WMA13 Pure Mathematics 3 (P3): January 2023 Paper 01

The functions f and g are defined by

(a)Write down the range of f[1]

(b)Find the value of fg (1.5)[2]

(c)Find g⁻¹[3]

(a)Express f (x) in the form where R and α are constants,

R > 0 and

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

(b)Using the answer to part (a),

(i)write down the exact maximum value of g (x),

(ii)find the smallest positive value of x for which this maximum value occurs, giving your answer to 2 decimal places.[3]

The line l in Figure 1 shows a linear relationship between log₁₀ y and x.

The line passes through the points (0, 1.5) and (−4.8, 0) as shown.

(a)Write down an equation for l.[2]

(b)Hence, or otherwise, express y in the form kb ^x , giving the values of the constants k and b to 3 significant figures.[3]

(a)Find the values of the constants A, B, C and D such that

[4]

(b)Hence find,

[3]

In this question you must show all stages of your working.

(a)Prove that[4]

(b)Hence solve, for

giving your answers to 2 decimal places.[5]

Figure 2 shows a sketch of the graph with equation

where a is a positive constant.

The graph

• cuts the y-axis at the point P • cuts the x-axis at the points Q and R • has a minimum point at S

(a)Find, in simplest form in terms of a, the coordinates of

(i)point P

(ii)points Q and R

(iii)point S[4]

(b)Find, in simplest form in terms of a, the values of x for which

[4]

The curve C has equation

(a)Show that

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

The point P with y coordinate lies on C.

Given that the tangent to C at P crosses the x-axis at the point Q.

(b)find, in simplest form, the exact x coordinate of Q.[5]

Find, in simplest form,

[5]

Figure 3 shows a sketch of part of the curve C with equation

(a)Find dydx , giving your answer in simplest form.[2]

The point P with x coordinate α lies on C.

Given that the tangent to C at P passes through the origin, as shown in Figure 3,

(b)show that x = α is a solution of the equation

[3]

(c)Hence show that α lies between 1 and 2[2]

(d)Show that the equation in part (b) can be written in the form

[1]

The iteration formula

with x₁ = 1 is used to find an approximation for α.

(e)Use the iteration formula to find, to 4 decimal places, the value of

(i)x₃

(ii)α[3]

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

A population of fruit flies is being studied.

The number of fruit flies, F, in the population, t days after the start of the study, is modelled by the equation

where k is a constant.

Use the equation of the model to answer parts (a), (b) and (c).

(a)Find the number of fruit flies in the population at the start of the study.[1]

Given that there are 200 fruit flies in the population 15 days after the start of the study,

(b)show that k = 115 ln12[3]

Given also that, when t = T, the number of fruit flies in the population is increasing at a rate of 10 per day,

(c)find the possible values of T, giving your answers to one decimal place.[5]