GUJCET cheat sheet

Dimensional formulae: Planck's constant and angular momentum both [M L² T⁻¹]; pressure and stress [M L⁻¹ T⁻²]; frequency [T⁻¹].

Projectile over level ground: R = u² sin2θ/g, H = u² sin²θ/(2g), T = 2u sinθ/g; range is maximum at 45° and equal for θ and 90° − θ.

Kirchhoff's rules: ΣI = 0 at a junction (conservation of charge); ΣΔV = 0 around a closed loop (conservation of energy).

Thin lens 1/f = 1/v − 1/u and mirror 1/f = 1/v + 1/u with Cartesian sign convention; power P = 1/f in dioptre, and P = P₁ + P₂ for thin lenses in contact.

Photoelectric effect: hν = φ₀ + K_max and eV₀ = hν − φ₀; a V₀-versus-ν graph has slope h/e for every metal and intercept −φ₀/e.

Capacitors: series 1/C = Σ1/Cᵢ, parallel C = ΣCᵢ, with dielectric C = κC₀; stored energy U = ½CV² = Q²/2C.

Series LCR: X_L = 2πfL, X_C = 1/(2πfC), Z = √(R² + (X_L − X_C)²); resonance at f = 1/(2π√(LC)) where Z = R, current is maximum and cos φ = 1.

Bohr hydrogen atom: E_n = −13.6/n² eV, r_n = 0.529 n² Å, angular momentum = nh/2π; ionisation energy from the ground state is 13.6 eV.

Gauss's law Φ = q_inside/ε₀ gives E = σ/2ε₀ near an infinite sheet, E = λ/(2πε₀r) near a line charge, and E = 0 inside a conductor in electrostatic equilibrium.

Molarity decreases when a solution is warmed because volume expands; molality, mole fraction and mass percentage are independent of temperature.

ΔG = ΔH − TΔS: the process is spontaneous when ΔG < 0, at equilibrium ΔG = 0, and ΔG° = −RT ln K = −2.303RT log K.

First order: k = (2.303/t) log[a/(a − x)] and t½ = 0.693/k; zero order: k = x/t and t½ = a/2k.

At 25 °C, K_w = 1 × 10⁻¹⁴ and pH + pOH = 14; for a weak acid pH = ½(pKa − log C); acidic buffer pH = pKa + log([salt]/[acid]).

Solubility product for A_xB_y is K_sp = xˣ yʸ s^(x+y); adding a common ion lowers solubility but leaves K_sp unchanged at fixed temperature.

Cubic unit cells: simple Z = 1 and 52% packing, bcc Z = 2 and 68% (coordination 8), fcc Z = 4 and 74% (coordination 12); density ρ = ZM/(N_A a³).

SN1 is first order via a carbocation and gives racemisation (3° > 2° > 1°); SN2 is second order with backside attack and inversion (1° > 2° > 3°).

Electrochemistry: E°_cell = E°_cathode − E°_anode; Nernst at 298 K E = E° − (0.0591/n) log Q; and ΔG° = −nFE°.

Hybridisation from steric number (σ bonds + lone pairs): 2 → sp, 180°; 3 → sp², 120°; 4 → sp³, 109.5°.

Maths - standard derivatives and integrals: d/dx(sin⁻¹x) = 1/√(1 − x²), d/dx(tan⁻¹x) = 1/(1 + x²), ∫dx/(1 + x²) = tan⁻¹x + C, ∫dx/√(1 − x²) = sin⁻¹x + C.

Maths - probability: P(A ∪ B) = P(A) + P(B) − P(A ∩ B), P(A|B) = P(A ∩ B)/P(B), and independence means P(A ∩ B) = P(A)P(B).

Maths - line ax + by + c = 0: distance from (x₁, y₁) is |ax₁ + by₁ + c|/√(a² + b²); between parallel lines it is |c₁ − c₂|/√(a² + b²).

Maths - matrices: A⁻¹ = adj A/|A| exists only when |A| ≠ 0; (AB)⁻¹ = B⁻¹A⁻¹; |kA| = kⁿ|A| for order n.

Maths - vectors: a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃, |a × b| = |a||b| sin θ, and a·(b × c) gives the signed volume of the parallelepiped.

Biology - mitosis order is prophase → metaphase → anaphase → telophase; DNA replicates in S phase; meiosis converts one diploid cell into four haploid cells.

Biology - Mendel: monohybrid F2 ratio 3:1, dihybrid F2 ratio 9:3:3:1, a test cross uses the homozygous recessive parent, and incomplete dominance gives 1:2:1 phenotypes.

Biology - blood groups: O negative is the universal donor and AB positive the universal recipient; an Rh-negative mother carrying an Rh-positive fetus risks erythroblastosis fetalis in later pregnancies.

Biology - photosynthesis: light reactions on thylakoid membranes (PSII → PSI, water splitting, ATP and NADPH production); Calvin cycle in the stroma, where RuBisCO fixes CO₂ into 3-PGA.