Mechanics - Kinematics & Dynamics
Motion, Forces, Newton's Laws
Kinematic Equations (Constant a)
v = u + at | s = ut + ½at² | v² = u² + 2as | sₙ = u + a(n - ½)
Relative Velocity
v_AB = v_A - v_B | River: v_resultant = √(v² + u² + 2vucosθ)
Projectile Motion
T = 2usinθ/g | H = u²sin²θ/2g | R = u²sin2θ/g | R_max = u²/g (θ=45°)
Newton's Laws
F_net = ma = dp/dt | Impulse: J = FΔt = Δp | F_12 = -F_21
Friction
f_s ≤ μ_sN | f_k = μ_kN | Angle of friction: tanλ = μ
Centripetal Force
F = mv²/r = mω²r | Banking: tanθ = v²/rg | v_max =
√(rg(μ+tanθ)/(1-μtanθ))
Work & Power
W = F·s·cosθ | P = dW/dt = F·v | Instantaneous P = Fvcosθ
Conservation Laws
ME = KE + PE = constant | ΔKE + ΔPE = 0 (conservative)
Collision
e = (v₂-v₁)/(u₁-u₂) | 1D elastic: v₁ = [(m₁-m₂)u₁ + 2m₂u₂]/(m₁+m₂)
Center of Mass
x_cm = Σmᵢxᵢ/Σmᵢ | v_cm = Σmᵢvᵢ/M | a_cm = F_ext/M
Mechanics - Rotation & Gravitation
Rigid Body, Moment of Inertia, Planetary Motion
Rotational Kinematics
ω = ω₀ + αt | θ = ω₀t + ½αt² | ω² = ω₀² + 2αθ
Moment of Inertia
I = Σmᵢrᵢ² | Ring: MR² | Disc: ½MR² | Rod: ML²/12 (center), ML²/3
(end)
Parallel Axis Theorem
I = I_cm + Md² | Perpendicular: I_z = I_x + I_y (planar)
Torque & Angular Momentum
τ = r×F = Iα | L = r×p = Iω | τ = dL/dt
Rolling Motion
v_cm = Rω | KE = ½Mv² + ½Iω² | a = gsinθ/(1 + I/MR²)
Gravitation
F = Gm₁m₂/r² | g = GM/R² | g' = g(1 - 2h/R) at height h<
Gravitational Potential Energy
U = -GMm/r | Escape velocity: vₑ = √(2GM/R) = √2gR ≈ 11.2 km/s
Orbital Mechanics
v₀ = √(GM/r) | T² = (4π²/GM)r³ (Kepler's 3rd) | E = -GMm/2a
Satellites
v_geo = √(gR²/(R+h)) | T_geo ≈ 24h at h ≈ 36000 km
Mechanics - Oscillations & Elasticity
SHM, Waves in Medium, Material Properties
SHM Basics
x = Asin(ωt+φ) | v = Aωcos(ωt+φ) | a = -ω²x | ω = 2πf = 2π/T
Spring Systems
T = 2π√(m/k) | Series: 1/k_eq = 1/k₁ + 1/k₂ | Parallel: k_eq = k₁
+ k₂
Simple Pendulum
T = 2π√(L/g) | Physical pendulum: T = 2π√(I/mgd)
Energy in SHM
E = ½kA² = ½mω²A² | KE_avg = PE_avg = E/4 | KE_max = PE_max = E/2
Damped Oscillations
x = A₀e^(-bt/2m)cos(ω't+φ) | ω' = √(ω₀² - (b/2m)²)
Stress & Strain
Stress = F/A | Strain = ΔL/L | Y = Stress/Strain (Young's modulus)
Elastic Moduli
Y = 3B(1-2σ) = 2η(1+σ) | B = bulk, η = shear, σ = Poisson's ratio
Elastic Energy
U = ½Y × (Strain)² × Volume | U = ½FΔL = ½kx²
Thermodynamics & Kinetic Theory
Laws of Thermodynamics, Heat Engines
Ideal Gas Law
PV = nRT = NkT | R = 8.314 J/mol·K | k = 1.38 × 10⁻²³ J/K
Gas Laws
Boyle: PV = const (T const) | Charles: V/T = const (P const) |
Gay-Lussac: P/T = const (V const)
Kinetic Theory
PV = ⅓Nm(v_rms)² | v_rms = √(3RT/M) = √(3kT/m)
Speeds Distribution
v_rms > v_avg > v_mp | v_mp = √(2RT/M) | v_avg = √(8RT/πM)
Degrees of Freedom
f = 3 (mono), 5 (diatomic), 6 (polyatomic) | U = (f/2)nRT
First Law
ΔU = Q - W (W by system) | dU = dQ - dW | C_p - C_v = R
Thermodynamic Processes
Isochoric: W=0, ΔU=Q | Isobaric: W=PΔV | Isothermal:
W=nRTln(V₂/V₁)
Adiabatic Process
PV^γ = const | TV^(γ-1) = const | W = (P₁V₁-P₂V₂)/(γ-1) =
nRΔT/(1-γ)
Heat Engines
η = W/Q₁ = 1 - Q₂/Q₁ | Carnot: η = 1 - T₂/T₁ | Refrigerator: COP =
Q₂/W
Entropy
ΔS = ∫dQ_rev/T | ΔS ≥ 0 (isolated system) | S = klnΩ
Fluid Mechanics & Heat Transfer
Hydrostatics, Surface Tension, Calorimetry
Hydrostatic Pressure
P = P₀ + ρgh | Gauge pressure = ρgh | Pascal's law: P transmitted
equally
Buoyancy
F_b = ρ_fluid V_displaced g | Float: ρ_obj < ρ_fluid | Apparent
weight=W - F_b
Continuity Equation
A₁v₁ = A₂v₂ | Mass flow: ρ₁A₁v₁ = ρ₂A₂v₂
Bernoulli's Equation
P + ½ρv² + ρgh = constant | Torricelli: v = √(2gh)
Viscosity
F = ηA(dv/dy) | Poiseuille: Q = πPr⁴/8ηL | Stokes: F = 6πηrv
Surface Tension
T = F/L | ΔP = 2T/R (bubble), 4T/R (soap bubble) | Capillary:
h = 2Tcosθ/ρgr
Calorimetry
Q = mcΔθ | Q = mL (phase change) | Principle: Heat lost = Heat
gained
Heat Transfer
Conduction: dQ/dt = -kA(dT/dx) | Radiation: P = σAeT⁴
(Stefan-Boltzmann)
Newton's Law of Cooling
dT/dt = -k(T - T₀) | T(t) = T₀ + (Tᵢ - T₀)e^(-kt)
Wien's Displacement Law
λ_max T = b = 2.898 × 10⁻³ m·K
Electrostatics
Coulomb's Law, Gauss's Law, Capacitors
Coulomb's Law
F = kq₁q₂/r² = q₁q₂/(4πε₀r²) | k = 9 × 10⁹ N·m²/C² | ε₀ = 8.85
× 10⁻¹² C²/N·m²
Electric Field
E = kQ/r² | E = σ/2ε₀ (infinite plane) | E = σ/ε₀ (conductor
surface)
Gauss's Law
∮E·dA = Q_enclosed/ε₀ | Φ_E = Q/ε₀ | For symmetric charge
distributions
Electric Potential
V = kQ/r | ΔV = -∫E·dr | E = -dV/dr | V = 0 at ∞
Potential Energy
U = kq₁q₂/r | U = qV | For system: U = ½ΣqᵢVᵢ
Dipole
p = qd | E_axial = 2kp/r³ | E_eq = kp/r³ | τ = p×E | U = -p·E
Capacitance
C = Q/V | C₀ = ε₀A/d | C = ε₀ε_r A/d | Energy: U = ½CV² = ½QV
= ½Q²/C
Capacitor Combinations
Series: 1/C = 1/C₁ + 1/C₂ | Parallel: C = C₁ + C₂ | Energy
density: u = ½ε₀E²
Dielectrics
