Kochi Entrance Science cheat sheet

Newton's second law: ΣF = ma; use components along chosen axes, and for circular motion take the radial component equal to mv²/r.

Kinematics with constant acceleration: v = u + at, s = ut + ½at², v² = u² + 2as; these fail when acceleration varies.

Projectile motion: horizontal velocity constant, vertical acceleration g; time of flight = 2u sinθ/g on level ground.

Momentum is conserved in any collision or explosion with no net external impulse; kinetic energy is conserved only in elastic collisions.

Work–energy theorem: net work done = ΔKE; for a conservative force, define potential energy so that F = −dU/dx.

Friction: limiting static friction f ≤ μsN, kinetic friction f = μkN; N is the normal reaction, not always equal to mg.

Torque about a point: τ = rF sinθ; rotational equilibrium requires zero net torque and zero net force.

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

Moment of inertia replaces mass in rotation: τ = Iα, KE_rot = ½Iω², angular momentum L = Iω is conserved when net torque is zero.

Wave speed relation: v = fλ; frequency is set by the source and is unchanged when a wave crosses into a new medium, while speed and wavelength change.

Transverse waves have displacement perpendicular to propagation; longitudinal waves (sound) have displacement parallel to propagation.

Superposition: displacements add algebraically; constructive interference needs path difference nλ, destructive needs (n + ½)λ.

Standing waves on a string fixed at both ends: λ = 2L/n, f = nv/(2L); a pipe open at both ends has the same set, a pipe closed at one end has odd harmonics only.

Doppler effect for sound: observed frequency rises as source and observer approach; f' = f(v ± v_o)/(v ∓ v_s) with signs chosen so approach gives f' > f.

Snell's law: n₁ sinθ₁ = n₂ sinθ₂; total internal reflection occurs only going from denser to rarer medium when θ exceeds the critical angle sinθ_c = n₂/n₁.

Thin lens and mirror equation: 1/f = 1/v + 1/u (sign conventions consistent); magnification m = −v/u for lenses, and real images are inverted.

Diffraction grating: d sinθ = nλ gives principal maxima; single-slit first minimum at a sinθ = λ.

Intensity of a wave is proportional to the square of its amplitude; doubling amplitude quadruples intensity.

Coulomb's law: F = kq₁q₂/r² with k ≈ 8.99 × 10⁹ N·m²/C²; force acts along the line joining the charges and is repulsive for like signs.

Electric field E = F/q; field lines start on positive charge and end on negative charge and never cross.

Ohm's law V = IR holds for ohmic conductors at constant temperature; resistivity ρ = RA/L is a material property.

Series: same current, resistances add. Parallel: same voltage, 1/R_total = Σ1/Rᵢ.

Electrical power P = VI = I²R = V²/R; energy consumed = Pt, billed in kilowatt-hours.

Kirchhoff's rules: junction rule (charge conservation) and loop rule (energy conservation) solve multi-loop circuits.

Capacitance C = Q/V; for parallel plates C = ε₀A/d; energy stored = ½CV²; capacitors in parallel add, in series reciprocals add.

Magnetic force on a moving charge F = qvB sinθ and on a current-carrying wire F = BIL sinθ; direction from Fleming's left-hand rule.

Faraday's law: induced emf = −dΦ/dt, where Φ = BA cosθ; Lenz's law fixes the induced current direction so it opposes the change in flux.

Ionic bonding transfers electrons between metal and non-metal; lattice energy increases with higher ionic charge and smaller ionic radius.

Covalent bonds share electron pairs; bond order, bond length and bond strength move inversely: higher order means shorter and stronger.

VSEPR: electron pairs repel and adopt maximum separation; lone pairs repel more than bonding pairs, compressing bond angles (CH₄ 109.5°, NH₃ 107°, H₂O 104.5°).

Hybridisation: sp = linear (180°), sp² = trigonal planar (120°), sp³ = tetrahedral (109.5°).

Electronegativity difference decides bond polarity; a polar bond plus asymmetric geometry gives a polar molecule (CO₂ is non-polar, H₂O is polar).

Intermolecular forces ranked: hydrogen bonding > dipole–dipole > London dispersion; stronger forces raise boiling point.

Aufbau and Hund's rules give electron configurations; Cr [Ar]3d⁵4s¹ and Cu [Ar]3d¹⁰4s¹ are exceptions caused by half-filled and filled subshell stability.

Periodic trends: atomic radius decreases across a period and increases down a group; ionisation energy and electronegativity show the reverse pattern.

Metallic bonding: delocalised electrons explain conductivity, malleability and ductility of metals.

Balance equations by conserving atoms and charge; the mole ratio from a balanced equation converts moles of one species to another.

Mole: n = m/M, and at STP one mole of ideal gas occupies 22.4 L; use n = PV/RT otherwise.

Limiting reagent is the reactant that produces the least product by mole ratio; excess reagent remains unreacted.

Percentage yield = (actual/theoretical) × 100; percentage atom economy = (mass of desired product/total mass of reactants) × 100.

