Cambridge IAL Mathematics common mistakes
Cambridge IAL Mathematics · Understand the traps and how to avoid them.
Differentiating or integrating without first rewriting the expression in power form, e.g. attempting to differentiate 3/x^2 or sqrt(x) directly and dropping a factor or the negative sign.
Candidates memorise d/dx(x^n)=nx^(n-1) but apply it to fractional and negative-index expressions without converting, so the index arithmetic is done mentally and slips.
What to do instead
Before differentiating, force every term into the form kx^n on a separate line: 3/x^2 becomes 3x^(-2), sqrt(x) becomes x^(1/2). Then apply the rule term by term and check that the new index is n-1, not n. Practise a set mixing negative and fractional indices until the rewrite is automatic.
Applying the chain rule only to the outer function, e.g. writing d/dx sin(3x) = cos(3x) or d/dx e^(x^2) = e^(x^2).
The outer derivative is the salient step and the inner derivative (the multiplier) is easy to forget, especially when the inner function is linear and 'looks' like it does nothing.
What to do instead
Adopt a fixed written routine: identify u, write dy/du and du/dx explicitly, multiply, then substitute back. For linear inner functions, deliberately write the constant multiplier even when it is 1, and always sanity-check by differentiating a simple case such as sin(2x) and confirming the factor of 2 appears.
Losing or mis-signing the constant of integration, or omitting +c entirely in indefinite integrals, and mis-evaluating limits in definite integrals by substituting in the wrong order.
Under time pressure the +c is treated as optional decoration, and in definite integrals the arithmetic of F(b)-F(a) is done quickly with sign errors when the lower limit is negative.
What to do instead
Treat +c as a mark-bearing part of the answer and write it every time, even mid-working. For definite integrals, write the bracket notation [F(x)] from a to b explicitly, then compute F(b)-F(a) on its own line with the substitution shown, so negative limits are handled carefully.
Solving trigonometric equations by giving only the principal-value solution, e.g. sin(x)=0.5 answered as x=30 degrees only, or losing solutions after substituting a double angle such as sin(2x)=0.5.
Candidates use the inverse-trig button and stop, forgetting the symmetry of the trig graphs and the extended range required by the question, and double-angle substitutions double the number of solutions needed.
What to do instead
Always sketch the relevant curve over the required interval, mark the line y=k, and read off all intersections. When the argument is 2x or 3x, first find the full range for the argument (e.g. 0 to 720 degrees for 2x over 0 to 360), solve, then divide at the end. State the general solution form and check each answer lies in range.
In binomial expansion, using the wrong sign or wrong power for the (a+bx)^n term, or forgetting the validity condition |bx/a|<1.
The general term formula is memorised imperfectly, and the validity range is seen as a formality rather than a required part of the answer, so it is omitted or stated as a bare inequality without justification.
What to do instead
Write the general term explicitly as C(n,r) a^(n-r) (bx)^r and substitute r values one at a time rather than pattern-matching. For validity, derive it from |bx/a|<1 and state the resulting interval clearly. When n is negative or fractional, confirm the expansion is only valid for the stated range and mention this in the answer.
Resolving forces without a consistent sign convention or correct component choice, e.g. using sin where cos is needed on an inclined plane, or treating weight as mg acting along the plane.
The component formulas are memorised as 'sin for the slope' without linking them to the geometry of the specific diagram, and the direction of weight (always vertically down) is confused with its component along the plane.
What to do instead
Draw a clear free-body diagram, mark the angle at the point where it actually occurs, and derive components from that triangle rather than from memory. State a positive direction in writing before summing forces, and check that weight has a component mg sin(theta) down the slope and mg cos(theta) perpendicular to it.
Using constant-acceleration (suvat) equations when acceleration is not constant, or mixing up which quantity each symbol represents in a multi-stage motion problem.
suvat is heavily practised and becomes the default tool, so candidates apply it to problems with variable acceleration or to separate stages of motion without resetting the initial conditions for each stage.
What to do instead
Before choosing a method, check explicitly whether acceleration is stated or implied to be constant; if not, use calculus (integrate a(t) for velocity, differentiate for acceleration). For multi-stage problems, draw a velocity-time sketch, label each stage separately, and write a fresh list of s, u, v, a, t for each stage with the final velocity of one stage as the initial velocity of the next.
Friction problems where the normal reaction is assumed equal to the weight regardless of other vertical forces, or where limiting friction is used when the body is not at the point of slipping.
The familiar case R=mg is over-generalised, and candidates do not distinguish between static equilibrium (F<=muR) and limiting equilibrium (F=muR).
What to do instead
Always resolve perpendicular to the surface to find R, including any component of an applied force or of tension that acts perpendicular to the plane. Then state explicitly whether the body is in limiting equilibrium; only then set F=mu*R, otherwise use the inequality or the actual equilibrium condition.
Probability questions where conditional probability is calculated as P(A)P(B) or where independent and mutually exclusive events are treated as the same thing.
The multiplication rule is memorised without its condition, and the everyday meaning of 'independent' and 'mutually exclusive' is confused with the technical definitions.
What to do instead
Write the formula with its condition attached: P(A and B)=P(A)P(B) only if independent, and P(A and B)=0 only if mutually exclusive. For conditional problems, redraw the sample space or use a tree diagram and restrict attention to the given event before computing. Test a suspected independence claim numerically rather than assuming it.
Hypothesis testing where the conclusion is stated as accepting the null hypothesis, or where the significance level is confused with the p-value, and the test is not set up with correctly defined H0 and H1.
Everyday language ('the claim is true') leaks into the conclusion, and the mechanics of comparing the test statistic with the critical value are done without interpreting what the comparison means.
What to do instead
Write H0 and H1 in symbols before any calculation, stating the parameter and its value. Compare the calculated probability or statistic with the significance level explicitly, then phrase the conclusion as 'there is sufficient/insufficient evidence to reject H0' rather than 'H0 is accepted'. Never conclude that H0 is true.