AEEE cheat sheet

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

Sum and product of roots of ax²+bx+c=0 are −b/a and c/a respectively; use these to build equations from given roots.

AP: nth term a+(n−1)d, sum (n/2)[2a+(n−1)d]. GP: nth term ar^(n−1), sum a(rⁿ−1)/(r−1) for r≠1, infinite sum a/(1−r) when |r|<1.

Standard limits: lim(x→0) sin x/x = 1, lim(x→0)(1−cos x)/x² = 1/2, lim(x→0)(eˣ−1)/x = 1, lim(x→0) ln(1+x)/x = 1.

Derivatives: d/dx(xⁿ)=nx^(n−1), d/dx(sin x)=cos x, d/dx(cos x)=−sin x, d/dx(tan x)=sec²x, d/dx(ln x)=1/x, d/dx(eˣ)=eˣ.

Integration by parts: ∫u dv = uv − ∫v du; choose u by the LIATE order (log, inverse trig, algebraic, trig, exponential).

Straight line: slope m=(y₂−y₁)/(x₂−x₁); point-slope y−y₁=m(x−x₁); perpendicular slopes satisfy m₁m₂=−1.

Circle x²+y²+2gx+2fy+c=0 has centre (−g,−f) and radius √(g²+f²−c); tangent length from (x₁,y₁) is √(x₁²+y₁²+2gx₁+2fy₁+c).

Probability: P(A∪B)=P(A)+P(B)−P(A∩B); conditional P(A|B)=P(A∩B)/P(B); Bayes' theorem reverses the conditioning.

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

Newton's second law in vector form F=ma; friction limit f_max=μN; on an incline, the component along the plane is mg sin θ.

Work-energy theorem: net work equals change in kinetic energy; conservative forces allow potential energy U with F=−dU/dx.

Projectile: time of flight 2u sin θ/g, range u² sin 2θ/g, maximum height u² sin²θ/(2g); range is maximum at θ=45°.

Momentum conservation holds when net external force is zero; for elastic collisions kinetic energy is also conserved and e=1.

Ohm's law V=IR; series resistance adds, parallel conductance adds; power P=VI=I²R=V²/R.

Kirchhoff's laws: junction rule (charge conservation) and loop rule (energy conservation); use them for multi-loop circuits.

Lens/mirror formula 1/f=1/v−1/u for lenses and 1/f=1/v+1/u for mirrors, with sign convention applied consistently.

Photoelectric effect: maximum kinetic energy = hν−φ, where φ is the work function; no emission below threshold frequency ν₀=φ/h.

Mole concept: 1 mol = 6.022×10²³ particles; molar mass in g/mol; 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 for a fixed amount.

pH=−log[H⁺]; pOH=−log[OH⁻]; pH+pOH=14 at 25 °C; strong acid/base calculations use complete dissociation.

Buffer (Henderson–Hasselbalch): pH=pKa+log([salt]/[acid]); buffers resist pH change on adding small amounts of acid or base.

Le Chatelier: adding reactant, removing product, or compressing a gas system shifts equilibrium toward fewer gas moles; catalysts change rate, not position.

Balancing redox: equalize electrons transferred, then combine half-reactions; in acidic medium balance O with H₂O and H with H⁺.

Hybridization from steric number: 2→sp, 3→sp², 4→sp³; VSEPR predicts geometry from electron-pair repulsion.

Bond order from molecular orbital theory = (bonding electrons − antibonding electrons)/2; higher bond order means shorter, stronger bond.

First law of thermodynamics ΔU=q+w; enthalpy change at constant pressure equals heat exchanged; Hess's law allows summing reaction enthalpies.

Percentage change = (new−old)/old ×100; successive changes of a% and b% give a net change of a+b+ab/100 percent.

Profit and loss: profit% = (SP−CP)/CP×100; discount% is calculated on marked price, not cost price.

Simple interest SI=PRT/100; compound interest A=P(1+R/100)ⁿ; CI−SI for two years equals P(R/100)².

Time-speed-distance: distance=speed×time; for a train crossing a platform, total distance = train length + platform length.

