Exam skills and problem solving

Exam technique in mathematics is the disciplined process of turning a question into a correct, clearly justified answer. It does not replace mathematical knowledge. It helps you recognise which knowledge is relevant, apply it efficiently and communicate enough reasoning to earn the available marks.

This page gives a complete workflow for unfamiliar and multi-step questions. The linked lessons develop each stage in greater depth.

What you should know first

You should be able to perform the main techniques from the topics you are studying. In particular, you need reasonable fluency with:

• algebraic manipulation;
• exact arithmetic and rounding;
• mathematical notation;
• calculator use.

If a solution makes sense when you read it but you cannot reproduce it without prompts, practise the underlying topic as well as the exam skill. Recognition is not yet recall.

The complete exam workflow

Use this cycle on every substantial question:

1. Read: identify the command, target, data, restrictions and required form.
2. Represent: translate words or diagrams into variables, equations, graphs, distributions or force models.
3. Select: choose a method whose conditions are satisfied.
4. Execute: write connected, exact working and use the calculator accurately.
5. Interpret: answer in the language and context of the question.
6. Check: test the result against the original problem.

The cycle is not rigid. A failed check may send you back to the representation or method. Strong problem solvers move between these stages deliberately.

1. Read the mathematical demand

Before calculating, identify exactly what the final answer must contain.

Command	What it requires
Find or calculate	A value, expression, equation or set of values, with sufficient working
Show that	A valid derivation of the stated result without assuming it
Prove	A general logical argument covering every allowed case
Sketch	Correct shape with relevant intercepts, asymptotes and labelled features
Hence	Use the preceding result, usually to shorten the next argument
Interpret	Explain the mathematical result in its stated context
State	A concise answer, usually with little or no derivation

Also mark any conditions such as

$$0\leq x<2\pi,\qquad x\in\mathbb R,\qquad n\in\mathbb N,$$

and any instruction about exact form, decimal places, significant figures or units.

Worked example 1: finishing the question asked

The curve

$$y=x^3-6x^2+9x+2$$

has two stationary points. Find the coordinates of the local maximum.

Differentiate:

$$\frac{dy}{dx}=3x^2-12x+9=3(x-1)(x-3).$$

Hence the stationary values of $x$ are $1$ and $3$. This is not yet an answer because the question asks for a coordinate and a classification.

Differentiate again:

$$\frac{d^2y}{dx^2}=6x-12.$$

At $x=1$,

$$\frac{d^2y}{dx^2}=-6<0,$$

so this point is a local maximum. Its $y$ coordinate is

$$y(1)=1-6+9+2=6.$$

Therefore the local maximum is

$$\boxed{(1,6)}.$$

The words coordinates and local maximum determine the extra steps after solving $dy/dx=0$.

2. Represent the information

A worded problem becomes manageable when each sentence is converted into mathematics. Define unknowns with units, draw a diagram if spatial relationships matter, and distinguish given facts from assumptions.

Common representations include:

• an equation or inequality in pure mathematics;
• a probability distribution and its parameters in statistics;
• a force diagram, positive direction and equations of motion in mechanics;
• a graph when roots, intersections, signs or rates of change matter.

Worked example 2: building a model

A rectangle has perimeter $40$ cm. Find its maximum possible area.

Let the side lengths be $x$ cm and $y$ cm. The perimeter condition gives

$$2x+2y=40,$$

so

$$y=20-x.$$

Because side lengths are positive,

$$0<x<20.$$

The area is therefore a function of one variable:

$$A=x(20-x)=20x-x^2.$$

Differentiate:

$$\frac{dA}{dx}=20-2x.$$

The stationary point occurs when

$$20-2x=0,$$

giving $x=10$. Also,

$$\frac{d^2A}{dx^2}=-2<0,$$

so this is a maximum. Then $y=10$ and

$$\boxed{A_{\max}=100\text{ cm}^2}.$$

The conceptual step is not the differentiation. It is using the perimeter condition to express the area in one variable. Study interpreting answers in context for more modelling examples.

3. Choose a method from structure

Do not ask only, “Which topic is this?” Ask:

1. What is the target?
2. What information is available?
3. Which definition, theorem or model connects them?
4. Are its conditions satisfied?

For example, a product in an integral may suggest integration by parts, but if one factor is the derivative of an inner function then substitution may be shorter. Repeated trials suggest a binomial model only if the number of trials is fixed, the trials are independent and the success probability is constant.

Worked example 3: use the strongest clue

Evaluate

$$\int_0^2 x\sqrt{x^2+5}\,dx.$$

The product alone is a weak clue. The factor $x$ is proportional to the derivative of the inner expression $x^2+5$, which is a strong clue for substitution.

