IMU CET cheat sheet

Quadratic roots: x = [-b ± √(b² - 4ac)] / (2a); discriminant D = b² - 4ac gives two real roots if D > 0, one if D = 0, complex if D < 0.

Sum and product of roots of ax² + bx + c = 0: α + β = -b/a and αβ = c/a.

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

sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ. Sine rule a/sin A = b/sin B = c/sin C = 2R; cosine rule a² = b² + c² - 2bc cos A.

Derivative rules: d/dx(xⁿ) = nx^(n-1); d/dx(sin x) = cos x; d/dx(eˣ) = eˣ; d/dx(ln x) = 1/x. Product rule (uv)' = u'v + uv'; quotient rule (u/v)' = (u'v - uv')/v².

Definite integral ∫ₐᵇ f(x) dx gives signed area under the curve; for a velocity function it equals net displacement over that interval.

Binomial theorem: (a + b)ⁿ = Σ C(n,r) a^(n-r) b^r for r = 0 to n; general term T(r+1) = C(n,r) a^(n-r) b^r.

Distance between points (x₁,y₁) and (x₂,y₂) is √[(x₂-x₁)² + (y₂-y₁)²]; slope m = (y₂-y₁)/(x₂-x₁); parallel lines have equal slopes, perpendicular slopes multiply to -1.

Standard limits: lim(x→0) sin x / x = 1; lim(x→0) (1 - cos x)/x² = 1/2; lim(n→∞) (1 + 1/n)ⁿ = e.

Newton's second law F = ma; weight W = mg where g ≈ 9.8 m/s². Momentum p = mv is conserved in the absence of external force.

Kinematics: v = u + at; s = ut + ½at²; v² = u² + 2as, valid only for constant acceleration.

Work-energy: work W = Fs cos θ; kinetic energy KE = ½mv²; gravitational potential energy PE = mgh; power P = W/t = Fv.

Archimedes' principle: buoyant force equals the weight of displaced fluid, F_b = ρ_fluid × V_displaced × g; a body floats when its average density is less than the fluid's.

Pascal's law: pressure applied to an enclosed fluid is transmitted undiminished; hydraulic press gives F₂ = F₁(A₂/A₁). Pressure in a fluid at depth h is P = P₀ + ρgh.

Ohm's law V = IR; electrical power P = VI = I²R = V²/R; resistors in series add, in parallel 1/R = 1/R₁ + 1/R₂.

Snell's law n₁ sin θ₁ = n₂ sin θ₂; refractive index n = c/v; total internal reflection occurs when the angle exceeds the critical angle sin θ_c = n₂/n₁.

Wave relation v = fλ; for a stretched string frequency f = (1/2L)√(T/μ) where μ is linear mass density.

First law of thermodynamics ΔQ = ΔU + ΔW; efficiency of a heat engine η = 1 - T_cold/T_hot (temperatures in kelvin).

Mole concept: 1 mole = 6.022 × 10²³ particles; moles n = mass / molar mass; at STP one mole of ideal gas occupies 22.4 L.

Ideal gas equation PV = nRT with R = 8.314 J/(mol·K); combined gas law P₁V₁/T₁ = P₂V₂/T₂.

pH = -log[H⁺]; pH + pOH = 14 at 25 °C; a strong acid fully dissociates, a weak acid only partially (Ka expression).

Oxidation is loss of electrons, reduction is gain; in a redox reaction the oxidising agent is itself reduced. Balancing uses half-reactions and electron count.

Periodic trends across a period (left to right): atomic radius decreases, ionisation energy and electronegativity increase; down a group radius increases and ionisation energy decreases.

Nomenclature: identify the longest carbon chain, number from the end nearest the substituent, and name by functional group suffix (-ol, -al, -oic acid, -amine).

Avogadro's law: equal volumes of gases at the same temperature and pressure contain equal numbers of molecules.