Boats and streams: downstream speed = boat + stream, upstream = boat − stream; average speed over equal distances is the harmonic mean 2xy/(x+y).

Work: if A finishes in a days and B in b days, together they take ab/(a+b) days; rate of work adds for simultaneous workers.

Ratio and proportion: if a:b=c:d then ad=bc; to divide amount N in ratio a:b:c, shares are N·a/(a+b+c) and so on.

Permutations and combinations: nPr=n!/(n−r)!, nCr=n!/[r!(n−r)!]; arrangements of a word with repeated letters divide by factorials of repeats.

Averages and mixtures: alligation gives ratio of quantities = (mean − cheaper):(dearer − mean) for two-component mixing.

Subject-verb agreement: a singular subject takes a singular verb; phrases between subject and verb do not change number ("The box of pens is").

Neither/nor and either/or take a verb agreeing with the nearer subject; "each", "every", "everyone" are grammatically singular.

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

Active to passive: object becomes subject, verb becomes be + past participle, agent introduced by "by"; tense of the original is preserved.

Reported speech shifts tense back one step and changes time/place words (now→then, today→that day, here→there) unless reporting verb is present.

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

Common confusions: affect (verb) vs effect (noun); principal (main/head) vs principle (rule); lie (recline) vs lay (place something).

Prepositions with fixed collocations: "comprise" takes no "of", "discuss" takes no "about", "superior to" not "superior than".

Reading comprehension: locate the claim in the passage before choosing an option; eliminate choices that are true in general but unsupported by the text.

For ax²+bx+c=0 the roots are [−b±√(b²−4ac)]/(2a); the sign of b²−4ac tells whether roots are real and distinct, equal, or complex.

The sum of roots is −b/a and the product is c/a, so a quadratic with given roots r₁, r₂ is x²−(r₁+r₂)x+r₁r₂=0.

Arithmetic progression nth term a+(n−1)d and sum (n/2)(first+last); geometric progression infinite sum a/(1−r) exists only when |r|<1.

The derivative of xⁿ is nx^(n−1), of sin x is cos x, of eˣ is eˣ, and of ln x is 1/x; use these before applying product or chain rules.

For integration by parts ∫u dv = uv−∫v du, pick u in the order logarithm, inverse trig, algebraic, trig, exponential.

A line perpendicular to one with slope m has slope −1/m; parallel lines share the same slope.

The circle x²+y²+2gx+2fy+c=0 has centre (−g,−f) and radius √(g²+f²−c).

Standard limits include sin x/x → 1 and (1−cos x)/x² → 1/2 as x → 0; these resolve many 0/0 forms without L'Hôpital.

P(A∪B)=P(A)+P(B)−P(A∩B); for independent events P(A∩B)=P(A)P(B).

Constant-acceleration equations v=u+at, s=ut+½at² and v²=u²+2as fail when acceleration varies; use energy or calculus then.

Maximum static friction equals μN, where N is the normal force; on an incline of angle θ the along-plane component of weight is mg sin θ.

For a projectile launched at speed u and angle θ: range u²sin2θ/g, maximum height u²sin²θ/(2g), time of flight 2u sinθ/g.

Work done by a net force equals the change in kinetic energy; for a conservative force the potential gradient satisfies F=−dU/dx.

Momentum is conserved when net external force is zero; kinetic energy is conserved only in elastic collisions (coefficient of restitution e=1).

In a series circuit current is common and resistances add; in parallel the voltage is common and 1/R_total=Σ1/R.

Kirchhoff's junction rule encodes charge conservation and the loop rule encodes energy conservation per cycle.

The photoelectric equation is hν=φ+KE_max; emission stops below threshold frequency ν₀=φ/h regardless of intensity.

The mirror formula 1/f=1/v+1/u and lens formula 1/f=1/v−1/u give correct results only with the sign convention applied consistently.

One mole contains 6.022×10²³ particles and, for an ideal gas at STP, occupies 22.4 L; molar mass converts grams to moles.

