Higher Applications of Maths 2022 Paper 1 — Questions & Video Solutions

1

Question 1

Finance

3 Marks

2022 Q1

Jill borrows £5650 from a bank.

The annual effective rate of interest on the loan is 9%.

Jill makes level monthly repayments of £186.01 at the end of each month.

Complete the following loan schedule for Jill's loan to show the loan outstanding at the end of month 2.

Time (months)	Repayment (£)	Loan outstanding (£)
0		5650
1	186.01
2	186.01

Show answer

Monthly interest rate: $\displaystyle 0.72...\%$ (or $\displaystyle 0.7\%$ )

Time (months)	Repayment (£)	Interest content of repayment (£)	Capital content of repayment (£)	Loan outstanding (£)
0				5650
1	186.01	40.72	145.29	5504.71
2	186.01	39.67	146.34	5358.37

2

Question 2

Planning & Decision Making

(6, 1, 3)10 Marks

2022 Q2

A company provides a team of workers to re-roof a house.

The table shows the list of tasks required to complete this job, the time required for each task, and the order in which the tasks are to be completed.

Task	Description	Preceding task	Time (hours)
A	Transport materials to location	None	3
B	Construct scaffolding	A	7
C	Remove old tiles	B	4
D	Take old tiles to recycling site	C	1
E	Inspect roofing	C	2
F	Purchase required number of roof tiles	D	1
G	Replace roofing felt	E	4
H	Replace roofing tiles	F, G	6
I	Remove scaffolding	H	3
J	Clean up	H	2

(a) Complete the PERT chart showing the earliest start time and the latest completion time for each task.

(b) The roofing company works for 9 hours each day on this job. State the minimum number of days that the company will require to complete this job.

(c) Construct a Gantt chart, without float times, for this job.

Show answer

(a) Correct forward and backward scan on PERT chart.

Forward scan values:
A: 0, 3
B: 3, 7
C: 10, 4
D: 14, 1
E: 14, 2
F: 15, 1
G: 16, 4
H: 20, 6
I: 26, 3
J: 26, 2

Backward scan values:
I: 26, 3, 29
J: 26, 2, 29
H: 20, 6, 26
F: 15, 1, 20
G: 16, 4, 20
D: 14, 1, 19
E: 14, 2, 16
C: 10, 4, 14
B: 3, 7, 10
A: 0, 3, 3

(b) 4 days

(c) Gantt chart plotted correctly. Critical path A-B-C-E-G-H-I.

3

Question 3

Statistics & Probability

(1, 1, 1, 1, 1)5 Marks

2022 Q3

You must refer to the information on 'Dogs and fleas' given in the pre-release material when answering this question.

After Study 1 the following bar chart was created. It shows the number of dogs requiring treatment for a flea infestation against the month of the year.

Bar chart of dogs requiring flea treatment by month

(a) State the type of data that best describes the number of dogs requiring treatment for a flea infestation.

The student vet claims this graph proves that flea infestations are more common in the warmer months across the UK.

(b) Explain why this claim is not valid.

The data gathered from Study 2 is shown in the table.

Area	Number of dogs inspected	Number of dogs with a flea infestation
Urban Clinic	213	30
Rural Clinic	156	43

(c) State a statistical test that would be appropriate to determine whether this data provides evidence of a difference in flea infestations between urban and rural areas.

(d) State one part of the design of Study 2 that would be needed to ensure that the assumptions required to perform the test in (c) are valid.

To fit in with the student vet's placements, the data from the urban clinic was collected from July to September, and the data from the rural clinic was collected from October to December.

(e) Explain how this may affect the conclusions of Study 2.

Show answer

(a) discrete numerical

(b) e.g., A claim cannot be made from a graph OR one clinic in Edinburgh is not representative of the UK.

(c) z-test for two proportions (a chi-squared test is also appropriate).

(d) e.g., Samples need to be chosen randomly.

(e) e.g., They are not comparable as the data is not collected over the same time periods (the seasonal variability could affect the results).

4

Question 4

Finance

(3, 2)5 Marks

2022 Q4

Zac deposits £500 into a bank account on 1 January 2018, 1 January 2019, and 1 January 2020.