Bond types: ionic bonds form between metals and non-metals by electron transfer; covalent bonds share electrons; metallic bonding involves delocalised electrons.

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

Active to passive: object becomes subject, verb becomes form of 'be' + past participle, agent introduced by 'by'. Example: 'The crew lowered the boat' → 'The boat was lowered by the crew'.

Reported speech shifts tense back one step (present → past, past → past perfect) and adjusts time words: now → then, today → that day, tomorrow → the next day.

Subject-verb agreement: a singular subject takes a singular verb; collective nouns like 'crew' take a singular verb when acting as one unit, plural when members act individually.

Articles: 'a/an' for non-specific countable singulars (an before vowel sounds); 'the' for specific or previously mentioned nouns; no article for uncountable general references.

Prepositions of place and time: in (enclosed space, months, years), on (surfaces, days, dates), at (points, exact times, addresses).

Conditionals: first (if + present, will + base) for real future; second (if + past, would + base) for unreal present; third (if + past perfect, would have + past participle) for unreal past.

Commonly confused pairs: affect (verb) / effect (noun); its (possessive) / it's (it is); principal (main) / principle (rule); stationary (not moving) / stationery (writing materials).

Punctuation: a semicolon joins two independent clauses; a colon introduces a list or explanation; commas set off non-defining relative clauses.

Synonyms and antonyms are tested by context: 'abandon' (forsake) vs 'retain' (keep); 'candid' (frank) vs 'evasive' (avoiding).

Direction sense: on a standard map, north is up; a turn of 90° clockwise from north gives east, 180° gives south, 270° gives west. Track net displacement with coordinates.

Number series: check differences (arithmetic), ratios (geometric), squares/cubes, and alternating patterns before assuming a single rule.

Percentage and profit: profit% = (SP - CP)/CP × 100; successive percentage changes of a% and b% give a net change of a + b + ab/100.

Time-speed-distance: distance = speed × time; to convert km/h to m/s multiply by 5/18; average speed for equal distances = 2xy/(x + y).

Work: if A finishes in x days and B in y days, together they take xy/(x + y) days; combined rate is the sum of individual rates.

Ratio and proportion: if a:b = c:d then ad = bc; to divide a quantity Q in ratio a:b:c, parts are Q × a/(a+b+c), etc.

Geography: the International Date Line roughly follows 180° longitude; crossing it eastward you subtract a day, westward you add a day.

Indian polity: the Constitution came into force on 26 January 1950; the President is head of state and the Prime Minister heads the government.

Nautical general knowledge: IMO is the UN agency for shipping safety; MARPOL addresses marine pollution; the ILO sets seafarer labour standards (MLC 2006).

Port is the left side and starboard the right side when facing forward; red marks port hand and green starboard hand in IALA Region A buoyage.

Cardinal marks indicate the safe side to pass: north (two cones up), east (cones base to base), south (two cones down), west (cones point to point); the topmarks show the direction.

Bearing is the direction of an object from the ship; heading is the direction the ship points; a relative bearing is measured from the ship's head clockwise 000°–360°.

Speed, distance, time: distance (nm) = speed (kn) × time (h); a nautical mile is 1,852 m and one knot is one nautical mile per hour.

Latitude is measured 0°–90° north or south of the equator; longitude 0°–180° east or west of Greenwich. Position is written latitude first, then longitude.

Knots and their uses: bowline makes a fixed loop; clove hitch ties to a post; reef knot joins two ropes of equal size; round turn and two half hitches secures to a ring.

COLREGs: in a head-on situation both vessels alter course to starboard; a power-driven vessel gives way to a sailing vessel, and the overtaking vessel keeps clear.

Anchoring: the scope of chain is the ratio of chain length to water depth; more scope improves holding. The anchor is let go when the vessel has minimal headway.

Firefighting: fires are classed A (solid), B (flammable liquid), C (gas), D (metal), and electrical; a CO₂ extinguisher suits electrical fires, water is never used on electrical or oil fires.

