WMA13 Pure Mathematics 3 (P3): January 2026 Paper 01

where x is measured in radians.

(a)Show that g(x) = 0 has a root α in the interval 0.7, 0.8[2]

(b)Show that the equation g(x) = 0 can be written in the form

where p is a constant to be found.[1]

Using the iteration formula

with the value of p found in part (b) and starting with x₁ = 0.7

(c)(i)find, to 4 decimal places, the value of x₂

(ii)find, by repeated iteration, the value of α, giving your answer to 4 decimal places.[3]

(i)Given y = x² ln 2x, find in simplest form, dydx[2]

(ii)Given

where a and b are constants to be found.[4]

(i)Find, in simplest form,

[2]

(ii)f(x)  x > 1

(a)Write f(x) in the form

where a, b and c are constants to be found.[4]

(b)Hence, using algebraic integration, find

writing the answer in simplest form.[4]

The resting heart rate, h beats per minute, of a particular group of mammals, can be modelled by the equation

where m kg is the mass of the mammal.

The mass of one of these mammals is 3 kg.

According to the model,

(a)find the resting heart rate of this mammal. Give your answer to 3 significant figures.[2]

(b)Show that the equation of the model can be written in the form

giving each of the values of the constants p and q to 3 significant figures where appropriate.[3]

(c)With reference to the model, interpret the value of the constant p.[1]

The function f is defined by

(a)Find ff(x), giving your answer as a single fraction in its simplest form.[3]

(b)Find f⁻¹(12)[2]

The function g is defined by

(c)Find the exact values of a such that

[3]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

(a)Show that the equation

can be written in the form

[4]

(b)Hence solve, for 0  x  180, the equation

giving your solutions to one decimal place.[3]

Figure 1 shows a sketch of part of the graph with equation y = f(x), where

The vertex of the graph is at the point P, as shown in Figure 1.

(a)State the coordinates of P.[2]

(b)Solve the equation f(x) = 8[3]

The line l has equation y = mx, where m is a constant.

Given that l intersects the graph of y = f(x) at exactly one point,

(c)find the possible values of m.[3]

The graph with equation y = f(x) is transformed onto the graph with equation y  a f (x  b), where a and b are constants.

Given that the vertex of the graph with equation y  a f (x  b) is 5, 16

(d)find the value of a and the value of b.[2]

A curve has equation

where a, b and n are integers to be found.[4]

(a)Write h(x) in the form

where P, R and α are positive constants, 0 <  < 2

Give the exact values of P and R and give the value of α, in radians, to 3 decimal places.[6]

Figure 2 is a sketch of the curve with equation y = h(x)

(b)State the range of h.[1]

The point M, shown in Figure 2, is the minimum point on the curve with the largest x coordinate.

(c)Find the x coordinate of M.

[2]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Two populations of insects are being studied over the same period of time.

The number of insects N A in population A, t years after the start of the study, is modelled by the equation

According to the model,

(a)find the rate of growth in the number of insects in population A exactly 5 years after the start of the study.[2]

The number of insects N B in population B, t years after the start of the study, is modelled by the equation

where P and k are positive constants.

Given that

• there are 10 000 insects in population B at the start of the study

• there are 11 570 insects in population B exactly 4 years after the start of the study

• there are the same number of insects in population A and population B after T years

find, according to the models,

(b)the exact value of P and the value of k to 3 decimal places,[4]

(c)the value of T, giving your answer to one decimal place.[3]