# WMA13 Pure Mathematics 3 (P3) January 2020 Paper 01

Pearson Edexcel International A Level WMA13 Pure Mathematics 3 (P3) past paper 01, January 2020. Open it online or download the question paper PDF free.

- Page: https://mathvsarham.com/past-papers/wma13/2020-january-paper-01/
- PDF: https://mathvsarham.com/papers/wma13/WMA13_january_2020_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 2020
- 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 population of a rare species of toad is being studied.
The number of toads, N, in the population, t years after the start of the study, is modelled by the equation
[Diagram or table: see page 2 of the paper]
According to this model,
(a) calculate the number of toads in the population at the start of the study, [1]
(b) find the value of t when there are 420 toads in the population, giving your answer to 2 decimal places. [4]
(c) Explain why, according to this model, the number of toads in the population can never reach 500 [1]

Question 2
The function f and the function g are defined by
[Diagram or table: see page 4 of the paper]
(a) Find, in simplest form, the value of fg(e^2) [2]
(b) Find f^–1 [3]
(c) Hence, or otherwise, find all real solutions of the equation
[Diagram or table: see page 4 of the paper]
[3]

Question 3
[Diagram or table: see page 8 of the paper]
Figure 1 shows a linear relationship between log_10 y and log_10 x
The line passes through the points (0, 4) and (6, 0) as shown.
(a) Find an equation linking log_10 y with log_10 x [2]
(b) Hence, or otherwise, express y in the form px ^q, where p and q are constants to be found. [3]
[Diagram or table: see page 10 of the paper]

Question 4
(i) f(x) = x ≠ 3
[Diagram or table: see page 10 of the paper]
(a) Find f ′(x) in the form P (x)/Q (x) where P(x) and Q(x) are fully factorised quadratic
expressions.
(b) Hence find the range of values of x for which f(x) is increasing. [6]
(ii) 
[Diagram or table: see page 10 of the paper]
The curve with equation y = g(x) has a maximum at the point M.
Show that the x coordinate of M satisfies the equation
[Diagram or table: see page 10 of the paper]
where k is a constant to be found. [5]

Question 5
(a) Use the substitution t = tan x to show that the equation
[Diagram or table: see page 14 of the paper]
can be written in the form
[Diagram or table: see page 14 of the paper]
[4]
(b) Hence solve, for 0  x < 360°, the equation
[Diagram or table: see page 14 of the paper]
Show each stage of your working and give your answers to one decimal place. [4]

Question 6
[Diagram or table: see page 18 of the paper]
Figure 2 shows part of the graph with equation y = f(x), where
The vertex of the graph is at point P as shown.
(a) State the coordinates of P. [2]
(b) Solve the equation f(x) = 3x – 2 [4]
Given that the equation
[Diagram or table: see page 18 of the paper]
where k is a constant, has exactly two roots,
(c) find the range of values of k. [3]

Question 7
[Diagram or table: see page 22 of the paper]
Figure 3 shows a sketch of part of the curve with equation
[Diagram or table: see page 22 of the paper]
where x is measured in radians.
The curve crosses the x‑axis at the point P, as shown in Figure 3.
Given that the x coordinate of P is α,
(a) show that α lies between 0.8 and 0.9 [2]
The iteration formula
[Diagram or table: see page 22 of the paper]
can be used to find an approximate value for α.
(b) Using this iteration formula with x_1= 0.8 find, to 4 decimal places, the value of
(i) x_2
(ii) x_5 [3]
The point Q and the point R are local minimum points on the curve, as shown in Figure 3.
Given that the x coordinates of Q and R are β and λ respectively, and that they are the two smallest values of x at which local minima occur,
(c) find, using calculus, the exact value of β and the exact value of λ. [6]

Question 8
(i) Find, using algebraic integration, the exact value of
[Diagram or table: see page 26 of the paper]
giving your answer in simplest form. [4]
[Diagram or table: see page 26 of the paper]
(ii) h(x) =_2 x > 1
[Diagram or table: see page 26 of the paper]
Given h(x) = Ax + B + C; (x − 1 )_2 where A, B and C are constants to be found, find
[Diagram or table: see page 26 of the paper]
[6]

Question 9
[Diagram or table: see page 30 of the paper]
(a) Express f(θ) in the form R cos (θ + 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. [3]
The curve with equation y = cos θ is transformed onto the curve with equation y = f(θ) by a sequence of two transformations.
Given that the first transformation is a stretch and the second a translation,
(b)(i) describe fully the transformation that is a stretch,
(ii) describe fully the transformation that is a translation. [2]
Given
[Diagram or table: see page 30 of the paper]
(c) find the range of g. [2]
[Diagram or table: see page 32 of the paper]