AP Intermediate Mathematics (BIEAP) common mistakes
AP Intermediate Mathematics (BIEAP) · Understand the traps and how to avoid them.
Applying Cramer's rule or cross-multiplication to a 3x3 system without first testing the coefficient determinant, then reporting a unique solution when the determinant is zero and the system is dependent or inconsistent.
Drilling on routine 'solve the system' problems trains the elimination steps but rarely the classification step, so candidates treat det = 0 as an arithmetic slip to be fixed rather than a case to be analysed.
What to do instead
Before solving any linear system, compute the coefficient determinant and write its value on the answer sheet. If it is nonzero, proceed to a unique solution; if it is zero, switch to row-reduction to decide between infinitely many solutions and no solution, and state that conclusion explicitly.
Dividing both sides of a trigonometric equation by a factor such as sin x or cos x, which silently discards the roots where that factor is zero.
The habit comes from algebraic equation solving where dividing by a common factor feels like simplification; the loss of roots is invisible because the remaining equation still solves cleanly.
What to do instead
Always transpose to the form (factor A)(factor B) = 0 and solve each factor separately. Reserve division for identities where the divisor is provably nonzero on the given domain, and check every discarded factor against the interval stated in the question.
Writing the general solution of sin θ = k as a single family, θ = nπ + (−1)^n α, when the required range is [0, 2π], and then failing to list every admissible value.
Textbook general solutions are memorised as one formula, and candidates do not test which integers n actually keep the angle inside the specified interval.
What to do instead
After writing the general solution, substitute n = 0, ±1, ±2 until the values fall outside the interval, and list all in-range values in ascending order. For sin θ = k also remember the second family π − α; for cos θ = k use 2nπ ± α.
Computing the slope of a line from Ax + By + C = 0 as −A/B instead of −A/B only after confirming B ≠ 0, and using the wrong sign for the perpendicular slope, giving m = B/A.
The standard form and slope-intercept form are mixed up under time pressure, and the negative reciprocal is applied to the wrong quantity.
What to do instead
Convert to y = mx + c first and read the slope directly, or memorise slope = −(coefficient of x)/(coefficient of y). For perpendicularity use m₁m₂ = −1 and verify by substituting the actual numbers back into that product.
Differentiating a composite function by ignoring the inner derivative, for example writing d/dx sin(x²) = cos(x²) or d/dx ln(3x + 1) = 1/(3x + 1).
Chain rule is learned as a formula but practised mostly on single-step examples, so the inner derivative is dropped the moment a second operation appears.
What to do instead
Name the inner function before differentiating: write u = x², so y = sin u, then dy/dx = cos u · du/dx. Do this explicitly for the first several practice items until the inner factor is produced automatically, and always sanity-check by asking what the derivative of the inside is.
Applying L'Hôpital's rule to a limit that is not of the form 0/0 or ∞/∞, or differentiating the numerator and denominator of a quotient that is not a limit at all.
The rule is memorised as 'differentiate top and bottom' without its hypotheses, and the 0/0 check is skipped because the substitution step feels redundant.
What to do instead
Before invoking the rule, substitute the limiting value and record the indeterminate form in writing. If the form is anything else, use algebraic manipulation, standard limits, or rationalisation instead. Never apply it to the derivative of a quotient; that requires the quotient rule.
Omitting the constant of integration, or substituting limits into the integrand before finding the antiderivative when evaluating a definite integral.
Indefinite and definite integrals are practised as separate skills, so under exam pressure the antiderivative step is compressed and the bracket notation with limits is applied to the wrong expression.
What to do instead
Write the antiderivative in square brackets with the limits attached, F(x)] from a to b, before any substitution of numbers. For indefinite integrals, make '+ C' the last thing written on every line until it becomes automatic.
Solving a first-order differential equation such as dy/dx = f(x)g(y) by integrating both sides with respect to x without separating, producing ∫g(y)dy incorrectly as ∫g(y)dx.
Separation of variables is taught as a sequence of symbol moves, and the fact that dy and dx track different variables is not internalised, so the variables are never actually separated.
What to do instead
Rewrite the equation so that every y appears on the left with dy and every x on the right with dx, then integrate each side with respect to its own variable. Verify by differentiating the implicit solution and recovering the original equation.
Computing a × b instead of b × a, or treating the dot product as a vector, when finding a unit vector perpendicular to two given vectors or the area of a parallelogram.
The two products look symmetric in notation, and the anticommutativity of the cross product is stated once and rarely tested in practice problems.
What to do instead
Fix the order from the question wording and write it down before expanding the determinant. Remember a × b = −(b × a) and a · b is a scalar. Check the direction with the right-hand rule and confirm the result is perpendicular by dotting it with both original vectors.
Using P(A ∪ B) = P(A) + P(B) for events that are not mutually exclusive, or multiplying P(A)P(B) for events that are not independent.
The addition and multiplication rules are memorised in their simplest forms, and the words 'mutually exclusive' and 'independent' are read as decoration rather than as conditions that change the formula.
What to do instead
Before substituting, write the general rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B) and P(A ∩ B) = P(A)P(B|A). Then decide from the problem statement whether the intersection term vanishes or whether P(B|A) equals P(B), and justify that decision in one line.