# WMA13 Pure Mathematics 3 (P3) Oct/Nov 2023 Paper 01

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

- Page: https://mathvsarham.com/past-papers/wma13/2023-oct-nov-paper-01/
- PDF: https://mathvsarham.com/papers/wma13/WMA13_oct-nov_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: Oct/Nov 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
A curve has equation y = f (x) where
[Diagram or table: see page 2 of the paper]
(a) Show that the equation f (x) = 0 has a root, α, in the interval [1, 2] [2]
The iterative formula
[Diagram or table: see page 2 of the paper]
with x_1= 1 is used to find an approximate value for the root α.
(b)(i) Find the value of x_2 to 4 decimal places.
(ii) Find, by repeated iteration, the value of α, giving your answer to 4 decimal places. [3]

Question 2
The function f is defined by
[Diagram or table: see page 4 of the paper]
(a) Find ff (6) [2]
(b) Find f^–1 [3]
The function g is defined by
[Diagram or table: see page 4 of the paper]
(c) Find the exact value of a for which
[Diagram or table: see page 4 of the paper]
[3]

Question 3
(a) Using the identity for cos (A + B) , prove that
[Diagram or table: see page 8 of the paper]
[2]
(b) Hence, using algebraic integration, find the exact value of
[Diagram or table: see page 8 of the paper]
[4]

Question 4
A new mobile phone is released for sale.
The total sales N of this phone, in thousands, is modelled by the equation
[Diagram or table: see page 10 of the paper]
where A is a constant and t is the time in months after the phone was released for sale.
Given that when t = 0 , N = 32
(a) state the value of A. [1]
Given that when t = T the total sales of the phone was 100 000
(b) find, according to the model, the value of T. Give your answer to 2 decimal places. [3]
(c) Find, according to the model, the rate of increase in total sales when t = 7, giving your answer to 3 significant figures.
[Diagram or table: see page 10 of the paper]
[2]
The total sales of the mobile phone is expected to reach 150 000
Using this information,
(d) give a reason why the given equation is not suitable for modelling the total sales of the phone. [1]

Question 5
The curve C has equation
[Diagram or table: see page 14 of the paper]
where k is a positive constant.
(a) Show that
[Diagram or table: see page 14 of the paper]
where A and B are constants to be found. [3]
Given that C has exactly three turning points,
(b) find the x coordinate of each of these points. Give your answer in terms of k where appropriate. [3]
(c) find the upper limit to the value for k. [1]

Question 6
An area of sea floor is being monitored.
The area of the sea floor, S km^2, covered by coral reefs is modelled by the equation
[Diagram or table: see page 16 of the paper]
where p and q are constants and t is the number of years after monitoring began.
Given that
[Diagram or table: see page 16 of the paper]
(a) find, according to the model, the area of sea floor covered by coral reefs when t = 2 [2]
(b) find a complete equation for the model in the form
[Diagram or table: see page 16 of the paper]
giving the value of p and the value of q each to 3 significant figures. [3]
(c) With reference to the model, interpret the value of the constant q [1]

Question 7
[Diagram or table: see page 18 of the paper]
Figure 1 shows a sketch of the curve C with equation y = f (x) where
[Diagram or table: see page 18 of the paper]
(a) Find the range of f [2]
(b) Show that
[Diagram or table: see page 18 of the paper]
where A and B are constants to be found. [4]
Given that the line y = k , where k is a constant, k > 0, intersects the curve at exactly two distinct points,
(c) find the exact range of values of k [4]

Question 8
(a) Prove that
[Diagram or table: see page 22 of the paper]
[4]
(b) Hence solve for 0 < x < 360° , where x ≠ (90n)° , n ∈  , the equation
[Diagram or table: see page 22 of the paper]
giving your answers to one decimal place.
[Diagram or table: see page 22 of the paper]
[4]

Question 9
In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
[Diagram or table: see page 26 of the paper]
Figure 2 shows a sketch of the curve with equation
[Diagram or table: see page 26 of the paper]
where k is a constant.
Given that the curve
• has an asymptote at x = k
• cuts the y-axis at point A
• meets the x-axis at point B
as shown in Figure 2,
(a) state the value of k [1]
(b)(i) find the y coordinate of A
(ii) find the exact x coordinate of B [3]
(c) Using algebra and showing your working, find the set of values of x such that
[Diagram or table: see page 26 of the paper]
[5]

Question 10
In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
A curve C has equation
[Diagram or table: see page 30 of the paper]
The point P with x coordinate 1/4 lies on C
(a) Find the exact y coordinate of P [2]
(b) Find dx/dy [2]
(c) Hence show that dy/dx can be written in the form
[Diagram or table: see page 30 of the paper]
where q, r and s are constants to be found. [3]
Using the answer to part (c),
(d)(i) state the x coordinate of the point where the value of dy/dx is a minimum,
(ii) state the value of dy/dx at this point. [2]