# WMA13 Pure Mathematics 3 (P3) May/June 2021 Paper 01

Pearson Edexcel International A Level WMA13 Pure Mathematics 3 (P3) past paper 01, May/June 2021. Open it online or download the question paper PDF free.

- Page: https://mathvsarham.com/past-papers/wma13/2021-may-june-paper-01/
- PDF: https://mathvsarham.com/papers/wma13/WMA13_may-june_2021_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 2021
- 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 curve C has equation
[Diagram or table: see page 2 of the paper]
The curve has a stationary point at the point P.
(a) Show, using calculus, that the x coordinate of P is a solution of the equation
[Diagram or table: see page 2 of the paper]
Using the iteration formula
[Diagram or table: see page 2 of the paper]
(b) find the value of x_2 and the value of x_6 , giving your answers to 3 decimal places.
[Diagram or table: see page 2 of the paper]
[7]

Question 2
(a) Show that
[Diagram or table: see page 4 of the paper]
where k is a constant to be found.
[Diagram or table: see page 4 of the paper]
(b) Hence solve, for 0 < θ < 90°
[Diagram or table: see page 4 of the paper]
giving your answers to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
[Diagram or table: see page 4 of the paper]
[9]

Question 3
(i) Find
[Diagram or table: see page 8 of the paper]
giving your answer in simplest form.
[Diagram or table: see page 8 of the paper]
(ii)(a) Write 4x + 3/x + 2 in the form
[Diagram or table: see page 8 of the paper]
(b) Hence find, using algebraic integration, the exact value of
[Diagram or table: see page 8 of the paper]
giving your answer in simplest form.
[Diagram or table: see page 8 of the paper]
[8]

Question 4
The functions f and g are defined by
[Diagram or table: see page 12 of the paper]
(a) Solve the equation
[Diagram or table: see page 12 of the paper]
(b) Find f^−1
[Diagram or table: see page 12 of the paper]
(c) Sketch and label, on the same axes, the curve with equation y = g(x) and the curve with equation y = g^−1(x) . Show on your sketch the coordinates of the points where each curve meets or cuts the coordinate axes.
[Diagram or table: see page 12 of the paper]
[10]

Question 5
[Diagram or table: see page 16 of the paper]
The growth of duckweed on a pond is being studied.
The surface area of the pond covered by duckweed, Am^2, at a time t days after the start of the study is modelled by the equation
[Diagram or table: see page 16 of the paper]
Figure 1 shows the linear relationship between log_10 A and t.
The points (0, 0.32) and (8, 0.56) lie on the line as shown.
(a) Find, to 3 decimal places, the value of p and the value of q.
[Diagram or table: see page 16 of the paper]
Using the model with the values of p and q found in part (a),
(b) find the rate of increase of the surface area of the pond covered by duckweed, in m^2/ day, exactly 6 days after the start of the study. Give your answer to 2 decimal places.
[Diagram or table: see page 16 of the paper]
[7]

Question 6
Given that k is a positive constant,
(a) on separate diagrams, sketch the graph with equation
(i) y = k − 2| x |
(ii) y = x − k/3
[Diagram or table: see page 20 of the paper]
Show on each sketch the coordinates, in terms of k, of each point where the graph meets or cuts the axes.
[Diagram or table: see page 20 of the paper]
(b) Hence find, in terms of k, the values of x for which
[Diagram or table: see page 20 of the paper]
giving your answers in simplest form.
[Diagram or table: see page 20 of the paper]
[8]

Question 7
Given that
[Diagram or table: see page 24 of the paper]
where A and B are integers to be found.
[Diagram or table: see page 24 of the paper]
[5]

Question 8
A scientist is studying a population of fish in a lake. The number of fish, N, in the population, t years after the start of the study, is modelled by the equation
[Diagram or table: see page 26 of the paper]
(a) Find the number of fish in the lake at the start of the study.
[Diagram or table: see page 26 of the paper]
(b) Find the upper limit to the number of fish in the lake.
[Diagram or table: see page 26 of the paper]
(c) Find the time, after the start of the study, when there are predicted to be 500 fish in the lake. Give your answer in years and months to the nearest month.
[Diagram or table: see page 26 of the paper]
(d) Show that
[Diagram or table: see page 26 of the paper]
where A is a constant to be found.
[Diagram or table: see page 26 of the paper]
Given that when t = T, dN/dt = 8
(e) find the value of T to one decimal place.
(Solutions relying entirely on calculator technology are not acceptable.)
[Diagram or table: see page 26 of the paper]
[13]

Question 9
(a) Express 12 sin x − 5 cos x in the form Rsin (x − α), where R and α are constants,
R > 0 and 0 < α < π/2 . Give the exact value of R and give the value of α in radians,
to 3 decimal places.
[Diagram or table: see page 30 of the paper]
The function g is defined by
[Diagram or table: see page 30 of the paper]
(b)(i) the minimum value of g(θ)
(ii) the smallest value of θ at which the minimum value occurs.
[Diagram or table: see page 30 of the paper]
The function h is defined by
[Diagram or table: see page 30 of the paper]
(c) Find the range of h.
[Diagram or table: see page 30 of the paper]
[Diagram or table: see page 32 of the paper]
[8]