Modulus questions in Additional Math cover solving equations and inequalities with a modulus, finding a modulus function from conditions, and drawing graphs of y = |f(x)|. The reports show that students often find one answer where two are needed, and that sketches lose marks for shape and labelling. 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
- Only one case. A modulus statement gives two possibilities. In one question, many students found one correct expression (often 2x − 1) and made no attempt to find the other, or found it and then stated the answer in an incorrect form such as ± (2x − 1) = 4. Others wrote down the two linear equations and made a sign error that left them with only one solution.
- Inequalities. Losing the final mark with a wrong final form, such as writing the two parts of the solution with "and" where "or" was needed, handling a modulus incorrectly so that the inequality ends up the wrong way round, or making an inequality error and attempting to correct it without justification. In another question, students who changed the sign of the constant only worked with one of the two terms.
- Squaring. When squaring both sides, forgetting to square the 7. In another question, students who tried to include a solution that does not belong destroyed their method.
- Critical values. Algebraic solutions often produced incorrect critical values. Reports say that students who chose to work with a pair of linear equations or inequalities were commonly successful.
- Finding the function from conditions. Listing the four possible linear equations in a and b and making no further progress, forming a quadratic such as (2.5a + b)² = 16 and expanding it incorrectly, or using only some of the possibilities.
- Graph shapes. Drawing ruled sections where a curve is needed (and the reverse), incorrect curvature, drawing cusps as stationary points, and responses that bore no resemblance to the graph of the modulus of a quadratic, such as a V-shape. Others placed a local maximum in the wrong quadrant or at the wrong point, such as (0, 12).
- Straight sections. For a graph made of straight lines, not using a straight edge, so the lines were not sufficiently straight or did not point at the vertex. Some drew a V-shaped graph that did not extend across the required range, or had the vertex on the y-axis.
- Labels. Omitting some or all of the intercepts, labelling the axes incorrectly, and not stating the y-intercept and vertex when the question asked for them. A vertex "between 1 and 2" is not enough to imply that it is at 1.4.
What examiners advise
- Write both cases of the modulus, and solve each one.
- Use two linear equations or inequalities, or square both sides carefully, and check the result.
- For graphs of y = |f(x)|, draw y = f(x) first, then reflect the part that lies below the x-axis. Reports say the reflection in the x-axis ensures the correct shape throughout.
- Rule straight sections with a straight edge and draw curves smoothly. Label the intercepts and the vertex or turning points with their coordinates, either on the graph or next to it.
- Refer back to the previous part of a question if it can help.
Habits that help
- After solving an equation with a modulus, substitute each answer back into the original equation.
- For an inequality, mark the critical values on a number line and test a point in each region.
- Before you draw a modulus graph, mark the x-intercepts of f(x) on your axes.
Questions students ask
How many cases does a modulus equation have?
Two. Write both and solve each one. In one question many students found one correct expression (often 2x − 1) and made no attempt to find the other.
How do I solve a modulus inequality?
Use two linear equations or inequalities, or square both sides carefully (including any number multiplying the modulus), and check the result. Mark critical values on a number line and test a point in each region. Watch "and" versus "or" in the final form.
How do I draw y = |f(x)|?
Draw y = f(x) first, then reflect the part below the x-axis. Reports say this ensures the correct shape throughout. Mark the x-intercepts of f(x) on your axes first.
What do I need to label on a modulus graph?
The intercepts and the vertex or turning points with their coordinates. A vertex "between 1 and 2" is not enough. Rule straight sections with a straight edge and draw curves smoothly.
How do I find a modulus function from conditions?
Do not stop at listing the four possible linear equations in a and b. Students formed a quadratic such as (2.5a + b)² = 16 and expanded it incorrectly. Work systematically.
Practice this topic
Ready to try some? Work through past paper questions on this topic one at a time. Attempt each question fully first, and open the mark scheme only afterwards.