4MA1 Mathematics A: May/June 2022 Paper 1HHigher Tier Paper 1

Here are the first five terms of an arithmetic sequence.

(a)Find an expression, in terms of n, for the nth term of this sequence.

[2]

The nth term of another arithmetic sequence is 3n + 5

(b)Find an expression, in terms of m, for the (2m)th term of this sequence.

[1]

Here is a biased 4-sided spinner.

The table gives the probabilities that, when the spinner is spun once, it will land on 1 or it will land on 3

The probability that the spinner will land on 2 is equal to the probability that the spinner will land on 4

Ravina is going to spin the spinner a number of times.

Ravina works out that an estimate for the number of times the spinner will land on 3 is 45

Work out an estimate for the number of times the spinner will land on 4

[4]

(a)Find the highest common factor (HCF) of 56 and 84 Show your working clearly.

[2]

(b)Find the lowest common multiple (LCM) of 60 and 72 Show your working clearly.

[2]

The diagram shows parts of three regular polygons, A, B and C, meeting at a point.

Polygon B has n sides.

Work out the value of n.

n = [4]

(a)Expand and simplify (n − 6)(n + 4)

[2]

(b)Solve 2x − 3 = 3x − 54

Show clear algebraic working.

x = [3]

Asha bought an apartment.

The table gives information about the value of apartments, in euros, and the annual service charge band.

Value (x euros) Service charge band

0 < x < 400 000 E

In 2021, the value of Asha’s apartment was 634 400 euros.

The value of Asha’s apartment had increased by 4% from its value in 2020

(a)Has the annual service charge band changed for Asha’s apartment? Show your working clearly.[3]

Pam bought a boat.

In each year after Pam bought the boat, the value of the boat depreciated by 15%

(b)Work out the total percentage by which the value of the boat had depreciated by the end of the second year after Pam bought the boat.

_%[3]

A cylinder is placed on the ground.

The height of the cylinder is 18 cm.

The force exerted by the cylinder on the ground is 72 newtons. The pressure on the ground due to the cylinder is 1.4 newtons / cm²

force

Work out the volume of the cylinder. Give your answer correct to 3 significant figures.

_cm³[4]

(a)Write 0.000 089 in standard form.

[1]

(b)Write 8.34 × 10⁴ as an ordinary number.

[1]

(a)Simplify 8 × (4t)⁰

[1]

x⁶÷ x⁻⁵ = x ^p

(b)Find the value of p

p = [1]

(c)Simplify fully (2k² m⁴)³

[2]

Two circles, C₁ and C₂ , are drawn on a centimetre grid, with a scale of 1 cm for 1 unit on each axis.

The centre of circle C₁ is at the point with coordinates (–1, 3) and the radius of C₁ is 13 cm.

The centre of circle C₂ is at the point with coordinates (7, 18) and the radius of C₂ is 6 cm.

(a)Work out the distance between the centre of C₁ and the centre of C₂

_cm[3]

(b)Explain why circle C₁ intersects circle C₂

[1]

(a)Factorise 9x² − 4y²

[2]

(b)Express 78 − x + 34x as a single fraction in its simplest form.

[3]

Rudolf goes to the gym.

The probability that he will use the treadmill is 0.8

When he uses the treadmill, the probability that he will use the cross trainer is 0.3 When he does not use the treadmill, the probability that he will use the cross trainer is 0.6

(a)Complete the probability tree diagram for this information.

does not use

does not use

[2]

(b)Work out the probability that Rudolf uses both the treadmill and the cross trainer.

[2]

Antoine is going on holiday. He makes 3 separate payments to cover the total cost of his holiday.

The following table shows how much money Antoine pays to the holiday company.

Payment 1 38 of the total cost

Payment 2 45% of the total cost

Payment 3 $406

Work out how much Antoine has to pay for Payment 2

$ [5]

The function f is defined as

(a)Find f(10)

[1]

(b)Express the inverse function f⁻¹ in the form f⁻¹ : x  …

f⁻¹ : x  [3]

Abraham is going to play a computer game. Abraham can win the game, draw the game or lose the game.

For any game that Abraham plays

the probability that he wins the game is 0.3 the probability that he draws the game is 0.5 the probability that he loses the game is 0.2

When Abraham wins a game, he scores +10 points. When Abraham draws a game, he scores 0 points. When Abraham loses a game, he scores −5 points.

Abraham plays 3 games and the points he scores in each of the 3 games are added together to get his total score.

Work out the probability that when he has played 3 games his total score is 0 points.

[4]

Without using a calculator, show that 122 − 1 − ( 2 = 2 32 + 12

Show your working clearly.[3]

A particle P moves along a straight line. The fixed point O lies on this line.

The displacement of P from O at time t seconds, t  1, is s metres where

The velocity of P at time t seconds, t  1, is v m / s

Work out the distance of P from O at the instant when v = 0

_m[5]

Here is triangle ABC

Work out the value of x Give your answer correct to 3 significant figures.

x = [5]

OAB is a sector S of a circle with centre O and radius (r + 7) metres. Angle AOB = 45°

A circle C has radius (r − 2) metres.

The area of sector S is twice the area of circle C

Find the value of r Show your working clearly.

r = [5]

The diagram shows a sketch of part of the curve with equation y = f (x)

There is one maximum point on this curve. The coordinates of this maximum point are (s, t)

Find, in terms of s and t, the coordinates of the maximum point on the curve with equation

(i)y = f (x − 2)

( , )[1]

(ii)y = 3f (x)

( , )[1]

The histogram shows information about the total time, m minutes, taken by each child in a school to walk to school every day for one week.

There are no children for whom m > 100 There are 10 children for whom m  20

Work out an estimate for the number of children for whom 50 < m  80

[3]

A solid is made from a cone and a hemisphere.

The circular plane face of the hemisphere coincides with the circular base of the cone. The radius of the hemisphere and the radius of the circular base of the cone are both 20 cm.

The curved surface area of the cone is 580π cm²

The volume of the solid is kπ cm³

Work out the exact value of k

k = [5]

A polygon has n sides, where n > 5

When arranged in order of size, starting with the largest number, the sizes of the interior angles of the polygon, in degrees, are the terms of an arithmetic sequence.

Here are the first five terms of this sequence.

Find the value of n Show clear algebraic working.

n = [6]

Express each of a, b and c in terms of q so that

can be written as a − b (x − c)²

a =

b =

c = [4]