# WMA13 Pure Mathematics 3 (P3) January 2023 Paper 01

Pearson Edexcel International A Level WMA13 Pure Mathematics 3 (P3) past paper 01, January 2023. Open it online or download the question paper PDF free.

- Page: https://mathvsarham.com/past-papers/wma13/2023-january-paper-01/
- PDF: https://mathvsarham.com/papers/wma13/WMA13_january_2023_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 2023
- 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
The functions f and g are defined by
(a) Write down the range of f [1]
(b) Find the value of fg (1.5) [2]
(c) Find g^−1 [3]

Question 2
[Diagram or table: see page 4 of the paper]
(a) Express f (x) in the form where R and α are constants,
R > 0 and
Give the exact value of R and give the value of α, in radians, to 3 decimal places. [3]
[Diagram or table: see page 4 of the paper]
(b) Using the answer to part (a),
(i) write down the exact maximum value of g (x),
(ii) find the smallest positive value of x for which this maximum value occurs, giving your answer to 2 decimal places. [3]

Question 3
[Diagram or table: see page 6 of the paper]
The line l in Figure 1 shows a linear relationship between log_10 y and x.
The line passes through the points (0, 1.5) and (−4.8, 0) as shown.
(a) Write down an equation for l. [2]
(b) Hence, or otherwise, express y in the form kb ^x , giving the values of the constants k and b to 3 significant figures. [3]
[Diagram or table: see page 8 of the paper]

Question 4
[Diagram or table: see page 8 of the paper]
(a) Find the values of the constants A, B, C and D such that
[Diagram or table: see page 8 of the paper]
[4]
(b) Hence find,
[Diagram or table: see page 8 of the paper]
[3]

Question 5
In this question you must show all stages of your working.
[Diagram or table: see page 12 of the paper]
(a) Prove that [4]
(b) Hence solve, for
giving your answers to 2 decimal places. [5]

Question 6
[Diagram or table: see page 16 of the paper]
Figure 2 shows a sketch of the graph with equation
[Diagram or table: see page 16 of the paper]
where a is a positive constant.
The graph
• cuts the y-axis at the point P • cuts the x-axis at the points Q and R • has a minimum point at S
(a) Find, in simplest form in terms of a, the coordinates of
(i) point P
(ii) points Q and R
(iii) point S [4]
(b) Find, in simplest form in terms of a, the values of x for which
[Diagram or table: see page 16 of the paper]
[4]

Question 7
The curve C has equation
(a) Show that
[Diagram or table: see page 20 of the paper]
where a and b are integers to be found. [4]
The point P with y coordinate lies on C.
Given that the tangent to C at P crosses the x-axis at the point Q.
(b) find, in simplest form, the exact x coordinate of Q. [5]

Question 8
Find, in simplest form,
[Diagram or table: see page 24 of the paper]
[5]

Question 9
[Diagram or table: see page 26 of the paper]
Figure 3 shows a sketch of part of the curve C with equation
[Diagram or table: see page 26 of the paper]
(a) Find dy/dx , giving your answer in simplest form. [2]
The point P with x coordinate α lies on C.
Given that the tangent to C at P passes through the origin, as shown in Figure 3,
(b) show that x = α is a solution of the equation
[Diagram or table: see page 26 of the paper]
[3]
(c) Hence show that α lies between 1 and 2 [2]
(d) Show that the equation in part (b) can be written in the form
[Diagram or table: see page 26 of the paper]
[1]
The iteration formula
[Diagram or table: see page 26 of the paper]
with x_1 = 1 is used to find an approximation for α.
(e) Use the iteration formula to find, to 4 decimal places, the value of
(i) x_3
(ii) α [3]

Question 10
In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
A population of fruit flies is being studied.
The number of fruit flies, F, in the population, t days after the start of the study, is modelled by the equation
[Diagram or table: see page 30 of the paper]
where k is a constant.
Use the equation of the model to answer parts (a), (b) and (c).
(a) Find the number of fruit flies in the population at the start of the study. [1]
Given that there are 200 fruit flies in the population 15 days after the start of the study,
(b) show that k = 1/15 ln12 [3]
Given also that, when t = T, the number of fruit flies in the population is increasing at a rate of 10 per day,
(c) find the possible values of T, giving your answers to one decimal place. [5]