CUSAT CAT B.Tech cheat sheet

Quadratic ax² + bx + c = 0 with a ≠ 0: x = (−b ± √(b² − 4ac))/2a; sum of roots = −b/a, product = c/a; discriminant D = b² − 4ac gives two distinct real (D > 0), equal real (D = 0) or complex conjugate (D < 0) roots.

Binomial expansion of (a + b)^n: general term T(r+1) = nCr · a^(n−r) · b^r for 0 ≤ r ≤ n; there are n + 1 terms; when n is even there is one middle term at r = n/2, when n is odd two middle terms at r = (n−1)/2 and (n+1)/2.

Standard limits (x in radians): lim(x→0) sin x / x = 1 = lim(x→0) tan x / x; lim(x→0) (1 − cos x)/x² = 1/2; lim(x→0) (e^x − 1)/x = 1; lim(x→∞) (1 + 1/x)^x = e; lim(x→0) ln(1 + x)/x = 1.

Derivatives to memorise: d/dx(arcsin x) = 1/√(1 − x²), d/dx(arccos x) = −1/√(1 − x²), d/dx(arctan x) = 1/(1 + x²), d/dx(a^x) = a^x ln a, d/dx(ln x) = 1/x, d/dx(|x|) undefined at x = 0.

Integration: ∫u dv = uv − ∫v du with u chosen by ILATE order (Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential); ∫dx/(x² + a²) = (1/a)arctan(x/a) + C; ∫dx/√(a² − x²) = arcsin(x/a) + C.

Conics: parabola y² = 4ax has e = 1; ellipse x²/a² + y²/b² = 1 (a > b) has e = √(1 − b²/a²) < 1; hyperbola x²/a² − y²/b² = 1 has e = √(1 + b²/a²) > 1 with asymptotes y = ±(b/a)x.

Probability: P(A|B) = P(A ∩ B)/P(B); Bayes gives P(A_i|B) = P(A_i)P(B|A_i) / Σ P(A_j)P(B|A_j); binomial distribution mean = np, variance = npq, standard deviation = √(npq).

Vectors: a·b = |a||b|cos θ (scalar); |a × b| = |a||b|sin θ equals the parallelogram area and a × b is perpendicular to both; scalar triple product a·(b × c) gives the parallelepiped volume and is zero exactly when the three vectors are coplanar.

Matrices: (AB)ᵀ = BᵀAᵀ; A⁻¹ = adj(A)/det(A) exists only when det A ≠ 0; det(AB) = det A · det B; 3×3 expansion det = a₁(b₂c₃ − b₃c₂) − a₂(b₁c₃ − b₃c₁) + a₃(b₁c₂ − b₂c₁).

Constant-acceleration kinematics: v = u + at, s = ut + ½at², v² = u² + 2as. Projectile on level ground: range R = u²sin2θ/g (maximum at 45°), maximum height H = u²sin²θ/(2g), time of flight T = 2u sinθ/g.

Work–energy theorem: W_net = ΔKE; instantaneous power P = F·v; spring force F = −kx with elastic PE = ½kx²; work by a conservative force is path independent and equals minus the change in potential energy.

Rotation: τ = Iα, L = Iω is conserved when net external torque is zero; parallel axis theorem I = I_cm + Md²; ring I = MR², disc I = ½MR² about the central axis; rolling KE = ½Mv²(1 + k²/R²).

SHM: acceleration a = −ω²x; speed v = ω√(A² − x²); total energy E = ½mω²A²; spring period T = 2π√(m/k); simple pendulum period T = 2π√(L/g) valid only for small angles.

Waves: v = fλ; string wave speed v = √(T/μ); beat frequency = |f₁ − f₂|; Doppler f' = f(v ± v_o)/(v ∓ v_s); pipes give f = nv/2L for both ends open and odd multiples nv/4L for one end closed.

Thermodynamics: first law ΔU = Q − W (W done by the gas); adiabatic process PV^γ = constant with γ = C_p/C_v; Carnot efficiency η = 1 − T_c/T_h using kelvin temperatures; refrigerator COP = T_c/(T_h − T_c).

Optics: mirror 1/f = 1/v + 1/u; thin lens 1/f = 1/v − 1/u with consistent sign convention; lens maker 1/f = (n − 1)(1/R₁ − 1/R₂); Young double-slit fringe width β = λD/d, so longer wavelengths give wider fringes.

Photoelectric effect and matter waves: hν = φ + K_max, threshold frequency ν₀ = φ/h, stopping potential eV₀ = K_max and intensity affects current but not K_max; de Broglie λ = h/p, and for electrons λ ≈ 12.27/√V ångström with V in volts.

Atomic and nuclear: Bohr E_n = −13.6 Z²/n² eV with r_n ∝ n²/Z; Rydberg 1/λ = RZ²(1/n₁² − 1/n₂²); radioactive decay N = N₀e^(−λt) with half-life t½ = 0.693/λ.

Mole relations: n = m/M = N/N_A, and V = 22.4 L per mole of ideal gas at STP; molarity = moles solute per litre of solution (temperature dependent); molality = moles solute per kg of solvent (temperature independent); normality = molarity × n-factor.

Gases: PV = nRT with R = 0.0821 L·atm·mol⁻¹·K⁻¹ = 8.314 J·mol⁻¹·K⁻¹; van der Waals (P + an²/V²)(V − nb) = nRT; Dalton partial pressures P_total = ΣP_i; Graham effusion rate ∝ 1/√M.

Equilibrium and ionic equilibria: Kp = Kc(RT)^Δn with Δn = moles gaseous products − reactants; weak acid Ka = Cα² when α is small; pH = −log[H⁺] and pH + pOH = 14 at 25 °C; buffer pH = pKa + log([salt]/[acid]).

Thermodynamics: ΔG = ΔH − TΔS, spontaneous when ΔG < 0 at constant T and P, equilibrium when ΔG = 0; ΔG° = −RT ln K; entropy increases solid → liquid → gas and when gaseous mole number rises.

Periodic trends: left to right across a period, ionisation enthalpy and electronegativity rise while atomic radius falls; down a group radius and metallic character rise and ionisation enthalpy falls; noble gases hold the highest ionisation enthalpy of each period.

VSEPR and hybridisation: steric number 2 linear 180°, 3 trigonal planar 120°, 4 tetrahedral 109.5°, 5 trigonal bipyramidal, 6 octahedral; lone pairs compress bond angles (NH₃ about 107°, H₂O about 104.5°); sp, sp², sp³ mean 2, 3, 4 electron domains.

Organic rules: Markovnikov addition puts H on the carbon bearing more hydrogens (more stable carbocation), peroxides reverse it; SN1 is two-step, first order, favoured by tertiary halides in polar protic solvents and gives racemisation; SN2 is one-step, second order, favoured by primary halides in aprotic solvents and inverts configuration.

Aromaticity (Hückel): a planar, cyclic, fully conjugated system with 4n + 2 π electrons is aromatic; 4n π electrons in the same geometry is antiaromatic; benzene has 6, cyclobutadiene 4; pyrrole's nitrogen lone pair joins the ring, pyridine's does not.

Electrochemistry and coordination: Nernst E = E° − (0.0591/n)log Q at 298 K; ΔG° = −nFE°; one faraday 96500 C deposits one equivalent weight; octahedral splitting gives CFSE = (−0.4 n_t2g + 0.6 n_eg)Δo, with high-spin favoured when Δo < pairing energy P.