# WMA13 Pure Mathematics 3 (P3) January 2021 Paper 01

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

- Page: https://mathvsarham.com/past-papers/wma13/2021-january-paper-01/
- PDF: https://mathvsarham.com/papers/wma13/WMA13_january_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: January 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
Find
[Diagram or table: see page 2 of the paper]
giving your answer in simplest form. [3]

Question 2
[Diagram or table: see page 4 of the paper]
Figure 1 shows a sketch of the curve with equation y = f(x), where x ∈  and f(x) is a polynomial.
The curve passes through the origin and touches the x-axis at the point (3, 0)
There is a maximum turning point at (1, 2) and a minimum turning point at (3, 0)
On separate diagrams, sketch the curve with equation
(i) y = 3f(2x) [3]
(ii) y = f(−x) − 1 [3]
On each sketch, show clearly the coordinates of
● the point where the curve crosses the y-axis ● any maximum or minimum turning points

Question 3
[Diagram or table: see page 6 of the paper]
(a) Show that
[Diagram or table: see page 6 of the paper]
where a, b, c and d are integers to be found. [4]
(b) Hence find f^−1(x) [2]
(c) Find the domain of f^−1 [2]

Question 4
[Diagram or table: see page 10 of the paper]
Figure 2 shows a sketch of the graph with equation y = f(x), where
[Diagram or table: see page 10 of the paper]
and where a is a positive constant.
The graph has a vertex at the point P, as shown in Figure 2.
(a) Find, in terms of a, the coordinates of P. [2]
(b) Sketch the graph with equation y = g(x), where
[Diagram or table: see page 10 of the paper]
On your sketch, show the coordinates, in terms of a, of each point where the graph cuts or meets the coordinate axes. [2]
The graph with equation y = g(x) intersects the graph with equation y = f(x) at two points.
(c) Find, in terms of a, the coordinates of the two points. [5]

Question 5
The temperature, θ °C, inside an oven, t minutes after the oven is switched on, is given by
[Diagram or table: see page 14 of the paper]
where A and k are positive constants.
Given that the temperature inside the oven is initially 18 °C,
(a) find the value of A. [2]
The temperature inside the oven, 5 minutes after the oven is switched on, is 90 °C.
(b) Show that k = p ln q where p and q are rational numbers to be found. [4]
Hence find
(c) the temperature inside the oven 9 minutes after the oven is switched on, giving your answer to 3 significant figures, [2]
(d) the rate of increase of the temperature inside the oven 9 minutes after the oven is switched on. Give your answer in °C min^–1 to 3 significant figures. [3]

Question 6
[Diagram or table: see page 18 of the paper]
(a) Find f ʹ(x) [2]
(b) Show that the equation f ʹ(x) = 0 can be written as
[Diagram or table: see page 18 of the paper]
where k is an integer to be found. [2]
(c) Starting with x_1= 2.5 use the iteration formula
[Diagram or table: see page 18 of the paper]
with the value of k found in part (b), to calculate the values of x_2 and x_6 giving your answers to 3 decimal places. [2]
(d) Using a suitable interval and a suitable function that should be stated, show that a root of fʹ (x) = 0 is 2.581 correct to 3 decimal places. [2]
[Diagram or table: see page 22 of the paper]

Question 7
(a) Prove that
[Diagram or table: see page 22 of the paper]
[3]
(b) Hence solve, for − < θ <
[Diagram or table: see page 22 of the paper]
giving your answers in radians to 3 significant figures where appropriate. [6]

Question 8
The percentage, P, of the population of a small country who have access to the internet, is modelled by the equation
[Diagram or table: see page 26 of the paper]
where a and b are constants and t is the number of years after the start of 2005
Using the data for the years between the start of 2005 and the start of 2010, a graph is plotted of log_10 P against t.
The points are found to lie approximately on a straight line with gradient 0.09 and intercept 0.68 on the log_10 P axis.
(a) Find, according to the model, the value of a and the value of b, giving your answers to 2 decimal places. [4]
(b) In the context of the model, give a practical interpretation of the constant a. [1]
(c) Use the model to estimate the percentage of the population who had access to the internet at the start of 2015 [2]

Question 9
Find
3x − 2
(i) _2 ∫ 3x − 4x + 5 [2]
[Diagram or table: see page 28 of the paper]
(ii) _2_xe/− 1)_3 ∫ (e [2]

Question 10
The curve C has equation
[Diagram or table: see page 30 of the paper]
(a) Find dx/dy in terms of y. [2]
(b) Hence show that
[Diagram or table: see page 30 of the paper]
where p is irrational and q is an integer, stating the values of p and q. [3]
(c) Find the equation of the normal to C at the point where y = π/12, giving your answer
in the form y = mx + c, giving m and c as exact irrational numbers. [5]
[Diagram or table: see page 32 of the paper]