Ideal gas law PV=nRT with R=8.314 J/(mol·K); for fixed n, P₁V₁/T₁=P₂V₂/T₂ with temperature in kelvin.

pH=−log[H⁺] and pH+pOH=14 at 25 °C; a tenfold increase in [H⁺] lowers pH by exactly one unit.

Henderson–Hasselbalch pH=pKa+log([salt]/[acid]) applies to buffer solutions; maximum buffering occurs when pH=pKa.

By Le Chatelier's principle, compressing a gaseous equilibrium shifts it toward the side with fewer gas moles; a catalyst does not shift it.

Steric number 2, 3, 4 gives sp, sp², sp³ hybridization; VSEPR then predicts linear, trigonal planar, or tetrahedral geometry.

Molecular orbital bond order = (bonding − antibonding electrons)/2; bond order zero means the molecule does not exist.

First law of thermodynamics ΔU=q+w; at constant pressure the enthalpy change equals the heat exchanged, enabling Hess's law summation.

In acidic redox balancing, oxygen is balanced with H₂O and hydrogen with H⁺ after electrons are equalized between half-reactions.

A percentage change is (new−old)/old×100; two successive changes a% then b% give a net change of a+b+ab/100 percent.

Profit percent is computed on cost price while discount percent is computed on marked price; SP=CP(1+profit/100).

Compound interest exceeds simple interest over two years by P(R/100)²; simple interest is PRT/100.

Average speed for equal distances at speeds x and y is 2xy/(x+y), the harmonic mean, not the arithmetic mean.

If A alone takes a days and B alone b days, working together they finish in ab/(a+b) days.

For a train crossing a platform, the distance travelled is train length plus platform length; for a pole it is train length only.

Downstream speed is boat speed plus stream speed; upstream is boat minus stream, so stream speed is half the difference of the two.

Alligation for two components gives quantity ratio (mean−cheaper):(dearer−mean).

nCr=n!/[r!(n−r)!] counts selections; nPr counts arrangements; permutations of a word with repeated letters divide by factorials of the repeats.

A verb agrees with its subject in number even when a prepositional phrase intervenes: "The list of items is long."

With either/or and neither/nor the verb agrees with the nearer subject; each, every, everyone, and nobody take singular verbs.

Present perfect (has/have + past participle) connects a past action to the present; past perfect marks the earlier of two past actions.

In passive voice the object becomes subject and the verb becomes a form of be plus past participle, keeping the original tense.

Reported speech shifts tense one step back and changes time and place words (now→then, here→there) when the reporting verb is past.

Third conditional: if + had + past participle, then would have + past participle; it describes an unreal past.

Affect is a verb, effect is usually a noun; principal means main or head, principle means a rule.

Fixed collocations: comprise takes no "of", discuss takes no "about", and superior is followed by "to", not "than".

In comprehension questions, reject options that are generally true but not stated or implied by the given passage.

The quadratic formula x=[−b±√(b²−4ac)]/(2a) solves ax²+bx+c=0; a negative discriminant gives a complex conjugate pair.

Vieta's relations give sum of roots −b/a and product c/a, useful for forming equations and checking answers.

For a geometric progression, sum to n terms is a(rⁿ−1)/(r−1) when r≠1; the infinite sum a/(1−r) requires |r|<1.

Differentiate xⁿ as nx^(n−1), sin x as cos x, cos x as −sin x, tan x as sec²x, and ln x as 1/x.

The chain rule states d/dx f(g(x)) = f′(g(x))·g′(x); the product rule is (uv)′=u′v+uv′.

The slope of a line through two points is (y₂−y₁)/(x₂−x₁); perpendicular lines multiply to slope −1.

For x²+y²+2gx+2fy+c=0, the centre is (−g,−f) and radius is √(g²+f²−c); the tangent length from an external point uses the same expression.

Conditional probability P(A|B)=P(A∩B)/P(B), and Bayes' theorem expresses P(A|B) in terms of P(B|A) and the priors.

Limits sin x/x→1 and (eˣ−1)/x→1 as x→0 are the standard tools for trigonometric and exponential indeterminate forms.