The bank pays interest into his account at the end of every year, using the following annual effective rates:

2018: 3.3%
2019: 2.4%
2020: 1.0%

He makes no withdrawals from his account.

(a) Calculate the balance in Zac's account at the end of 2020.

On 1 January 2021 Zac deposits another £500 into his account. He makes no further deposits into his account in 2021.

(b) Calculate the annual effective rate of interest needed in 2021 for the account balance to be £2100 by the end of the year.

Show answer

(a) £1556.30

(b) $\displaystyle 2.125...\%$

5

Question 5

Mathematical ModellingExcel

(3, 1, 1, 2, 1)8 Marks

2022 Q5

You must refer to the spreadsheet file 'Q5 School Roll.xlsx' when answering this question.

A school is planning a new building as it is approaching its maximum capacity.

The school roll in August 2021 was 650 pupils.

Approximately 18% of pupils leave by the end of each school year. 140 new S1 pupils join the roll in August each year.

(a)(i) Complete the 'School Roll' worksheet to predict the school roll in August 2031.

(a)(ii) Comment on the precision of this prediction.

(b) Comment on the relationship between time and the predicted school roll up to August 2031.

The school moves forward with plans for a new building. This will increase the capacity of the school to 800 pupils.

(c)(i) Extend the table in your worksheet to construct a graph to show what is predicted to happen to the school roll in the long term. You must consider what happens to the school roll beyond August 2031.

(c)(ii) Use your graph to determine whether the new capacity is suitable.

Q5 School Roll.xlsx

Show answer

(a)(i) 761 (pupils)

(a)(ii) e.g., the number of pupils leaving each year is approximate.

(b) the school roll gradually increases each year.

(c)(i) Evidence of extending spreadsheet to at least August 2042 and constructing an appropriate graph.

(c)(ii) Yes, the population is not expected to exceed 800 pupils.

6

Question 6

Finance

(1, 2)3 Marks

2022 Q6

The average price of petrol increased at the following annual effective rates:

March 2018 to 2019: 2.1%
March 2019 to 2020: 0.5%
March 2020 to 2021: 2.0%

(a) Calculate the overall percentage increase in the average price of petrol over the three years from March 2018 to March 2021.

The average price of petrol in March 2021 was 136.4 pence per litre.

(b) Hence calculate the average cost of filling a 45-litre tank with petrol in March 2018.

Show answer

(a) $\displaystyle 4.66...\%$

(b) £58.64

7

Question 7

Statistics & ProbabilityRStudio

(2, 1, 2, 2, 2, 1)10 Marks

2022 Q7

You must refer to the information on 'Strength and conditioning' given in the pre-release material when answering this question.
You must also refer to the spreadsheet file 'Q7 Jump.csv' for the data, and the word processing file 'Q7 Jump Answers.docx' when answering this question.
You must complete parts (a) (i), (b) (i), (b) (ii), and (c) using appropriate statistical software.
You must include all output from statistical software, and your answers in the word processing file 'Q7 Jump Answers.docx'.

A strength and conditioning coach wants to increase vertical jump height performance in their trainees. The data in the spreadsheet file shows back squat weight (kg) and vertical jump height (cm).

(a)(i) Construct a scatter plot of vertical jump height on back squat weight for the data.

(a)(ii) Make an appropriate comment about the relationship between vertical jump height and back squat weight.

(b)(i) Find the correlation coefficient between back squat weight and vertical jump height.

(b)(ii) Find the equation of the regression line of vertical jump height on back squat weight.

(c) Use your statistical software to estimate the vertical jump height for a trainee who can back squat 165 kg, and comment on the accuracy of the predicted value.

Based on the correlation, the coach advises the trainees that increasing their back squat weight will increase their vertical jump height.

(d) Explain why the statistical analysis does not support this advice.

Q7 Jump.csv Q7 Jump Answers.docx

Show answer

(a)(i) Generate scatterplot from software with appropriate title and axis labels.

(a)(ii) Approximately a positive linear relationship (or as back squat weight increases, vertical jump height increases).