Quadratic roots: x = [-b ± √(b² - 4ac)] / (2a); discriminant D = b² - 4ac gives two real roots if D > 0, one if D = 0, complex if D < 0.

Sum and product of roots of ax² + bx + c = 0: α + β = -b/a and αβ = c/a.

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

sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ. Sine rule a/sin A = b/sin B = c/sin C = 2R; cosine rule a² = b² + c² - 2bc cos A.

Derivative rules: d/dx(xⁿ) = nx^(n-1); d/dx(sin x) = cos x; d/dx(eˣ) = eˣ; d/dx(ln x) = 1/x. Product rule (uv)' = u'v + uv'; quotient rule (u/v)' = (u'v - uv')/v².

Definite integral ∫ₐᵇ f(x) dx gives signed area under the curve; for a velocity function it equals net displacement over that interval.

Binomial theorem: (a + b)ⁿ = Σ C(n,r) a^(n-r) b^r for r = 0 to n; general term T(r+1) = C(n,r) a^(n-r) b^r.

Distance between points (x₁,y₁) and (x₂,y₂) is √[(x₂-x₁)² + (y₂-y₁)²]; slope m = (y₂-y₁)/(x₂-x₁); parallel lines have equal slopes, perpendicular slopes multiply to -1.

Standard limits: lim(x→0) sin x / x = 1; lim(x→0) (1 - cos x)/x² = 1/2; lim(n→∞) (1 + 1/n)ⁿ = e.

Newton's second law F = ma; weight W = mg where g ≈ 9.8 m/s². Momentum p = mv is conserved in the absence of external force.

Kinematics: v = u + at; s = ut + ½at²; v² = u² + 2as, valid only for constant acceleration.

Work-energy: work W = Fs cos θ; kinetic energy KE = ½mv²; gravitational potential energy PE = mgh; power P = W/t = Fv.

Archimedes' principle: buoyant force equals the weight of displaced fluid, F_b = ρ_fluid × V_displaced × g; a body floats when its average density is less than the fluid's.

Pascal's law: pressure applied to an enclosed fluid is transmitted undiminished; hydraulic press gives F₂ = F₁(A₂/A₁). Pressure in a fluid at depth h is P = P₀ + ρgh.

Ohm's law V = IR; electrical power P = VI = I²R = V²/R; resistors in series add, in parallel 1/R = 1/R₁ + 1/R₂.

Snell's law n₁ sin θ₁ = n₂ sin θ₂; refractive index n = c/v; total internal reflection occurs when the angle exceeds the critical angle sin θ_c = n₂/n₁.

Wave relation v = fλ; for a stretched string frequency f = (1/2L)√(T/μ) where μ is linear mass density.

First law of thermodynamics ΔQ = ΔU + ΔW; efficiency of a heat engine η = 1 - T_cold/T_hot (temperatures in kelvin).

Mole concept: 1 mole = 6.022 × 10²³ particles; moles n = mass / molar mass; at STP one mole of ideal gas occupies 22.4 L.

Ideal gas equation PV = nRT with R = 8.314 J/(mol·K); combined gas law P₁V₁/T₁ = P₂V₂/T₂.

pH = -log[H⁺]; pH + pOH = 14 at 25 °C; a strong acid fully dissociates, a weak acid only partially (Ka expression).

Oxidation is loss of electrons, reduction is gain; in a redox reaction the oxidising agent is itself reduced. Balancing uses half-reactions and electron count.

Periodic trends across a period (left to right): atomic radius decreases, ionisation energy and electronegativity increase; down a group radius increases and ionisation energy decreases.

Nomenclature: identify the longest carbon chain, number from the end nearest the substituent, and name by functional group suffix (-ol, -al, -oic acid, -amine).

Avogadro's law: equal volumes of gases at the same temperature and pressure contain equal numbers of molecules.