Le Chatelier's principle: raising pressure favours the side with fewer gas moles; raising temperature favours the endothermic direction.

Equilibrium constant K depends only on temperature; Q < K means the forward reaction is favoured, Q > K means the reverse.

For an acid, pH = −log[H⁺]; strong acids dissociate fully, weak acids use Ka, and pH + pOH = 14 at 25 °C.

Rate of reaction increases with concentration, temperature, surface area and catalysts; a catalyst lowers activation energy without shifting K.

Oxidation is loss of electrons (increase in oxidation number), reduction is gain; balance redox half-reactions by adding H₂O and H⁺ or OH⁻.

Prokaryotes lack a membrane-bound nucleus and organelles; eukaryotes have a nucleus, mitochondria and (in plants) chloroplasts.

Mitochondria are the site of aerobic respiration and ATP production; ribosomes synthesise proteins; the rough ER transports them.

Mitosis produces two genetically identical diploid cells for growth and repair; meiosis produces four haploid gametes with crossing over and independent assortment.

DNA is a double helix with complementary base pairing: A–T and G–C, held by hydrogen bonds between strands.

Transcription makes mRNA from DNA in the nucleus; translation at the ribosome reads codons and assembles amino acids into protein.

Monohybrid cross of heterozygotes gives a 3:1 phenotype and 1:2:1 genotype ratio; a test cross with a recessive reveals an unknown genotype.

Sex-linked traits carried on the X chromosome appear more often in males, who have only one X.

Mutations are heritable changes in DNA sequence; substitutions may be silent, missense or nonsense, while frameshifts usually disrupt protein function.

Mammalian heart: right side pumps deoxygenated blood to lungs, left side pumps oxygenated blood to the body; the left ventricle has the thickest wall.

Gas exchange in alveoli relies on a large surface area, thin walls and a concentration gradient; haemoglobin carries oxygen and releases it in low-pH, warm tissue.

Blood glucose homeostasis: insulin lowers glucose by promoting uptake and glycogen formation; glucagon raises it by breaking down glycogen.

Nephrons filter blood at the glomerulus and reabsorb glucose and most water in the proximal tubule; ADH increases water reabsorption.

Neurone impulse: depolarisation opens Na⁺ channels, repolarisation opens K⁺ channels; the refractory period enforces one-way conduction.

Photosynthesis: 6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂, driven by light energy captured by chlorophyll; limiting factor is the one in shortest supply.

Food chains transfer about 10% of energy between trophic levels, so pyramids of energy are steep and top predators are few.

Population growth: exponential when resources are unlimited, logistic when carrying capacity K is approached; density-dependent factors intensify with crowding.

Carbon cycle: photosynthesis removes CO₂, respiration and combustion release it; decomposers return nutrients to the soil.

Quadratic formula: x = (−b ± √(b² − 4ac))/2a; discriminant b² − 4ac > 0 gives two real roots, = 0 one repeated root, < 0 complex roots.

For ax² + bx + c, the sum of roots is −b/a and the product is c/a.

Factor theorem: (x − a) is a factor of P(x) exactly when P(a) = 0; remainder theorem gives remainder P(a) on division by (x − a).

Laws of logarithms: log(xy) = log x + log y, log(x/y) = log x − log y, log xⁿ = n log x; change of base log_b x = log x / log b.

Exponential and logarithm are inverses: a^(log_a x) = x and log_a(aˣ) = x; domain of log is x > 0.

Composite and inverse functions: (f∘g)(x) = f(g(x)); f⁻¹ exists only if f is one-to-one, and its graph is the reflection in y = x.

Arithmetic sequence: uₙ = a + (n − 1)d, Sₙ = n/2[2a + (n − 1)d]; geometric: uₙ = arⁿ⁻¹, Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1.

Binomial expansion: (a + b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳbʳ, with the general term ⁿCᵣ aⁿ⁻ʳbʳ.

Solving inequalities: multiplying or dividing by a negative number reverses the inequality sign.

Derivative from first principles: f'(x) = lim(h→0)[f(x + h) − f(x)]/h; the derivative is the gradient of the tangent.

Power rule d/dx(xⁿ) = nxⁿ⁻¹; d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x.

Chain rule: dy/dx = dy/du × du/dx; product rule (uv)' = u'v + uv'; quotient rule (u/v)' = (u'v − uv')/v².

Stationary points occur where f'(x) = 0; f''(x) > 0 gives a minimum, f''(x) < 0 a maximum, f''(x) = 0 needs further testing.

Integration reverses differentiation: ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ −1; definite integral gives signed area under the curve.

Area between two curves is ∫(upper − lower) dx between intersection points; find intersections by setting the functions equal.

Circle: (x − h)² + (y − k)² = r² has centre (h, k) and radius r; a tangent is perpendicular to the radius at the point of contact.

Straight line: gradient m = (y₂ − y₁)/(x₂ − x₁); parallel lines share m, perpendicular lines satisfy m₁m₂ = −1.

Distance between points = √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).