The equations v=u+at, s=ut+½at², and v²=u²+2as hold only for uniform acceleration; otherwise integrate acceleration over time.

Friction force is at most μN; kinetic friction is approximately μ_k N and acts opposite to relative motion.

Projectile range is u²sin2θ/g, which is maximum at 45°; complementary launch angles give equal ranges.

The work-energy theorem states net work equals ΔKE; for conservative forces mechanical energy is conserved.

Momentum conservation requires zero net external force; perfectly inelastic collisions conserve momentum but lose maximum kinetic energy.

Series resistors add directly; parallel resistors combine as 1/R=1/R₁+1/R₂, giving a total smaller than the smallest branch.

Kirchhoff's loop rule sums potential differences to zero around any closed path; the junction rule sums currents at a node to zero.

Photoelectric emission requires frequency above ν₀=φ/h; increasing intensity above threshold raises current, not maximum electron energy.

The lens formula 1/f=1/v−1/u and mirror formula 1/f=1/v+1/u require consistent real/virtual sign conventions.

Molar mass in g/mol links mass to moles; one mole of any ideal gas at STP occupies 22.4 L and contains 6.022×10²³ particles.

PV=nRT with R=8.314 J/(mol·K); temperatures must be in kelvin in all gas-law calculations.

pH=−log[H⁺] and pOH=−log[OH⁻]; at 25 °C their sum is 14, so a pH of 3 implies pOH 11.

A buffer's pH follows pH=pKa+log([salt]/[acid]) and changes little on adding small amounts of strong acid or base.

Le Chatelier's principle: raising pressure shifts a gaseous equilibrium toward fewer moles of gas; adding a catalyst leaves the position unchanged.

Hybridization follows steric number: 2→sp (linear), 3→sp² (trigonal planar), 4→sp³ (tetrahedral).

Bond order = (bonding − antibonding electrons)/2; O₂ has bond order 2 with two unpaired electrons in antibonding π* orbitals.

ΔU=q+w is the first law; at constant pressure q equals ΔH, so Hess's law lets you add enthalpy changes of steps.

Balancing redox in acid: equalize electrons, balance O with H₂O, then H with H⁺, and finally check charge balance.

Successive percentage changes of a% and b% produce a net change of a+b+ab/100 percent, not simply a+b.

Profit percent uses cost price as base; discount percent uses marked price, so SP=MP(1−discount/100).

Simple interest is PRT/100; compound amount is P(1+R/100)ⁿ, and CI−SI over two years equals P(R/100)².

Average speed over two equal distances at speeds x and y is 2xy/(x+y); over unequal distances use total distance over total time.

Combined work rate adds: if A does 1/a of a job per day and B does 1/b, together they take ab/(a+b) days.

Train crossing problems: distance is train length plus platform length for a platform, but only train length for a pole or signal.

Boat problems: downstream = boat+stream, upstream = boat−stream; stream speed is half the difference of downstream and upstream speeds.

Alligation gives the mixing ratio of two components as (mean−cheaper):(dearer−mean).

Combinations nCr=n!/[r!(n−r)!] count selections where order is irrelevant; permutations nPr count ordered arrangements.

A singular subject takes a singular verb regardless of intervening phrases: "The bouquet of roses smells sweet."

Either/or and neither/nor agree with the nearer subject; indefinite pronouns each, everyone, and nobody are singular.

Present perfect (has/have + past participle) links a past action to the present; past perfect marks the earlier of two past events.

To form the passive, make the object the subject and use be + past participle, preserving the original tense.

Reported speech after a past reporting verb shifts tense back one step and changes now→then, today→that day, here→there.

Second conditional uses if + past simple with would + base verb; third conditional uses if + had + participle with would have + participle.

Affect is a verb meaning to influence; effect is usually a noun meaning result; principal means main, principle means rule.

Comprise takes no "of", discuss takes no "about", and superior, inferior, and senior take "to" rather than "than".

For reading comprehension, verify each option against the passage; generally true statements unsupported by the text are wrong answers.

The discriminant b²−4ac decides the nature of quadratic roots: positive for two real roots, zero for a repeated root, negative for complex conjugates.

