Edexcel A Level Maths Paper 2: Pure Mathematics 2 June 2024 question paper 9MA0/02

y 4x³ 7x² 5x 10

Find in simplest form

Hence find the exact value of x when = 0

Jamie takes out an interest‑free loan of £8100

Jamie makes a payment every month to pay back the loan.

Jamie repays £400 in month 1, £390 in month 2, £380 in month 3, and so on, so that the amounts repaid each month form an arithmetic sequence.

Show that Jamie repays £290 in month 12

After Jamie’s Nth payment, the loan is completely paid back.

Show thatN² – 81N + 1620 = 0

Hence find the value of N.

The point P(3, –2) lies on the curve with equation y = f(x), x

Find the coordinates of the point to which P is mapped when the curve with equation y = f(x) is transformed to the curve with equation

y = f(x – 2)

y = f(2x)

y = 3f(–x) + 5

A sequence u₁, u₂, u₃,... is defined by

where k is a positive constant.

Given that u₃ = –1

show that

Hence

find the value of k,

Given that θ is small and in radians, use the small angle approximations to find an approximate numerical value of

Figure 1 shows a sketch of the curves with equations y = f(x) and y = g(x) where

Find

f′(x)

g′(x)

Given that f′(x) = g′(x) at x = α

show that α satisfies the equation

The iterative formula

is used withx₁ = 0.6 to find an approximate value for α

Calculate, giving each answer to 4 decimal places,

the value of x₂

the value of α

Figure 2 shows a sketch of the straight line l.

Line l passes through the points A and B.

Relative to a fixed origin O

• the point A has position vector 2 i – 3 j + 5 k

• the point B has position vector 5 i + 6 j + 8 k

Given that a point P lies on l such that

find the possible position vectors of P.

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

Solutions relying entirely on calculator technology are not acceptable.

Prove that

Hence solve, for 0 < x < 90, the equation

Give each answer, in degrees, to one decimal place.

The graph in Figure 3 shows the path of a small ball.

The ball travels in a vertical plane above horizontal ground.

The ball is thrown from the point represented by A and caught at the point represented by B.

The height, H metres, of the ball above the ground has been plotted against the horizontal distance, x metres, measured from the point where the ball was thrown.

With respect to a fixed origin O, the point A has coordinates (0, 2) and the point B has coordinates (20, 0.8), as shown in Figure 3.

The ball reaches its maximum height when x = 9

A quadratic function, linking H with x, is used to model the path of the ball.

Find H in terms of x.

Give one limitation of the model.

Chandra is standing directly under the path of the ball at a point 16 m horizontally from O.

Chandra can catch the ball if the ball is less than 2.5 m above the ground.

Use the model to determine if Chandra can catch the ball.

Figure 4 shows a sketch of the curve C with parametric equations

The point P with coordinates (4, 2) lies on C.

Using parametric differentiation, show that the tangent to C at P has equation

The curve C is used to model the profile of a slide at a water park.

Units are in metres, with y being the height of the slide above water level.

Find, according to the model, the greatest height of the slide above water level.

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

Solutions relying entirely on calculator technology are not acceptable.

Figure 5 shows a sketch of part of the curve C with equation

The finite region R, shown shaded in Figure 5, is bounded by

• the curve C

• the line with equation x = 1

• the x-axis

Find the exact area of R, giving your answer in the form

where A and B are rational numbers to be found.

The volume, V microlitres, of a plant cell t hours after the plant is watered is modelled by the differential equation

The plant cell has an initial volume of 20 microlitres.

Find, according to the model, the time taken, in minutes, for the volume of the plant cell to reach 24 microlitres.

Show that

where A, B and k are constants to be found.

The model predicts that there is an upper limit, L microlitres, on the volume of the plant cell.

Find the value of L, giving a reason for your answer.

The world human population, P billions, is modelled by the equation

where a and b are constants and t is the number of years after 2004

Using the estimated population figures for the years from 2004 to 2007, a graph is plotted of log₁₀P against t. The points lie approximately on a straight line with

• gradient 0.0054

• intercept 0.81 on the log₁₀P axis

Estimate, to 3 decimal places, the value of a and the value of b.

In the context of the model,

interpret the value of the constant a,

interpret the value of the constant b.

Use the model to estimate the world human population in 2030

Comment on the reliability of the answer to part (c).

The circle C₁ has equation

Find

the coordinates of the centre of C₁

the radius of C₁

A different circle C₂

• has centre with coordinates (–6, –8)

• has radius k, where k is a constant

Given that C₁ and C₂ intersect at 2 distinct points,

find the range of values of k, writing your answer in set notation.

The curve C has equation

Find an expression for in terms of x and y.

The point P(1, 0) lies on C.

Show that the normal to C at P has equation

Prove that the normal to C at P does not meet C again.

You should use algebra for your proof and make your reasoning clear.