C = KC₀ | E = E₀/K | U = U₀/K | K = dielectric constant
Special Capacitors
Spherical: C = 4πε₀R | Cylindrical: C = 2πε₀L/ln(b/a)
Current Electricity
Ohm's Law, Circuits, Kirchhoff's Laws
Ohm's Law & Resistance
V = IR | R = ρL/A | ρ = ρ₀(1 + αΔT) | J = σE (microscopic
Ohm's law)
Drift Velocity
v_d = eEτ/m | I = neAv_d | Mobility μ = v_d/E = eτ/m
Resistor Combinations
Series: R = R₁ + R₂ | Parallel: 1/R = 1/R₁ + 1/R₂ | Wheatstone
bridge: P/Q = R/S
EMF & Internal Resistance
V = ε - Ir | I = ε/(R + r) | Terminal p.d. = IR = εR/(R+r)
Power in Circuits
P = VI = I²R = V²/R | Max power transfer: R = r | P_max =
ε²/4r
Kirchhoff's Laws
Junction: ΣI = 0 | Loop: ΣIR + ΣEMF = 0 | Sign convention
essential
Potentiometer
V ∝ l | ε₁/ε₂ = l₁/l₂ | Internal resistance: r = R(l₁-l₂)/l₂
Meter Bridge
R/S = l/(100-l) | Null point when galvanometer shows zero
deflection
Galvanometer Conversion
Ammeter: S = G/(n-1) where n = I/I_g | Voltmeter: R = G(n-1)
where n = V/V_g
RC Circuits
Charging: q = Cε(1-e^(-t/RC)) | Discharging: q = q₀e^(-t/RC) |
τ = RC
Magnetism & EMI
Magnetic Fields, Induction, AC Circuits
Biot-Savart Law
dB = (μ₀/4π)(Idl×r)/r³ | μ₀ = 4π × 10⁻⁷ T·m/A
Ampere's Law
∮B·dl = μ₀I_enclosed | For long wire: B = μ₀I/2πr | Solenoid:
B = μ₀nI
Magnetic Force
F = q(v×B) | F = I(L×B) | Lorentz: F = q(E + v×B)
Motion in B-field
Radius: r = mv/qB | Period: T = 2πm/qB | Pitch: p = v_∥T =
2πmv_∥/qB
Magnetic Dipole
M = IA | τ = M×B | U = -M·B | B_axial = (μ₀/4π)(2M/r³)
Faraday's Law
ε = -dΦ/dt | Φ = BAcosθ | Motional EMF: ε = Blv
Inductance
Self: ε = -L(dI/dt) | Mutual: ε₂ = -M(dI₁/dt) | Energy: U =
½LI²
LR & LC Circuits
LR: I = (ε/R)(1-e^(-t/τ)) τ=L/R | LC: ω = 1/√(LC) | T =
2π√(LC)
AC Fundamentals
V = V₀sin(ωt) | I = I₀sin(ωt+φ) | X_L = ωL | X_C = 1/ωC
AC Circuits
Z = √(R² + (X_L-X_C)²) | tanφ = (X_L-X_C)/R | P_avg = VIcosφ =
I²R
Resonance
ω₀ = 1/√(LC) | Q-factor = ω₀L/R = 1/(ω₀CR) | Bandwidth = R/L
Transformer
V_s/V_p = N_s/N_p = I_p/I_s | Efficiency η = (V_sI_s)/(V_pI_p)
× 100%
Ray Optics
Reflection, Refraction, Optical Instruments
Mirror Formula
1/f = 1/v + 1/u | f = R/2 | Sign convention: Distances
measured from pole
Magnification
m = -v/u = f/(f-u) | m = h_i/h_o | For mirror: m = -v/u
Lens Formula
1/f = 1/v - 1/u | Lensmaker: 1/f = (μ-1)(1/R₁ - 1/R₂)
Lens Combinations
1/F = 1/f₁ + 1/f₂ | P = P₁ + P₂ (diopters) | Effective focal
length
Refraction
n₁sinθ₁ = n₂sinθ₂ | n = c/v = λ₀/λ | Critical angle: sinθ_c =
n₂/n₁ (n₁>n₂)
Apparent Depth
d' = d/n | Shift = d(1 - 1/n) | Normal shift for near-normal
viewing
Prism
r₁ + r₂ = A | δ = i + e - A | δ_min = 2i - A | μ =
sin((A+δ_m)/2)/sin(A/2)
Dispersion
ω = (μ_v - μ_r)/(μ_y - 1) = δ_v - δ_r/δ_y | Angular dispersion
Microscope
Simple: m = 1 + D/f | Compound: M = (v₀/u₀)(1 + D/f_e)
Telescope
Astronomical: M = -f₀/f_e | Length = f₀ + f_e | Terrestrial: M
= f₀/f_e
Wave Optics & Modern Physics
Interference, Diffraction, Quantum Physics
Huygens' Principle
Every point on wavefront is source of secondary wavelets
YDSE (Interference)
Δx = dsinθ = nλ (bright) | β = λD/d | I = I₁ + I₂ +
2√(I₁I₂)cosφ
Diffraction (Single Slit)
asinθ = nλ (minima) | Central maxima width = 2λD/a | Angular
width = 2λ/a
Polarization
Brewster's: tanθ_p = n₂/n₁ | Malus: I = I₀cos²θ | Polarization
by reflection
Photoelectric Effect
hν = φ + K_max | K_max = eV₀ | ν₀ = φ/h | Einstein's equation
de Broglie Wavelength
λ = h/p = h/√(2mK) = h/√(2meV) | Matter waves
Bohr's Model
mvr = nh/2π | E_n = -13.6Z²/n² eV | r_n = 0.529n²/Z Å | 1/λ =
R(1/n₁² - 1/n₂²)
X-rays
λ_min = hc/eV = 1240/V nm | Moseley's law: √ν = a(Z-b)
Nuclear Physics
E = mc² | 1 u = 931.5 MeV/c² | B.E./A = [Zm_p + Nm_n -
M_nucleus]c²/A
Radioactivity
N = N₀e^(-λt) | A = A₀e^(-λt) | t½ = 0.693/λ | Mean life τ =
1/λ = t½/0.693
Decay Laws
α-decay: Z↓2, A↓4 | β-decay: Z↑1, A same | γ-decay: Z, A same
Physical Chemistry - Mole Concept & Solutions
Stoichiometry, Concentration Terms
Mole Concept
n = w/M = N/N_A = V/22.4 (STP) | N_A = 6.022 × 10²³ mol⁻¹
Concentration Terms
M = n/V(L) | m = n/w(kg) | %w/w = (w_solute/w_solution) × 100
| ppm = (w_solute/w_solution) × 10⁶
Mole Fraction & Relations
X_A = n_A/(n_A + n_B) | M = (1000ρX)/(MX + 1000) | m =
(1000M)/(1000ρ - MM_solute)
Vapor Pressure
Raoult's: P = X_AP°_A + X_BP°_B | Relative lowering: (P°-P)/P°
= X_solute
Colligative Properties
ΔT_b = iK_bm | ΔT_f = iK_fm | π = iCRT | i = van't Hoff factor
Abnormal Molar Mass
i = (Normal molar mass)/(Observed molar mass) | For
dissociation: i = 1 + (n-1)α
Henry's Law
p = K_H·X | C = k·p | Solubility of gas ∝ pressure
Azeotropes
Minimum boiling: +ve deviation | Maximum boiling: -ve
deviation | Cannot separate by simple distillation
Physical Chemistry - Thermodynamics & Equilibrium
