UPMSP Intermediate Mathematics common mistakes
UPMSP Intermediate Mathematics · Understand the traps and how to avoid them.
When finding the inverse of a 2×2 or 3×3 matrix, the candidate computes the adjoint but forgets to divide by the determinant, or divides by the determinant instead of multiplying by its reciprocal, producing an inverse that fails A·A⁻¹ = I.
The formula A⁻¹ = (1/|A|)·adj(A) is memorised as a single string, and under time pressure candidates drop the scalar factor entirely or confuse the cofactor matrix C with adj(A) = Cᵀ, so the transposition step silently disappears.
What to do instead
After every inverse, multiply your answer by A and check the product is I. Also write the formula on rough work as three separate labelled steps (cofactor → transpose → divide by |A|), and verify |A| ≠ 0 before starting, since a zero determinant means no inverse exists.
While evaluating a definite integral by substitution, the candidate substitutes u = g(x) but keeps the original limits of x, or forgets to change dx to du, giving a numerically wrong value even though the antiderivative is correct.
The substitution is done mechanically on the integrand only, and the limits are copied down unchanged because they sit outside the expression being rewritten; the dx is treated as decoration rather than part of the measure.
What to do instead
Adopt a fixed layout: write the new limits u(a) and u(b) explicitly under the integral sign before integrating, and always write 'du = g′(x) dx' as its own line. If you prefer not to change limits, back-substitute to x before applying the original bounds, and never mix the two approaches in one problem.
In a word problem on maxima and minima, the candidate differentiates, sets f′(x) = 0, solves for x, and declares that value the maximum without testing the second derivative or checking the endpoints of the stated interval.
The procedure is learned as a two-step ritual (differentiate, equate to zero), and the second-derivative test plus endpoint comparison are treated as optional extras rather than part of the answer.
What to do instead
Finish every optimisation with a sign check: compute f″(x) at each critical point and state clearly 'f″ < 0 so maximum' or 'f″ > 0 so minimum'. When the domain is a closed interval, tabulate f at the critical points and at both endpoints and pick the largest and smallest from that table.
The candidate writes the general solution of a differential equation such as dy/dx = y as y = eˣ, omitting the arbitrary constant, or drops the constant of integration midway through a multi-step solution.
Indefinite integration is practised so often without constants in drill exercises that the constant becomes invisible, and in separable equations it is absorbed into a logarithm and then lost during exponentiation.
What to do instead
Treat '+ C' as part of the integral sign itself: never close an indefinite integral without writing it. For separable equations, keep the constant until the final line, then rename combined constants (for example e^C becomes a single new C) and state the order of the equation to confirm how many arbitrary constants the general solution must contain.
When asked for the angle between two vectors or the projection of one vector on another, the candidate uses the cross-product magnitude in place of the dot product, or forgets to divide by |b| when projecting a onto b.
Both formulas involve the same three ingredients (two vectors and their magnitudes), and the geometric picture of 'angle' feels closer to the perpendicular cross product, so the formulas get swapped under recall pressure.
What to do instead
Anchor the formulas to meaning: dot product answers 'how much in the same direction', cross product answers 'how much perpendicular / area'. Write cos θ = (a·b)/(|a||b|) and projection = (a·b)/|b| side by side, and sanity-check that cos θ lies between −1 and 1 and that the projection cannot exceed |a|.
While solving a system of linear equations by Cramer's rule or matrix inversion, the candidate builds the determinant for x by replacing a column but replaces the wrong column, or forgets that the constant column replaces the column of the variable being found.
Column order is not labelled, so under exam pressure the candidate replaces the first column by habit regardless of which variable is requested, and inconsistent ordering of variables between the coefficient matrix and the constant vector goes unnoticed.
What to do instead
Before computing, write the system as a labelled table with variable names above the columns. For each variable, state in words 'replace the column of x with the constants' and cross out that column visually. Alternatively solve by row reduction, which removes the column-indexing risk entirely, and substitute the answer back into the original equations.
In probability, the candidate adds probabilities of overlapping events without subtracting the intersection, or treats two events as independent merely because they are described in the same sentence.
The addition rule is remembered as P(A) + P(B), and independence is assumed from narrative proximity rather than verified from P(A∩B) = P(A)P(B); the distinction between mutually exclusive and independent is blurred.
What to do instead
Always write the general rule P(A∪B) = P(A) + P(B) − P(A∩B) and decide explicitly whether the events can occur together. Test independence numerically by computing P(A)P(B) and comparing with the given or computed P(A∩B); remember mutually exclusive events with nonzero probabilities are never independent.
When finding the equation of a tangent or normal to a circle, parabola or ellipse, the candidate uses the slope of the radius as the slope of the tangent, or forgets that the normal slope is the negative reciprocal of the tangent slope.
The point-slope form is applied immediately after computing dy/dx, and the geometric relationship between radius, tangent and normal is not held in mind, so a sign or reciprocal error passes unnoticed.
What to do instead
Sketch the curve and the point, then mark the radius and tangent at right angles before computing. Write down explicitly: tangent slope = dy/dx at the point; normal slope = −1/(dy/dx). If the tangent is vertical, handle it as a special case rather than dividing by zero, and check the final line passes through the given point.
The candidate evaluates a limit or checks continuity at a point by substituting the value into the expression, and when it yields 0/0 declares the function discontinuous or the limit nonexistent instead of resolving the indeterminate form.
Substitution works for most routine limits, so the 0/0 outcome is read as a final answer rather than as a signal to factorise, rationalise, or apply L'Hôpital's rule; the three continuity conditions are also not checked separately.
What to do instead
Treat 0/0 as an instruction, not a result: try factorisation, rationalisation, or a standard limit first. For continuity at x = a, write all three quantities explicitly - f(a), lim(x→a⁻) f(x), lim(x→a⁺) f(x) - and state that all three are equal. For differentiability, check left and right derivatives separately.
In linear programming, the candidate graphs the constraints but tests the objective function only at the intersection points of constraint lines, ignoring vertices where a constraint meets an axis, and does not check whether the feasible region is bounded.
The corner-point method is remembered as 'solve pairs of constraint equations', which misses axis intercepts, and the boundedness condition that justifies the corner-point theorem is never stated.
What to do instead
After shading the feasible region, list every vertex by walking the boundary: axis intercepts, intersections with each axis, and constraint-constraint intersections. Evaluate Z at each listed vertex in a table, state whether the region is bounded, and if unbounded explain why the maximum or minimum still occurs at a listed vertex or does not exist.