PAU Matemáticas II (Extremadura) common mistakes
PAU Matemáticas II (Extremadura) · 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 asked for the relative position of two lines or a line and a plane in space, the candidate computes the determinant of the three direction/connecting vectors and stops, declaring 'they intersect' or 'they are skew' from that single number without checking the parallel/contained cases separately.
The candidate collapses the classification into one test, forgetting that the determinant being zero covers three different situations (parallel distinct, coincident, coplanar-intersecting), and that the determinant being nonzero only rules out coplanarity.
What to do instead
Use a decision tree and write it out each time: compare direction vectors first (proportional or not). If proportional, test whether a point of one satisfies the other (coincident vs parallel distinct). If not proportional, compute the mixed product; zero means they intersect in a point (find it), nonzero means they are skew. For line–plane: check n·v; if zero, test whether the line's point lies in the plane (contained vs parallel). State the case name explicitly in the answer.
When computing a distance from a point to a plane or between skew lines, the candidate substitutes into the formula without normalising, using n·PQ instead of |n·PQ|/|n|, or forgetting the absolute value, and reports a negative distance or one scaled by the norm of the normal vector.
The formula is recalled as 'dot product over something' and the denominator (the modulus of the normal, or the cross product for skew lines) is dropped because it feels like a formality; the absolute value is dropped because the arithmetic 'came out positive anyway' in practice exercises.
What to do instead
Memorise the distance formula as a fraction with a visible denominator: d = |Ax₀+By₀+Cz₀+D| / √(A²+B²+C²), and for skew lines d = |(P₂−P₁)·(v₁×v₂)| / |v₁×v₂|. After computing, sanity-check the magnitude against the coordinates (a distance cannot exceed the distance between the two named points). Write the absolute value bars in the very first line of working, before substituting numbers, so they cannot be omitted later.
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.
When computing a definite integral by substitution or by parts, the candidate changes the variable but keeps the original limits, substituting them into the new variable's expression, or forgets to return to the original variable before evaluating.
The mechanics of the antiderivative dominate attention, and the limits are copied down at the start of the line and never revisited; the candidate also treats the change of limits as an optional refinement.
What to do instead
Choose one of two disciplined formats and stick to it: either (a) change the limits at the same time as the variable and never write x again after the substitution, or (b) keep indefinite-integral notation, find the antiderivative in x, and only then apply the original limits. Mark the limits with a bracket [F(x)] from a to b so the evaluation step is visually forced. After evaluating, check the sign and rough magnitude (e.g. an area must be positive).
When computing the area between two curves or the volume of a solid of revolution, the candidate integrates from the x-coordinates of the intersection points without checking which curve is on top, and reports a negative area, or uses the function value instead of the actual distance to the rotation axis when the axis of revolution is not a coordinate axis.
The intersection points are found correctly and the integral is set up from them automatically, but the ordering of the curves and the geometry of the radius are not examined; the candidate assumes 'area = integral of the difference' regardless of sign and 'radius = f(x)' regardless of the axis.
What to do instead
Sketch the region (even roughly) and mark which curve is upper. Set up the integral as ∫(upper − lower) and, if the curves cross inside the interval, split the integral at the crossing point and add absolute values. For volumes, write the radius explicitly as a distance: for rotation about y=k, r(x)=|f(x)−k|; about x=h with horizontal slices, r(y)=|g(y)−h|. State the radius formula in words before integrating.
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].