KUUMAT Nepal cheat sheet

Kinematics: v = u + at, s = ut + ½at², v² = u² + 2as; for free fall take a = g ≈ 9.8 m/s² downward and set sign convention before substituting.

Projectile: horizontal range R = u²sin2θ/g is maximum at θ = 45°; time of flight T = 2u sinθ/g; horizontal and vertical motions are independent.

Newton's second law F = ma; friction limit f_max = μN; on an incline the component along the plane is mg sinθ and the normal force is mg cosθ.

Work–energy: W_net = ΔKE = ½mv² − ½mu²; gravitational PE near Earth = mgh; power P = W/t = Fv.

Momentum: p = mv is conserved in collisions with no external force; perfectly inelastic collision loses maximum KE while conserving momentum.

Circular motion: centripetal acceleration a = v²/r = ω²r; angular momentum L = Iω; for a rolling body KE = ½mv² + ½Iω².

Ohm's law V = IR; series resistance adds (R = R₁ + R₂), parallel gives 1/R = 1/R₁ + 1/R₂; power P = VI = I²R = V²/R.

Lens/mirror equation 1/f = 1/v − 1/u (sign conventions matter); magnification m = v/u; real images from converging lenses form beyond f.

Modern physics: E = hf = hc/λ; photoelectric threshold gives K_max = hf − φ; half-life t½ = 0.693/λ for radioactive decay.

Mole concept: 1 mol = 6.022×10²³ particles = molar mass in grams; at STP one mole of ideal gas occupies 22.4 L.

Ideal gas law PV = nRT with R = 0.0821 L·atm/(mol·K) or 8.314 J/(mol·K); convert Celsius to kelvin by adding 273.

Molarity M = moles solute / litres solution; dilution obeys M₁V₁ = M₂V₂; molality uses kg of solvent.

pH = −log[H⁺]; pH + pOH = 14 at 25 °C; strong acid fully dissociates while weak acid needs Ka.

Periodic trends: atomic radius decreases across a period and increases down a group; electronegativity and ionisation energy increase across a period.

Oxidation number rules: free element = 0, oxygen usually −2, hydrogen usually +1, sum equals charge on a polyatomic ion.

Le Chatelier: raising pressure shifts equilibrium toward fewer gas moles; adding heat shifts toward the endothermic direction.

Hybridisation from steric number: 2 → sp (linear), 3 → sp² (trigonal planar), 4 → sp³ (tetrahedral); lone pairs compress bond angles.

Organic nomenclature: suffix -ane (single bonds), -ene (double), -ol (alcohol), -oic acid (carboxylic acid); functional group determines homologous series.

Photosynthesis: 6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂; light reactions in thylakoid membranes, Calvin cycle in stroma; O₂ comes from water photolysis.

Cellular respiration: glycolysis in cytoplasm yields 2 ATP and 2 pyruvate; Krebs cycle in mitochondrial matrix; oxidative phosphorylation on the inner membrane.

Mendelian genetics: monohybrid F₂ ratio 3:1, dihybrid 9:3:3:1; test cross uses a homozygous recessive parent to reveal genotype.

Blood circulation: double circulation in mammals; arteries carry blood away from the heart, veins carry it back; SA node sets heart rhythm.

Plant tissue and transport: xylem carries water and minerals upward, phloem translocates sucrose; transpiration pull drives xylem flow.

Taxonomy hierarchy: Kingdom, Phylum, Class, Order, Family, Genus, Species; binomial names italicised with genus capitalised.

Human digestion: starch begins digestion in the mouth (salivary amylase), proteins in the stomach (pepsin, acidic pH), fats emulsified by bile in the small intestine.

Hormones: insulin lowers blood glucose, glucagon raises it; thyroid hormone regulates basal metabolic rate; ADH increases water reabsorption in kidneys.

Nervous system: action potential is depolarisation by Na⁺ influx followed by repolarisation by K⁺ efflux; synapse uses neurotransmitters.

Quadratic ax² + bx + c = 0: roots x = [−b ± √(b² − 4ac)]/2a; discriminant D > 0 two real roots, D = 0 equal roots, D < 0 complex roots.

Sum and product of roots: α + β = −b/a and αβ = c/a; use these to build equations without solving.

Arithmetic progression: nth term a + (n−1)d, sum Sₙ = n/2[2a + (n−1)d]; geometric progression nth term ar^(n−1), sum a(rⁿ−1)/(r−1).

Trigonometry: sin²θ + cos²θ = 1; values sin30° = ½, cos30° = √3/2, tan45° = 1; sine rule a/sinA = b/sinB = 2R.

Mensuration: circle area πr², circumference 2πr; sphere volume 4/3πr³ and surface 4πr²; cone volume ⅓πr²h.

Probability of an event = favourable outcomes / total equally likely outcomes; P(A or B) = P(A) + P(B) − P(A and B).

Permutation vs combination: nPr = n!/(n−r)! orders items, nCr = n!/[r!(n−r)!] does not; use combinations when order is irrelevant.

Percentage and ratio shortcuts: successive percentage changes multiply factors (e.g. 20% rise then 10% fall gives ×1.2×0.9 = ×1.08); average speed for equal distances is 2v₁v₂/(v₁+v₂).

Divisibility rules: by 3 if digit sum divisible by 3, by 4 if last two digits divisible by 4, by 9 if digit sum divisible by 9, by 11 if alternating digit-sum difference is 0 or a multiple of 11.