Energetics, Chemical Equilibrium, Electrochemistry
First Law
ΔU = q + w | ΔH = ΔU + Δn_gRT | At constant P: q_p = ΔH
Heat Capacities
C_p - C_v = R | ΔH = ∫C_pdT | For ideal gas: C_p = (f/2+1)R,
C_v = fR/2
Hess's Law
ΔH_reaction = ΣΔH_f(products) - ΣΔH_f(reactants) | Path
independent
Entropy & Gibbs
ΔS = q_rev/T | ΔG = ΔH - TΔS | ΔG° = -RTlnK | ΔG = ΔG° + RTlnQ
Equilibrium Constant
K_p = K_c(RT)^Δn | K = [Products]/[Reactants] | K_eq = k_f/k_b
Le Chatelier's Principle
System shifts to counteract disturbance | Δn_g > 0: P↑ shifts
left, T↑ shifts right (endothermic)
Ionic Equilibrium
K_w = [H⁺][OH⁻] = 10⁻¹⁴ at 25°C | pH + pOH = 14 | pH =
-log[H⁺]
Buffer Solutions
pH = pK_a + log([A⁻]/[HA]) | Buffer capacity maximum when [A⁻]
= [HA]
Solubility Product
K_sp = [A⁺]^m[B⁻]^n | For A_mB_n | Solubility s = (K_sp/m^m
n^n)^(1/(m+n))
Electrochemistry
E°_cell = E°_cathode - E°_anode | ΔG° = -nFE° | E = E° -
(RT/nF)lnQ
Nernst Equation
E = E° - (0.059/n)logQ at 25°C | For cell: E_cell = E°_cell -
(0.059/n)logQ
Conductivity
κ = 1/ρ = G* | Λ_m = κ/C | Λ°_m = Λ°_+ + Λ°_- (Kohlrausch)
Physical Chemistry - Kinetics & Surface Chemistry
Reaction Rates, Catalysis, Adsorption
Rate of Reaction
Rate = -d[R]/dt = +d[P]/dt | Instantaneous rate = slope at
point
Rate Law
Rate = k[A]^m[B]^n | Order = m + n | Units of k: (mol/L)^(1-n)
time⁻¹
Integrated Rate Laws
Zero: [A] = [A]₀ - kt | First: ln[A] = ln[A]₀ - kt | Second:
1/[A] = 1/[A]₀ + kt
Half-life
t½ = [A]₀/2k (zero) | t½ = 0.693/k (first) | t½ = 1/k[A]₀
(second)
Arrhenius Equation
k = Ae^(-E_a/RT) | ln(k₂/k₁) = (E_a/R)(1/T₁ - 1/T₂) | log form
available
Activation Energy
E_a = RT²(dlnk/dT) | Threshold energy for reaction
Collision Theory
Rate = PZ_ABe^(-E_a/RT) | P = steric factor | Z = collision
frequency
Adsorption
Freundlich: x/m = kp^(1/n) | Langmuir: x/m = ap/(1+bp)
Catalyst
Lowers E_a | Same ΔH | Equilibrium reached faster | Does not
affect K_eq
Inorganic Chemistry - Periodic Table & Chemical Bonding
Atomic Structure, Bonding, Coordination
de Broglie
λ = h/mv = h/p | Heisenberg: Δx·Δp ≥ h/4π
Quantum Numbers
n = 1,2,3... | l = 0 to n-1 | m = -l to +l | s = ±½ | Max e⁻
in shell = 2n²
Bohr's Model
r_n = 0.529n²/Z Å | E_n = -13.6Z²/n² eV | v_n = 2.188×10⁶Z/n
m/s
div class="formula-name">Effective Nuclear Charge
Z_eff = Z - σ | Slater's rules for shielding constant σ
Bond Order
BO = ½(N_b - N_a) | Higher BO → shorter bond, higher bond energy
Dipole Moment
μ = q × d | % ionic character = (μ_obs/μ_theo) × 100
Fajan's Rules
Polarization ∝ charge/size | Small cation + large anion → covalent
VSEPR
2BP: linear | 3BP: trigonal planar | 4BP: tetrahedral | 5BP:
trigonal bipyramidal | 6BP: octahedral
Hybridization
sp (180°) | sp² (120°) | sp³ (109.5°) | sp³d (90°, 120°) | sp³d²
(90°)
Molecular Orbital
σ1s < σ*1s < σ2s < σ*2s < π2p_x=π2p_y < σ2p_z < π*2p_x=π*2p_y <
σ*2p_z (O₂, F₂)
Crystal Field Theory
Δ_o = 10Dq (octahedral) | Δ_t = 4/9 Δ_o (tetrahedral) | CFSE =
(-0.4n_t2g + 0.6n_eg)Δ_o
EAN Rule
EAN = Z - O.N. + 2×C.N. | Stable when EAN = 36, 54, 86 (next
noble gas)
Inorganic Chemistry - s,p,d,f Block & Metallurgy
Element Properties, Extraction Processes
Diagonal Relationship
Li-Mg, Be-Al, B-Si show similar properties due to comparable
charge/size ratio
Inert Pair Effect
ns² electrons remain unshared in heavy p-block elements | Tl⁺
> Tl³⁺ stability
Lanthanide Contraction
Steady decrease in atomic/ionic size from La to Lu due to poor
shielding by 4f
Ellingham Diagram
ΔG° vs T | Lower line reduces oxide of upper line | C reduces
most metal oxides at high T
Van Arkel Method
Metal + I₂ → MI₂ (volatile) → Decomposition on hot wire → Pure
metal
Mond's Process
Ni + 4CO → Ni(CO)₄ (330-350K) → Ni + 4CO (450-470K)
Magnetic Moment
μ = √(n(n+2)) BM | n = unpaired electrons | Spin-only formula
Color of Complexes
Complementary color absorbed | d-d transitions | Charge
transfer spectra
Isomerism
Structural: ionization, linkage, coordination |
Stereoisomerism: geometrical, optical
Organic Chemistry - General & Isomerism
Nomenclature, Stereochemistry, Reactions
Degree of Unsaturation
DU = C - H/2 - X/2 + N/2 + 1 | One ring or one double bond = 1
DU
IUPAC Nomenclature Priority
-COOH > -SO₃H > -COOR > -COCl > -CONH₂ > -CN > -CHO > -CO- >
-OH > -NH₂ > -OR
Optical Activity
[α] = α/(l×c) | Specific rotation | Enantiomers: mirror
images, non-superimposable
R/S Configuration
Cahn-Ingold-Prelog priority | Clockwise = R, Anticlockwise = S
(lowest priority at back)
E/Z Isomerism
E (entgegen/trans): high priority groups opposite | Z
(zusammen/cis): same side
Resonance Energy
Difference between experimental and calculated heat of
hydrogenation | Benzene: 150.4 kJ/mol
Aromaticity
Hückel: (4n+2)π e⁻ | Planar, cyclic, conjugated | n = 0,1,2...
