HKDSE Maths M2 common mistakes
HKDSE Maths M2 · Understand the traps and how to avoid them.
Writing a general solution of a trigonometric equation as a single value, e.g. answering sin 2x = 1/2 with only x = π/12 and dropping x = 5π/12 + kπ.
Candidates solve the reference angle, then forget that the given interval (often 0 ≤ x ≤ 2π or 0 ≤ x < π) forces them to enumerate every solution inside it, and they mishandle the doubled argument 2x when converting the interval for x.
What to do instead
Before solving, rewrite the required interval in terms of the inner argument (for 0 ≤ x ≤ 2π and sin 2x, use 0 ≤ 2x ≤ 4π), list all solutions for 2x, then divide back and keep only those in the original range. Always state the number of solutions you expect from the period.
Treating a matrix equation as if matrices commuted: solving AX = B as X = A⁻¹B when the equation is actually XA = B, or cancelling A from AB = AC to conclude B = C.
Candidates transfer scalar algebra habits (dividing both sides, cancelling a common factor) to matrices, ignoring that order matters in multiplication and that A may be singular.
What to do instead
Decide explicitly whether you need a left inverse or a right inverse: for AX = B multiply on the left by A⁻¹, for XA = B multiply on the right. Never cancel a matrix; instead check det A ≠ 0 and multiply by the inverse on the correct side.
Proving a statement by mathematical induction without using the induction hypothesis, or verifying P(1) and then jumping straight to P(k+1) with no link.
Candidates treat induction as a formality: they expand the (k+1) case from scratch and either never invoke the assumed P(k) result or use it without stating the assumption, so the logical chain is broken.
What to do instead
Use a fixed three-part template: (1) verify the base case numerically; (2) write 'Assume the statement is true for n = k, i.e. '; (3) show the algebra where the k-case result is substituted into the k+1 expression. Underline the exact place the hypothesis is used.
Applying the binomial theorem to (a + b)^n with the wrong sign or index, e.g. writing the general term of (2 − 3x)^n as C(n,r) 2^r (−3x)^(n−r) instead of C(n,r) 2^(n−r) (−3x)^r.
Candidates memorise the term loosely and swap which factor carries r, and they lose the negative sign when the second term is subtracted, giving the wrong coefficient or the wrong power of x.
What to do instead
Always write the general term explicitly as C(n,r) a^(n−r) b^r before substituting, put the minus sign inside the bracket of b, and then match the power of x to the required term to solve for r.
Differentiating a composite or product function by handling only one layer, e.g. d/dx sin(x²) given as cos(x²) or d/dx x²e^(3x) given as 2x·3e^(3x).
Under time pressure candidates pattern-match the outer function and stop, and they apply the chain rule and product rule separately instead of together when both are needed.
What to do instead
Name the structure before differentiating: write 'outer = sin, inner = x²' or 'product of u = x² and v = e^(3x)'. Then apply each rule in writing, keeping the inner derivative as an explicit factor rather than doing it mentally.
Computing a definite integral of a function with a sign change or a vertical asymptote without splitting the interval, e.g. evaluating ∫ from −1 to 1 of 1/x² as [−1/x] and getting −2.
Candidates apply the fundamental theorem mechanically without checking that the integrand is continuous on the closed interval, and they do not notice that an area below the axis makes a signed contribution.
What to do instead
Before integrating, list the points where the integrand is undefined or zero inside the interval. If any exist, split the integral at those points; if the integrand is unbounded there, state that the improper integral must be treated as a limit and test convergence.
Setting up a volume of revolution about the y-axis with disks (∫ πy² dy) instead of shells, or mixing the variable of integration with the wrong radius.
Candidates default to the disk formula they practised most, without checking whether the slice is perpendicular to the axis of rotation, and they confuse the radius with the function value.
What to do instead
Draw the region and one representative strip. If the strip is parallel to the axis of rotation use shells (2π × radius × height); if perpendicular use disks/washers (π × (outer² − inner²)). Write the radius as a distance from the axis, not automatically as y.
Losing the constant of integration, or writing '+ C' after a definite integral, and omitting the modulus in ∫ (1/x) dx = ln|x| + C.
Candidates treat '+ C' as decoration to add only when reminded, and they copy the standard integral of 1/x without the absolute value, which is wrong for negative x.
What to do instead
Adopt the habit of writing '+ C' in the same stroke as the antiderivative, and never after a definite integral (substitute the limits instead). For 1/x, always write ln|x| and check the sign of the domain.
Finding the equation of a tangent or normal to a curve at a point without verifying the point lies on the curve, or using the gradient of the tangent as the gradient of the normal.
Candidates plug the x-coordinate into the derivative and the original equation interchangeably, and they forget that perpendicular gradients multiply to −1.
What to do instead
First substitute the point into the curve equation to confirm it lies on the curve. Then compute dy/dx at that x for the tangent gradient; for the normal use −1/(dy/dx). Write the final line in the requested form and check the point satisfies it.
Substituting a limit directly when it gives 0/0 or ∞/∞, or using L'Hôpital's rule on a form that is not indeterminate, such as (x² + 1)/x as x → 0.
Candidates see a fraction and reach for L'Hôpital's rule as a reflex, without first checking the form, and they apply it repeatedly even after the expression becomes determinate.
What to do instead
Always evaluate the form first and write it down (e.g. '0/0'). Only then apply L'Hôpital's rule, differentiating numerator and denominator separately, and stop as soon as the limit is determinate. If the form is not indeterminate, use direct substitution or one-sided analysis instead.