Let

$$u=x^2+5, \qquad du=2x\,dx.$$

Change the limits:

$$x=0\Rightarrow u=5, \qquad x=2\Rightarrow u=9.$$

Then

$$\begin{aligned} \int_0^2 x\sqrt{x^2+5}\,dx &=\frac12\int_5^9 u^{1/2}\,du\\ &=\frac12\left[\frac23u^{3/2}\right]_5^9\\ &=\frac13\left(27-5\sqrt5\right). \end{aligned}$$

Thus

$$\boxed{\int_0^2 x\sqrt{x^2+5}\,dx=9-\frac{5\sqrt5}{3}}.$$

See how to choose a method for a systematic approach to unfamiliar questions.

4. Communicate enough reasoning

Mathematical working should make the argument reconstructable. A useful principle is:

Write the equation or fact that justifies the next important conclusion.

Good working usually includes:

• the formula before numerical substitution;
• definitions of introduced variables;
• intermediate unrounded values where later accuracy depends on them;
• reasons for conclusions, such as $f''(a)<0$, a probability comparison or a sign change;
• exact equality signs only between exactly equal quantities.

Use

$$=$$

for exact equality and

$$\approx$$

for a rounded value. For example,

$$\frac{2\pi}{7}\approx0.898,$$

not $2\pi/7=0.898$.

Worked example 4: a statistical conclusion

Under a null hypothesis, $X\sim\operatorname{Bin}(20,0.3)$. An observed result is $x=11$. The test is upper tailed at the $5\%$ significance level.

A complete argument contains the probability, comparison and contextual decision:

$$P(X\geq11)=0.0171\text{, approximately}.$$

Since

$$0.0171<0.05,$$

the result is significant at the $5\%$ level. Reject the null hypothesis. There is sufficient evidence, at that level, that the probability of success is greater than $0.3$.

Writing only “reject $H_0$” hides both the numerical basis and the meaning of the decision. Writing ”$H_0$ is false” overstates what a significance test establishes.

The lesson on communicating mathematical reasoning develops proofs, explanations and mark worthy conclusions.

5. Keep exact values and round once

Unless a question requires decimals, exact values are usually clearer and safer:

$$\frac37,\qquad \sqrt{13},\qquad \frac{5\pi}{12},\qquad \ln 2.$$

If a final decimal is required, retain full calculator precision during the calculation and round only the final answer. Premature rounding can move an answer outside an accepted tolerance or change a later conclusion.

Worked example 5: avoid premature rounding

A particle travels at constant speed

$$v=\frac{20}{3}\text{ m s}^{-1}$$

for $17$ seconds. Its distance is

$$s=vt=\frac{20}{3}\times17=\frac{340}{3}\text{ m}.$$

Therefore, to $3$ significant figures,

$$\boxed{s=113\text{ m}}.$$

Rounding the speed first to $6.67$ gives $113.39$ m. That does not affect this particular final rounding, but in a longer calculation the error accumulates. Store exact values or full calculator values.

6. Interpret the result in context

A mathematically valid root may be impossible in the model. Check:

• units and dimensions;
• sign and plausible size;
• whether a quantity must be an integer;
• domain restrictions;
• whether the model’s assumptions remain reasonable;
• what a statistical quantity actually says.

Worked example 6: reject an inadmissible solution

The height of a ball above the ground is modelled by

$$h(t)=1+12t-4.9t^2,$$

where $t\geq0$ is measured in seconds. Find when the ball reaches the ground.

Set $h(t)=0$:

$$4.9t^2-12t-1=0.$$

The quadratic formula gives

$$t=\frac{12\pm\sqrt{144+19.6}}{9.8} =\frac{12\pm\sqrt{163.6}}{9.8}.$$

Thus

$$t\approx2.53 \qquad\text{or}\qquad t\approx-0.0808.$$

The negative root is algebraically valid but outside the model’s domain $t\geq0$. Therefore the ball reaches the ground after

$$\boxed{2.53\text{ s}}$$

to $3$ significant figures.

7. Check independently

Repeating the same calculator entry is not a strong check because it may repeat the same mistake. Use a check that approaches the answer differently.

Type of answer	Useful check
Solution of an equation	Substitute into the original equation
Derivative	Differentiate by another form or compare local gradients
Indefinite integral	Differentiate the answer
Probability	Confirm $0\leq P\leq1$ and inspect complements or totals
Mechanics result	Check units, direction, limiting cases and physical plausibility
Numerical answer	Estimate its sign and order of magnitude
Proof or derivation	Check every implication and all relevant cases

Worked example 7: check an integral

Suppose

$$\int xe^{2x}\,dx =e^{2x}\left(\frac{x}{2}-\frac14\right)+C.$$

Differentiate the proposed answer:

$$\begin{aligned} \frac{d}{dx}\left[e^{2x}\left(\frac{x}{2}-\frac14\right)\right] &=2e^{2x}\left(\frac{x}{2}-\frac14\right)+\frac12e^{2x}\\ &=e^{2x}\left(x-\frac12+\frac12\right)\\ &=xe^{2x}. \end{aligned}$$

The derivative reproduces the integrand, so the antiderivative is correct. Learn a wider range of tests in checking answers.