Sum and product of roots of ax²+bx+c=0 are −b/a and c/a; a quadratic with roots r₁ and r₂ is x²−(r₁+r₂)x+r₁r₂.

AP sum is (n/2)(2a+(n−1)d); GP sum is a(rⁿ−1)/(r−1) for r≠1 and a/(1−r) for |r|<1.

Derivatives to memorize: d/dx(xⁿ)=nx^(n−1), d/dx(sin x)=cos x, d/dx(cos x)=−sin x, d/dx(tan x)=sec²x, d/dx(ln x)=1/x.

Integration by parts ∫u dv=uv−∫v du with u chosen by the LIATE order: log, inverse trig, algebraic, trig, exponential.

Parallel lines have equal slopes; perpendicular lines satisfy m₁m₂=−1; the point-slope form is y−y₁=m(x−x₁).

Circle x²+y²+2gx+2fy+c=0: centre (−g,−f), radius √(g²+f²−c); tangent length from (x₁,y₁) is √(x₁²+y₁²+2gx₁+2fy₁+c).

Standard limits: sin x/x→1, (1−cos x)/x²→1/2, (eˣ−1)/x→1, and ln(1+x)/x→1 as x→0.

Probability addition rule P(A∪B)=P(A)+P(B)−P(A∩B); for independent events P(A∩B)=P(A)·P(B).

Kinematic equations v=u+at, s=ut+½at², v²=u²+2as apply only when acceleration is constant.

Friction is bounded by f_max=μN; on an incline of angle θ the weight component along the plane is mg sinθ and the normal force is mg cosθ.

Projectile motion: time of flight 2u sinθ/g, maximum height u²sin²θ/(2g), range u²sin2θ/g, maximum range at 45°.

Work-energy theorem: the net work on a body equals its change in kinetic energy; conservative forces define potential energy via F=−dU/dx.

Momentum is conserved when net external force is zero; elastic collisions conserve kinetic energy and have coefficient of restitution 1.

Series resistances add; parallel resistances satisfy 1/R=1/R₁+1/R₂, giving a total below the smallest resistor; power is I²R or V²/R.

Kirchhoff's junction rule expresses charge conservation and the loop rule expresses energy conservation around a closed circuit.

The photoelectric equation hν=φ+KE_max means no emission below threshold frequency ν₀=φ/h, whatever the intensity.

Lens formula 1/f=1/v−1/u and mirror formula 1/f=1/v+1/u give correct results only with a consistent sign convention.

One mole equals 6.022×10²³ particles and, for an ideal gas at STP, 22.4 L; divide mass by molar mass to obtain moles.

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

pH=−log[H⁺] and pH+pOH=14 at 25 °C; lowering pH by one unit means ten times more H⁺.

Henderson–Hasselbalch pH=pKa+log([salt]/[acid]) gives buffer pH; buffering is best when pH is near pKa.

Le Chatelier's principle: adding reactant or removing product shifts forward; compressing a gas equilibrium favors the side with fewer gas moles.

Steric number 2, 3, and 4 correspond to sp, sp², and sp³ hybridization with linear, trigonal planar, and tetrahedral shapes.

Bond order in molecular orbital theory is (bonding−antibonding)/2; higher bond order means a shorter and stronger bond.

First law ΔU=q+w; at constant pressure heat exchanged equals ΔH, and Hess's law allows summing step enthalpies.

Redox balancing in acid: equalize electrons, balance oxygen with H₂O, balance hydrogen with H⁺, then verify charge balance.

Percentage change = (new−old)/old×100; successive changes a% and b% give a+b+ab/100 percent net change.

Profit percent is on cost price and discount percent is on marked price; SP=CP(1+profit/100)=MP(1−discount/100).

Simple interest SI=PRT/100; compound amount A=P(1+R/100)ⁿ; over two years CI−SI=P(R/100)².

Average speed for equal distances at speeds x and y is 2xy/(x+y); for unequal distances use total distance divided by total time.

Work rates add: A in a days and B in b days together take ab/(a+b) days.

