Assam CEE cheat sheet

Projectile: range R = u²sin2θ/g on level ground; maximum range occurs at θ = 45°, and complementary angles θ and (90° − θ) give equal ranges.

Uniform circular motion: speed constant, but acceleration a = v²/r = ω²r points to the centre; centripetal force does zero work.

Work–energy theorem: net work W_net = ΔKE = ½mv² − ½mu²; work by a conservative force equals minus the change in potential energy.

Elastic collision with a stationary target: v₁′ = (m₁ − m₂)u/(m₁ + m₂) and v₂′ = 2m₁u/(m₁ + m₂); equal masses exchange velocities.

Rotational analogues: τ = Iα, L = Iω, KE_rot = ½Iω²; moment of inertia of a rod about its centre is ML²/12, about an end ML²/3.

Simple harmonic motion: T = 2π√(m/k) for a spring, T = 2π√(l/g) for a pendulum; acceleration a = −ω²x is maximum at the extremes, zero at mean position.

Lens and mirror formula: 1/v − 1/u = 1/f for a thin lens (Cartesian sign convention); 1/v + 1/u = 1/f for a spherical mirror.

Photoelectric effect: hν = φ + KE_max, with threshold frequency ν₀ = φ/h; stopping potential depends on frequency, not on intensity.

Radioactive decay: N = N₀e^(−λt), half-life T½ = 0.693/λ; activity A = λN falls by half every half-life.

Ideal gas: PV = nRT with R = 8.314 J mol⁻¹K⁻¹; at STP one mole of an ideal gas occupies 22.4 L.

Le Chatelier: raising pressure shifts equilibrium toward the side with fewer gaseous moles; raising temperature favours the endothermic direction.

pH = −log[H⁺]; for a strong acid pH = −log C, and pOH = 14 − pH at 25 °C; a buffer follows the Henderson–Hasselbalch relation pH = pKa + log([salt]/[acid]).

First law of thermodynamics: ΔU = q + w; for an isothermal ideal-gas process ΔU = 0, and for an adiabatic process q = 0.

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

Benzene undergoes electrophilic substitution (nitration, halogenation, Friedel–Crafts) rather than addition, because the aromatic sextet is preserved.

Aldehydes are oxidised by Tollens' reagent to a silver mirror and by Fehling's solution to a red Cu₂O precipitate; ketones do not respond.

SN1 proceeds via a carbocation with first-order kinetics and is favoured by tertiary halides and polar protic solvents; SN2 is a single-step backside attack favoured by primary halides.

Colligative properties depend on particle count: ΔTb = i·Kb·m and ΔTf = i·Kf·m, where the van't Hoff factor i counts dissociated ions.

Quadratic ax² + bx + c = 0: roots sum to −b/a and multiply to c/a; the discriminant D = b² − 4ac decides real (D ≥ 0) versus complex roots.

Limits: lim(x→0) sin x/x = 1 and lim(x→0) (1 + x)^(1/x) = e; L'Hôpital's rule applies to 0/0 and ∞/∞ indeterminate forms.

Derivatives: 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.

Integration by parts: ∫u dv = uv − ∫v du; choose u by the ILATE order (inverse trig, log, algebraic, trig, exponential).

Definite integral property: ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx, useful for evaluating symmetric trigonometric integrals.

Straight line: slope m = (y₂ − y₁)/(x₂ − x₁); perpendicular lines satisfy m₁m₂ = −1, parallel lines have equal slopes.

Distance from point (x₁, y₁) to line ax + by + c = 0 is |ax₁ + by₁ + c|/√(a² + b²).

Binomial theorem: (a + b)ⁿ = Σ C(n,r) aⁿ⁻ʳ bʳ; the general term is C(n,r) aⁿ⁻ʳ bʳ, and C(n,r) = C(n, n − r).

Probability: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for independent events P(A ∩ B) = P(A)P(B), and conditional P(A|B) = P(A ∩ B)/P(B).

Photosynthesis: light reactions in thylakoid membranes split water, release O₂ and yield ATP and NADPH; the Calvin cycle in the stroma fixes CO₂ via RuBisCO.

Cellular respiration: glycolysis in cytoplasm (net 2 ATP, 2 NADH), Krebs cycle in the mitochondrial matrix, and oxidative phosphorylation on the inner membrane giving most ATP.

Mendel's law of segregation: alleles separate during gamete formation; the law of independent assortment applies to genes on different chromosomes, giving a 9:3:3:1 F₂ ratio in a dihybrid cross.

DNA structure: antiparallel double helix with A–T and G–C base pairs; A pairs with T by two hydrogen bonds, G with C by three.

Central dogma: DNA → RNA (transcription) → protein (translation); the genetic code is triplet and degenerate, with AUG as the start codon.

Human blood groups follow multiple alleles (Iᴬ, Iᴵ, i); Iᴬ and Iᴵ are codominant, and type O is the recessive genotype ii.

Nephron function: glomerular filtration in the Bowman's capsule, selective reabsorption of glucose and most water in the proximal tubule, and concentration of urine in the loop of Henle.

Hormones: insulin lowers blood glucose by promoting cellular uptake and glycogenesis; glucagon and adrenaline raise it by promoting glycogenolysis.

Ecosystem energy flow: only about 10% of energy passes to the next trophic level, so food chains rarely exceed four or five levels.