# WMA13 Pure Mathematics 3 (P3) May/June 2023 Paper 01

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

- Page: https://mathvsarham.com/past-papers/wma13/2023-may-june-paper-01/
- PDF: https://mathvsarham.com/papers/wma13/WMA13_may-june_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: May/June 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
[Diagram or table: see page 2 of the paper]
(a) Show that g(x) = 0 has a root α in the interval [3, 4] [2]
Using the iteration formula
[Diagram or table: see page 2 of the paper]
(b)(i) find, to 4 decimal places, the value of x_2
(ii) find, by repeated iteration, the value of α. Give your answer to 4 decimal places. [3]

Question 2
[Diagram or table: see page 4 of the paper]
Figure 1 shows the linear relationship between log_6 T and log_6 x
The line passes through the points (0, 4) and (2, 0) as shown.
(a)(i) Find an equation linking log_6 T and log_6 x
(ii) Hence find the exact value of T when x = 216 [3]
(b) Find an equation, not involving logs, linking T with x [3]

Question 3
(i) Find d/dx ln sin^23x writing your answer in simplest form.
[2]
(ii)(a) Find d/dx 3x^2  4
[2]
(b) Hence show that
[Diagram or table: see page 6 of the paper]
where R is an integer to be found.
[Diagram or table: see page 6 of the paper]
[3]

Question 4
The function f is defined by
[Diagram or table: see page 8 of the paper]
(a) State the range of f [1]
On the following page there is a diagram, labelled Diagram 1, which shows a sketch of the curve with equation y = f(x).
(b) On Diagram 1, sketch the curve with equation y = f^–1(x). [2]
The curve with equation y = f(x) meets the curve with equation y = f^–1(x) at the point P
Using algebra and showing your working,
(c) find the exact x coordinate of P [3]
[Diagram or table: see page 9 of the paper]
y = f (x)
[Diagram or table: see page 9 of the paper]

Question 5
In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
(i) Solve, for 0 < x < π
[Diagram or table: see page 10 of the paper]
[3]
(ii) Solve, for 0 < θ < 360°
[Diagram or table: see page 10 of the paper]
[4]

Question 6
[Diagram or table: see page 14 of the paper]
Figure 2 shows a sketch of the graph y = f(x), where
The vertex of the graph is at point P, shown in Figure 2.
(a) Find the coordinates of P [2]
(b) Find ff(0) [2]
(c) Solve the inequality [2]
(d) Solve the equation [3]

Question 7
A scientist is studying two different populations of bacteria.
The number of bacteria N in the first population is modelled by the equation
[Diagram or table: see page 18 of the paper]
where A and k are positive constants and t is the time in hours from the start of the study.
Given that • there were 2500 bacteria in this population at the start of the study • there were 10 000 bacteria 8 hours later
(a) find the exact value of A and the value of k to 4 significant figures. [3]
The number of bacteria N in the second population is modelled by the equation
[Diagram or table: see page 18 of the paper]
where t is the time in hours from the start of the study.
(b) Find the rate of decrease of bacteria in this population exactly 5 hours from the start of the study. Give your answer to 3 significant figures. [2]
When t = T, the number of bacteria in the two different populations was the same.
(c) Find the value of T, giving your answer to 3 significant figures.
[Diagram or table: see page 18 of the paper]
[3]

Question 8
[Diagram or table: see page 22 of the paper]
Figure 3 shows a sketch of the curve C with equation y = f(x) , where
[Diagram or table: see page 22 of the paper]
(a) Show that
[Diagram or table: see page 22 of the paper]
where A is a constant to be found. [4]
(b) Hence find the exact coordinates of the two stationary points on C. [3]
The function g is defined by
[Diagram or table: see page 22 of the paper]
(c) Find the coordinates of the maximum stationary point on the curve with equation y = g(x) . [2]

Question 9
In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
(a) Show that
[Diagram or table: see page 26 of the paper]
[3]
(b) Hence solve, for 0 < θ < π/2
[Diagram or table: see page 26 of the paper]
giving your answers to 3 significant figures as appropriate. [5]
(c) Using the result from part (a), or otherwise, find the exact value of
[Diagram or table: see page 26 of the paper]
[2]
[Diagram or table: see page 29 of the paper]

Question 10
[Diagram or table: see page 30 of the paper]
Figure 4 shows a sketch of the curve with equation
[Diagram or table: see page 30 of the paper]
(a) Find giving your answer as a fully simplified fraction. dy [4]
The tangents at points P and Q on the curve are parallel to the y-axis, as shown in Figure 4.
(b) Use the answer to part (a) to find the equations of these two tangents. [4]