A train crossing a platform covers train length plus platform length; crossing a pole covers only the train length.

Downstream speed = boat+stream and upstream = boat−stream; stream speed is half the difference of the two.

Alligation gives the ratio of mixed quantities as (mean−cheaper):(dearer−mean).

nCr=n!/[r!(n−r)!] counts unordered selections; nPr=n!/(n−r)! counts ordered arrangements; repeated letters divide by factorials of repeats.

Subject-verb agreement is unaffected by phrases between subject and verb: "The box of pens is heavy."

Neither/nor and either/or take a verb matching the nearer subject; each, every, everyone, and nobody are singular.

Present perfect (has/have + past participle) links past to present; past perfect (had + participle) marks the earlier past action.

Passive voice: object becomes subject, verb becomes be + past participle, and the original tense is retained.

Reported speech after a past verb shifts tense back one step and changes time and place words such as now→then and here→there.

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

Affect is a verb, effect is a noun; principal means main, principle means rule; lie means recline, lay means place.

Fixed usage: comprise takes no "of", discuss takes no "about", and superior takes "to" rather than "than".

In comprehension, eliminate options that are generally true but not supported by the passage before choosing.

Quadratic formula: x=[−b±√(b²−4ac)]/(2a); the discriminant b²−4ac gives real distinct, equal, or complex roots.

Sum of roots −b/a and product c/a follow from Vieta's relations and let you reconstruct a quadratic from its roots.

AP: nth term a+(n−1)d, sum (n/2)[2a+(n−1)d]; GP: nth term ar^(n−1), sum a(rⁿ−1)/(r−1) for r≠1, infinite sum a/(1−r) for |r|<1.

Derivatives: xⁿ→nx^(n−1), sin x→cos x, cos x→−sin x, tan x→sec²x, ln x→1/x, eˣ→eˣ; apply the chain rule for composites.

Integration by parts ∫u dv=uv−∫v du, choosing u by LIATE order (log, inverse trig, algebraic, trig, exponential).

Line slope m=(y₂−y₁)/(x₂−x₁); point-slope y−y₁=m(x−x₁); perpendicular slopes satisfy m₁m₂=−1.

Circle x²+y²+2gx+2fy+c=0 has centre (−g,−f) and radius √(g²+f²−c).

Standard limits: sin x/x→1, (1−cos x)/x²→1/2, (eˣ−1)/x→1, ln(1+x)/x→1 as x→0.

Probability: P(A∪B)=P(A)+P(B)−P(A∩B); conditional P(A|B)=P(A∩B)/P(B); independent events satisfy P(A∩B)=P(A)P(B).

Constant-acceleration kinematics: v=u+at, s=ut+½at², v²=u²+2as; these fail if acceleration changes with time.

Friction limit f_max=μN; on an incline the along-plane weight component is mg sinθ while the normal force is mg cosθ.

Projectile: time of flight 2u sinθ/g, range u²sin2θ/g, maximum height u²sin²θ/(2g); range peaks at 45°.

Net work equals change in kinetic energy; for conservative forces F=−dU/dx and mechanical energy is conserved.

Momentum conservation applies when net external force is zero; elastic collisions also conserve kinetic energy with e=1.

Series resistors add; parallel resistors satisfy 1/R=1/R₁+1/R₂; electrical power is P=VI=I²R=V²/R.

Kirchhoff's junction rule encodes charge conservation; the loop rule encodes energy conservation around a closed path.

Photoelectric effect: KE_max=hν−φ, so no emission occurs below the threshold frequency ν₀=φ/h regardless of intensity.

Lens formula 1/f=1/v−1/u and mirror formula 1/f=1/v+1/u are valid only with a consistent sign convention.

One mole is 6.022×10²³ particles; one mole of ideal gas occupies 22.4 L at STP; molar mass converts mass to moles.

Ideal gas law PV=nRT with R=8.314 J/(mol·K); for fixed amount, P₁V₁/T₁=P₂V₂/T₂ with T in kelvin.

pH=−log[H⁺], pOH=−log[OH⁻], and pH+pOH=14 at 25 °C; strong acids dissociate completely.

