Numerical solution of equations in A Level Math Paper 3: common mistakes and examiner advice

This topic combines sketching, a change-of-sign test and an iterative formula, and it often starts with a "show that" derivation. The reports show that students usually know what an iteration is. Marks are lost in the sketches, in the justification, and in the accuracy of the final answer. 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

• Sketches that do not show the key features. Reports describe graphs that did not resemble the functions at all. Examples: a logarithm graph that does not cross the x-axis at 1, a cosecant graph without its minimum, a line y = x + 1 that does not pass through (0, 1), an exponential drawn as a straight line without an intercept or asymptote, and a y = x line that does not pass through the origin.
• Not showing why there is only one root. Even on correct sketches, students often did not say how the diagram shows that the equation has a single root.
• A change of sign that is not justified. Reports describe students who compared the x values with each other, or compared the f values between the wrong pairs, instead of showing f(a) and f(b) have opposite signs. Some left out the comparison entirely or drew the wrong conclusion from correct work.
• Degrees instead of radians. This is listed in several reports, even when the question gave an interval such as 0 < x < π, which indicates radians.
• Calculator keying. Missing brackets when typing a function into the calculator.
• Showing the given equation. Slips in differentiating, such as ln 5x differentiating to 1/(5x) instead of 1/x, missing the factor eˣ in the product rule, and errors in taking out common factors. In a "show that" question, every step must be shown.
• Not tracing an error. Very few students who could not reach the given answer went back to find their error.
• Accuracy and presentation. Students did not work to the required accuracy in the iterations, gave an answer in decimal form when an exact form was requested, or did not state the conclusion of the iteration.
• Not using the right function. Reports note that many other functions could be used at the values, but some students did not consider an appropriate one.

What examiners advise

• Show the main features of the functions over nearly the whole interval: intercepts, asymptotes, the minimum point, and where the curves cross.
• Say what the diagram shows about the number of roots.
• Show both function values and their signs when testing for a root, and end with a conclusion.
• Check you are in radians before you start.
• Meet the demands of the question. One report says an answer that does not match the requirement of the question does not earn the mark.
• Work to the required accuracy and keep extra figures between iterations.

Habits that help

• Sketch before you calculate. A sketch usually gives the interval for the root.
• Write a short sentence under every sketch: "The curves meet once, so there is one root."
• Store each iteration on the calculator and use the stored value.

Questions students ask

What must my sketch show in a numerical solution question?

The main features over nearly the whole interval: intercepts, asymptotes, minimum points and where the curves cross. Reports describe a logarithm graph that does not cross the x-axis at 1, a cosecant graph without its minimum, and a line y = x + 1 that misses (0, 1).

How do I show there is only one root?

Say what the diagram shows. A short sentence under the sketch, such as "the curves meet once, so there is one root", is what examiners are looking for. Students often drew correct sketches and did not say how they show a single root.

How do I show that a root lies between two values?

Work out f(a) and f(b), show both values and their signs, and write a conclusion. Reports record students comparing the x values with each other, using the wrong pairs, or leaving out the comparison.

Why do my iteration answers come out wrong?

Check the calculator is in radians (degrees are a listed error, even when the question gives an interval such as 0 < x < π), type brackets correctly, and keep extra figures between iterations. Store each iteration on the calculator and use the stored value.

What should I do if I cannot reach the printed answer in a "show that" question?

Go back and find your error. Reports note that very few students who could not reach the given answer traced their mistake. Slips such as differentiating ln 5x as 1/(5x) instead of 1/x are common.