ETEA Engineering Test cheat sheet

Constant-acceleration kinematics: v = u + at; s = ut + ½at²; v² = u² + 2as. These hold only for straight-line motion with uniform acceleration, not for changing force.

Newton's second law: F_net = ma = Δp/Δt. Action–reaction pairs act on two different bodies, so they never cancel each other on a single body.

Impulse and momentum: J = FΔt = Δp. In an isolated system the total momentum before and after a collision is equal, whether the collision is elastic or inelastic.

Work–energy theorem: W_net = ΔKE = ½mv² − ½mu². Friction converts mechanical energy to heat, so mechanical energy is conserved only when conservative forces alone act.

Circular motion: centripetal acceleration a = v²/r = ω²r directed to the centre; for a frictionless banked track the design angle satisfies tan θ = v²/(rg).

Simple harmonic motion: mass–spring T = 2π√(m/k); simple pendulum T = 2π√(L/g). Both periods are independent of amplitude for small oscillations (below about 10°).

Waves: v = fλ; speed of sound in air ≈ 331 + 0.6T °C m/s; moving-source frequency f′ = f(v ± v_o)/(v ∓ v_s), with signs chosen so approach raises pitch.

D.C. circuits: V = IR; P = VI = I²R = V²/R. Series resistors add, parallel conductances add, and Kirchhoff's junction and loop rules follow from charge and energy conservation.

Optics: mirror/lens relation 1/f = 1/v + 1/u with the standard sign convention; Snell's law n₁sin θ₁ = n₂sin θ₂; total internal reflection begins at sin θ_c = n₂/n₁ when n₁ > n₂.

Mole relations: n = m/M = N/N_A = V/22.4 for a gas in dm³ at STP, where N_A = 6.022 × 10²³ mol⁻¹.

Ideal gas equation PV = nRT requires absolute temperature; R = 8.314 J K⁻¹ mol⁻¹ or 0.0821 dm³ atm K⁻¹ mol⁻¹.

Solution concentration: molarity = moles of solute per dm³ of solution; dilution obeys M₁V₁ = M₂V₂ because the moles of solute stay fixed.

Equilibrium: Kp = Kc(RT)^Δn, where Δn is moles of gaseous product minus reactant. If the reaction quotient Q < K the reaction proceeds forward.

Acids and bases: pH = −log[H⁺], pOH = −log[OH⁻], pH + pOH = 14 at 25 °C; strong acids and bases ionise essentially completely.

Thermochemistry: ΔH = H(products) − H(reactants); a negative ΔH means exothermic, and Hess's law allows step enthalpies to be added to obtain the overall value.

Periodic trends: atomic radius falls across a period and rises down a group; ionisation energy and electronegativity rise across a period and fall down a group.

Redox: oxidation is loss of electrons (oxidation number increases) and reduction is gain; balance complex reactions by the half-reaction method in acidic or basic medium.

Hybridisation and shape: sp linear 180°, sp² trigonal planar 120°, sp³ tetrahedral 109.5°. VSEPR ranks lone-pair repulsion above bond-pair repulsion, compressing bond angles.

Quadratic: for ax² + bx + c = 0, x = (−b ± √(b² − 4ac))/2a; sum of roots = −b/a, product of roots = c/a; D = b² − 4ac decides the nature of the roots.

Binomial expansion: (a + b)ⁿ = Σ ⁿC_r a^(n−r) b^r; general term T_(r+1) = ⁿC_r a^(n−r) b^r, with ⁿC_r = n!/[r!(n−r)!].

Sequences: arithmetic S_n = n/2[2a + (n−1)d]; geometric S_n = a(rⁿ − 1)/(r − 1) and S∞ = a/(1 − r) valid only when |r| < 1.

Straight lines: m = (y₂ − y₁)/(x₂ − x₁); point-slope form y − y₁ = m(x − x₁); parallel lines share m, perpendicular lines satisfy m₁m₂ = −1.

Differentiation: d/dx(xⁿ) = nx^(n−1); apply the product, quotient and chain rules; L'Hôpital's rule applies only to 0/0 and ∞/∞ indeterminate forms.

Integration: ∫xⁿ dx = x^(n+1)/(n+1) + C for n ≠ −1; a definite integral gives signed area; integration by parts uses ∫u dv = uv − ∫v du.

Trigonometry: sin²θ + cos²θ = 1; law of sines a/sin A = b/sin B = c/sin C; law of cosines a² = b² + c² − 2bc cos A.

Matrices: det[[a,b],[c,d]] = ad − bc; inverse = (1/det)[[d,−b],[−c,a]] exists only when det ≠ 0; A(adj A) = det A · I.

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

Subject–verb agreement: intervening prepositional phrases do not change the number of the subject; each, every, either and neither take a singular verb.

Present perfect (has/have + past participle) links a past action to the present, and pairs with since for a point in time and for with a duration.

Conditionals: first = if + present, will + base; second = if + past, would + base; third = if + past perfect, would have + past participle.

Passive voice = form of be + past participle; only transitive verbs can be made passive, and the agent introduced by by is dropped when unknown or unimportant.

Articles: a/an with singular countable nouns on first mention; the for specific, unique or superlative nouns; no article for general plurals and uncountables.

Reported speech: after a past reporting verb, backshift one tense (present → past, past → past perfect) and adjust pronouns and time words such as now → then.

Prepositions: at + clock time, on + day or date, in + month, year or longer period; since + starting point, for + length of time.

Clause punctuation: a comma cannot join two independent clauses on its own - use a full stop, semicolon or coordinating conjunction; non-defining relative clauses are set off by commas.