Concours Commun Mines-Ponts - independent subject practice cheat sheet

Control quantifiers and the domain in pointwise versus uniform convergence.

Verify hypotheses before exchanging limits, integrals or derivatives.

A positive Hessian direction rules out a local maximum.

Theorem normalizations, constraints and topology cannot be omitted.

Distinguish similarity of maps from congruence of forms.

Projection coefficients follow residual orthogonality and the actual Gram matrix.

Repeated eigenvalues do not automatically imply diagonalizability.

Norm-spectral-radius equality is not an if-and-only-if test for normality.

Include covariance in the variance of a linear combination.

Independence is stronger than zero covariance.

Bounds are not exact probabilities.

Limit, testing and interval conclusions require their stated assumptions.

Define coordinates, frames, constraints and the system boundary.

Small-oscillation frequencies depend on mass as well as stiffness.

Momentum and kinetic-energy conservation have different conditions.

Fluid and rotating-frame formulas require their actual hypotheses.

Use geometry and boundary conditions, not symmetry by assertion.

An ideal magnetic force changes direction without doing work.

Reversible adiabatic and arbitrary adiabatic processes are different.

Thermal efficiency limits use absolute temperature and stated reservoir assumptions.

n = m/M, n = cV (V in dm3) and n = V/24.0 for gases at RTP are the three key conversions.

The limiting reagent is the one with the smallest moles ÷ coefficient; product amount depends on it.

Percentage yield measures practical success; atom economy measures how much reactant mass is useful.

Dilution uses c1V1 = c2V2 because the moles of solute do not change.

Always balance the equation before applying any molar ratio.

ΔH is negative for exothermic reactions; standard enthalpy of formation of an element in its standard state is zero.

q = mcΔT uses the total mass of solution, and ΔH per mole = −q/n.

Hess's law lets you combine any route; combustion data are used in the reverse order to formation data.

ΔH = Σ bonds broken − Σ bonds formed, and mean bond enthalpies are averages.

Feasibility is determined by ΔG = ΔH − TΔS, not by ΔH alone.

Orders are determined experimentally, not from stoichiometric coefficients.

Rate constant units depend on overall order: for second order overall, dm3 mol−1 s−1; for third order overall, dm6 mol−2 s−1.

Relative atomic mass is an abundance-weighted mean of isotope masses, calculated from mass-spectrum data.

Across a period effective nuclear charge rises, so radius falls and ionisation energy and electronegativity rise.

Large jumps in successive ionisation energies reveal the number of electrons in the outer shell.

Cr and Cu adopt 3d5 4s1 and 3d10 4s1 configurations; 4s electrons are removed first on ionisation.

Transition metals require a partially filled d subshell in at least one common oxidation state.

VSEPR: count all electron domains, then apply lone pair–lone pair > lone pair–bond pair > bond pair–bond pair repulsion.

Molecular polarity requires both polar bonds and an asymmetric arrangement of those bonds.

Higher bond order gives a shorter, stronger bond; resonance spreads the order over equivalent positions.

Only period 3 elements and below can expand their octet; N, O, F and C cannot.

Ionic lattice strength increases with ion charge and decreases with ion radius.

Convert to K and m3 before using PV = nRT; R = 8.314 J mol−1 K−1.

Real gases deviate from ideality at high pressure and low temperature.

Functional group priority determines the suffix and numbering in IUPAC names.

2,4-DNP is used for suitable aldehyde/ketone carbonyl tests; reducing tests have exceptions and must be interpreted with the supplied structure and conditions.

Primary alcohols oxidise to aldehyde then acid; secondary to ketone; tertiary alcohols are resistant.

For acyclic-alkene E/Z isomerism, each carbon needs two different groups; a tetrahedral carbon with four different groups is one common, not the only, source of chirality.

Markovnikov: the hydrogen adds to the carbon already carrying more hydrogens.

A linear noncyclic n-residue peptide has n−1 peptide links in the stated model.

Protein folding changes do not necessarily hydrolyze the primary peptide backbone.

Triglycerides contain ester links; saponification gives glycerol and carboxylate salts in excess base.

A permanent crosslinked network does not simply remelt like an ordinary thermoplastic.

Bio-based origin, fragmentation and verified biodegradation are different claims.

Recursion needs a base case and progress toward that base case.

Overloading uses different signatures; overriding replaces an inherited method.

Unit tests isolate units; integration tests combine them.

Short-circuit evaluation can skip the right operand.

Pass-by-value copies arguments; pass-by-reference can mutate the caller's object.

Decomposition breaks problems down; abstraction removes irrelevant detail.

Binary search requires sorted data and is O(log n).

Merge sort is O(n log n) worst case; bubble sort is O(n^2).

Trace tables prevent guessing about loop output.

Update search bounds to mid + 1 or mid - 1, not mid.

Sensitivity = Δoutput/Δinput; resolution = smallest detectable change

Closed-loop error compares reference and measurement; integral action can remove appropriate steady-state error only under suitable stability and operating assumptions.

PLC scan: read inputs → run program → update outputs

4–20 mA live zero: (I − 4)/16 of the span

Repeatability ≠ accuracy, resolution ≠ sensitivity

Robot DOF come from its joints; workspace is the reachable volume

Read modals, conditions, negation and logical connectors carefully.

Keep units, denominators and percentage-point distinctions.

Paraphrases must preserve qualification and scope.

A written technical exercise does not certify practical or oral competence.