Henderson–Hasselbalch pH=pKa+log([salt]/[acid]) describes buffers, which resist pH change on adding small amounts of acid or base.

Le Chatelier's principle: adding reactant or removing product shifts forward; compressing a gas equilibrium favors fewer gas moles; catalysts do not shift it.

Hybridization from steric number: 2→sp, 3→sp², 4→sp³; VSEPR then predicts linear, trigonal planar, or tetrahedral geometry.

Molecular orbital bond order=(bonding−antibonding electrons)/2; higher bond order means shorter, stronger bonds.

First law ΔU=q+w; at constant pressure q=ΔH, and Hess's law allows summing enthalpy changes of individual steps.

Acidic redox balancing: equalize electrons, balance O with H₂O, balance H with H⁺, then check that charges balance.

Percentage change=(new−old)/old×100; two successive changes a% and b% give a net change of a+b+ab/100 percent.

Profit percent uses cost price; discount percent uses marked price; SP=CP(1+profit/100) and SP=MP(1−discount/100).

Simple interest SI=PRT/100; compound amount A=P(1+R/100)ⁿ; the two-year gap CI−SI equals P(R/100)².

Average speed over equal distances at speeds x and y is 2xy/(x+y); otherwise use total distance over total time.

If A alone takes a days and B alone b days, together they take ab/(a+b) days because their rates add.

Train crossing a platform covers train length plus platform length; crossing a pole covers only train length.

Boat speed downstream is boat+stream and upstream is boat−stream; stream speed is half the difference.

Alligation gives the mixing ratio (mean−cheaper):(dearer−mean) for two components.

nCr=n!/[r!(n−r)!] counts selections; nPr=n!/(n−r)! counts arrangements; divide by factorials of repeated letters for word permutations.

A verb agrees with its subject in number even when a prepositional phrase intervenes: "The list of items is long."

Either/or and neither/nor agree with the nearer subject; each, every, everyone, and nobody take singular verbs.

Present perfect (has/have + past participle) connects past action to present; past perfect marks the earlier of two past events.

Passive voice makes the object the subject and uses be + past participle while preserving the original tense.

Reported speech after a past reporting verb shifts tense back one step and changes now→then, today→that day, here→there.

Third conditional: if + had + past participle, then would have + past participle; it describes an unreal past situation.

Affect is a verb meaning to influence; effect is a noun meaning result; principal means main, principle means rule.

Comprise takes no "of", discuss takes no "about", and superior, inferior, and senior take "to", not "than".

In comprehension, reject options that are generally true but not stated or implied by the passage.

For ax²+bx+c=0, roots are [−b±√(b²−4ac)]/(2a); the discriminant sign determines real distinct, repeated, or complex roots.

Vieta: sum of roots=−b/a and product=c/a, so a quadratic from roots r₁, r₂ is x²−(r₁+r₂)x+r₁r₂.

AP nth term a+(n−1)d and sum (n/2)(first+last); GP infinite sum a/(1−r) exists only when |r|<1.

Derivative rules: d/dx(xⁿ)=nx^(n−1), d/dx(sin x)=cos x, d/dx(cos x)=−sin x, d/dx(tan x)=sec²x, d/dx(ln x)=1/x.

Integration by parts ∫u dv=uv−∫v du with u selected by LIATE: log, inverse trig, algebraic, trig, exponential.

Perpendicular lines have slopes whose product is −1; parallel lines have equal slopes; point-slope form is y−y₁=m(x−x₁).

Circle x²+y²+2gx+2fy+c=0 has centre (−g,−f) and radius √(g²+f²−c); tangent length from (x₁,y₁) is √(x₁²+y₁²+2gx₁+2fy₁+c).

Limits sin x/x→1, (1−cos x)/x²→1/2, (eˣ−1)/x→1, and ln(1+x)/x→1 as x→0 resolve common indeterminate forms.

