STPM Mathematics 954 common mistakes
STPM Mathematics 954 · 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, or multiplies both sides by (x-2) without splitting into cases x-2 > 0 and x-2 < 0.
The candidate treats inequalities as if they were equations, carrying over the habit of solving each factor separately and joining with 'and'. The sign-reversal rule when multiplying or dividing by a negative quantity is memorised as a slogan but not applied to expressions whose sign is unknown.
What to do instead
Always draw a sign chart: mark the critical values on a number line, test one value in each interval, and read off the intervals whose sign satisfies the inequality. For rational inequalities, first move everything to one side so the comparison is against zero, then use the sign chart on numerator and denominator separately, excluding values that make the denominator zero. Check the final answer by substituting one value from each stated interval.
When finding the inverse of a function, the candidate writes f^{-1}(x) = 1/f(x), or produces an inverse expression without restricting the domain, so that f^{-1} is not actually a function.
The notation f^{-1} looks like a power, and the candidate has not internalised that inversion means solving y = f(x) for x and swapping the variables. The one-to-one requirement is known in theory but not checked in practice, especially for quadratics and trigonometric functions.
What to do instead
Use the procedure: set y = f(x), solve algebraically for x in terms of y, then interchange x and y. Before declaring the inverse, state the domain restriction that makes f one-to-one (for example x >= 0 for f(x) = x^2 + 1), and give the domain and range of f^{-1} explicitly. Verify by computing f(f^{-1}(x)) = x on the stated domain.
In partial fractions with a repeated linear factor, the candidate writes a single term A/(x-1) instead of A/(x-1) + B/(x-1)^2, or uses a quadratic denominator (x^2+1) with a constant numerator only.
The candidate memorises one template and applies it to every denominator. The reason each power of a repeated factor needs its own term, and why an irreducible quadratic needs a linear numerator, is not understood.
What to do instead
Before decomposing, fully factorise the denominator over the reals and list every factor with its multiplicity. For each factor (ax+b)^k include terms A_1/(ax+b) + ... + A_k/(ax+b)^k; for each irreducible quadratic include (Bx+C)/(quadratic). Then substitute convenient values of x and also compare coefficients to solve for all constants, and check by recombining.
When differentiating a composite function such as sin(3x^2) or ln(x^2+1), the candidate differentiates the outer function only, giving cos(3x^2) or 1/(x^2+1), omitting the inner derivative.
The chain rule is recognised as a rule but applied only when the inner function is a single variable. Under exam pressure the candidate pattern-matches to the standard derivative table and stops.
What to do instead
Adopt an explicit substitution habit: let u = inner expression, write dy/dx = dy/du * du/dx, and only then substitute back. For products and quotients inside a composite, identify the outermost operation first and work inward. Practise mixed expressions such as e^{sin x}, (2x+5)^7 and tan(ln x) until the inner derivative is written automatically.
When evaluating a definite integral by substitution, the candidate changes the variable to u but keeps the original x-limits, or omits the constant of integration in an indefinite integral and then uses it in a later step.
The substitution is treated as a cosmetic rewriting rather than a change of variable. The link between limits and the variable actually being integrated is not made explicit.
What to do instead
Whenever you substitute, immediately convert the limits using the substitution equation and write the new integral entirely in u with u-limits, or alternatively integrate and convert back to x before applying the original limits. Never mix variables and limits in the same bracket. For indefinite integrals, write '+ C' on the same line as the answer.
When finding the area between a curve and the x-axis, the candidate computes the integral across a region where the curve crosses the axis and reports the signed value as the area, or uses vertical strips for a region more naturally described horizontally.
The definite integral is identified with area without checking whether the integrand is negative on part of the interval. Sketching is skipped to save time.
What to do instead
Always sketch the curve and locate the x-intercepts before integrating. Split the interval at each intercept and integrate the absolute value on each subinterval, adding the magnitudes. For regions bounded by a curve and a line, integrate (upper - lower) with respect to x, or convert to x = g(y) and integrate with respect to y when horizontal strips are simpler. State the strip direction in your working.
In vector problems, the candidate computes the scalar product and sets it equal to the product of magnitudes, or uses the scalar product formula to find an angle but forgets to check the direction of the vectors, giving an obtuse angle when an acute one is required.
The cosine formula is remembered but the geometric meaning of the scalar product, and the role of direction, is not. The candidate does not distinguish between the angle between two vectors and the angle between a line and a plane.
What to do instead
Write the formula a·b = |a||b|cos(theta) explicitly and solve for cos(theta), then interpret the sign: negative cosine means the angle between the vectors is obtuse. For the angle between a line and a plane, use the direction vector of the line and a normal to the plane, then take the complement (90 degrees minus the angle between line and normal). Draw a small diagram showing the two vectors and the angle you are actually finding.
In hypothesis testing, the candidate states H0 and H1 after looking at the sample mean, chooses a one-tailed or two-tailed test to fit the data, or reports a p-value without comparing it to the significance level.
The logic of the test is treated as a calculation exercise. The candidate has not internalised that the alternative hypothesis must be fixed by the wording of the question before any data are examined.
What to do instead
Write H0 and H1 first, in symbols, directly from the question wording, and decide one- or two-tailed before computing anything. State the significance level and the test statistic with its distribution under H0. Compute the p-value or critical region, then write an explicit comparison sentence ('since p < 0.05, reject H0') and a conclusion in the context of the problem.
When working with the normal distribution, the candidate uses the raw value x instead of the standardised z = (x - mu)/sigma, or adds probabilities for a 'between' question instead of subtracting, and ignores continuity correction when approximating a binomial or Poisson by a normal distribution.
Standardisation is memorised as a formula but not linked to the table of standard normal probabilities. The geometry of the area under the curve is not visualised, so tail probabilities are combined incorrectly.
What to do instead
Sketch the normal curve for every problem, shade the required region, and mark the mean and the x-values. Standardise each boundary, then express the required probability in terms of z and use symmetry: P(Z < -a) = 1 - P(Z < a). When approximating a discrete distribution, apply the continuity correction (for example P(X <= 7) becomes P(Y < 7.5)) and state the conditions np > 5 and n(1-p) > 5.
In probability, the candidate assumes two events are independent and multiplies their probabilities without justification, or confuses P(A|B) with P(B|A) in a conditional probability question.
Independence is treated as a default rather than a property to be checked. The conditional probability notation is read casually, and the candidate does not rewrite the condition in words.
What to do instead
Before multiplying, check independence using P(A and B) = P(A)P(B) or the problem statement; otherwise use the general multiplication rule P(A and B) = P(A)P(B|A). For conditional questions, rewrite P(A|B) in words as 'probability of A given that B has occurred' and restrict the sample space to B. Use a two-way table or tree diagram to keep the conditioning direction visible.