Inductive Effect
-I: electron withdrawing (NO₂, CN, X) | +I: electron donating
(alkyl) | Decreases with distance
Resonance/Mesomeric
+R: electron donating (OH, OR, NH₂) | -R: electron withdrawing
(C=O, NO₂, CN)
Hyperconjugation
σ-π conjugation | α-H atoms | Stability: 3° > 2° > 1°
carbocation
Organic Chemistry - Reaction Mechanisms
Substitution, Addition, Elimination, Rearrangements
SN1 vs SN2
SN1: 3° > 2° > 1°, polar protic, carbocation intermediate,
racemization | SN2: 1° > 2° > 3°, polar aprotic, inversion
E1 vs E2
E1: carbocation, Zaitsev product | E2: concerted,
anti-periplanar geometry, bulky base → Hofmann
Markovnikov's Rule
H adds to C with more H | OH adds to C with less H |
Anti-Markovnikov: peroxide effect (HBr only)
Zaitsev's Rule
More substituted alkene is major product | Hofmann: less
substituted with bulky base
Carbocation Stability
3° > 2° > 1° > methyl | Resonance stabilized >
allylic/benzylic > simple
Grignard Reagent
RMgX | Reacts with C=O, CO₂, epoxides | Strong nucleophile and
base
Aldol Condensation
Req: α-H | Dilute base | β-hydroxy carbonyl → α,β-unsaturated
carbonyl
Cannizzaro Reaction
No α-H aldehydes | Conc. base | Disproportionation: 2RCHO →
RCH₂OH + RCOOH
Haloform Reaction
CH₃CO- group | X₂/OH⁻ → CHX₃ + RCOO⁻ | Iodoform: yellow ppt
with NaOI
Diels-Alder
[4+2] cycloaddition | Diene + dienophile → Cyclohexene |
Stereospecific, concerted
Reimer-Tiemann
Phenol + CHCl₃/OH⁻ → o-hydroxybenzaldehyde (salicylaldehyde)
Friedel-Crafts
Alkylation: RCl/AlCl₃ | Acylation: RCOCl/AlCl₃ | Deactivating
groups prevent reaction
Algebra - Complex Numbers & Quadratic
Complex Plane, Roots, Equations
Complex Number Forms
z = x + iy = r(cosθ + isinθ) = re^(iθ) | r = |z| = √(x²+y²), θ
= arg(z)
Euler's Formula
e^(iθ) = cosθ + isinθ | e^(iπ) + 1 = 0 | De Moivre: (cosθ +
isinθ)^n = cos(nθ) + isin(nθ)
Roots of Unity
x^n = 1 has roots e^(2πik/n) for k = 0,1,...,n-1 | 1 + ω + ω²
= 0 where ω = e^(2πi/3)
Quadratic Equations
x = (-b ± √(b²-4ac))/2a | D > 0: real distinct | D = 0: equal
| D < 0: complex conjugate
Vieta's Formulas
α + β = -b/a | αβ = c/a | For ax² + bx + c = 0
Transformation of Roots
Roots kα, kβ: replace x by x/k | Roots 1/α, 1/β: replace x
by 1/x (then ×x²)
Cubic Equations
α + β + γ = -b/a | αβ + βγ + γα = c/a | αβγ = -d/a
Condition for Common Roots
Two quadratics: (c₁a₂-c₂a₁)² = (a₁b₂-a₂b₁)(b₁c₂-b₂c₁) |
One common root condition
Algebra - Sequences & Series
AP, GP, HP, Special Series
Arithmetic Progression
a_n = a + (n-1)d | S_n = n/2[2a + (n-1)d] = n/2(a + l) |
A.M. = (a+b)/2
Geometric Progression
a_n = ar^(n-1) | S_n = a(r^n-1)/(r-1) | S_∞ = a/(1-r) for
|r| < 1 | G.M.=√(ab)
Harmonic Progression
Reciprocals in AP | H.M. = 2ab/(a+b) | A.M. ≥ G.M. ≥
H.M.
Arithmetico-Geometric
S_n = a + (a+d)r + (a+2d)r² + ... | S_n = a/(1-r) +
dr(1-r^(n-1))/(1-r)² - [a+(n-1)d]r^n/(1-r)
Sum of Powers
Σn = n(n+1)/2 | Σn² = n(n+1)(2n+1)/6 | Σn³ =
[n(n+1)/2]²
Method of Differences
If T_n = f(n) - f(n-1), then S_n = f(n) - f(0) |
Telescoping series
Exponential & Log Series
e^x = 1 + x + x²/2! + x³/3! + ... | ln(1+x) = x - x²/2
+ x³/3 - ... for |x| < 1
Binomial Theorem
(a+b)^n = Σ ⁿC_r a^(n-r)b^r | T_(r+1) = ⁿC_r
a^(n-r)b^r | ⁿC_r = n!/(r!(n-r)!)