When you get stuck

Being stuck usually means that one stage of the workflow is incomplete. Use this recovery routine.

1. Rewrite the target in precise mathematical language.
2. List the relevant facts, including information from earlier parts.
3. Draw or sketch the situation.
4. Write a definition or standard formula connected to the target.
5. Simplify expressions before using a major technique.
6. Try a special case to expose structure, but do not mistake it for a proof.
7. If you abandon a route, preserve any result that remains valid.

For example, if asked for a tangent but unsure how to begin, write

$$y-y_1=m(x-x_1).$$

This reveals the two missing ingredients: a point $(x_1,y_1)$ and gradient $m$. The target itself now guides the method.

Common misconceptions

”A calculator answer is enough”

Not usually. A decimal without a model, equation or method may not show the reasoning being assessed. Write the mathematical setup before using the calculator.

”More working always earns more marks”

Only relevant, valid working helps. Long unstructured calculations can obscure the argument. Show the decisive steps and preserve logical continuity.

”If the algebra gives two roots, both are answers”

Roots are candidates until checked against the original equation, interval and context.

”Show that means substitute the printed answer”

Substitution may verify a result but does not normally derive it. Begin from known information and reach the stated result through valid steps.

”A non-significant test proves there is no effect”

It does not. It says the sample does not provide sufficient evidence at the chosen significance level to reject the null hypothesis.

See common A level Mathematics exam mistakes for corrected examples across the course.

Self-check

Attempt each question before opening its answer.

1. Read and complete

For $f(x)=x^2e^{-x}$, solving $f'(x)=0$ gives $x=0$ and $x=2$. The question asks for the maximum value of $f$ for $x\geq0$. What further work is needed?

Answer

Find function values and classify the candidates. Since

$$f(0)=0, \qquad f(2)=4e^{-2},$$

and $f(x)\to0$ as $x\to\infty$, the maximum value is

$$\boxed{4e^{-2}}.$$

The answer is a value, not the coordinate $x=2$.

2. Choose a method

Which method is most efficient for

$$\int \frac{3x^2}{x^3+4}\,dx?$$

Answer

Use substitution because the numerator is the derivative of the denominator. Let $u=x^3+4$, so $du=3x^2\,dx$. Then

$$\int \frac{3x^2}{x^3+4}\,dx =\int\frac1u\,du =\ln|u|+C =\boxed{\ln|x^3+4|+C}.$$

3. Interpret a model

A model for the number of items produced gives $n=38.7$. What should be reported if $n$ must be a whole number and production must meet a minimum target?

Answer

Report

$$\boxed{n=39}.$$

Ordinary rounding to $39$ happens to agree, but the reason is the constraint: $38$ items would not meet the minimum target. Always state the contextual reason for rounding up.

4. Check a proposed solution

A student claims that the solutions of

$$\sqrt{x+2}=x$$

are $x=2$ and $x=-1$. Decide whether the claim is correct.

Answer

Squaring gives

$$x+2=x^2,$$

so $(x-2)(x+1)=0$. However, candidates must be checked in the original equation. For $x=2$,

$$\sqrt4=2,$$

so it works. For $x=-1$,

$$\sqrt1=1\ne-1.$$

Therefore the only solution is

$$\boxed{x=2}.$$

Squaring introduced the extraneous root.

5. Communicate a mechanics result

Taking upwards as positive, a calculation gives the acceleration of a lift as $a=-1.4\text{ m s}^{-2}$. State the result in context.

Answer

The lift’s acceleration is

$$\boxed{1.4\text{ m s}^{-2}\text{ downwards}}.$$

The negative sign records direction relative to the chosen positive axis. It does not mean that the magnitude is negative.

Build reliable exam habits

After completing a timed question, diagnose the earliest cause of any lost mark:

• knowledge gap;
• misread demand;
• weak representation;
• unsuitable method;
• algebra or calculator error;
• incomplete reasoning;
• missing interpretation;
• absent check.

Then practise that cause. Copying the correct final solution without identifying why your route failed gives little protection against the next unfamiliar question.

Continue with choosing a method, communicating mathematical reasoning, interpreting answers in context, checking answers and common exam mistakes.