# WMA13 Pure Mathematics 3 (P3) January 2025 Paper 01

Pearson Edexcel International A Level WMA13 Pure Mathematics 3 (P3) past paper 01, January 2025. Open it online or download the question paper PDF free.

- Page: https://mathvsarham.com/past-papers/wma13/2025-january-paper-01/
- PDF: https://mathvsarham.com/papers/wma13/WMA13_january_2025_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 2025
- 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]
The equation f (x) = 0 has a single root α
(a) Show that 0.1 < α < 0.2 [2]
(b) Show that α is a solution of
[Diagram or table: see page 2 of the paper]
[1]
The iterative formula
[Diagram or table: see page 2 of the paper]
is used to find α
(c) Starting with x_1= 0.15 and using the iterative formula,
(i) find, to 4 decimal places, the value of x_2
(ii) find, to 4 decimal places, the value of α [3]

Question 2
The weed on the surface of a pond is being monitored.
The surface area of the pond covered by the weed, A m^2, is modelled by the equation
[Diagram or table: see page 4 of the paper]
where t is the number of weeks after monitoring began.
Use the equation of the model to answer parts (a) and (b).
(a) Find the surface area of the pond initially covered by the weed. [1]
After T weeks, 25 m^2 of the pond is covered by the weed.
(b) Find the value of T, giving your answer to 2 decimal places. [2]

Question 3
In this question you must show all stages of your working.
[Diagram or table: see page 6 of the paper]
A curve has equation
[Diagram or table: see page 6 of the paper]
Use calculus to find the range of values of x for which y is increasing. [6]

Question 4
Given that
[Diagram or table: see page 8 of the paper]
(a)(i) find the values of the constants A, B and C
(ii) show that D = 0 [4]
(b) Hence, using algebraic integration, find
[Diagram or table: see page 8 of the paper]
giving your answer in the form p + q ln 2, where p and q are integers. [5]

Question 5
A hot piece of metal is cooled by dropping it into water. The temperature, H °C, of the metal, t minutes after it is dropped into the water, is modelled by the equation
[Diagram or table: see page 12 of the paper]
(a) Find the initial temperature of the piece of metal. [1]
(b) On Diagram 1, sketch the graph of H against t. On your sketch, state the equation of the asymptote to the curve. [2]
(c) Find the value of t for which H = 144, giving your answer to 2 decimal places.
[Diagram or table: see page 12 of the paper]
[3]
(d) Show by differentiation that
[Diagram or table: see page 12 of the paper]
where a and b are constants to be found. [3]
[Diagram or table: see page 13 of the paper]

Question 6
The function f is defined by
[Diagram or table: see page 16 of the paper]
(a) Find f^–1 [3]
(b) Show that
[Diagram or table: see page 16 of the paper]
where a, b, c and d are integers to be found. [3]
The point P (3, 15) lies on the curve with equation y = f (x) .
(c) Find the point to which P is mapped when y = f (x) is transformed to the curve with equation y = 2f (3x) + 8 [2]

Question 7
Given that a and b are positive constants with a > b ,
(a) sketch, on separate diagrams, the graph with equation
(i) y = |3x – a|
(ii) y = |3x – a| – b
Show on each sketch
• the coordinates of the minimum point on the graph • the coordinates of the point at which the graph crosses the y-axis [6]
(b) Solve the equation
[Diagram or table: see page 20 of the paper]
giving any solution for x in terms of a and b. [2]

Question 8
In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
(i) Solve, for 0 < θ < π
[Diagram or table: see page 22 of the paper]
giving your answers, in radians, to 3 significant figures. [5]
(ii) Solve, for 0 < x < 180°
[Diagram or table: see page 22 of the paper]
giving your answers, in degrees, to one decimal place. [4]

Question 9
[Diagram or table: see page 26 of the paper]
Figure 1 shows a sketch of part of the curve C with equation y = f (x) where
[Diagram or table: see page 26 of the paper]
The curve cuts the x-axis at point P
(a) State the x coordinate of P [1]
The point Q, shown in Figure 1, is the stationary point on C
(b) Use calculus to find the exact coordinates of Q [5]
(c) Hence find the range of the function g (x) where
[Diagram or table: see page 26 of the paper]
[2]

Question 10
[Diagram or table: see page 30 of the paper]
Figure 2 shows a sketch of the curve with equation
[Diagram or table: see page 30 of the paper]
(a) Find in terms of y. dy [2]
(b) Hence show that
[Diagram or table: see page 30 of the paper]
where k is a constant to be found. [3]
The point P (a, b) lies on the curve and is shown in Figure 2.
Given that
• the gradient of the curve at P is – 1/4
• both a and b are positive
(c) find the exact values of a and b. [4]