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

n is an integer.

(a)Write down all the values of n such that –2  n < 3

[2]

(b)On the number line, represent the inequality y  1

[1]

Each time John plays a game, the probability that he wins the game is 0.65

John is going to play the game 300 times.

Work out an estimate for the number of games that John wins.

[2]

The shaded shape is made using three identical right-angled triangles and a square.

Work out the perimeter of the shaded shape.

_cm[4]

(a)Complete the table of values for y = x²– 4x + 3

[2]

(b)On the grid, draw the graph of y = x²– 4x + 3 for values of x from –2 to 4

[2]

Yusuf sat 8 examinations.

Here are his marks for 5 of the examinations.

For his results in all 8 examinations

the mode of his marks is 80 the median of his marks is 74 the range of his marks is 16

Find Yusuf’s marks for each of the other 3 examinations.

[4]

(a)Work out the lowest common multiple (LCM) of 36 and 120

[2]

A = 5 × 7 × 11^p B = 5^2/m × 7^4/n–5 × 11

m, n and p are integers such that m > 2 n > 10 p > 1

(b)Find the highest common factor (HCF) of A and B Give your answer as a product of powers of its prime factors.

[2]

Milly went on a car journey. She travelled from Anesey to Breigh to Clando and then to Duckbridge.

For Anesey to Breigh, Milly drove the 245 km in 2.5 hours. For Breigh to Clando, Milly drove the 220 km at an average speed of 80 km/h For Clando to Duckbridge, Milly drove at an average speed of 72 km/h in 50 minutes.

Work out Milly’s average speed, in km/h, for the journey from Anesey to Duckbridge. Give your answer correct to one decimal place.

_km/h[4]

(a)Write 5 × 10⁴ as an ordinary number.

[1]

(b)Write 0.000 06 in standard form.

[1]

(c)Work out (4 × 10⁵¹²) ÷ (1.6 × 10⁷⁰⁰) Give your answer in standard form.

[2]

(a)Simplify x⁴ × x⁵

[1]

(b)Simplify (4y²)³

[2]

(c)Factorise n²– 7n + 12

[2]

Jonty has a storage container in the shape of a cuboid, as shown in the diagram.

Jonty is going to paint the outside of his storage container, apart from the base which is shown shaded in the diagram. He needs enough paint to cover the four sides and the top.

Each tin of paint covers an area of 15 m²

The cost of each tin of paint recently increased by 10% After the increase, the cost of each tin of paint is £26.95

Jonty says

“Before the increase, I could have bought enough paint for less than £200”

Show that Jonty is correct. Show your working clearly.[6]

The diagram shows sector OPQ of a circle, centre O

OP = OQ = 7.1 cm Angle POQ = 110°

Calculate the area of sector OPQ Give your answer correct to one decimal place.

_cm²[2]

(a)Expand and simplify n(n – 4)(3n + 5)

[2]

(b)Express

3 x + 2 1

as a single fraction in its simplest form.

[3]

Hector has a bag that contains 12 counters. There are 7 green counters and 5 red counters in the bag.

Hector takes at random a counter from the bag. He looks at the counter and puts the counter back into the bag.

Hector then takes at random a second counter from the bag. He looks at the counter and puts the counter back into the bag.

(a)Complete the probability tree diagram.

Green

[2]

(b)Work out the probability that both counters are red.

[2]

Meghan has a jar containing 15 counters. There are only blue counters, green counters and red counters in the jar.

Hector is going to take at random one of the counters from his bag of 12 counters. He will look at the counter and put the counter back into the bag.

Hector is then going to take at random a second counter from his bag. He will look at the counter and put the counter back into the bag.

Meghan is then going to take at random one of the counters from her jar of counters. She will look at the counter and put the counter back into the jar.

The probability that the 3 counters each have a different colour is 724

(c)Work out how many blue counters there are in the jar.

[3]

A, B, C and D are points on a circle, centre O AOC is a diameter of the circle.

Angle BAC = 55°

Work out the size of angle ADB Give a reason for each stage of your working.

[4]

Using algebra, prove that, given any 3 consecutive whole numbers, the sum of the square of the smallest number and the square of the largest number is always 2 more than twice the square of the middle number.[3]

An arithmetic series has first term 1 and common difference 4

Find the sum of all terms of the series from the 41st term to the 100th term inclusive.

[4]

The Venn diagram shows a universal set E and three sets A, B and C.

E

6, 3, 8, 2, 5 and 4 represent the numbers of elements.

Find

(i)n(A ∪ B)

[1]

(ii)n(A ∩ C)

[1]

(iii)n(B ∩ C' )

[1]

(iv)n(A' ∪ B' ∪ C' )

[1]

The three solids A, B and C are similar such that

the surface area of A : the surface area of B = 4 : 9 and the volume of B : the volume of C = 125 : 343

Work out the ratio

the height of A : the height of C

Give your ratio in its simplest form.

[4]

Given that ³ 1x   x^m    

(a)find the value of m

m = [1]

Given that a, b and c are integers,

(b)express 3x²+ 12x + 19 in the form a(x + b)²+ c

[2]

The curve with equation y = f(x) has one turning point.

The coordinates of this turning point are (– 6, – 4)

(a)Write down the coordinates of the turning point on the curve with equation

(i)y = f(x) + 5

( , )

(ii)y = f(3x)

( , )[2]

The graph of y = g(x) is shown on the grid below.

(b)On the grid, sketch the graph of y = 2g(x) for –1  x  7[2]

The graph of y = h(x) intersects the x-axis at two points. The coordinates of the two points are (–1, 0) and (6, 0)

The graph of y = h(x + a) passes through the point with coordinates (2, 0), where a is a constant.

(c)Find the two possible values of a

_, [2]

Frequency density

The histogram gives information about the weights, in kg, of all the watermelons in a field.

There are 16 watermelons with a weight between 8 kg and 8.5 kg

Work out the total number of watermelons in the field.

[3]

The diagram shows triangle ABC

c = 11.5 correct to one decimal place x = 80 correct to the nearest whole number y = 75 correct to the nearest whole number

Calculate the upper bound for the value of b Show your working clearly. Give your answer correct to 3 significant figures.

[4]

Two particles, P and Q, move along a straight line. The fixed point O lies on this line.

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

The displacement of Q from O at time t seconds is x metres, where

Find the range of values of t where t > 1 for which both particles are moving in the same direction along the straight line.

[6]

OXYZ is a trapezium.

M is the midpoint of OX N is the point such that MNZ and ONY are straight lines.

Given that ON : OY = λ : 1

use a vector method to find the value of λ

λ = [5]