Binomial Properties
ⁿC_r = ⁿC_(n-r) | ⁿC_r + ⁿC_(r-1) = ⁿ⁺¹C_r | Σ
ⁿC_r = 2^n | Sum of even = Sum of odd = 2^(n-1)
Algebra - Permutations, Combinations & Probability
Counting Principles, Probability Rules
Fundamental Principle
AND → Multiply | OR → Add | n(P) ways for P, n(Q)
for Q → n(P)×n(Q) for P and Q
Permutations
ⁿP_r = n!/(n-r)! | Circular: (n-1)! | Necklace:
½(n-1)! for n > 2
Combinations
ⁿC_r = n!/(r!(n-r)!) | ⁿC_r = ⁿC_(n-r) | ⁿC_r =
ⁿ⁻¹C_r + ⁿ⁻¹C_(r-1) (Pascal)
Divisions into Groups
(mn)!/(n!)^m m! for m groups of n each |
(m+n)!/(m!n!) for two groups
Derangements
D_n = n!(1 - 1/1! + 1/2! - 1/3! + ... + (-1)^n/n!)
| D_n ≈ n!/e for large n
Basic Probability
P(E) = n(E)/n(S) | 0 ≤ P(E) ≤ 1 | P(E') = 1 - P(E)
Addition Rule
P(A∪B) = P(A) + P(B) - P(A∩B) | Mutually
exclusive: P(A∪B) = P(A) + P(B)
Conditional Probability
P(A|B) = P(A∩B)/P(B) | Independent: P(A∩B) =
P(A)P(B) | Bayes': P(A|B) = P(B|A)P(A)/P(B)
Total Probability
P(A) = Σ P(E_i)P(A|E_i) where E_i are mutually
exclusive and exhaustive
Binomial Distribution
P(X=r) = ⁿC_r p^r q^(n-r) | Mean = np | Variance =
npq | Standard deviation = √(npq)
Algebra - Matrices & Determinants
Matrix Algebra, System of Equations
Matrix Operations
(AB)^T = B^T A^T | (AB)^-1 = B^-1 A^-1 | tr(AB) =
tr(BA)
Determinant Properties
|AB| = |A||B| | |A^T| = |A| | |kA| = k^n|A| (n×n
matrix) | Row swap: sign change
Inverse Matrix
A^-1 = adj(A)/|A| | A(adj A) = |A|I | (adj A)^-1 =
adj(A^-1) = A/|A|
Cramer's Rule
x = Δ₁/Δ, y = Δ₂/Δ, z = Δ₃/Δ where Δ is
coefficient determinant
System Consistency
|A| ≠ 0: unique solution | |A| = 0, (adj A)B ≠ 0:
no solution | |A| = 0, (adj A)B = 0: infinite
Special Determinants
Vandermonde: Π_{i
Eigenvalues
|A - λI| = 0 | Product = |A | | Sum = tr(A) |
A^-1 has eigenvalues 1/λ
Calculus - Limits, Continuity &
Differentiability
Fundamental Concepts, Standard Limits
Standard Limits
lim(x→0) sinx/x = 1 | lim(x→0) tanx/x = 1 |
lim(x→0) (1-cosx)/x² = 1/2 | lim(x→0) e^x-1/x = 1
Exponential & Log Limits
lim(x→0) ln(1+x)/x = 1 | lim(x→0) (a^x-1)/x =
ln(a) | lim(x→∞) (1+1/x)^x = e
L'Hospital's Rule
lim(x→a) f(x)/g(x) = lim(x→a) f'(x)/g'(x) for
0/0 or ∞/∞ | Can apply repeatedly
Continuity
f continuous at a if lim(x→a) f(x) = f(a) |
LHL = RHL = f(a)
Differentiability
f'(a) = lim(h→0) [f(a+h)-f(a)]/h | LHD = RHD |
Differentiable ⇒ Continuous, converse false
Sandwich Theorem
If g(x) ≤ f(x) ≤ h(x) and lim g(x) = lim h(x)
= L, then lim f(x) = L
Greatest Integer Function
[x] = n for n ≤ x < n+1 | Discontinuous at
integers | f(x)=x - [x] is periodic with period 1
Calculus - Differentiation & Applications
Rules, Mean Value Theorems, Maxima/Minima
Basic Derivatives
d(x^n)/dx = nx^(n-1) | d(e^x)/dx = e^x |
d(lnx)/dx = 1/x | d(sinx)/dx = cosx | d(cosx)/dx = -sinx
Product & Quotient
(uv)' = u'v + uv' | (u/v)' = (u'v -
uv')/v² | (u/v/w)' formula available
Chain Rule
dy/dx = dy/du × du/dx | Parametric: dy/dx
= (dy/dt)/(dx/dt) | d²y/dx² = d/dt(dy/dx) / (dx/dt)
Inverse Functions
d(sin⁻¹x)/dx = 1/√(1-x²) | d(cos⁻¹x)/dx =
-1/√(1-x²) | d(tan⁻¹x)/dx = 1/(1+x²)
Logarithmic Differentiation
For f(x)^g(x) or complex products | Take
ln, differentiate, solve for dy/dx
Leibniz Rule
(uv)^(n) = Σ ⁿC_r u^(n-r) v^(r) | nth
derivative of product
Rolle's Theorem
f continuous [a,b], differentiable (a,b),
f(a)=f(b) ⇒ ∃c∈(a,b): f'(c) = 0
LMVT & CMVT
LMVT: f'(c) = [f(b)-f(a)]/(b-a) | CMVT:
f'(c)/g'(c) = [f(b)-f(a)]/[g(b)-g(a)]
Maxima & Minima
f'(x) = 0 (critical point) | f''(x) > 0:
local min | f''(x) < 0: local max | f''(x)=0: test fails
Tangents & Normals
Slope of tangent = dy/dx | Slope of
normal = -dx/dy | Length of tangent, sub-tangent, normal,
sub-normal
Angle of Intersection
tanθ = |(m₁-m₂)/(1+m₁m₂)| | Orthogonal
curves: m₁m₂ = -1
Calculus - Integration
Indefinite & Definite Integrals,
Properties
Basic Integrals
∫x^n dx = x^(n+1)/(n+1) + C (n≠-1) |
∫1/x dx = ln|x| + C | ∫e^x dx = e^x + C
Trigonometric Integrals
∫sinx dx = -cosx | ∫cosx dx = sinx |
∫tanx dx = -ln|cosx| | ∫sec²x dx = tanx
Standard Forms
∫1/(x²+a²) dx = (1/a)tan⁻¹(x/a) |
∫1/√(a²-x²) dx = sin⁻¹(x/a) | ∫1/√(x²+a²) dx =
ln|x+√(x²+a²)|
Integration by Parts
∫u dv = uv - ∫v du | ILATE priority:
Inverse, Log, Algebraic, Trigonometric, Exponential
Special Integrals
∫e^x[f(x)+f'(x)]dx = e^xf(x) + C |
∫[f'(x)/f(x)]dx = ln|f(x)| + C
Partial Fractions
Linear factors: A/(x-a) + B/(x-b) |
Repeated: A/(x-a) + B/(x-a)² | Quadratic: (Ax+B)/(ax²+bx+c)
Definite Integral Properties
∫_a^b f(x)dx = ∫_a^b f(a+b-x)dx |
∫_0^a f(x)dx = ∫_0^a f(a-x)dx | ∫_(-a)^a f(x)dx = 2∫_0^a
f(x) (even) or 0 (odd)
Walli's Formula
∫_0^(π/2) sin^n x dx = ∫_0^(π/2) cos^n
x dx | For even n: [(n-1)/n][(n-3)/(n-2)]...[1/2][π/2] | For
