OCR GCSE Maths Paper 6: Calculator Higher June 2024 question paper J560/06

Below are six graphs, numbered 1 to 6, that show different relationships between x and y.

Graph 1
Graph 2
Graph 3

Graph 4
Graph 5
Graph 6

y y y

For each description below, write down the number of the graph that best represents the relationship between x and y.

x: temperature in °C y: temperature in °F, where y = 1.8 x + 32. Graph........

x: average speed when running 200 m y: time taken to run 200 m. Graph........

x: mass of a solid object made from clay y: volume of the same solid object. Graph........

The graph shows a household’s average daily electricity usage, in kilowatt hours (kWh).

Give two different reasons why this graph is misleading.

The word MATHEMATICS is spelt using tiles.

The tiles are put into an empty bag.

One at a time, 99 children each take a tile at random from the bag. They record the letter and then put the tile back in the bag.

Work out how many times the letter M is expected to be taken from the bag.

1 cm on a map represents 150 km in real life.

Below is part of the map showing City A and City B.

Heidi is planning to walk from City A to City B for charity.

Heidi measures the straight-line distance on the map from City A to City B accurately as 5 cm. Heidi says she will walk from City A to City B at an average speed of 2.5 km / h for 10 hours per day.

Use this information to work out how many days Heidi needs to complete the walk from City A to City B.

Explain why the information used in part (i) is likely to give an underestimate for the number of days Heidi needs to complete the walk.

Yoshi writes the scale of the map, 1 cm represents 150 km, as the ratio 1: 150.

Explain Yoshi’s error and write the scale correctly in the form 1: n.

The correct answer is 1:........

The diagram shows a regular pentagon made using ten congruent right-angled triangles. The length of one side of the pentagon is 12 cm.

Show that the area of the pentagon is 247.75 cm², correct to 2 decimal places.

The regular pentagon is the base of a pyramid.

The pyramid has volume 450 cm³. The perpendicular height of the pyramid is h cm.

Calculate the value of h. [The volume of a pyramid is × area of base × perpendicular height.]

Two numbers, A and B, are written as the product of their prime factors.

Find the lowest common multiple (LCM) of A and B. Give your answer as an ordinary number.

A number, R, is written as the product of its prime factors.

R = 2 × 3² × 5 × k, where k is a prime number.

The highest common factor (HCF) of R and another number, P, is 26.

Find the value of k.

Two bags of fruit contain only apples and bananas. In bag X, the ratio of apples to bananas is 5: 7. In bag Y, of the fruit are apples.

Finley says

Bag X and bag Y contain the same number of apples.

Tick the correct statement.

Show how you decided.

Finley adds 4 apples to bag X. The ratio of apples to bananas is now 11: 14.

How many bananas are in bag X?

Complete this table for y = x³ - 3x².

Draw the graph of y = x³ - 3x² for values of x from 2^- to 4.

Use the graph to solve the equation x³ - 3x² = 5. Give your answer to 1 decimal place.

The diagram shows a shaded sector inside a parallelogram. The sector has an angle of 30°. The parallelogram, ABCD, has length BC = 20 cm and AB = 12 cm. The perpendicular distance between BC and AD is 6 cm.

Show that the area of the sector is 37.7 cm², correct to 3 significant figures.

Work out the percentage of the parallelogram that is not shaded.

Eight consecutive numbers are written in ascending order in this grid, starting from the top and working left to right.

Kareem writes the numbers 5 to 12 in the grid.

Show that for Kareem’s grid, the sum of the numbers in the top half of the grid is 16 less

than the sum of the numbers in the bottom half of the grid.

Use algebra to prove that for any set of eight consecutive numbers written in this grid in the same way, the sum of the numbers in the top half of the grid is 16 less than the sum of the

numbers in the bottom half of the grid.

Jack travels to work each day by train.

He records whether

• the train is on time or late

• there are seats available or no seats available.

Jack’s results are shown on this partly completed frequency tree.

Find the relative frequency of there being no seats available on Jack’s train journey.

Jack says

If the train is late, travellers are less likely to find seats available than if the train was on time.

Does Jack’s data suggest he is correct? Show how you decide.

The region R is shown on this grid.

The region R is defined by three inequalities. The first inequality is given below.

Complete the second inequality and write down the third inequality needed to define region R.

y ⩾ 0

2 y........ 3 x + 6

A group of students record the time taken to travel to school. All students in the group took less than an hour to travel to school. Some of their results are recorded on this histogram.

16 students took less than 5 minutes to travel to school.

How many students took less than 20 minutes to travel to school?

9 students took between 30 and 45 minutes to travel to school.

Add these students’ results to the histogram.

The sketch shows the coordinate axes and the point (0, 10).

The distance from the point (0, 10) to a point (12, t) is 12.5 units.

Work out the two possible values of t. You must show your working and you may use the sketch to help.

y is inversely proportional to x³. y = 4 when x = 12.

Find a formula linking x and y.

Without using a calculator, show that can be written as

Explain how = can be used to find as a decimal and write down its value.

The diagram shows two triangles ABC and ADC.

Angle ABC = 60 °, angle ACB = 20 ° and angle ACD = 55 °. AB = 8 cm and DC = 6 cm.

Calculate length AD. You must show your working.

The diagram shows a circle with centre (0, 0) and a tangent at (12, 5^-). The tangent at (12, 5^-) crosses the x-axis at (p, 0).

Find the exact value of p. You must show your working.

Write as a single fraction in its simplest form.