PAU Applied Mathematics for Social Sciences II - Balearic Islands / UIB common mistakes
PAU Applied Mathematics for Social Sciences II - Balearic Islands / UIB · Understand the traps and how to avoid them.
When computing a determinant or the inverse of a 3×3 matrix, the candidate expands the cofactors using the wrong sign pattern, e.g. writing +,+,+ or applying the alternating signs starting from the wrong corner, producing a determinant that differs only in sign from the correct value.
The checkerboard sign pattern (+ − + / − + − / + − +) is memorised loosely as 'alternating' without anchoring it, and under exam pressure the candidate reproduces the signs by eye rather than from the (−1)^(i+j) rule, so a single misplaced sign flips the whole result.
What to do instead
Before every expansion, write the sign grid explicitly next to the matrix and derive each sign from (−1)^(i+j) for the row or column chosen. Then verify with an independent route: expand along a different row and confirm the same value, or check the determinant of a matrix with a zero row/column pattern. Practise five 3×3 determinants per session expanding along row 1, row 2, column 3, checking each against expansion along another line.
In a system with a parameter, the candidate solves only the generic case (dividing by an expression in the parameter) and reports one solution, omitting the special parameter values where the determinant vanishes and the system becomes incompatible or has infinitely many solutions.
Rouché–Capelli is applied mechanically as 'compute det(A) and if nonzero, Cramer', and the discussion of the degenerate cases is treated as an optional extra rather than the point of the question, so the candidate never returns to the values excluded during division.
What to do instead
Always structure the answer in three explicit stages: (1) det(A) as a polynomial in the parameter, (2) solve det(A)=0 and list those values, (3) for each such value substitute into the augmented matrix and compare rank(A) with rank(A|b). Write the final answer as a case table (parameter value → type of system → solution). Before dividing by anything containing the parameter, note the excluded values in the margin so they cannot be forgotten.
When computing a limit that leads to an indeterminate form, the candidate applies L'Hôpital directly without first checking the form, or applies it to a quotient that is not 0/0 or ∞/∞, and differentiates numerator and denominator as if differentiating a quotient (using the quotient rule instead of separate derivatives).
L'Hôpital is over-practised as a reflex, and the quotient rule is so recently drilled that it contaminates the procedure; the candidate also skips the substitution step that would reveal the form.
What to do instead
Adopt a fixed routine: substitute the value first and write the resulting form (0/0, ∞/∞, 0·∞, ∞−∞, 1^∞…) explicitly. Only for 0/0 and ∞/∞ apply L'Hôpital, and write the derivatives of numerator and denominator as two separate expressions, never as a quotient. For other forms, transform first (common denominator, conjugate, or take logarithms for 1^∞). Verify the final limit by a quick numerical check with a value close to the limit point.
In an optimisation problem, the candidate finds f′(x)=0 and reports those critical points as the answer, without verifying that the point lies in the feasible interval, without classifying it as a maximum or minimum, and without answering the actual question asked (the dimensions, the maximum area, the minimum cost).
The algebraic step (solve f′=0) is the part that feels like 'the maths', while translating back to the problem statement and checking endpoints is treated as cosmetic; the candidate also forgets that a constrained problem can have its optimum at a boundary.
What to do instead
End every optimisation with a four-line checklist written in the answer: domain of the variable from the geometry; critical points inside that domain; classification via the second derivative or a sign table of f′; evaluation of f at the critical point and at the endpoints. Then write a sentence in words giving the requested quantity with units. If the question asks for dimensions, the final line must state the dimensions, not just the value of the function.
In a probability problem, the candidate uses the binomial formula for a situation where draws are without replacement from a small population, or treats events as independent when the question describes sampling without replacement, and multiplies probabilities that should be conditional.
The binomial is the most recent and most drilled model, so any 'number of successes in n trials' phrasing triggers it, and the distinction between with and without replacement is buried in a single word of the statement that the candidate reads past.
What to do instead
Before choosing a model, answer two questions in writing: are the trials independent (with replacement, or a large population), and is the number of trials fixed? If sampling is without replacement from a small population, use the hypergeometric/combinatorial count, not the binomial. When events are not independent, write the conditional probability P(B|A) explicitly and use the multiplication rule P(A∩B)=P(A)P(B|A). Circle the words 'with/without replacement' and 'independent' in the statement before starting.
When working with the normal distribution, the candidate uses the raw value x in the standard normal table instead of standardising with z=(x−μ)/σ, or misuses the symmetry/complement to obtain a probability greater than one or a negative probability, and in inference problems confuses the standard deviation of the sample with the standard error σ/√n.
The standardisation step is treated as an optional preliminary, and the table is consulted from memory of 'look up the number'; the sample-versus-population distinction is blurred because both are called 'standard deviation' in the statement.
What to do instead
Write the standardisation as an explicit line every time: P(X≤a)=P(Z≤(a−μ)/σ), and only then consult the table. Draw the bell curve and shade the required region before reading the table, so the complement or symmetry is decided visually. In inference, write the standard error as σ/√n in the first line and label it 'standard error', keeping σ (population) and s (sample) visually distinct. Check that every probability lies in [0,1].
When computing a definite integral with substitution, the candidate keeps the antiderivative in the substituted variable but evaluates it at the original x-limits, or changes the limits to u-values and then substitutes x back before evaluating, producing a numerically wrong value with correct-looking work.
The two valid procedures (change limits and stay in u, or keep limits and back-substitute) are each individually simple, but mixing them is easy because the back-substitution step feels like part of 'finishing the problem'. The error is silent: the algebra is all correct, only the evaluation is wrong.
What to do instead
Choose one convention and write it at the top of the solution. If changing limits, compute the new limits immediately after defining u and cross out the old ones on the integral sign. If keeping limits, write 'back-substitute, then evaluate at x = a and x = b' as a literal step. Sanity-check the sign and rough magnitude against an area estimate.
The candidate states a limit exists and equals a finite number when the one-sided limits disagree, when the function is unbounded, or when the expression is an indeterminate form such as 0/0 that was never resolved; alternatively, the candidate differentiates a function to check continuity.
Limit, continuity and differentiability are studied in adjacent chapters and their definitions blur together. Indeterminate forms tempt the candidate to cancel incorrectly or to declare 'undefined' as if it were a limit value, and the distinction between a limit existing and a function being defined at a point is subtle in notation.
What to do instead
For every limit question, first compute the left-hand and right-hand limits separately and compare them explicitly. Classify the form before doing algebra, and only then choose a technique (factor and cancel, rationalise, or compare growth rates). Write the formal criteria as a three-line checklist: limit exists, value equals f(a), and the two-sided limit matches f(a), for continuity; derivative from the limit definition for differentiability.
The candidate interprets a p-value as the probability that the null hypothesis is true, or concludes that a non-significant result proves the null hypothesis, and describes a confidence interval as having a 95% chance of containing the true parameter for this particular sample.
The conditional direction of the p-value is reversed in ordinary speech ('the probability the result is due to chance'), and textbook phrasing is compressed. The correct statement is conditional on the null hypothesis and concerns data, not hypotheses, which is the opposite of what intuition supplies.
What to do instead
Write out the definition in full sentences with the conditional made explicit: the p-value is the probability of data at least as extreme as observed, assuming the null model. Practise rewriting wrong interpretations into correct ones, and phrase confidence intervals as a property of the procedure over repeated sampling, not of the single computed interval.