odd n: end with 2/3
Leibniz Integral Rule
d/dx ∫_(a(x))^(b(x)) f(t)dt =
f(b(x))b'(x) - f(a(x))a'(x)
Gamma Function
Γ(n) = ∫_0^∞ e^(-x) x^(n-1) dx |
Γ(n+1) = nΓ(n) | Γ(n+1) = n! for positive integer n
Calculus - Differential Equations & Area
ODEs, Area Under Curves, Applications
Variable Separable
dy/dx = f(x)g(y) ⇒ ∫dy/g(y) = ∫f(x)dx
| Separate and integrate both sides
Homogeneous Equations
dy/dx = f(y/x) | Substitute y = vx,
dy/dx = v + x(dv/dx) | Then separate variables
Linear First Order
dy/dx + P(x)y = Q(x) | I.F. = e^(∫Pdx)
| Solution: y(I.F.) = ∫Q(I.F.)dx + C
Bernoulli's Equation
dy/dx + Py = Qy^n | Divide by y^n,
substitute v = y^(1-n) | Convert to linear
Exact Equations
Mdx + Ndy = 0 is exact if ∂M/∂y =
∂N/∂x | Solution: ∫Mdx + ∫(terms of N not containing x)dy =
C
Area Under Curve
A = ∫_a^b y dx (above x-axis) | A =
∫_c^d x dy (right of y-axis) | Take absolute value for area
Area Between Curves
A = ∫_a^b [f(x) - g(x)] dx where f(x)
≥ g(x) | Find intersection points for limits
Curve Tracing
Symmetry, intercepts, asymptotes,
tangents at origin, region of existence, maxima/minima
Coordinate Geometry - Straight Lines &
Circles
2D Geometry, Conic Sections Basics
Distance & Section
d = √[(x₂-x₁)² + (y₂-y₁)²] | Internal:
((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)) | External: use -n
Area Formulas
Triangle: ½|x₁(y₂-y₃) + x₂(y₃-y₁) +
x₃(y₁-y₂)| | Centroid: ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)
Line Equations
Slope-intercept: y = mx + c |
Point-slope: y-y₁ = m(x-x₁) | Two-point: (y-y₁)/(y₂-y₁) =
(x-x₁)/(x₂-x₁)
Angle & Distance
tanθ = |(m₁-m₂)/(1+m₁m₂)| | Distance
from point: |ax₁+by₁+c|/√(a²+b²) | Distance between parallel
lines: |c₁-c₂|/√(a²+b²)
Concurrency & Collinearity
Three lines concurrent if determinant
of coefficients = 0 | Three points collinear if area of
triangle = 0
Pair of Lines
ax² + 2hxy + by² = 0 represents pair
through origin | Angle: tanθ = |2√(h²-ab)/(a+b)|
Circle Basics
Standard: (x-h)² + (y-k)² = r² |
General: x² + y² + 2gx + 2fy + c = 0 | Center (-g,-f),
radius √(g²+f²-c)
Tangent & Normal
Tangent at (x₁,y₁): xx₁ + yy₁ +
g(x+x₁) + f(y+y₁) + c = 0 | Length of tangent from (x₁,y₁):
√(S₁)
Chord of Contact
T = 0: xx₁ + yy₁ = r² for x²+y²=r² |
From external point, equation of chord joining contact
points
Radical Axis
S₁ - S₂ = 0 | Locus of points with
equal tangent lengths to two circles | Perpendicular to line
of centers
Coordinate Geometry - Conic Sections
Parabola, Ellipse, Hyperbola
Parabola y² = 4ax
Focus: (a,0) | Directrix: x = -a |
Latus rectum: 4a | Parametric: (at², 2at) | Tangent: ty = x
+ at²
Parabola Properties
Subtangent = 2x₁ | Subnormal = 2a |
Tangent at t₁: t₁y = x + at₁² | Point of intersection of
tangents: (at₁t₂, a(t₁+t₂))
Ellipse x²/a² + y²/b² = 1
a > b: major axis 2a | e = √(1-b²/a²)
| Foci: (±ae, 0) | Latus rectum: 2b²/a
Ellipse Properties
Sum of focal distances = 2a | Tangent:
(xx₁)/a² + (yy₁)/b² = 1 | Director circle: x² + y² = a² + b²
Hyperbola x²/a² - y²/b² = 1
e = √(1+b²/a²) > 1 | Foci: (±ae, 0) |
Latus rectum: 2b²/a | Difference of focal distances = 2a
Hyperbola Asymptotes
y = ±(b/a)x | Rectangular hyperbola: a
= b, e = √2 | xy = c² is standard rectangular form
Director Circle
Ellipse: x² + y² = a² + b² |
Hyperbola: x² + y² = a² - b² (only if a > b)
Auxiliary Circle
x² + y² = a² for both ellipse and
hyperbola | Eccentric angle θ: (acosθ, bsinθ) for ellipse
Chord of Contact
T = 0 for all conics | For parabola
y²=4ax: yy₁ = 2a(x+x₁) | For ellipse: T = (xx₁)/a² +
(yy₁)/b² - 1 = 0
Polar & Pole
Polar of (x₁,y₁) w.r.t. conic S = 0 is
T = 0 | Pole of line lx + my + n = 0 can be found by
comparing
Vectors & 3D Geometry
Vector Algebra, Lines, Planes
Vector Operations
a⃗·b⃗ = |a||b|cosθ = a₁b₁ + a₂b₂ +
a₃b₃ | a⃗×b⃗ = |i j k; a₁ a₂ a₃; b₁ b₂ b₃|
Cross Product Magnitude
|a⃗×b⃗| = |a||b|sinθ | Area of
parallelogram = |a⃗×b⃗| | Area of triangle = ½|a⃗×b⃗|
Scalar Triple Product
[a⃗ b⃗ c⃗] = a⃗·(b⃗×c⃗) = determinant
| Volume of parallelepiped = |[a⃗ b⃗ c⃗]| | Coplanar if [a⃗
b⃗ c⃗] = 0
Vector Triple Product
a⃗×(b⃗×c⃗) = (a⃗·c⃗)b⃗ - (a⃗·b⃗)c⃗ |
(a⃗×b⃗)×c⃗ = (a⃗·c⃗)b⃗ - (b⃗·c⃗)a⃗
Direction Cosines
l = cosα, m = cosβ, n = cosγ | l² + m²
+ n² = 1 | Line with DCs: (x-x₁)/l = (y-y₁)/m = (z-z₁)/n
Line in 3D
Vector: r⃗ = a⃗ + λb⃗ | Cartesian:
(x-x₁)/a = (y-y₁)/b = (z-z₁)/c | Symmetric form
Angle Between Lines
cosθ = |a₁a₂ + b₁b₂ +
c₁c₂|/√(a₁²+b₁²+c₁²)√(a₂²+b₂²+c₂²) | Vector: cosθ =
|b⃗₁·b⃗₂|/|b⃗₁||b⃗₂|
Shortest Distance
Between skew lines: d =
|(a⃗₂-a⃗₁)·(b⃗₁×b⃗₂)|/|b⃗₁×b⃗₂| | Between parallel lines:
perpendicular distance
Plane Equation
Vector: (r⃗ - a⃗)·n⃗ = 0 or r⃗·n⃗ = d
| Cartesian: ax + by + cz + d = 0 | Normal: (a,b,c)
