Basque PAU Mathematics II common mistakes
Basque PAU Mathematics II · Understand the traps and how to avoid them.
When a system is given with a parameter (for example a 3×3 system whose coefficient matrix contains a term like a or a−1), the candidate divides by the parameter or by a determinant expression such as a(a−2) without ever stating the values of a for which that division is illegal, and then reports a single solution as if the discussion were complete.
The candidate treats the parameter as an ordinary number because the algebra looks routine, and the habit of 'solve for x, y, z' carries over from numerical systems. The discussion of cases is the part of the question that carries most of the marks, and it is the part that requires a deliberate pause that routine practice never trains.
What to do instead
Before solving, compute the determinant of the coefficient matrix, factor it fully, and write down the values that make it zero. Structure the answer in explicit cases: for each exceptional value substitute it back into the augmented matrix, row-reduce, and classify the system as incompatible or compatible indeterminate by comparing ranks. Only in the remaining case use Cramer's rule or row reduction. End with a one-line summary table of parameter values and system type.
The candidate computes a determinant of order 3 by expanding along a row but drops the alternating signs, writing every cofactor with a plus sign, or applies a row operation such as adding one row to another to a determinant without noting that the operation leaves the value unchanged while multiplying a row by a scalar does not.
The sign pattern (−1)^(i+j) is memorised as a picture rather than as a rule, and under exam pressure the candidate reconstructs it from memory and gets it wrong. Row-operation properties are learned as manipulation tricks for matrices and are not transferred to determinants.
What to do instead
Always write the cofactor sign matrix explicitly next to the expansion before substituting numbers. For row operations, maintain a written ledger: 'R2 ← R2 + 3R1: determinant unchanged', 'R1 ← 2R1: determinant multiplied by 2, so divide the final result by 2'. Verify any 3×3 determinant by a second expansion along a different row or by Sarrus' rule and check that the two values agree.
Asked whether three points are collinear or four points are coplanar, the candidate computes a determinant of a matrix whose rows are the coordinates of the points themselves, including the constant terms, instead of the vectors between them, and concludes collinearity or coplanarity from a nonzero result that has no geometric meaning.
The candidate has memorised 'determinant equal to zero means dependent' without tracking what the rows of the matrix represent. The distinction between position vectors and direction vectors is glossed over because both are written as triples of numbers.
What to do instead
Before writing any determinant, name the vectors: for collinearity of A, B, C use AB and AC; for coplanarity of A, B, C, D use AB, AC and AD. State that the determinant of these three vectors is the scalar triple product and equals the volume of the parallelepiped, so zero means the vectors are linearly dependent, which is exactly coplanarity. Check the answer against a simple geometric picture when the coordinates allow it.
In a problem about a line and a plane, the candidate uses the direction vector of the line as the normal vector of the plane, or takes the normal of a plane as the direction of a line contained in it, producing a line that is not parallel to the plane or a plane that does not contain the line.
Both objects are described by a triple of numbers and the candidate stops distinguishing 'direction' from 'normal'. The condition for a line to lie in a plane (direction perpendicular to normal, plus one point of the line satisfying the plane equation) is a two-part condition that is often reduced to one part.
What to do instead
Annotate every vector as you extract it: write 'd = direction of r' or 'n = normal of π' beside it. For a line in a plane, verify both d·n = 0 and that substituting the line's point into the plane equation gives 0. For perpendicularity, verify d is parallel to n. Never substitute one for the other without checking the geometric relation the problem states.
When computing a limit that leads to an indeterminate form, the candidate substitutes the value, writes '0/0', and then either declares the limit does not exist or applies L'Hôpital's rule to a quotient that is not of the form 0/0 or ∞/∞, for example after the first application has already resolved the indeterminacy.
The candidate has learned L'Hôpital as a universal tool and does not re-check the hypothesis after each step. The distinction between an indeterminate form and an undefined expression is not secure, so '0/0' is read as 'no limit'.
