Pearson Edexcel International GCSE (9–1) · MATHEMATICS A · Unit
Unit 9
105 practice questions
Topics covered
Practice questions
101–105 of 105
Question 101
8 marks · Normal
Answer and see the solution →A small cinema sells two types of snacks: Popcorn and Nachos.
(a)[3 marks]Popcorn is sold in tubs of two sizes: - Small Tub: 150 g for $2.85 - Large Tub: 420 g for $7.20 Work out which tub offers the better value for money by calculating the cost per gram for both tubs. Give your answer correct to 3 decimal places.
(b)[5 marks]A student buys one of the better-value tubs and some nachos. The total weight of food bought is 420 g and the total cost is $8.50.
The cost of Nachos (in dollars) can be modelled as a quadratic function of their mass (in grams):
where is the mass of nachos in grams, and is the cost in dollars of grams of nachos.
Set up simultaneous equations using your result from (a) and solve them to find the mass of nachos and the cost of nachos bought by the student. Give both answers correct to 2 decimal places.
Question 102
9 marks · Normal
Answer and see the solution →A shop in the UK is selling a calculator for £28.50. In another country, the same calculator costs $36.
The exchange rate is £1 = $1.20.
(a)[2 marks]Convert the UK price of the calculator into dollars. Give your answer to 2 decimal places.
(b)[3 marks]Hence, prove that buying the calculator in the UK is a better deal, and explain your reasoning.
(c)[4 marks]Let be the price in pounds, and be the price in dollars, with the exchange rate .
Write a general proof that shows for any UK price and other country price , the UK is the better deal if .
Question 103
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 104
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 105
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.