# WMA13 Pure Mathematics 3 (P3) Oct/Nov 2022 Paper 01

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

- Page: https://mathvsarham.com/past-papers/wma13/2022-oct-nov-paper-01/
- PDF: https://mathvsarham.com/papers/wma13/WMA13_oct-nov_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: Oct/Nov 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
In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
[Diagram or table: see page 2 of the paper]
(a) Show that
[Diagram or table: see page 2 of the paper]
where A, B and C are integers to be found. [4]
(b) Hence, find
[Diagram or table: see page 2 of the paper]
giving your answer in the form p + ln q , where p and q are integers. [5]

Question 2
The functions f and g are defined by
[Diagram or table: see page 6 of the paper]
(a) Find the range of f [2]
(b)(i) Find f^−1(x)
(ii) Write down the domain of f^−1 [3]
(c) Find fg(−π) [2]

Question 3
In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
[Diagram or table: see page 8 of the paper]
Figure 1 shows a sketch of part of the curve with equation y = f(x) where
[Diagram or table: see page 8 of the paper]
The curve has a maximum turning point at A and a minimum turning point at (2, 0)
(a) Use calculus to find the exact coordinates of A. [5]
Given that the equation f(x) = k , where k is a constant, has at least two distinct roots,
(b) state the range of possible values for k. [2]

Question 4
[Diagram or table: see page 10 of the paper]
(a) Express x in terms of y. [2]
(b) Hence, giving your answer in terms of x, find dy/dx [3]

Question 5
[Diagram or table: see page 12 of the paper]
The profit made by a company, £ P million, t years after the company started trading, is modelled by the equation
[Diagram or table: see page 12 of the paper]
The graph of P against t is shown in Figure 2.
According to the model,
(a) show that exactly one year after it started trading, the company had made a loss of approximately £ 830 000 [2]
A manager of the company wants to know the value of t for which P = 0
(b) Show that this value of t occurs in the interval [6, 7] [2]
(c) Show that the equation P = 0 can be expressed in the form
[Diagram or table: see page 12 of the paper]
[2]
(d) Using the iteration formula
[Diagram or table: see page 12 of the paper]
find the value of t_2 and the value of t_6 , giving your answers to 3 decimal places. [3]
(e) Hence find, according to the model, how many months it takes in total, from when the company started trading, for it to make a profit. [2]

Question 6
[Diagram or table: see page 16 of the paper]
where a, b and c are integers to be found. [6]

Question 7
[Diagram or table: see page 20 of the paper]
Figure 3 shows a sketch of the graph of C_1 with equation
[Diagram or table: see page 20 of the paper]
(a) Write down the coordinates of
(i) the vertex of C_1
(ii) the intersection of C_1 with the y-axis. [2]
(b) Find the x coordinates of the intersections of C_1 with the x-axis. [2]
Diagram 1, shown on page 21, is a copy of Figure 3.
(c) On Diagram 1, sketch the curve C_2 with equation
[Diagram or table: see page 20 of the paper]
Identify clearly the coordinates of any points of intersection of C_2 with the coordinate axes. [3]
(d) Find the coordinates of the points of intersection of C_1 and C_2 (Solutions relying entirely on calculator technology are not acceptable.) [5]
[Diagram or table: see page 21 of the paper]

Question 8
In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
(a) Express 8 sin x − 15 cos x in the form R sin (x − α) , where R > 0 and 0 < α < π/2
Give the exact value of R, and give the value of α, in radians, to 4 significant figures. [3]
[Diagram or table: see page 24 of the paper]
(b) Find
(i) the minimum value of f(x)
(ii) the smallest value of x at which this minimum value occurs. [4]
(c) State the y coordinate of the minimum points on the curve with equation
[Diagram or table: see page 24 of the paper]
[1]
(d) State the smallest value of x at which a maximum point occurs for the curve with equation
[Diagram or table: see page 24 of the paper]
[1]

Question 9
In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
Given that cos 2θ − sin 3θ ≠ 0
(a) prove that
[Diagram or table: see page 28 of the paper]
[4]
(b) Hence solve, for 0 < θ  360°
[Diagram or table: see page 28 of the paper]
Give your answers to one decimal place. [5]
[Diagram or table: see page 32 of the paper]