Concours Commun Mines-Ponts - independent subject practice common mistakes
Concours Commun Mines-Ponts - independent subject practice · Understand the traps and how to avoid them.
Interchanging a limit and integral from pointwise convergence alone
Pointwise convergence does not supply the uniform or integrable control needed by the relevant theorem.
What to do instead
Name a sufficient convergence theorem and verify its assumptions on the exact domain before exchanging the operations.
Treating a semidefinite Hessian as permitting every extremum
A positive quadratic direction can already rule out a maximum even when a zero direction leaves a minimum or saddle unresolved.
What to do instead
Check known directions, then analyze higher-order behavior in the null directions without contradicting the nonzero curvature.
Using a sufficient spectral condition as a characterization
Normality gives Euclidean-norm/spectral-radius equality, but the equality can hold for some non-normal matrices.
What to do instead
State the implication direction and test a block-matrix counterexample before asserting an if-and-only-if theorem.
Computing a projection with the wrong inner product
Euclidean coordinate intuition can differ from a weighted or integral inner product.
What to do instead
Build the correct Gram system and verify that the residual is orthogonal to every basis vector.
Assuming Newton converges from every starting point
A small derivative, multiple root or unsuitable initialization can invalidate an expected local-convergence argument.
What to do instead
Check the hypotheses, use a bracket or safeguarded method where appropriate and assess a meaningful residual and error criterion.
Equating a stable numerical step with an accurate solution
Stability controls growth but does not itself make truncation error acceptable.
What to do instead
Check the stability restriction, compare mesh refinements and evaluate accuracy against the specified tolerance.
Applying conservation outside its system boundary
External forces, torque, heat or work can change the conserved quantity for the selected system.
What to do instead
Define the system and frame, identify exchanges and state the assumptions supporting each conservation relation.
Assuming adiabatic always means constant entropy
Irreversible processes can generate entropy even when heat transfer is absent.
What to do instead
Distinguish no heat transfer from reversibility and track entropy generation under the actual model.
In a limiting-reactant problem, the candidate calculates moles of product from each reactant separately and then averages the two results, or selects the reactant with the smaller initial mass as the limiting reactant.
Averaging is a mathematical habit from other contexts; selecting by mass ignores molar mass and stoichiometry. The candidate may also skip the step of dividing moles by the stoichiometric coefficient.
What to do instead
For each reactant, convert mass to moles, then divide by its coefficient in the balanced equation to get the 'reaction extent'. The smallest extent identifies the limiting reactant. Calculate product moles from that reactant only. Always check that the other reactant is in excess.
When using Hess's law or calorimetry data, the candidate forgets to reverse the sign of ΔH when reversing a reaction, or fails to multiply ΔH by the stoichiometric factor when scaling a reaction.
The candidate treats ΔH as an ordinary number rather than a state function that depends on direction and amount. They may also confuse ΔH of formation with ΔH of reaction and use tabulated values without considering the coefficients.
What to do instead
Write each reaction with its ΔH value. When reversing, change the sign; when multiplying by a factor, multiply ΔH by the same factor. Then add the equations and their ΔH values. For formation enthalpies, use ΔH°rxn = Σ nΔH°f(products) − Σ mΔH°f(reactants), being careful with coefficients.
The candidate calculates an equilibrium constant using initial concentrations instead of equilibrium concentrations, or uses the reaction quotient Q and calls it K without evaluating at equilibrium.
The candidate confuses the initial state with the equilibrium state, especially in ICE table problems where they forget to subtract the change. They may also assume that K can be calculated directly from stoichiometry without experimental data.
What to do instead
Always set up an ICE table: write initial concentrations, define the change in terms of x, and express equilibrium concentrations. Substitute only the equilibrium expressions into the K expression. For Q, use current concentrations; compare Q to K to determine direction. Practice problems where the change is not negligible.
When calculating the pH of a weak acid solution, the candidate uses the initial concentration directly in the Ka expression without accounting for the degree of ionization, or assumes pH = pKa for any buffer regardless of concentration ratio.
The candidate overgeneralizes the approximation that weak acids ionize only slightly, forgetting that for moderately weak acids the x cannot be neglected. For buffers, they may misapply the Henderson–Hasselbalch equation without checking the ratio of concentrations or the validity of the approximation.
What to do instead
For weak acids, write Ka = x²/(C − x) and solve the quadratic if x > 5% of C. For buffers, use pH = pKa + log([A⁻]/[HA]) and verify that the ratio is between 0.1 and 10 and that concentrations are much larger than Ka. Always check assumptions.