Edexcel A Level Maths Paper 2: Pure Mathematics 2 June 2025 question paper 9MA0/02

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

Find

giving the answer in simplest form.

Given that

find the exact value of

The shape ABCD, shown in Figure 1, consists of a triangle ABD containing a sector ABC of a circle with centre B. Given that

• AD = 6.4 cm

• BD = 13 cm

• BA = BC = 8 cm

show that angle ABC = 0.394 radians to 3 significant figures. The region R, shown shaded in Figure 1, is bounded by the line CD, the line DA and the arc AC.

Find the area of R, giving the answer in cm² to 3 significant figures. You must make your method clear.

The curve C has parametric equations

The point P with x coordinate –3 lies on C.

Find the y coordinate of P.

Find a Cartesian equation for C, giving the answer in the form y = f (x)

Hence, or otherwise, find the gradient of C at the point P.

In this question you must show detailed reasoning.

Given that x is small and in radians, use the small angle approximation for cosθ to show that

Given that x is small and in radians, use

• the answer to part (a)

• the small angle approximations for sin θ and tan θ

to show that

where a and b are constants to be found.

Hence, given that x is very small, deduce an approximate value for

giving a reason for your answer.

Given that

find the value of each of the constants A, B, C and D.

where a is a constant. Figure 2 shows a sketch of the curve C with equation y = f (x) Given that C has a local maximum at x = –

show that a = –4

find the exact y coordinate of the local maximum. The equation f (x) = k, where k is a constant, has 4 distinct solutions.

Using algebra and showing all stages of your working, find the range of values of k. Give the answer using set notation. (Solutions relying on calculator technology are not acceptable.)

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. A new type of car is released for sale. The total number of this type of car sold, N, in a particular region, t months after the cars were released for sale, is modelled by the equation

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

Find the total number of cars sold in the first 3 months. Given that N = 3000 when t = T

find the value of T giving the answer to 2 decimal places.

Find the rate of increase in the total number of cars sold when t = 3, giving the answer to 3 significant figures. After a marketing campaign, the total number of cars sold is expected to rise and have an upper limit of 6500

Using this information, suggest one refinement to the model.

Water flows at a constant rate into a large container. There is a tap at the bottom of the container. At time t hours after the tap was opened

• the volume of water in the container is Vm³

• water is flowing into the container at a constant rate of 0.45 m³ per hour

• water is leaving the container through the tap at a rate of 0.3Vm³ per hour

Show that

Given that when the tap was opened, there was 0.25 m³ of water in the container,

solve the differential equation to show that

where P, Q and k are positive constants to be found. Given that

• the capacity of the container is 2 m³

• the tap remains open

• the water continues to flow into the tank at the same rate

determine whether the container will ever become full, giving a reason for your answer.

A company is trying to determine the most profitable selling price for a new toy. Given that

• if the selling price of each toy is £30, the company expects to sell 1500 toys in one year

• if the selling price of each toy is £50, the company expects to sell 300 toys in one year

Using a linear model, with y being the expected number of toys sold in one year and x pounds being the selling price of the toy,

find an equation for y in terms of x. Given that

• the cost of making each toy is £10

• the company has additional costs of £8 000 per year

show that, according to the model, the yearly profit, P, in thousands of pounds, is given by

Use the model given in part (b) to answer parts (c), (d) and (e). Given that the company wishes to make a profit on sales of the toy,

find the range of possible selling prices of the toy.

Hence, or otherwise, deduce the selling price of the toy that maximises the profit. In one particular year, the company sold the toy for £35 and made £21 750 profit.

Use this information to evaluate the suitability of the model.

Figure 3 shows a sketch of the graph with equation y = f (x) where f (x) = 4 x – 3 – 5 x Given that a is a constant and a = 1

find the possible values of f (a) The function g is defined by g (x) = 2x + 17 x

Find the range of g f (x) The function h is defined by h (x) = kx x where k is a constant. Given that the equation f (x) = h (x) has no solutions,

find the range of values of k.

Given that for all values of k, where 0 < k < 1, the equation

has exactly 6 solutions in the interval 0 ≤ x < 2π

deduce the value of n

deduce the number of solutions of the equation

in the interval 0 ≤ x < 5π, justifying your answer.

In this question you must show detailed reasoning. Figure 4 shows a trapezium ABCD where AD is parallel to BC Given that

• AB = 2a + 3b

• BC = 15a – 5b

• DB = –4a + kb where k is an integer

show that k = 5 Given also that

• the point N lies on BC such that BN: NC = 1: 4

• AN intersects BD at X

find BX: XD

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Figure 5 shows a sketch of part of the curve with equation y = f (x), where

The curve

• intersects the x ‑axis at –1 and 1

• has minimum turning points at P and Q

as shown in Figure 5.

Use calculus to find the exact coordinates of P.

Using the substitution x = tanθ show that

where α and β are constants to be found. The finite region R, shown shaded in Figure 5, is bounded by the x‑axis and the curve.

Use algebraic integration to find the area of R.