What to do instead
Write the form explicitly at each stage, for example 'form 0/0, hypotheses of L'Hôpital satisfied'. After each application, substitute again and stop as soon as a finite value or an infinite limit appears. Where algebraic simplification, a known equivalent infinitesimal, or factoring is faster, use it first and reserve L'Hôpital for genuine 0/0 or ∞/∞ quotients.
In a piecewise function with a parameter, the candidate imposes continuity at the join point and solves for the parameter, but then claims the function is differentiable there without checking that the two one-sided derivatives coincide, or computes the derivative of each branch and compares their values at the join without using the limit definition of the derivative.
Continuity and differentiability are treated as a single step because in most textbook examples the parameter chosen for continuity also works for differentiability. The logical implication 'differentiable implies continuous' is confused with its converse.
What to do instead
Treat the two questions as separate parts. First impose equality of the one-sided limits for continuity and record the parameter value. Then, with that value fixed, compute the derivative from the definition on each side, that is the limit of the difference quotient as h tends to 0 from below and from above, and require equality. State explicitly that continuity is necessary but not sufficient for differentiability.
When computing a definite integral by substitution, the candidate changes the variable in the integrand but leaves the original limits in place, or forgets to multiply by the derivative of the substitution when the new variable is not simply the old one, for example writing ∫e^(x²)·2x dx as ∫e^u du only after dropping the factor 2x.
The substitution is performed mechanically on the integrand while the limits are copied from the previous line. The differential du = g′(x) dx is treated as decoration rather than as the bookkeeping that makes the substitution valid.
What to do instead
Write the full substitution line: 'u = g(x), du = g′(x) dx, and when x = a, u = g(a); when x = b, u = g(b)'. Convert the limits in the same line as the substitution, never afterwards. As a check, differentiate the antiderivative you obtain and confirm you recover the original integrand.
In an area-between-curves problem, the candidate integrates f(x) − g(x) over an interval without first finding the intersection points, or splits the integral at a crossing point but assigns the wrong sign to one piece, so that a region partly above and partly below the axis is reported as a smaller area or as zero.
The formula 'integral of the difference' is applied as a template, and the geometric meaning of the definite integral as a signed area is not brought to bear. Intersection points are seen as optional extra work rather than as the boundaries of the region.
What to do instead
Begin by solving f(x) = g(x) to find all intersection points, and sketch the two curves to see which is above on each subinterval. Set up the area as a sum of integrals, each with the upper function minus the lower function on that subinterval. For areas involving the y-axis, integrate with respect to y and use right minus left. Sanity-check that the result is positive and plausible against the sketch.
In a normal-distribution problem, the candidate standardises by computing z = (x − μ)/σ but then uses the wrong tail of the standard normal table, or forgets to convert a 'greater than' probability into 1 minus the tabulated cumulative value, and reports a probability that is the complement of the correct one.
Tables are read by habit and the candidate does not pause to ask what the tabulated entry represents. Symmetry of the normal curve is known as a fact but is not used as a computational tool, so tail probabilities are guessed rather than derived.
What to do instead
State in words what the tabulated value gives, for example 'Φ(z) = P(Z ≤ z)'. Draw a small sketch of the normal curve and shade the required region before touching the table. Use P(Z ≥ z) = 1 − Φ(z) and P(Z ≤ −z) = 1 − Φ(z) explicitly, writing the identity you are applying. Check that the final probability lies between 0 and 1 and is consistent with the sketch.
In a confidence-interval or hypothesis-test problem, the candidate uses the sample standard deviation in place of the standard error, omitting the factor 1/√n, or compares a test statistic with the critical value for the wrong significance level or the wrong number of tails.
The formula is recalled as 'mean plus or minus something times sigma' without the sample-size correction, and the phrase '95% confidence' is mapped to a memorised number without checking whether the test is one-sided or two-sided.
What to do instead
Write the standard error explicitly as σ/√n (or s/√n) and substitute the sample size before any arithmetic. For hypothesis tests, write H0 and H1 symbolically first, decide from H1 whether the test is one- or two-tailed, and only then select the critical value, quoting it with its tail convention. Interpret the conclusion in the context of the problem rather than as a bare comparison of numbers.