(b)(i) $\displaystyle 0.828...$

(b)(ii) jump height = $\displaystyle 0.36$ back squat + $\displaystyle 7.2$

(c) $\displaystyle 66.5$ cm. Since $\displaystyle 165$ kg is within the range of the data used to make the model and the model is a strong linear model, the prediction is fairly accurate.

(d) Correlation is not causation.

8

Question 8

Finance

(3, 3, 1)7 Marks

2022 Q8

You must refer to the information on 'Deductions from salaries' given in the pre-release material when answering this question.

Sophie starts a job selling mobile phones. Her company provides the following financial benefits at the end of each month:

Salary: £1000 per month
Commission: £20 for each mobile phone sold during the month
Pension scheme: Sophie pays 2.5% of her monthly earnings (salary and commission) before tax into her pension fund. The company contributes another 6.5% of the same monthly earnings.

Sophie expects to sell 150 mobile phones per month.

(a) Calculate how much income tax Sophie expects to pay each year.

Sophie's pension fund earns an annual effective rate of interest of 10%.

(b) Calculate the expected value of Sophie's pension fund immediately after the third pension contribution is made.

Sophie has the choice to leave the company pension scheme.

(c) State one reason why Sophie would not leave the company pension scheme.

Show answer

(a) Taxable income: £46,800 per year. Tax payable: £7667.67.

(b) £1088.63

(c) e.g., It is important to save to provide an income in retirement when Sophie is not working OR the employer's pension contribution is an extra part of the financial package which Sophie will not receive if she does not join the scheme.

9

Question 9

FinanceExcel

(4, 3, 2)9 Marks

2022 Q9

You must refer to the spreadsheet file 'Q9 Car Repayments.xlsx' when answering this question.

Maria is buying a car for £15,000. She has arranged a loan for the full amount from the bank, to be paid back with level monthly repayments for 48 months. The annual effective interest rate is 9.5%.

Open the 'Bank Loan' worksheet.

(a) Complete the 'Loan Repayment Schedule' to determine the level monthly repayment amount, and the final repayment amount.

The car dealership has their own finance options.

Option 1: Pay £300 per month for 48 months, and return the car to the dealer. Additional charges will apply if the mileage exceeds 24000 miles when the car is returned.

Option 2: Pay £300 per month for 48 months, and keep the car by making an additional payment of £5000 in the final month.

Open the 'Car Finance' worksheet.

(b) Complete the 'Finance Repayment Schedule' to find the annual effective interest rate of Option 2.

(c) State two reasons why Maria might decide to purchase the car using the bank loan instead of the car dealership finance options.

Q9 Car Repayments.xlsx

Show answer

(a) Level monthly repayment: £374.06. Final repayment: £373.99.

(b) $\displaystyle 11.04...\%$

(c) e.g., Maria will own the car outright after 48 months without needing to find a lump sum of £5000. The finance deal is risky - if Maria travels more than 24,000 miles she will pay more due to penalty payments.

10

Question 10

Mathematical Modelling

(2, 3)5 Marks

2022 Q10

You must refer to the information on 'Mountain gorillas' given in the pre-release material when answering this question.

The 2020 study found that the population of mountain gorillas had increased to 1004.

An expert has stated that if the mountain gorilla population in the Virunga Mountains continues to grow exponentially there will be 1600 gorillas by the year 2032.

(a) Determine if the expert's statement is correct. Give a reason for your answer.

A typical adult mountain gorilla eats 30 kg of food per day.

(b) Estimate the maximum amount of termites and ants (in kg) that a typical mountain gorilla will eat during their adult lifetime.

State any assumptions you have made.

Show answer

(a) The expert is incorrect since the predicted population $\displaystyle 1482 < 1600.$ (Using multiplying factor $\displaystyle \frac{1004}{680}$ ).

(b) Estimate total amount of food: $\displaystyle 30 \times 365 \times 25 = 273,750$ kg. Assume 49% as a maximum percentage of diet. Amount $\displaystyle = 49\% \times 273,750 \approx 134,000$ kg.