IOE Entrance Nepal cheat sheet

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

Arithmetic progression: nth term a+(n−1)d; sum Sₙ = n/2[2a+(n−1)d]. Geometric progression: nth term ar^(n−1); sum Sₙ = a(rⁿ−1)/(r−1) for r≠1; infinite sum a/(1−r) only when |r|<1.

Binomial theorem: (a+b)ⁿ = Σ C(n,k)a^(n−k)b^k; general term T_(k+1) = C(n,k)a^(n−k)b^k; number of terms n+1.

Trigonometric identities: sin²θ+cos²θ=1; sin2θ=2sinθcosθ; cos2θ=1−2sin²θ; tan(A±B)=(tanA±tanB)/(1∓tanA·tanB).

Derivative rules: d/dx(xⁿ)=nx^(n−1); product (uv)'=u'v+uv'; quotient (u/v)'=(u'v−uv')/v²; chain d/dx f(g(x))=f'(g(x))g'(x).

Definite integrals: ∫ₐᵇ f(x)dx = F(b)−F(a); area under curve between x=a and x=b equals that integral when f≥0 on the interval.

Straight line: slope m=(y₂−y₁)/(x₂−x₁); point-slope y−y₁=m(x−x₁); perpendicular slopes satisfy m₁m₂=−1; distance from (x₁,y₁) to ax+by+c=0 is |ax₁+by₁+c|/√(a²+b²).

Circle: x²+y²+2gx+2fy+c=0 has centre (−g,−f) and radius √(g²+f²−c); tangent at (x₁,y₁) is xx₁+yy₁+g(x+x₁)+f(y+y₁)+c=0.

Limits and L'Hôpital: if substitution gives 0/0 or ∞/∞, lim f/g = lim f'/g'; standard limits lim(x→0) sinx/x = 1, lim(x→0)(1−cosx)/x² = 1/2, lim(x→0)(eˣ−1)/x = 1.

Kinematics: v=u+at; s=ut+½at²; v²=u²+2as; for constant acceleration only. Projectile range R=u²sin2θ/g, max height H=u²sin²θ/(2g).

Newton's laws: F=ma in inertial frames; friction limit f_max=μN; on an incline, component along slope = mg sinθ, normal = mg cosθ.

Work–energy theorem: W_net=ΔKE; gravitational PE near Earth = mgh; spring PE = ½kx²; power P=F·v.

Momentum: p=mv conserved when net external force is zero; perfectly inelastic collision loses KE; elastic collision conserves both p and KE.

Circular motion: centripetal acceleration a=v²/r=ω²r; angular momentum L=Iω; torque τ=Iα.

Gravitation: F=Gm₁m₂/r²; orbital speed v=√(GM/r); escape speed v_e=√(2GM/R)=√(2gR); Kepler's third law T²∝r³.

Simple harmonic motion: T=2π√(m/k) for a spring; T=2π√(L/g) for a simple pendulum; acceleration a=−ω²x directed toward equilibrium.

Thermodynamics: first law ΔU=Q−W; efficiency of Carnot engine η=1−T_c/T_h with temperatures in kelvin; ideal gas PV=nRT.

Electricity: Ohm's law V=IR; series R=R₁+R₂; parallel 1/R=1/R₁+1/R₂; power P=VI=I²R=V²/R; Coulomb force F=kq₁q₂/r².

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

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

pH = −log[H⁺]; pOH = −log[OH⁻]; at 25 °C pH+pOH=14; strong acid fully dissociates, weak acid partially (use Ka).

Ideal gas equation PV=nRT with R=0.0821 L·atm/(mol·K) or 8.314 J/(mol·K); use kelvin.

Periodic trends: atomic radius decreases left→right and increases top→bottom; electronegativity increases left→right and up a group; ionisation energy follows the same direction.

Oxidation number rules: free element 0; O usually −2; H usually +1; sum of oxidation numbers equals charge of the species; oxidation is loss of electrons.

Balancing redox in acidic medium: balance atoms, add H₂O to balance O, add H⁺ to balance H, then balance charge with electrons.

Bond types: ionic from large electronegativity difference with metal–nonmetal; covalent shares electrons; hydrogen bonding explains high boiling point of water, HF, NH₃.

Organic basics: alkanes CₙH₂ₙ₊₂ saturated; alkenes CₙH₂ₙ with C=C; alkynes CₙH₂ₙ₋₂ with C≡C; alcohols contain −OH; carboxylic acids −COOH; IUPAC suffix -ane/-ene/-yne/-ol/-oic acid.

Subject–verb agreement: singular subject takes singular verb; compound subjects joined by 'and' take a plural verb; 'either…or' and 'neither…nor' agree with the nearer subject.

Articles: 'a/an' for indefinite singular countable nouns ('an' before vowel sound); 'the' for specific or previously mentioned nouns; no article with most proper nouns and uncountables in general sense.

Tenses: present perfect (have/has + past participle) links past action to present; past perfect (had + participle) marks the earlier of two past actions; future perfect 'will have + participle'.

Passive voice: form of 'be' + past participle; convert object of active sentence into subject; only transitive verbs take passive.

Conditionals: first (if + present, will + base), second (if + past, would + base), third (if + had + participle, would have + participle).

Direct to indirect speech: shift tense one step back, change pronouns to match reporter, and adjust time/place words (now→then, today→that day, here→there).

Prepositions of time: 'in' for months, years, seasons; 'on' for days and dates; 'at' for clock times and night/noon.

Common confusions: 'affect' verb vs 'effect' noun; 'its' possessive vs 'it's' contraction; 'lie/lay', 'fewer' (countable) vs 'less' (uncountable).

Reading comprehension strategy: locate the topic sentence, distinguish stated fact from inference, and match paraphrase to passage meaning rather than exact wording.

SI base units: metre (length), kilogram (mass), second (time), ampere (current), kelvin (temperature), mole (amount), candela (luminous intensity).

Derived units: newton = kg·m/s²; joule = N·m; watt = J/s; pascal = N/m²; coulomb = A·s; volt = W/A; ohm = V/A; hertz = s⁻¹.

Unit prefixes: k=10³, M=10⁶, G=10⁹, m=10⁻³, μ=10⁻⁶, n=10⁻⁹, p=10⁻¹².

Number series and coding: check differences, ratios, squares/cubes, and alternating patterns; letter codes use position numbers A=1 to Z=26.

Logical reasoning: contrapositive of 'if P then Q' is 'if not Q then not P'; converse and inverse are not logically equivalent to the original.

Ratio and percentage: percentage change = (new−old)/old ×100; successive percentage changes multiply factors (1+a)(1+b), not add.

Time–work and speed: work rate adds when working together (1/a+1/b); speed = distance/time; average speed = total distance / total time, not mean of speeds.

Nepal context: Kathmandu is the capital; Mount Everest (Sagarmatha) 8,848.86 m is the highest peak; Nepal Standard Time is UTC+5:45.

Engineering milestones: Newton's Principia (1687) established classical mechanics; Faraday's electromagnetic induction (1831) underpins generators; Watt's steam engine improvements drove the Industrial Revolution.