold frequency ν₀ = φ/h regardless of intensity.

Mole relations: number of moles n = mass/molar mass = N/N_A = volume of gas at STP/22.4 L; molarity M = moles of solute per litre of solution.

Ideal gas law applies approximately to real gases at low pressure and high temperature; van der Waals equation corrects for molecular volume (b) and attraction (a).

Le Chatelier's principle: increasing pressure shifts equilibrium toward the side with fewer gas moles; raising temperature shifts an exothermic reaction toward reactants.

pH = −log[H⁺]; for a strong acid pH = −log C, for a weak acid pH = ½(pK_a − log C); buffer pH = pK_a + log([salt]/[acid]).

First law of thermodynamics for chemistry: ΔU = q + w; at constant pressure q_p = ΔH, and ΔH = ΔU + Δn_gRT for reactions involving gases.

Hybridisation from steric number: 2 → sp (linear), 3 → sp² (trigonal planar), 4 → sp³ (tetrahedral), 5 → sp³d, 6 → sp³d².

Aromaticity (Hückel rule): a planar, cyclic, fully conjugated system with 4n + 2 π electrons is aromatic; 4n π electrons in the same geometry is antiaromatic.

SN1 versus SN2: SN1 proceeds via a carbocation, is favoured by tertiary substrates and polar protic solvents, and gives racemisation; SN2 is a single-step backside attack favoured by primary substrates and polar aprotic solvents.

Electrochemistry: cell EMF E°_cell = E°_cathode − E°_anode; ΔG° = −nFE°_cell; one faraday F ≈ 96500 C per mole of electrons.

Quadratic equation ax² + bx + c = 0: roots sum to −b/a and multiply to c/a; discriminant D = b² − 4ac decides real distinct (D > 0), equal (D = 0) or complex (D < 0) roots.

Limits: L'Hôpital's rule applies only to 0/0 or ∞/∞ indeterminate forms; check by substitution and algebraic simplification first.

Derivatives to memorise: d/dx(xⁿ) = nxⁿ⁻¹, d/dx(sin x) = cos x, d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x; chain rule for composites, product and quotient rules for products and ratios.

Definite integral property: ∫ₐᵇ f(x) dx = ∫ₐᵇ f(a + b − x) dx, useful for symmetric trigonometric integrands; area between curves is ∫|f − g| dx.

Straight line: slope m = (y₂ − y₁)/(x₂ − x₁); point-slope form y − y₁ = m(x − x₁); perpendicular slopes satisfy m₁m₂ = −1.

Conic sections: circle x² + y² = r²; parabola y² = 4ax with focus (a, 0) and directrix x = −a; ellipse x²/a² + y²/b² = 1 with eccentricity e = √(1 − b²/a²).

Probability: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); conditional probability P(A|B) = P(A ∩ B)/P(B); Bayes' theorem reverses the conditioning.

Binomial theorem: (x + y)ⁿ = Σ C(n, r) xⁿ⁻ʳ yʳ; general term T_(r+1) = C(n, r) xⁿ⁻ʳ yʳ; middle term exists only when n is even.

Vectors: scalar (dot) product a·b = |a||b|cosθ gives zero for perpendicular vectors; vector (cross) product magnitude |a||b|sinθ gives area of the parallelogram they span.

Cell cycle: interphase comprises G1, S (DNA replication) and G2; mitosis has prophase, metaphase, anaphase and telophase, with chromosomes aligning at the metaphase plate.

Photosynthesis: light reactions occur in the thylakoid membranes producing ATP and NADPH and releasing O₂ from water splitting; the Calvin cycle in the stroma fixes CO₂ via RuBisCO.

Cellular respiration ATP yield per glucose (aerobic): glycolysis 2 net ATP, Krebs cycle 2 ATP, oxidative phosphorylation about 26–28 ATP; anaerobic fermentation yields only 2 ATP.

Mendelian genetics: monohybrid F2 phenotypic ratio 3:1; dihybrid F2 ratio 9:3:3:1 for independently assorting unlinked genes; test cross reveals an unknown genotype.

DNA structure: antiparallel double helix with adenine pairing thymine (2 hydrogen bonds) and guanine pairing cytosine (3 hydrogen bonds); replication is semi-conservative.

Human blood circulation: oxygenated blood travels from lungs via pulmonary veins to the left atrium, then left ventricle, then aorta; the right side handles deoxygenated blood.

Nervous system: action potential depolarisation is driven by Na⁺ influx and repolarisation by K⁺ efflux; the resting potential is maintained near −70 mV by the Na⁺/K⁺ pump.

Kidney function: filtration at the glomerulus, selective reabsorption in the proximal tubule, and concentration in the loop of Henle; ADH increases water reabsorption in the collecting duct.

Ecology: in a food chain only about 10% of energy transfers to the next trophic level; gross primary productivity minus respiration gives net primary productivity.