Bond types: ionic bonds form between metals and non-metals by electron transfer; covalent bonds share electrons; metallic bonding involves delocalised electrons.

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

Active to passive: object becomes subject, verb becomes form of 'be' + past participle, agent introduced by 'by'. Example: 'The crew lowered the boat' → 'The boat was lowered by the crew'.

Reported speech shifts tense back one step (present → past, past → past perfect) and adjusts time words: now → then, today → that day, tomorrow → the next day.

Subject-verb agreement: a singular subject takes a singular verb; collective nouns like 'crew' take a singular verb when acting as one unit, plural when members act individually.

Articles: 'a/an' for non-specific countable singulars (an before vowel sounds); 'the' for specific or previously mentioned nouns; no article for uncountable general references.

Prepositions of place and time: in (enclosed space, months, years), on (surfaces, days, dates), at (points, exact times, addresses).

Conditionals: first (if + present, will + base) for real future; second (if + past, would + base) for unreal present; third (if + past perfect, would have + past participle) for unreal past.

Commonly confused pairs: affect (verb) / effect (noun); its (possessive) / it's (it is); principal (main) / principle (rule); stationary (not moving) / stationery (writing materials).

Punctuation: a semicolon joins two independent clauses; a colon introduces a list or explanation; commas set off non-defining relative clauses.

Synonyms and antonyms are tested by context: 'abandon' (forsake) vs 'retain' (keep); 'candid' (frank) vs 'evasive' (avoiding).

Direction sense: on a standard map, north is up; a turn of 90° clockwise from north gives east, 180° gives south, 270° gives west. Track net displacement with coordinates.

Number series: check differences (arithmetic), ratios (geometric), squares/cubes, and alternating patterns before assuming a single rule.

Percentage and profit: profit% = (SP - CP)/CP × 100; successive percentage changes of a% and b% give a net change of a + b + ab/100.

Time-speed-distance: distance = speed × time; to convert km/h to m/s multiply by 5/18; average speed for equal distances = 2xy/(x + y).

Work: if A finishes in x days and B in y days, together they take xy/(x + y) days; combined rate is the sum of individual rates.

Ratio and proportion: if a:b = c:d then ad = bc; to divide a quantity Q in ratio a:b:c, parts are Q × a/(a+b+c), etc.

Geography: the International Date Line roughly follows 180° longitude; crossing it eastward you subtract a day, westward you add a day.

Indian polity: the Constitution came into force on 26 January 1950; the President is head of state and the Prime Minister heads the government.

Nautical general knowledge: IMO is the UN agency for shipping safety; MARPOL addresses marine pollution; the ILO sets seafarer labour standards (MLC 2006).

Port is the left side and starboard the right side when facing forward; red marks port hand and green starboard hand in IALA Region A buoyage.

Cardinal marks indicate the safe side to pass: north (two cones up), east (cones base to base), south (two cones down), west (cones point to point); the topmarks show the direction.

Bearing is the direction of an object from the ship; heading is the direction the ship points; a relative bearing is measured from the ship's head clockwise 000°–360°.

Speed, distance, time: distance (nm) = speed (kn) × time (h); a nautical mile is 1,852 m and one knot is one nautical mile per hour.

Latitude is measured 0°–90° north or south of the equator; longitude 0°–180° east or west of Greenwich. Position is written latitude first, then longitude.

Knots and their uses: bowline makes a fixed loop; clove hitch ties to a post; reef knot joins two ropes of equal size; round turn and two half hitches secures to a ring.

COLREGs: in a head-on situation both vessels alter course to starboard; a power-driven vessel gives way to a sailing vessel, and the overtaking vessel keeps clear.

Anchoring: the scope of chain is the ratio of chain length to water depth; more scope improves holding. The anchor is let go when the vessel has minimal headway.

Firefighting: fires are classed A (solid), B (flammable liquid), C (gas), D (metal), and electrical; a CO₂ extinguisher suits electrical fires, water is never used on electrical or oil fires.

