MATSEC AM Pure Mathematics common mistakes
MATSEC AM Pure Mathematics · Understand the traps and how to avoid them.
When solving an inequality such as (x-1)(x+3) < 0, the candidate writes x < 1 and x < -3, treating the factors as if each must independently be negative.
The candidate transfers a rule learned for simultaneous conditions onto a product inequality, without testing sign on each interval or drawing a sign chart.
What to do instead
Always find the critical values, mark them on a number line, test one value in each interval, and state the solution as a union of intervals; check the final answer by substituting a sample point.
When finding an inverse function, the candidate swaps x and y but then leaves the expression unsolved, writing f⁻¹(x) = x + 3 instead of solving y + 3 = x for y.
The candidate remembers 'swap x and y' as the whole method and stops there, forgetting that the inverse must be expressed as y in terms of x.
What to do instead
After swapping, explicitly rearrange to make y the subject, then verify by computing f(f⁻¹(x)) and confirming it returns x.
In an arithmetic or geometric series problem the candidate uses n instead of n-1 in the nth-term formula, or uses the sum-to-n formula when the question asks for the sum from the 5th term onward.
The candidate pattern-matches on the formula sheet without identifying which term is the first term of the required range and how many terms are actually being summed.
What to do instead
Before substituting, write down the first term of the required range, the common difference or ratio, and count the number of terms explicitly; for a range from term k to term n the count is n - k + 1.
When asked whether a geometric series converges, the candidate checks whether the terms tend to zero rather than checking |r| < 1, and concludes that a series with r = 2 converges because the terms 'get small eventually'.
The candidate conflates the necessary condition for convergence of the terms with the condition on the common ratio, and does not distinguish r > 1 from |r| < 1.
What to do instead
State the convergence condition |r| < 1 explicitly, compute r by dividing consecutive terms, and if |r| ≥ 1 state divergence; when |r| < 1 use S∞ = a/(1-r) with a the first term.
When finding the equation of a tangent or normal, the candidate substitutes the point into y = mx + c using the gradient of the original curve at a different point, or uses the curve's gradient as the normal's gradient.
The candidate computes the derivative but does not re-evaluate it at the specified x-coordinate, and forgets that the normal gradient is the negative reciprocal.
What to do instead
Write three lines before substituting: the x-coordinate, the corresponding y-coordinate from the original equation, and the gradient from the derivative evaluated at that x; for a normal take m_normal = -1/m_tangent.
In a chain-rule or product-rule differentiation the candidate differentiates the outer function but leaves the inner function unchanged, writing d/dx sin(3x) = cos(3x) or d/dx (2x+1)⁵ = 5(2x+1)⁴.
The candidate has memorised the standard derivatives but not internalised that a composite argument contributes a multiplicative factor.
What to do instead
Adopt the habit of naming u = inner function, writing dy/du and du/dx separately, then multiplying; for (ax+b)ⁿ always check the answer differentiates back to the original.
When evaluating a definite integral by substitution, the candidate changes the integrand to the new variable but keeps the original limits, or omits the constant of integration on an indefinite integral.
The candidate treats the limits as attached to the integral sign rather than to the variable of integration, and treats '+ c' as optional decoration.
What to do instead
Either convert the limits to the new variable at the same time as the substitution, or integrate fully back to x before applying the original limits; write '+ c' immediately after the integral sign as a prompt.
When solving a trigonometric equation such as sin 2x = 0.5 over 0° ≤ x ≤ 360°, the candidate gives only the principal value and misses the second solution, or forgets to divide the adjusted range by 2.
The candidate uses the inverse-trig button on the calculator as if it produced all solutions, and does not account for the change of range caused by the double angle.
What to do instead
First transform the range for the new argument (here 0° ≤ 2x ≤ 720°), then find all solutions in that range using symmetry about 90°/270° or the CAST diagram, and finally divide each by 2.
In a vector problem the candidate computes a·b to test perpendicularity but uses it to test parallelism, or forgets that the cross product (or the ratio of components) is what detects parallel vectors.
The candidate has two tests with similar-looking formulas and does not attach each to its geometric meaning.
What to do instead
Memorise the pairing: a·b = 0 means perpendicular, a = kb (equal component ratios) means parallel; before computing, state which geometric relationship the question asks about.
When solving an equation numerically by Newton-Raphson or fixed-point iteration, the candidate uses a starting value that causes divergence or a zero derivative, and reports the first iterate as the answer.
The candidate treats the method as a single formula to apply once, without checking convergence or iterating to the required accuracy.
What to do instead
Choose a starting value near the root (justified by a sign change of f), check f'(x₀) ≠ 0, iterate until successive values agree to the required number of decimal places, and state the final answer to that accuracy.