OCR GCSE Maths Paper 6: Calculator Higher June 2023 question paper J560/06

Alex draws a bar chart to show the age of the young people attending a youth club.

Make one criticism of Alex’s bar chart.

Rearrange this formula to make u the subject.

A rocket accelerates at 90 m / s² and travels 270 km. The rocket’s final velocity is 8000 m / s.

Using part (a), or otherwise, calculate the rocket’s initial velocity in m / s.

A bag contains 150 counters. The counters are either red or yellow.

Riley picks a counter from the bag, records its colour, and replaces it. They do this nine times.

Here are Riley’s results.

Use Riley’s results to work out how many red counters are likely to be in the bag.

(a)........ red counters

Ling uses the same bag of counters and picks the counters in the same way.

Here are Ling’s results.

Use Ling’s results to estimate the probability of choosing a red counter from the bag. Give your answer as a fraction in its simplest form.

Explain why Ling’s results are likely to give a better estimate of the probability of choosing a red counter from the bag than Riley’s results.

The time taken to complete a journey halves as the speed doubles.

On the axes below, sketch a graph to show this relationship.

It takes 40 minutes to fill a garden pond using water from 5 identical hose pipes.

Assuming the rate of flow of water from each hose pipe is the same, work out how many minutes it would take to fill the same garden pond using 2 of these hose pipes.

The diagram represents a coastline.

A, B and C are lighthouses.

A boat is

• the same distance from A and B

• the same distance from AB and BC.

Using a ruler and compasses only, construct the position of the boat.

Label the position of the boat clearly.

At the end of each year, a driver records how many kilometres they have driven.

In 2021, they drove 18% more kilometres than in 2020. In 2022, they drove 25% more kilometres than in 2020.

In 2022, they drove 3500 km.

Kai says

I can work out how many kilometres were driven in 2020 by reducing 3500 by 25%. 3500 × 0.75 = 2625 km.

Explain why 2625 is not the correct number of kilometres driven in 2020.

Calculate the number of kilometres driven in 2021.

The diagram shows the graph of y = kx - x² + 2, where k is an integer.

Show that k = 3.

Use the graph to solve 3x - x² + 2 = 3. Give your answers to 1 decimal place.

Taylor designs a logo using isosceles triangles joined at a central point, P.

This is the start of Taylor’s design.

The completed design will have rotational symmetry, order 60 about point P.

Each triangle has base, b, and height, h, measured in mm.

Calculate h when b = 40 mm.

Give your answer correct to 1 decimal place.

On Heidi’s bookcase, the ratio of fiction to non-fiction books is 2: 3. Heidi removes 2 fiction books from the bookcase. The ratio of fiction to non-fiction books is then 5: 8.

How many books are left on the bookcase in total?

Show that 95 is not a prime number.

2000 and 8750 are written below as the product of their prime factors.

Find the highest common factor (HCF) of 2000 and 8750.

Write 2 × 10¹² as a product of its prime factors.

The diagram shows a quadrilateral, PQRS.

PS = 10 cm. Angle QPS = Angle PSR = 90°.

SR is 6 cm longer than PQ. The area of quadrilateral PQRS is A cm².

Write a simplified expression for the length PQ in terms of A. You must show your working.

A box contains 200 matches, correct to the nearest ten matches.

Complete the error interval for n, the number of matches in the box.

The box is a cuboid with

• length 7 cm, correct to the nearest cm

• width 5 cm, correct to the nearest cm

• volume 248 cm³, correct to the nearest cm³.

Show that the smallest possible height of the box is 6 cm.

A running club records the distances run by each member during December. The results are shown in this histogram.

18 members run less than 20 km.

Work out the number of members who run more than 30 km.

Finley says

To estimate the range, I subtracted the smallest possible value from the largest possible value. So, 80 – 0 = 80 km.

Explain why Finley’s method is likely to overestimate the true value of the range.

This box plot shows the distribution of the distance run by each member of the running club during July.

During December,

• the median distance run was 30 km

• the interquartile range of the distance run was 20 km.

Make two comparisons between the distances run during December and the distances run during July. Include values to support your comparisons.

The diagram shows a square-based pyramid ABCDE. O is the centre of the base.

The pyramid has base length 20 cm and each sloping edge has length 14.5 cm.

Draw the plan view of the pyramid on the one-centimetre grid below.

Scale: 1 cm represents 4 cm.

Calculate the volume of the pyramid. You must show your working. [The volume of a pyramid is × area of base × perpendicular height]

Two bottles are mathematically similar.

The small bottle holds 0.5 litres and has a height of 35 cm. The large bottle holds 2 litres.

Calculate the height of the large bottle.

The price of a seat on a flight, £ P, is given by

P = 49 × 1.009ⁿ

where n is the number of seats already sold on this flight.

Write down the percentage increase in price of the second seat sold compared to the first seat sold.

Show that the price of the 40th seat sold is less than £70.

The kth term of a sequence is r^k, where r ≠ 0. The sixth term is equal to three times the second term.

Find the value of r, giving your answer correct to 3 decimal places.

Describe fully the graph of x² + y² = 20.

The graph of y = 3x + 10 intersects the graph of x² + y² = 20 at two points.

Use an algebraic method to work out the coordinates of the two points. You must show your working.

(b) (........,........) and (........,........)

Write (2x - 5)(x + 4) in the form 2(x + a)² - b.

You must show your working.

Charlie, Dev and Eve all attempt to sketch the graph of y = (2 x - 5)(x + 4).

Charlie‛s sketch
Dev‛s sketch Eve‛s sketch

Whose sketch is the most accurate? Write down the properties of the graph that you used in making your decision.