Quadratic roots: x = [-b ± √(b² - 4ac)] / (2a); discriminant D = b² - 4ac gives two real roots if D > 0, one if D = 0, complex if D < 0.

Sum and product of roots of ax² + bx + c = 0: α + β = -b/a and αβ = c/a.

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

sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ. Sine rule a/sin A = b/sin B = c/sin C = 2R; cosine rule a² = b² + c² - 2bc cos A.

Derivative rules: d/dx(xⁿ) = nx^(n-1); d/dx(sin x) = cos x; d/dx(eˣ) = eˣ; d/dx(ln x) = 1/x. Product rule (uv)' = u'v + uv'; quotient rule (u/v)' = (u'v - uv')/v².

Definite integral ∫ₐᵇ f(x) dx gives signed area under the curve; for a velocity function it equals net displacement over that interval.

Binomial theorem: (a + b)ⁿ = Σ C(n,r) a^(n-r) b^r for r = 0 to n; general term T(r+1) = C(n,r) a^(n-r) b^r.

Distance between points (x₁,y₁) and (x₂,y₂) is √[(x₂-x₁)² + (y₂-y₁)²]; slope m = (y₂-y₁)/(x₂-x₁); parallel lines have equal slopes, perpendicular slopes multiply to -1.

Standard limits: lim(x→0) sin x / x = 1; lim(x→0) (1 - cos x)/x² = 1/2; lim(n→∞) (1 + 1/n)ⁿ = e.

Newton's second law F = ma; weight W = mg where g ≈ 9.8 m/s². Momentum p = mv is conserved in the absence of external force.

Kinematics: v = u + at; s = ut + ½at²; v² = u² + 2as, valid only for constant acceleration.

Work-energy: work W = Fs cos θ; kinetic energy KE = ½mv²; gravitational potential energy PE = mgh; power P = W/t = Fv.

Archimedes' principle: buoyant force equals the weight of displaced fluid, F_b = ρ_fluid × V_displaced × g; a body floats when its average density is less than the fluid's.

Pascal's law: pressure applied to an enclosed fluid is transmitted undiminished; hydraulic press gives F₂ = F₁(A₂/A₁). Pressure in a fluid at depth h is P = P₀ + ρgh.

Ohm's law V = IR; electrical power P = VI = I²R = V²/R; resistors in series add, in parallel 1/R = 1/R₁ + 1/R₂.

Snell's law n₁ sin θ₁ = n₂ sin θ₂; refractive index n = c/v; total internal reflection occurs when the angle exceeds the critical angle sin θ_c = n₂/n₁.

Wave relation v = fλ; for a stretched string frequency f = (1/2L)√(T/μ) where μ is linear mass density.

First law of thermodynamics ΔQ = ΔU + ΔW; efficiency of a heat engine η = 1 - T_cold/T_hot (temperatures in kelvin).

Mole concept: 1 mole = 6.022 × 10²³ particles; moles n = mass / molar mass; at STP one mole of ideal gas occupies 22.4 L.

Ideal gas equation PV = nRT with R = 8.314 J/(mol·K); combined gas law P₁V₁/T₁ = P₂V₂/T₂.

pH = -log[H⁺]; pH + pOH = 14 at 25 °C; a strong acid fully dissociates, a weak acid only partially (Ka expression).

Oxidation is loss of electrons, reduction is gain; in a redox reaction the oxidising agent is itself reduced. Balancing uses half-reactions and electron count.

Periodic trends across a period (left to right): atomic radius decreases, ionisation energy and electronegativity increase; down a group radius increases and ionisation energy decreases.

Nomenclature: identify the longest carbon chain, number from the end nearest the substituent, and name by functional group suffix (-ol, -al, -oic acid, -amine).

Avogadro's law: equal volumes of gases at the same temperature and pressure contain equal numbers of molecules.