Plane Forms
Intercept: x/a + y/b + z/c = 1 |
Normal form: lx + my + nz = p where p is perpendicular
distance from origin
Angle Between Planes
cosθ = |a₁a₂ + b₁b₂ +
c₁c₂|/√(a₁²+b₁²+c₁²)√(a₂²+b₂²+c₂²) | Line ⊥ to plane:
parallel to normal
Distance from Point
|ax₁ + by₁ + cz₁ + d|/√(a²+b²+c²) |
Foot of perpendicular and image formulas available
Statistics & Probability Distributions
Measures of Central Tendency,
Dispersion, Correlation
Mean
x̄ = Σxᵢ/n | Combined: x̄₁₂ = (n₁x̄₁ +
n₂x̄₂)/(n₁+n₂) | Shortcut: x̄ = A + Σfᵢdᵢ/Σfᵢ
Variance & SD
σ² = Σ(xᵢ - x̄)²/n = (Σxᵢ²/n) - x̄² |
σ = √σ² | CV = (σ/x̄) × 100%
Median & Mode
Median: (n+1)/2 th term (odd), average
of n/2 and (n/2+1) (even) | Mode: maximum frequency |
Empirical: Mode = 3Median - 2Mean
Correlation
r = Cov(x,y)/(σ_x σ_y) = [nΣxy -
(Σx)(Σy)]/√[nΣx²-(Σx)²][nΣy²-(Σy)²] | -1 ≤ r ≤ 1
Regression Lines
y on x: y - ȳ = r(σ_y/σ_x)(x - x̄) |
x on y: x - x̄ = r(σ_x/σ_y)(y - ȳ) | Angle between lines:
tanθ = |(1-r²)/r|(σ_xσ_y)/(σ_x²+σ_y²)
Probability Distributions
Mean = Σxᵢpᵢ | Variance = Σxᵢ²pᵢ -
(Σxᵢpᵢ)² | For binomial: mean = np, variance = npq
Cell Biology & Biomolecules
Cell Structure, Enzymes, Metabolism
Cell Cycle Timing
G₁: variable | S: DNA synthesis | G₂:
preparation for mitosis | M: mitosis (shortest) | G₀:
quiescent
Surface Area to Volume
SA:V = 4πr²/(4/3πr³) = 3/r | As cell
grows, SA:V decreases, limiting diffusion efficiency
Water Potential
Ψ = Ψ_s + Ψ_p | Ψ_s = -iCRT (solute
potential) | Ψ_p = pressure potential | Pure water: Ψ = 0
Osmotic Relations
Hypertonic: cell loses water |
Hypotonic: cell gains water | Isotonic: no net movement |
Plasmolysis begins at Ψ_cell = Ψ_external
Michaelis-Menten
V = (V_max [S])/(K_m + [S]) | K_m =
[S] at V_max/2 | 1/V = (K_m/V_max)(1/[S]) + 1/V_max
(Lineweaver-Burk)
Enzyme Inhibition
Competitive: V_max unchanged, K_m
increases | Non-competitive: V_max decreases, K_m unchanged
Respiratory Quotient
RQ = CO₂ produced/O₂ consumed |
Carbohydrate: 1.0 | Fat: ~0.7 | Protein: ~0.8 | Anaerobic: ∞
ATP Yield
Aerobic: ~36-38 ATP/glucose |
Substrate level: 4 ATP (glycolysis 2, Krebs 2) | Oxidative
phosphorylation: ~32-34
Photosynthetic Efficiency
Quantum yield: O₂ evolved per quantum
absorbed | Theoretical max: 0.125 (8 photons/CO₂) | C4 more
efficient in hot climates
Genetics & Molecular Biology
Inheritance Patterns, DNA Technology
Hardy-Weinberg Equilibrium
p + q = 1 (alleles) | p² + 2pq + q² =
1 (genotypes) | p = freq(dominant), q = freq(recessive)
Allele Frequency
p = (2N_AA + N_Aa)/2N | q = (2N_aa +
N_Aa)/2N | N = total individuals
Recombination Frequency
RF = (Number of recombinants/Total
progeny) × 100% | 1% RF = 1 map unit (cM) = 1 centimorgan
Chi-Square Test
χ² = Σ[(O-E)²/E] | df = n-1 | Compare
to critical value at desired significance level (usually
0.05)
DNA Content
1C = haploid genome content | 2C =
diploid | G₁: 2C, S: 2C→4C, G₂/M: 4C | Gametes: 1C
Chargaff's Rules
%A = %T, %G = %C | Purines =
Pyrimidines | A+G = T+C = 50% | Ratio (A+T)/(G+C) varies by
species
Melting Temperature
T_m ≈ 2°C × (A+T) + 4°C × (G+C) for
short oligos | Higher GC content → higher T_m
PCR Amplification
DNA copies = 2^n where n = cycles |
After 30 cycles: ~1 billion copies | Efficiency rarely 100%
Transformation Efficiency
CFU/μg DNA = (Colonies × Dilution
factor)/Amount of DNA plated (in μg)
Codon Usage
64 codons, 20 amino acids |
Degeneracy: multiple codons per amino acid (except Met, Trp)
| Start: AUG, Stop: UAA, UAG, UGA
Biotechnology & Genetic Engineering
Cloning, Sequencing, CRISPR
DNA Quantification
1 A260 unit = 50 μg/mL dsDNA = 33
μg/mL ssDNA = 40 μg/mL RNA | A260/A280: pure DNA ~1.8, pure
RNA ~2.0
Plasmid Copy Number
High copy: 100-300/cell (pUC origin) |
Low copy: <20 /cell (pSC101 origin) | pBR322: ~15-20
copies
Ligation Ratio
Insert:Vector molar ratio = 3:1 to
10:1 | Total DNA < 100 ng | Sticky ends more efficient
than blunt
Restriction Enzymes
6-cutter: 1 site per 4^6 =
4096 bp (average) | 4-cutter: 1 per 256 bp |
8-cutter: 1 per 65536 bp
DNA Insert Capacity
Plasmid: <10 kb | λ phage:
9-23 kb | Cosmid: 35-45 kb | BAC: 100-300 kb |
YAC: 0.2-2 Mb
Sequencing Coverage
Coverage = (Read length ×
Number of reads)/Genome size | Human genome: 30×
for reliable variant calling
qPCR Efficiency
E = 10^(-1/slope) - 1 |
Ideal: 90-110% | Slope of standard curve: ~-3.32
for 100% efficiency
Ct Value
Interpretation
Lower Ct = Higher initial
template amount | ΔCt method: 2^(-ΔΔCt) for
relative quantification
CRISPR Efficiency
Indel frequency estimated
by T7E1 assay or sequencing | NHEJ: error-prone,
indels | HDR: precise, requires template
Plant Physiology
Photosynthesis,
Respiration, Transport
