# WMA13 Pure Mathematics 3 (P3) January 2026 Paper 01

Pearson Edexcel International A Level WMA13 Pure Mathematics 3 (P3) past paper 01, January 2026. Open it online or download the question paper PDF free.

- Page: https://mathvsarham.com/past-papers/wma13/2026-january-paper-01/
- PDF: https://mathvsarham.com/papers/wma13/WMA13_january_2026_paper_01_qp.pdf
- Board: Edexcel
- Qualification: Pearson Edexcel International A Level
- Subject: WMA13 Pure Mathematics 3 (P3) (https://mathvsarham.com/past-papers/wma13/)
- Session: January 2026
- Paper: 01
- Component code: WMA13/01
- Time allowed: 90 minutes
- Total marks: 75
- Pages: 32

## Question paper text

Extracted automatically from the PDF; diagrams are missing and notation may be imperfect.

Question 1
[Diagram or table: see page 2 of the paper]
where x is measured in radians.
(a) Show that g(x) = 0 has a root α in the interval 0.7, 0.8 [2]
(b) Show that the equation g(x) = 0 can be written in the form
[Diagram or table: see page 2 of the paper]
where p is a constant to be found. [1]
Using the iteration formula
[Diagram or table: see page 2 of the paper]
with the value of p found in part (b) and starting with x_1 = 0.7
(c)(i) find, to 4 decimal places, the value of x_2
(ii) find, by repeated iteration, the value of α, giving your answer to 4 decimal places. [3]

Question 2
(i) Given y = x^2 ln 2x, find in simplest form, dy/dx [2]
(ii) Given
[Diagram or table: see page 4 of the paper]
where a and b are constants to be found. [4]

Question 3
(i) Find, in simplest form,
[Diagram or table: see page 6 of the paper]
[2]
[Diagram or table: see page 6 of the paper]
(ii) f(x)  x > 1
[Diagram or table: see page 6 of the paper]
(a) Write f(x) in the form
[Diagram or table: see page 6 of the paper]
where a, b and c are constants to be found. [4]
(b) Hence, using algebraic integration, find
[Diagram or table: see page 6 of the paper]
writing the answer in simplest form. [4]

Question 4
The resting heart rate, h beats per minute, of a particular group of mammals, can be modelled by the equation
[Diagram or table: see page 10 of the paper]
where m kg is the mass of the mammal.
The mass of one of these mammals is 3 kg.
According to the model,
(a) find the resting heart rate of this mammal. Give your answer to 3 significant figures. [2]
(b) Show that the equation of the model can be written in the form
[Diagram or table: see page 10 of the paper]
giving each of the values of the constants p and q to 3 significant figures where appropriate. [3]
(c) With reference to the model, interpret the value of the constant p. [1]

Question 5
The function f is defined by
[Diagram or table: see page 12 of the paper]
(a) Find ff(x), giving your answer as a single fraction in its simplest form. [3]
(b) Find f^−^1(12) [2]
The function g is defined by
[Diagram or table: see page 12 of the paper]
(c) Find the exact values of a such that
[Diagram or table: see page 12 of the paper]
[3]

Question 6
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
(a) Show that the equation
can be written in the form
[Diagram or table: see page 16 of the paper]
[4]
(b) Hence solve, for 0  x  180, the equation
[Diagram or table: see page 16 of the paper]
giving your solutions to one decimal place. [3]

Question 7
[Diagram or table: see page 20 of the paper]
Figure 1 shows a sketch of part of the graph with equation y = f(x), where
[Diagram or table: see page 20 of the paper]
The vertex of the graph is at the point P, as shown in Figure 1.
(a) State the coordinates of P. [2]
(b) Solve the equation f(x) = 8 [3]
The line l has equation y = mx, where m is a constant.
Given that l intersects the graph of y = f(x) at exactly one point,
(c) find the possible values of m. [3]
The graph with equation y = f(x) is transformed onto the graph with equation y  a f (x  b), where a and b are constants.
Given that the vertex of the graph with equation y  a f (x  b) is 5, 16
(d) find the value of a and the value of b. [2]

Question 8
A curve has equation
[Diagram or table: see page 24 of the paper]
where a, b and n are integers to be found. [4]

Question 9
[Diagram or table: see page 26 of the paper]
(a) Write h(x) in the form
[Diagram or table: see page 26 of the paper]
where P, R and α are positive constants, 0 <  < /2
Give the exact values of P and R and give the value of α, in radians, to 3 decimal places. [6]
[Diagram or table: see page 26 of the paper]
Figure 2 is a sketch of the curve with equation y = h(x)
(b) State the range of h. [1]
The point M, shown in Figure 2, is the minimum point on the curve with the largest x coordinate.
(c) Find the x coordinate of M.
[Diagram or table: see page 26 of the paper]
[2]

Question 10
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
Two populations of insects are being studied over the same period of time.
The number of insects N A in population A, t years after the start of the study, is modelled by the equation
[Diagram or table: see page 30 of the paper]
According to the model,
(a) find the rate of growth in the number of insects in population A exactly 5 years after the start of the study. [2]
The number of insects N B in population B, t years after the start of the study, is modelled by the equation
[Diagram or table: see page 30 of the paper]
where P and k are positive constants.
Given that
• there are 10 000 insects in population B at the start of the study
• there are 11 570 insects in population B exactly 4 years after the start of the study
• there are the same number of insects in population A and population B after T years
find, according to the models,
(b) the exact value of P and the value of k to 3 decimal places, [4]
(c) the value of T, giving your answer to one decimal place. [3]