Bond types: ionic bonds form between metals and non-metals by electron transfer; covalent bonds share electrons; metallic bonding involves delocalised electrons.

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

Active to passive: object becomes subject, verb becomes form of 'be' + past participle, agent introduced by 'by'. Example: 'The crew lowered the boat' → 'The boat was lowered by the crew'.

Reported speech shifts tense back one step (present → past, past → past perfect) and adjusts time words: now → then, today → that day, tomorrow → the next day.

Subject-verb agreement: a singular subject takes a singular verb; collective nouns like 'crew' take a singular verb when acting as one unit, plural when members act individually.

Articles: 'a/an' for non-specific countable singulars (an before vowel sounds); 'the' for specific or previously mentioned nouns; no article for uncountable general references.

Prepositions of place and time: in (enclosed space, months, years), on (surfaces, days, dates), at (points, exact times, addresses).

Conditionals: first (if + present, will + base) for real future; second (if + past, would + base) for unreal present; third (if + past perfect, would have + past participle) for unreal past.

Commonly confused pairs: affect (verb) / effect (noun); its (possessive) / it's (it is); principal (main) / principle (rule); stationary (not moving) / stationery (writing materials).

Punctuation: a semicolon joins two independent clauses; a colon introduces a list or explanation; commas set off non-defining relative clauses.

Synonyms and antonyms are tested by context: 'abandon' (forsake) vs 'retain' (keep); 'candid' (frank) vs 'evasive' (avoiding).

Direction sense: on a standard map, north is up; a turn of 90° clockwise from north gives east, 180° gives south, 270° gives west. Track net displacement with coordinates.

Number series: check differences (arithmetic), ratios (geometric), squares/cubes, and alternating patterns before assuming a single rule.

Percentage and profit: profit% = (SP - CP)/CP × 100; successive percentage changes of a% and b% give a net change of a + b + ab/100.

Time-speed-distance: distance = speed × time; to convert km/h to m/s multiply by 5/18; average speed for equal distances = 2xy/(x + y).

Work: if A finishes in x days and B in y days, together they take xy/(x + y) days; combined rate is the sum of individual rates.

Ratio and proportion: if a:b = c:d then ad = bc; to divide a quantity Q in ratio a:b:c, parts are Q × a/(a+b+c), etc.

Geography: the International Date Line roughly follows 180° longitude; crossing it eastward you subtract a day, westward you add a day.

Indian polity: the Constitution came into force on 26 January 1950; the President is head of state and the Prime Minister heads the government.

Nautical general knowledge: IMO is the UN agency for shipping safety; MARPOL addresses marine pollution; the ILO sets seafarer labour standards (MLC 2006).

Port is the left side and starboard the right side when facing forward; red marks port hand and green starboard hand in IALA Region A buoyage.

Cardinal marks indicate the safe side to pass: north (two cones up), east (cones base to base), south (two cones down), west (cones point to point); the topmarks show the direction.

Bearing is the direction of an object from the ship; heading is the direction the ship points; a relative bearing is measured from the ship's head clockwise 000°–360°.

Speed, distance, time: distance (nm) = speed (kn) × time (h); a nautical mile is 1,852 m and one knot is one nautical mile per hour.

Latitude is measured 0°–90° north or south of the equator; longitude 0°–180° east or west of Greenwich. Position is written latitude first, then longitude.

Knots and their uses: bowline makes a fixed loop; clove hitch ties to a post; reef knot joins two ropes of equal size; round turn and two half hitches secures to a ring.

COLREGs: in a head-on situation both vessels alter course to starboard; a power-driven vessel gives way to a sailing vessel, and the overtaking vessel keeps clear.

Anchoring: the scope of chain is the ratio of chain length to water depth; more scope improves holding. The anchor is let go when the vessel has minimal headway.

Firefighting: fires are classed A (solid), B (flammable liquid), C (gas), D (metal), and electrical; a CO₂ extinguisher suits electrical fires, water is never used on electrical or oil fires.