Photosynthesis
Equation
6CO₂ + 6H₂O → C₆H₁₂O₆ +
6O₂ | ΔG°' = +2870 kJ/mol (endergonic) | Light
reaction: H₂O split, O₂ released
Z-scheme of
Photosynthesis
PSII (680 nm) → Cyt b6f →
PSI (700 nm) | Non-cyclic photophosphorylation:
ATP + NADPH produced
C3 vs C4 vs CAM
C3: 3 PGA, no Kranz,
photorespiration | C4: 4C acids, Kranz anatomy,
PEP carboxylase | CAM: temporal separation,
succulents
Photorespiration Cost
No ATP or sugar produced |
Consumes O₂, releases CO₂ | 25% of
photosynthetic carbon lost in C3 plants
Transpiration Rates
C3 plants: 450-950 H₂O per
g dry matter | C4: 250-350 | CAM: 50-100 | Most
water lost via stomata
Mineral Nutrients
Mobile: N, P, K, Mg, S
(deficiency in old leaves) | Immobile: Ca, Fe, B
(deficiency in young leaves)
Nitrogen Fixation
N₂ + 8H⁺ + 8e⁻ + 16ATP →
2NH₃ + H₂ + 16ADP + 16Pi | Nitrogenase enzyme,
anaerobic conditions required
Plant Growth Analysis
RGR = (lnW₂ - lnW₁)/(t₂ -
t₁) | NAR = (1/L)(dW/dt) | LAR = L/W | RGR = NAR
× LAR
Phytochrome
Pr (red, 660 nm) ↔ Pfr
(far-red, 730 nm) | Pfr is active form |
Photostationary state depends on light quality
Human Physiology
Circulation, Respiration,
Excretion, Neural
Cardiac Output
CO = HR × SV | Normal: ~5
L/min | HR: 60-100 bpm, SV: 70 mL/beat | CO
increases with exercise
Stroke Volume & EF
SV = EDV - ESV | Ejection
Fraction (EF) = (SV/EDV) × 100% | Normal EF:
55-70%
Blood Pressure
MAP = DP + (SP - DP)/3 =
DP + PP/3 | Normal: 120/80 mmHg | MAP ≈ 93 mmHg
| PP = SP - DP
Respiratory Volumes
TV: 500 mL | IRV: 3000 mL
| ERV: 1100 mL | RV: 1200 mL | VC = 4600 mL |
TLC = 5800 mL
Alveolar Gas Equation
P_AO₂ = P_IO₂ - (P_aCO₂/R)
| R = respiratory quotient (~0.8) | A-a gradient
= P_AO₂ - P_aO₂
GFR & Renal Clearance
GFR = K_f × (P_GC - P_BS -
π_GC) | ~125 mL/min | Clearance = (U×V)/P |
Inulin clearance = GFR
BMI & BMR
BMI =
weight(kg)/height²(m²) | Categories: <18.5
underweight, 18.5-24.9 normal,>30 obese |
BMR: Harris-Benedict equation
Nerve Conduction
Velocity ∝ diameter ×
myelination | Saltatory conduction at nodes of
Ranvier | Myelinated: 120 m/s, Unmyelinated:
0.5-2 m/s
Action Potential
Phases
Resting: -70 mV (K⁺ leak)
| Depolarization: Na⁺ influx | Repolarization:
K⁺ efflux | Hyperpolarization: K⁺ continues
Synaptic Transmission
EPSP: Na⁺ influx,
depolarization | IPSP: Cl⁻ influx/K⁺ efflux,
hyperpolarization | Summation: temporal and
spatial
Ecology & Evolution
Population Dynamics,
Community, Ecosystem
Exponential Growth
dN/dt = rN | N_t =
N₀e^(rt) | J-shaped curve | Unlimited resources,
no predators
Logistic Growth
dN/dt = rN((K-N)/K) |
S-shaped curve | K = carrying capacity | N =
K/2: maximum dN/dt
Net Reproductive Rate
R₀ = Σ l_x m_x | l_x =
survivorship, m_x = fecundity | R₀ > 1:
population growing | R₀ = 1: stable | R₀ < 1:
declining
Generation Time
T = Σ x l_x m_x / R₀ |
Intrinsic rate of increase: r ≈ ln(R₀)/T |
Doubling time: t_d = ln(2)/r ≈ 0.693/r
Diversity Indices
Shannon: H' = -Σ p_i
ln(p_i) | Simpson: D = 1 - Σ p_i² | Higher
values = more diverse
Trophic Efficiency
Lindeman's 10% rule:
~10% energy transferred between levels | 90%
lost as heat, metabolism, undigested
material
Selection
Coefficient
s = 1 - w | w =
relative fitness | s = 0: neutral | 0 < s <
1: deleterious | s=1: lethal
Mutation-Selection Balance
q = √(μ/s) for
recessive lethal | q = μ/s for dominant
lethal | μ = mutation rate, q = allele
frequency
Heterozygote
Advantage
w_Aa > w_AA, w_aa
| Maintains polymorphism | Classic
example: Sickle cell trait in malaria
regions
Genetic Drift
Stronger in small
populations | Fixation probability =
initial frequency | Founder effect and
bottleneck
Reproduction &
Development
Life Cycles,
Hormonal Control, Embryology
Menstrual
Cycle
Follicular phase:
FSH stimulates follicle | Ovulation: LH
surge, day 14 | Luteal phase:
progesterone from corpus luteum
HCG Doubling
Time
hCG doubles every
48-72 hours in early pregnancy | Peak at
8-11 weeks | Used in pregnancy tests
Spermatogenesis Duration
~74 days from stem
cell to mature sperm | Continuous
process | Millions produced daily
Oogenesis
Timeline
Begins in fetal
life, arrests at prophase I (birth to
puberty) | Completes meiosis I at
ovulation | Meiosis II at fertilization
Cleavage
Patterns
Holoblastic:
complete (microlecithal eggs) |
Meroblastic: incomplete (macrolecithal
eggs) | Radial vs spiral cleavage
Gastrulation
Forms three germ
layers | Ectoderm: skin, nervous system
| Mesoderm: muscle, bone, circulatory |
Endoderm: gut, respiratory
Organizer &
Induction
Spemann organizer:
dorsal lip of blastopore | Primary
embryonic induction: neural tube
formation | Morphogen gradients
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