P(A∪B)=P(A)+P(B)−P(A∩B); conditional P(A|B)=P(A∩B)/P(B); independence means P(A∩B)=P(A)P(B).

v=u+at, s=ut+½at², and v²=u²+2as hold only for constant acceleration; variable acceleration requires integration.

Friction is at most μN; on an incline of angle θ, the along-plane weight component is mg sinθ and the normal force is mg cosθ.

Projectile range is u²sin2θ/g, maximum at 45°; maximum height is u²sin²θ/(2g) and time of flight 2u sinθ/g.

The work-energy theorem equates net work to ΔKE; conservative forces allow potential energy with F=−dU/dx.

Momentum conservation needs zero net external force; perfectly inelastic collisions conserve momentum but lose the most kinetic energy.

Parallel resistors combine as 1/R=1/R₁+1/R₂, giving a total smaller than the smallest branch; series resistors add directly.

Kirchhoff's junction rule is charge conservation and the loop rule is energy conservation around a closed circuit path.

Photoelectric emission requires ν>ν₀=φ/h; raising intensity above threshold increases current, not maximum electron kinetic energy.

Lens formula 1/f=1/v−1/u and mirror formula 1/f=1/v+1/u need consistent real/virtual sign conventions.

Molar mass in g/mol converts mass to moles; one mole contains 6.022×10²³ particles and occupies 22.4 L as an ideal gas at STP.

PV=nRT with R=8.314 J/(mol·K); all gas-law temperatures must be in kelvin.

pH=−log[H⁺] and pOH=−log[OH⁻]; at 25 °C pH+pOH=14, so a pH of 3 corresponds to pOH 11.

Buffer pH follows pH=pKa+log([salt]/[acid]); buffering is most effective when pH is close to pKa.

Le Chatelier: increasing pressure shifts a gaseous equilibrium toward fewer gas moles; adding a catalyst does not shift the position.

Steric number 2, 3, 4 gives sp, sp², sp³ hybridization with linear, trigonal planar, and tetrahedral geometry.

Bond order=(bonding−antibonding electrons)/2; O₂ has bond order 2 and two unpaired electrons in antibonding π* orbitals.

First law ΔU=q+w; at constant pressure q=ΔH, and Hess's law permits summing the enthalpies of individual steps.

Balancing redox in acid: equalize electrons, balance O with H₂O, balance H with H⁺, and verify charge balance.

Successive percentage changes of a% and b% yield a net change of a+b+ab/100 percent, not a simple sum.

Profit percent is based on cost price while discount percent is based on marked price, so SP=CP(1+profit/100)=MP(1−discount/100).

Simple interest PRT/100 versus compound amount P(1+R/100)ⁿ; over two years compound interest exceeds simple interest by P(R/100)².

Average speed for two equal distances at speeds x and y is 2xy/(x+y), the harmonic mean, not (x+y)/2.

Combined work: if A takes a days and B takes b days, together they take ab/(a+b) days since rates add.

Train crossing problems use train length plus platform length for a platform, but only train length for a pole or signal.

Downstream speed=boat+stream and upstream=boat−stream; stream speed is half the difference of the two speeds.

Alligation gives the ratio of two mixed quantities as (mean−cheaper):(dearer−mean).

nCr=n!/[r!(n−r)!] counts unordered selections while nPr=n!/(n−r)! counts ordered arrangements.

Subject-verb agreement ignores intervening prepositional phrases: "The bouquet of roses smells sweet."

Either/or and neither/nor agree with the nearer subject; each, everyone, and nobody are singular.

Present perfect (has/have + past participle) links a past action to the present; past perfect marks the earlier of two past events.

To form the passive, promote the object to subject and use be + past participle, preserving the original tense.

Reported speech after a past reporting verb shifts tense back one step and changes now→then, today→that day, here→there.

Second conditional is if + past simple with would + base verb; third conditional is if + had + participle with would have + participle.

Affect is a verb, effect is a noun; principal means main, principle means rule; lie means recline, lay means place.

Comprise takes no "of", discuss takes no "about", and superior, inferior, and senior take "to" rather than "than".

For comprehension, check every option against the passage; statements that are generally true but unsupported are wrong.