Pearson Edexcel International GCSE (9–1) · MATHEMATICS A · Unit
Unit 9
105 practice questions
Topics covered
Practice questions
11–19 of 19 · Hard
Question 11
4 marks · Hard
Answer and see the solution →A cuboid has sides of lengths 7 cm, 9 cm, and 15 cm. Calculate the length of the space diagonal (from one corner to the farthest opposite corner), giving your answer to 3 significant figures.
Cuboid: length = 7 cm, width = 9 cm, height = 15 cm; diagonal joins vertex A (0, 0, 0) to vertex F (7, 9, 15). Question 12
4 marks · Hard
Answer and see the solution →A square-based pyramid has a base of side 10 cm. The vertex V is vertically above the centre of the base and is 14 cm above the centre. Explain, step by step, how trigonometry can be used to calculate the angle between a slant edge (V to one corner of the base) and the base. You may use a diagram to support your answer.
Square-based pyramid with base PQRS, side 10 cm, vertex V above base centre, height 14 cm, slant edge VP. Question 13
5 marks · Hard
Answer and see the solution →The diagram shows a cuboid with length 12 cm, width 7 cm, and height 20 cm.
Cuboid: Length = 12 cm (from A to B), width = 7 cm (from B to C), height = 20 cm (from A to E vertically above A). (a) Calculate the length of the diagonal from A to the opposite vertex G (the space diagonal).
(b) Calculate the angle between the diagonal AG and the base ABCD. Give your answer to 3 significant figures.
(a)[2 marks]Calculate the length of the space diagonal AG.
(b)[3 marks]Calculate the angle between AG and the base ABCD. Give your answer to 3 significant figures.
Question 14
2 marks · Hard
Answer and see the solution →The grouped frequency table below shows the ages of people attending a workshop.
Age (years) Frequency 10 < a \leq 20 8 20 < a \leq 30 16 30 < a \leq 50 12 What is the frequency density for the group 20 < a \leq 30?
Question 15
4 marks · Hard
Answer and see the solution →The histogram shows the times taken by 60 students to finish a computer game.
Calculate the total number of students who finished the game in more than 10 minutes but no more than 25 minutes.
Question 16
4 marks · Hard
Answer and see the solution →The histograms below show the distribution of journey times (in minutes) to school for two local schools, Alpha and Beta. Both schools had the same total number of students surveyed.
Journey Times to School Compare the two distributions. Give two differences and what they show about the journey times for Alpha and Beta.
Question 17
11 marks · Hard
Answer and see the solution →A scientist records the cooling of a metal rod and the time it takes to drop temperature. The results are grouped below.
(a)The grouped results appear below.
Temperature drop times Time, t (seconds) Frequency 0 < t ≤ 10 6 10 < t ≤ 20 14 20 < t ≤ 30 9 30 < t ≤ 50 11 (i)[2 marks]Complete the table by finding the class width and frequency density for each group.
(ii)[1 mark]Use the information from (i) to draw a histogram representing the data (you may make a rough sketch).
(b)[2 marks]A function $f$ is defined by $f(x) = e^{-x/10}$ for $0 \leq x \leq 50$. The curve $y = f(x)$ models the cooling process. On the axes where your histogram is drawn, sketch the curve $y = f(x)$, matching the scales.
(c)[2 marks]Describe two key differences between the histogram and the curve $y = f(x)$ in the context of the cooling process.
(d)[2 marks]The scientist shifts the experiment by 10 seconds, so she now measures $y = f(x-10)$. Describe, algebraically and using your sketches, the transformation applied to the curve.
(e)[2 marks]Suppose all times in the experiment are doubled. Write down the new function (in terms of $x$) and describe the graphical effect.
Question 18
10 marks · Hard
Answer and see the solution →Students study the ages of participants in a marathon and compare this with a graph transformation.
(a)The table shows the ages of participants.
Ages of marathon runners Age, $a$ (years) Frequency 10 < a ≤ 20 6 20 < a ≤ 30 22 30 < a ≤ 45 31 45 < a ≤ 60 14 60 < a ≤ 80 7 (i)[2 marks]Calculate the class widths and frequency densities for each interval.
(ii)[1 mark]Draw a histogram for the data (rough sketch acceptable).
(b)[2 marks]The function $g(x) = (x-30)^2$ is sketched for $x$ in the interval $10 \leq x \leq 80$ on the same axes as your histogram.
(c)[2 marks]Describe the transformation that maps $y = x^2$ onto $y = (x-30)^2$. Explain how this can represent a change in the age profile.
(d)[2 marks]From your histogram, estimate the modal class and comment on how this relates to the minimum point of the curve $g(x)$.
(e)[1 mark]Suppose the data is adjusted so every age is reduced by 10 years. Write down the equation for the new function and state the transformation applied.
Question 19
11 marks · Hard
Answer and see the solution →A curved graph is shown below. The graph shows the distance, metres, travelled by a cyclist after seconds.
Graph of \( s \) against \( t \) for a cyclist (a)[4 marks]Use the graph to estimate the gradient of the curve at seconds. Give your answer in .
(b)[4 marks]A group of students all cycle the same route. The frequency table below shows the time each student takes to finish the route.
Times taken to finish the route Time, t (seconds) Frequency 0 < t \leq 4 4 4 < t \leq 8 7 8 < t \leq 10 9 10 < t \leq 16 10 Draw a histogram to represent the data in the table. Your histogram should have frequency density on the vertical axis.
(c)[3 marks]By referring to your histogram and the graph in part (a), compare how the cyclist's time at seconds relates to the overall distribution of times for the group. Write two facts comparing these results.