# WMA13 Pure Mathematics 3 (P3) Oct/Nov 2021 Paper 01

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

- Page: https://mathvsarham.com/past-papers/wma13/2021-oct-nov-paper-01/
- PDF: https://mathvsarham.com/papers/wma13/WMA13_oct-nov_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: Oct/Nov 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 function f is defined by
[Diagram or table: see page 2 of the paper]
(a) Show that f (x) = 5x/x + 3 [3]
(b) Find f ^–1 [3]
(c)(i) Find, in simplest form, f ′(x).
(ii) Hence, state whether f is an increasing or a decreasing function, giving a reason for your answer. [3]

Question 2
[Diagram or table: see page 6 of the paper]
Figure 1 shows a sketch of part of the graph with equation y = f(x), where
The vertex of the graph is at point P, as shown in Figure 1.
(a) State the coordinates of P. [2]
(b)(i) State the range of f.
(ii) Find the value of ff(4) [2]
(c) Solve, using algebra and showing your working, [4]
The graph with equation y = f(x) is transformed onto the graph with equation y = af(x + b)
The vertex of the graph with equation y = af(x + b) is (4, 20)
Given that a and b are constants,
(d) find the value of a and the value of b. [2]

Question 3
[Diagram or table: see page 10 of the paper]
The total mass of gold, G tonnes, extracted from a mine is modelled by the equation
[Diagram or table: see page 10 of the paper]
where t is the number of years after 1st January 1800.
Figure 2 shows a sketch of G against t.
[Diagram or table: see page 10 of the paper]
(a)(i) Find the value of k.
(ii) Hence find the year and month in which gold started being extracted from the mine. [3]
(b) Find the total mass of gold extracted from the mine up to 1st January 1870. [2]
There is a limit to the mass of gold that can be extracted from the mine.
(c) State the value of this limit. [1]

Question 4
[Diagram or table: see page 12 of the paper]
(a) Show that the equation
[Diagram or table: see page 12 of the paper]
can be written in the form
[Diagram or table: see page 12 of the paper]
[4]
(b) Hence, or otherwise, solve for 0  x  360°
[Diagram or table: see page 12 of the paper]
giving your answers to one decimal place. [3]

Question 5
(i) Find, by algebraic integration, the exact value of
[Diagram or table: see page 16 of the paper]
[4]
(ii) Find, in simplest form,
[Diagram or table: see page 16 of the paper]
[2]

Question 6
(i) The curve C_1 has equation
[Diagram or table: see page 18 of the paper]
Show that C_1 has a stationary point at x = p/2 where p is a constant to be found. [4]
(ii) A different curve C_2 has equation
[Diagram or table: see page 18 of the paper]
(a) Show that, for this curve,
[Diagram or table: see page 18 of the paper]
where A and B are constants to be found.
(b) Hence, state the maximum gradient of this curve. [4]

Question 7
[Diagram or table: see page 20 of the paper]
where r cm is the base radius of the tree.
The base radius of a particular tree of this species is 45 cm.
According to the model,
(a) find the mass of this tree, giving your answer to 2 significant figures. [2]
(b) Show that the equation of the model can be written in the form
[Diagram or table: see page 20 of the paper]
giving the values of the constants p and q to 3 significant figures. [3]
(c) With reference to the model, interpret the value of the constant p. [1]

Question 8
A curve C has equation y = f(x), where
[Diagram or table: see page 22 of the paper]
(a) Sketch C. [1]
(b) Given x = 2 sin y, show that
[Diagram or table: see page 22 of the paper]
where A is a constant to be found. [3]
The point P lies on C and has y coordinate π/4
(c) Find the equation of the tangent to C at P. Write your answer in the form y = mx + c, where m and c are constants to be found. [3]

Question 9
[Diagram or table: see page 26 of the paper]
Figure 3 shows a sketch of part of the curve with equation y = f(x), where
[Diagram or table: see page 26 of the paper]
(a) Find f ′(x). [2]
The line l is the normal to the curve at O and meets the curve again at the point P.
The point P lies in the 3rd quadrant, as shown in Figure 3.
(b) Show that the x coordinate of P is a solution of the equation
[Diagram or table: see page 26 of the paper]
[4]
(c) Using the iterative formula
[Diagram or table: see page 26 of the paper]
find, to 4 decimal places,
(i) the value of x_2
(ii) the x coordinate of P. [3]

Question 10
[Diagram or table: see page 30 of the paper]
Figure 4 shows a sketch of part of the curve with equation
[Diagram or table: see page 30 of the paper]
(a) Show that
[Diagram or table: see page 30 of the paper]
where p, q and r are constants to be found. [2]
The curve touches the positive x‑axis for the second time when x = a, as shown in Figure 4.
The regions bounded by the curve, the y‑axis and the x‑axis up to x = a are shown shaded in Figure 4.
(b) Find, using algebraic integration and making your method clear, the exact total area of the shaded regions. Write your answer in simplest form. [5]
[Diagram or table: see page 32 of the paper]