MATSEC IM Pure Mathematics common mistakes
MATSEC IM Pure Mathematics · Understand the traps and how to avoid them.
Solving x^2 = 4x by dividing both sides by x and giving only x = 4, losing x = 0.
Candidates treat cancellation as always safe and forget that division by a variable expression is invalid when that expression can be zero.
What to do instead
Rearrange to x^2 - 4x = 0, factor as x(x - 4) = 0, and state both solutions. Never divide by an expression unless it is known to be nonzero; check for lost zero roots.
Applying graph transformations with the wrong scale, such as saying y = f(2x) stretches the graph horizontally by factor 2 instead of compressing it by factor 1/2.
Candidates memorise '2 means stretch' without distinguishing inside-bracket changes from outside changes.
What to do instead
Use a point-mapping table: for y = f(ax), a point (x, y) moves to (x/a, y); for y = af(x), it moves to (x, ay). Test one known point before sketching.
Using log(a + b) = log a + log b, or solving a log equation and accepting a root that makes an argument zero or negative.
The addition-to-multiplication law is overgeneralised from linearity, and log domain restrictions are ignored after algebra.
What to do instead
State that log laws apply only to products, quotients and powers: log(ab) = log a + log b. After solving, substitute each root into every logarithm and reject any with argument <= 0.
Using an arithmetic-series formula for a geometric sequence, or applying S_infinity = a/(1 - r) when |r| >= 1.
Formula recall is mixed up under time pressure, and the convergence condition for an infinite geometric series is omitted.
What to do instead
Write the defining feature first: common difference for AP, common ratio for GP. Check the infinite-sum condition |r| < 1 before using a/(1 - r), and test the formula with n = 1.
Solving sin x = 1/2 over 0 <= x <= 360 and giving only x = 30, missing x = 150.
Candidates rely on the calculator's principal-value inverse sine and do not use the symmetry of the trigonometric circle.
What to do instead
Find the reference angle, then use a CAST diagram or sketch to locate all angles in the required interval with the correct sign and period. List every solution before moving on.
Using equal gradients for perpendicular lines, or using the negative reciprocal for parallel lines.
The two gradient conditions are memorised as similar-looking rules and become swapped under pressure.
What to do instead
Write the conditions explicitly: parallel means m1 = m2; perpendicular means m1*m2 = -1. Verify with a simple pair such as y = 2x and y = -x/2 before applying.
Calculating the magnitude of a vector as the sum of its components, e.g. treating |(3,4)| as 7, or using a·b = 0 as the condition for parallel vectors.
Candidates confuse vector addition with magnitude and mix up the dot-product conditions for perpendicular and parallel.
What to do instead
Use |v| = sqrt(x^2 + y^2 + z^2). State that a·b = 0 means perpendicular, while parallel means one vector is a scalar multiple of the other. Check with a numerical example.
Dividing complex numbers by multiplying only the numerator by the conjugate, or simplifying i^2 as +1.
The conjugate method is recalled incompletely, and the definition i^2 = -1 is lost during expansion.
What to do instead
Multiply numerator and denominator by the conjugate of the denominator: (a+bi)/(c+di) * (c-di)/(c-di). Expand carefully and replace every i^2 with -1 before collecting real and imaginary parts.
Differentiating sin(3x) as cos(3x), or differentiating x^2 sin x as 2x cos x by omitting one product-rule term.
Candidates recognise the outer function but forget the inner derivative, or apply the product rule as a simple product of derivatives.
What to do instead
For composites, write the inner function u and multiply by du/dx. For products, write u, v, u', v' and use u'v + uv'. Practise one chain-rule and one product-rule example side by side.
Omitting +C from an indefinite integral, or evaluating a definite integral after substitution without changing the limits.
Antidifferentiation is treated as a mechanical reverse rule, and substitution limits are forgotten once the integrand is transformed.
What to do instead
Write +C immediately after every indefinite integral. For definite integrals by substitution, either change the limits to u-values or substitute back to x before applying the original limits; state which method is being used.