Quadratic roots: x = [-b ± √(b² - 4ac)] / (2a); discriminant D = b² - 4ac gives two real roots if D > 0, one if D = 0, complex if D < 0.

Sum and product of roots of ax² + bx + c = 0: α + β = -b/a and αβ = c/a.

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

sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ. Sine rule a/sin A = b/sin B = c/sin C = 2R; cosine rule a² = b² + c² - 2bc cos A.

Derivative rules: d/dx(xⁿ) = nx^(n-1); d/dx(sin x) = cos x; d/dx(eˣ) = eˣ; d/dx(ln x) = 1/x. Product rule (uv)' = u'v + uv'; quotient rule (u/v)' = (u'v - uv')/v².

Definite integral ∫ₐᵇ f(x) dx gives signed area under the curve; for a velocity function it equals net displacement over that interval.

Binomial theorem: (a + b)ⁿ = Σ C(n,r) a^(n-r) b^r for r = 0 to n; general term T(r+1) = C(n,r) a^(n-r) b^r.

Distance between points (x₁,y₁) and (x₂,y₂) is √[(x₂-x₁)² + (y₂-y₁)²]; slope m = (y₂-y₁)/(x₂-x₁); parallel lines have equal slopes, perpendicular slopes multiply to -1.

Standard limits: lim(x→0) sin x / x = 1; lim(x→0) (1 - cos x)/x² = 1/2; lim(n→∞) (1 + 1/n)ⁿ = e.

Newton's second law F = ma; weight W = mg where g ≈ 9.8 m/s². Momentum p = mv is conserved in the absence of external force.

Kinematics: v = u + at; s = ut + ½at²; v² = u² + 2as, valid only for constant acceleration.

Work-energy: work W = Fs cos θ; kinetic energy KE = ½mv²; gravitational potential energy PE = mgh; power P = W/t = Fv.

Archimedes' principle: buoyant force equals the weight of displaced fluid, F_b = ρ_fluid × V_displaced × g; a body floats when its average density is less than the fluid's.

Pascal's law: pressure applied to an enclosed fluid is transmitted undiminished; hydraulic press gives F₂ = F₁(A₂/A₁). Pressure in a fluid at depth h is P = P₀ + ρgh.

Ohm's law V = IR; electrical power P = VI = I²R = V²/R; resistors in series add, in parallel 1/R = 1/R₁ + 1/R₂.

Snell's law n₁ sin θ₁ = n₂ sin θ₂; refractive index n = c/v; total internal reflection occurs when the angle exceeds the critical angle sin θ_c = n₂/n₁.

Wave relation v = fλ; for a stretched string frequency f = (1/2L)√(T/μ) where μ is linear mass density.

First law of thermodynamics ΔQ = ΔU + ΔW; efficiency of a heat engine η = 1 - T_cold/T_hot (temperatures in kelvin).

Mole concept: 1 mole = 6.022 × 10²³ particles; moles n = mass / molar mass; at STP one mole of ideal gas occupies 22.4 L.

Ideal gas equation PV = nRT with R = 8.314 J/(mol·K); combined gas law P₁V₁/T₁ = P₂V₂/T₂.

pH = -log[H⁺]; pH + pOH = 14 at 25 °C; a strong acid fully dissociates, a weak acid only partially (Ka expression).

Oxidation is loss of electrons, reduction is gain; in a redox reaction the oxidising agent is itself reduced. Balancing uses half-reactions and electron count.

Periodic trends across a period (left to right): atomic radius decreases, ionisation energy and electronegativity increase; down a group radius increases and ionisation energy decreases.

Nomenclature: identify the longest carbon chain, number from the end nearest the substituent, and name by functional group suffix (-ol, -al, -oic acid, -amine).

Avogadro's law: equal volumes of gases at the same temperature and pressure contain equal numbers of molecules.

Bond types: ionic bonds form between metals and non-metals by electron transfer; covalent bonds share electrons; metallic bonding involves delocalised electrons.

