# WMA13 Pure Mathematics 3 (P3) January 2022 Paper 01

Pearson Edexcel International A Level WMA13 Pure Mathematics 3 (P3) past paper 01, January 2022. Open it online or download the question paper PDF free.

- Page: https://mathvsarham.com/past-papers/wma13/2022-january-paper-01/
- PDF: https://mathvsarham.com/papers/wma13/WMA13_january_2022_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 2022
- 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
Find, using calculus, the x coordinate of the stationary point on the curve with equation
[Diagram or table: see page 2 of the paper]
[4]

Question 2
(a) Show that the equation
[Diagram or table: see page 4 of the paper]
can be written in the form
[Diagram or table: see page 4 of the paper]
where k is a constant to be found. [3]
(b) Hence find the smallest positive solution of the equation
[Diagram or table: see page 4 of the paper]
giving your answer, in degrees, to one decimal place. [2]

Question 3
(i) Find, in simplest form,
[Diagram or table: see page 6 of the paper]
[2]
(ii) Show, by algebraic integration, that
[Diagram or table: see page 6 of the paper]
where a is a rational constant to be found. [4]

Question 4
The growth of a weed on the surface of a pond is being studied.
The surface area of the pond covered by the weed, A m^2, is modelled by the equation
[Diagram or table: see page 8 of the paper]
where p is a positive constant and t is the number of days after the start of the study.
Given that
• 30 m^2 of the surface of the pond was covered by the weed at the start of the study
• 50 m^2 of the surface of the pond was covered by the weed T days after the start of the study
(a) show that p = 2.4 [2]
(b) find the value of T, giving your answer to one decimal place.
[Diagram or table: see page 8 of the paper]
[4]
The weed grows until it covers the surface of the pond.
(c) Find, according to the model, the maximum possible surface area of the pond. [1]
[Diagram or table: see page 10 of the paper]

Question 5
[Diagram or table: see page 10 of the paper]
Figure 1 shows a sketch of part of the curve with equation
[Diagram or table: see page 10 of the paper]
The curve cuts the negative x‑axis at the point P, as shown in Figure 1.
(a) Show that the x coordinate of P lies in the interval [–1.25, –1.2] [2]
The curve cuts the positive x‑axis at the point Q, also shown in Figure 1.
Using the iterative formula
[Diagram or table: see page 10 of the paper]
(b)(i) find, to 4 decimal places, the value of x_2
(ii) find, by continued iteration, the x coordinate of Q. Give your answer to 4 decimal places. [3]
The curve has a maximum turning point at M, as shown in Figure 1.
(c) Using calculus and showing each stage of your working, find the x coordinate of M. [4]

Question 6
The function f is defined by
[Diagram or table: see page 14 of the paper]
(a) Show, by using calculus, that f is a decreasing function. [3]
(b) Find f^–1 [3]
(c)(i) Show that ff(x) = where a, b and c are constants to be found.
(ii) Deduce the range of ff. [5]
[Diagram or table: see page 18 of the paper]

Question 7
[Diagram or table: see page 18 of the paper]
Figure 2 shows a sketch of part of the graph with equation y = f(x), where
[Diagram or table: see page 18 of the paper]
(a) State the coordinates of the vertex, V, of the graph. [2]
(b) Solve, using algebra,
[Diagram or table: see page 18 of the paper]
[4]
(c) Sketch the graph with equation
stating the coordinates of the local maximum point and each local minimum point. [4]

Question 8
A dose of antibiotics is given to a patient.
The amount of the antibiotic, x milligrams, in the patient's bloodstream t hours after the dose was given, is found to satisfy the equation
[Diagram or table: see page 22 of the paper]
(a) Show that this equation can be written in the form
[Diagram or table: see page 22 of the paper]
where p and q are constants to be found. Give the value of p to the nearest whole number and the value of q to 2 significant figures. [4]
(b) With reference to the equation in part (a), interpret the value of the constant p. [1]
When a different dose of the antibiotic is given to another patient, the values of x and t satisfy the equation
[Diagram or table: see page 22 of the paper]
(c) Use calculus to find, to 2 significant figures, the value of dx/dt when t = 5 [3]

Question 9
[Diagram or table: see page 26 of the paper]
Solutions relying entirely on calculator technology are not acceptable.
(i) Solve, for 0 < x  π, the equation
[Diagram or table: see page 26 of the paper]
giving the answers, as appropriate, to 3 significant figures. [4]
(ii) Prove that
[Diagram or table: see page 26 of the paper]
[4]
[Diagram or table: see page 30 of the paper]

Question 10
[Diagram or table: see page 30 of the paper]
Figure 3 shows a sketch of the curve C with equation
[Diagram or table: see page 30 of the paper]
(a) Show that
[Diagram or table: see page 30 of the paper]
[4]
Given that the straight line with equation x = k, where k is a constant, cuts C at exactly two points,
(b) find the range of possible values for k. [3]
[Diagram or table: see page 32 of the paper]