4MA1 Mathematics A: January 2022 Paper 1HHigher Tier Paper 1

(a)Simplify a⁷ × a⁴

[1]

(b)Simplify w¹⁵÷ w³

[1]

(c)Simplify (8x⁵ y³)²

[2]

(d)Make t the subject of c = t³ – 8v

[2]

Danil, Gabriel and Hadley share some money in the ratios 3 : 5 : 9 The difference between the amount of money that Gabriel receives and the amount of money that Hadley receives is 196 euros.

Work out the amount of money that Danil receives.

_euros[3]

The diagram shows triangle ABC

Work out the length of the side AB Give your answer correct to 3 significant figures.

_cm[3]

Sarah makes and sells mugs.

One day she makes 150 mugs. Her total cost for making these mugs is £1140

Of these mugs

and the rest are large mugs

Here is Sarah’s price list for selling each mug.

Sarah sells all 150 mugs.

Work out her percentage profit. Give your answer correct to the nearest whole number.

%[5]

Jenny has six cards.

Each card has a whole number written on it so that

the smallest number is 5 the largest number is 24 the median of the six numbers is 14 the mode of the six numbers is 8

Jenny arranges her cards so that the numbers are in order of size.

5 24

(a)For the remaining four cards, write on each dotted line a number that could be on the card.[3]

A basketball team plays 6 games. After playing 5 games, the team has a mean score of 21 points per game. After playing 6 games, the team has a mean score of 23 points per game.

(b)Work out the number of points the team scored in its 6th game.

[3]

(a)Solve the inequality 5x – 7  2

[2]

(b)(i)Factorise y²– 2y – 35

[2]

(ii)Hence, solve y²– 2y – 35 = 0

[1]

A ∩ B = {5, 10, 15}

B¢ = {7, 8, 9, 11, 12, 13, 14}

A¢ = {4, 6, 7, 8, 14}

Complete the Venn diagram for this information.

[3]

Work out the value of a × b Give your answer in standard form.

[2]

The diagram shows isosceles triangle ABC

AB = AC = 17.5 cm BC = 28 cm

Calculate the area of triangle ABC

_cm²[4]

The straight line L has equation 2y + 7x = 10

(a)Find the gradient of L

[2]

(b)Find the coordinates of the point where L crosses the y‑axis.

( , )[1]

Himari invests 200 000 yen for 3 years in a savings account paying compound interest.

The rate of interest is 1.8% for the first year and x% for each of the second year and the third year.

The value of the investment at the end of the third year is 209 754 yen.

Work out the value of x Give your answer correct to one decimal place.

x = [3]

The table gives information about the times, in minutes, taken by 80 customers to do their shopping in a supermarket.

Time taken (t minutes) Frequency

(a)Complete the cumulative frequency table.

Time taken (t minutes) frequency

[1]

(b)On the grid opposite, draw a cumulative frequency graph for your table.

80

70

60

50

[2]

(c)Use your graph to find an estimate for the median time taken.

_minutes[1]

One of the 80 customers is chosen at random.

(d)Use your graph to find an estimate for the probability that the time taken by this customer was more than 42 minutes.

[2]

(a)Expand and simplify 5x(x + 2)(3x – 4)

[3]

(b)Simplify completely

[3]

Aika has 2 packets of seeds, packet A and packet B

There are 12 seeds in packet A and 7 of these are sunflower seeds. There are 15 seeds in packet B and 8 of these are sunflower seeds.

Aika is going to take at random a seed from packet A and a seed from packet B

(a)Complete the probability tree diagram.

[2]

(b)Calculate the probability that Aika will take two sunflower seeds.

[2]

A is inversely proportional to C²

A = 40 when C = 1.5

Calculate the value of C when A = 1000

C = [3]

The diagram shows a circle with centre O

A, B and C are points on the circle so that the length of the arc ABC is 5 cm.

Given that angle AOC = 55°

work out the area of the circle. Give your answer correct to one decimal place.

cm²[4]

A and B are two similar vases.

Vase A has height 10 cm. Vase B has height 15 cm.

The difference between the volume of vase A and the volume of vase B is 1197 cm³

Calculate the volume of vase A

cm³

[4]

A = w – y

w = 3.45 correct to 2 decimal places. x = 1.9 correct to 1 decimal place. y = 5 correct to the nearest whole number.

Work out the lower bound of the value of A Show your working clearly.

[3]

Solve the simultaneous equations

Show clear algebraic working.

[5]

(a)Express 7 + 12x – 3x² in the form a + b(x + c)² where a, b and c are integers.

[3]

C is the curve with equation y = 7 + 12x – 3x² The point A is the maximum point on C

(b)Use your answer to part (a) to write down the coordinates of A

( , )[1]

The diagram shows the prism ABCDEFGHJK with horizontal base AEFG

ABCDE is a cross section of the prism where ABDE is a square BCD is an equilateral triangle

EF = 2 × AE M is the midpoint of GF so that JM is vertical.

Angle MAJ = y°

Given that tan y° = T

are integers.

T = [5]

The diagram shows triangle OAB

OA = 8a OB = 6b

M is the point on OB such that OM : MB = 1 : 2 N is the midpoint of AB P is the point of intersection of ON and AM

Using a vector method, find OP as a simplified expression in terms of a and b Show your working clearly.

OP = [5]

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

There is only one maximum point on the curve. The coordinates of this maximum point are (5, 7)

Write down the coordinates of the maximum point on the curve with equation

(i)y = f(x + 9)

( , )

(ii)y = f(x) + 3

( , )[2]

The curve C has equation y = ax³+ bx²– 12x + 6 where a and b are constants.

The point A with coordinates (2, –6) lies on C The gradient of the curve at A is 16

Find the y coordinate of the point on the curve whose x coordinate is 3 Show clear algebraic working.

y = [6]