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

Active to passive: object becomes subject, verb becomes form of 'be' + past participle, agent introduced by 'by'. Example: 'The crew lowered the boat' → 'The boat was lowered by the crew'.

Reported speech shifts tense back one step (present → past, past → past perfect) and adjusts time words: now → then, today → that day, tomorrow → the next day.

Subject-verb agreement: a singular subject takes a singular verb; collective nouns like 'crew' take a singular verb when acting as one unit, plural when members act individually.

Articles: 'a/an' for non-specific countable singulars (an before vowel sounds); 'the' for specific or previously mentioned nouns; no article for uncountable general references.

Prepositions of place and time: in (enclosed space, months, years), on (surfaces, days, dates), at (points, exact times, addresses).

Conditionals: first (if + present, will + base) for real future; second (if + past, would + base) for unreal present; third (if + past perfect, would have + past participle) for unreal past.

Commonly confused pairs: affect (verb) / effect (noun); its (possessive) / it's (it is); principal (main) / principle (rule); stationary (not moving) / stationery (writing materials).

Punctuation: a semicolon joins two independent clauses; a colon introduces a list or explanation; commas set off non-defining relative clauses.

Synonyms and antonyms are tested by context: 'abandon' (forsake) vs 'retain' (keep); 'candid' (frank) vs 'evasive' (avoiding).

Direction sense: on a standard map, north is up; a turn of 90° clockwise from north gives east, 180° gives south, 270° gives west. Track net displacement with coordinates.

Number series: check differences (arithmetic), ratios (geometric), squares/cubes, and alternating patterns before assuming a single rule.

Percentage and profit: profit% = (SP - CP)/CP × 100; successive percentage changes of a% and b% give a net change of a + b + ab/100.

Time-speed-distance: distance = speed × time; to convert km/h to m/s multiply by 5/18; average speed for equal distances = 2xy/(x + y).

Work: if A finishes in x days and B in y days, together they take xy/(x + y) days; combined rate is the sum of individual rates.

Ratio and proportion: if a:b = c:d then ad = bc; to divide a quantity Q in ratio a:b:c, parts are Q × a/(a+b+c), etc.

Geography: the International Date Line roughly follows 180° longitude; crossing it eastward you subtract a day, westward you add a day.

Indian polity: the Constitution came into force on 26 January 1950; the President is head of state and the Prime Minister heads the government.

Nautical general knowledge: IMO is the UN agency for shipping safety; MARPOL addresses marine pollution; the ILO sets seafarer labour standards (MLC 2006).

Port is the left side and starboard the right side when facing forward; red marks port hand and green starboard hand in IALA Region A buoyage.

Cardinal marks indicate the safe side to pass: north (two cones up), east (cones base to base), south (two cones down), west (cones point to point); the topmarks show the direction.

Bearing is the direction of an object from the ship; heading is the direction the ship points; a relative bearing is measured from the ship's head clockwise 000°–360°.

Speed, distance, time: distance (nm) = speed (kn) × time (h); a nautical mile is 1,852 m and one knot is one nautical mile per hour.

Latitude is measured 0°–90° north or south of the equator; longitude 0°–180° east or west of Greenwich. Position is written latitude first, then longitude.

Knots and their uses: bowline makes a fixed loop; clove hitch ties to a post; reef knot joins two ropes of equal size; round turn and two half hitches secures to a ring.

COLREGs: in a head-on situation both vessels alter course to starboard; a power-driven vessel gives way to a sailing vessel, and the overtaking vessel keeps clear.

Anchoring: the scope of chain is the ratio of chain length to water depth; more scope improves holding. The anchor is let go when the vessel has minimal headway.

Firefighting: fires are classed A (solid), B (flammable liquid), C (gas), D (metal), and electrical; a CO₂ extinguisher suits electrical fires, water is never used on electrical or oil fires.