Modulus in A Level Math Paper 3: common mistakes and examiner advice

Modulus questions in Paper 3 are short and usually come near the start of the paper, so they are marks that students should expect to collect. They are also marked tightly. The reports describe the same two or three slips every session. This guide collects what Cambridge examiners say students get wrong, based on recent reports.

Summarised in our own words from a reading of the examiner reports.

Mistakes that cost marks

• Not squaring everything. This is the most repeated mistake. When both sides of an equation or inequality are squared, any number multiplying the modulus, or on the other side, must be squared too. Students forgot to square the 3 (or the 2) on the right-hand side, or squared only the modulus side. One of the popular incorrect starting equations in a past question was 2(x − 1) = 3x instead of 4(x − 1)² = 9x².
• Errors in squaring brackets, and in simplifying to a quadratic equation or inequality.
• Only one of the two linear equations. Students using the linear method often found one relevant equation and did not consider the combination of signs that gives the second one. Reports record sign slips such as 1 + 3x = 4x + 2 instead of 1 + 3x = −(4x + 2).
• Rejecting a critical value. Having found two critical values, some students rejected one of them and wrote only part of the solution, or rejected the negative root in error.
• Wrong inequality. The final answer was given as an impossible statement such as "x < −4 and x > −2/5", or as the wrong region (an inside interval when two outside regions were needed, or the reverse). Several students did not present the final answer in a suitable form, or did not state the final range clearly.
• Sketching graphs. The most common errors were a graph of the wrong shape, the right shape in the wrong place, or missing or incorrect markings on the axes. A modulus graph of this type is a V shape, with straight lines, a vertex in the right place (one report notes that the vertex was often placed on the vertical axis), and the intercepts on each axis. Reports also note that a single straight line was very common as an answer.
• Not using the sketch. A minority of students used their sketch to see what the solution should look like.
• Solving by inspection. Some students found one root by inspection and did not look for the second, or relied on trial and error instead of an algebraic method.

What examiners advise

• Square carefully: write the whole of each side in brackets before squaring.
• Check both critical values against the original inequality before rejecting any.
• Label the sketch: show the vertex, the intercepts on both axes, and a suitable domain.
• Present the final answer in a standard form, and say which regions are included.
• Show your method, since a response with limited working is only creditworthy if the solution is correct.

Habits that help

• Sketch both sides of the inequality before you start, and use the sketch to check your final answer.
• Test a value from each region in the original inequality.
• When you square, state the step in words: "squaring both sides".

Questions students ask

Why do I lose marks when I square both sides of a modulus equation?

Because not everything gets squared. Any number multiplying the modulus, or on the other side, must be squared too, and the whole of each side should be in brackets before squaring. This is the most repeated mistake.

How do I solve a modulus inequality without missing a region?

Find both critical values, check them in the original inequality before rejecting any, and test a value from each region. Reports record students rejecting a valid critical value and writing only part of the solution.

How should I write the final answer to a modulus inequality?

In a standard form, saying which regions are included. An impossible statement such as "x < −4 and x > −2/5", or an inside interval when two outside regions were needed, loses marks.

What must a modulus graph show?

A V shape made of straight lines with the vertex in the right place, the intercepts on both axes, and sensible markings. A minority of students used their sketch to see what the solution should look like.

Can I solve a modulus equation by inspection?

Not safely. Students who found one root by inspection often missed the second, or relied on trial and error instead of an algebraic method. A response with limited working is only creditworthy if the solution is correct.