Mock Exam Set 1 - Paper 1

[Maximum mark: 5]

The manager of a small movie theatre recorded the number of tickets purchased each day for $14$ days. The number of tickets purchased each day were:

31, 36, 37, 17, 25, 27, 32, 54, 37, 21, 19, 26, 26, 29

For these data, the lower quartile is $25$ and the upper quartile is $36$.

1. Show that the value 54 would be considered an outlier.[3]
   

The box and whisker diagram for tickets purchased is given below.

The manager of a second small theatre also recorded the number of tickets purchased for the same 14 days, giving the following box and whisker diagram:

The manager of the second theatre claims that, in general, his theatre has more customers than the first theatre.

2. By comparing the two box and whisker diagrams, give one piece of evidence that supports this claim and one that may refute it. [2]
   

[Maximum mark: 6]

Edward wants to buy a new car, and he decides to take out a loan of $70\hspace{0.15em}000$ Australian dollars from a bank. The loan is for $6$ years, with a nominal annual interest rate of $7.2\%$, compounded monthly. Edward will pay the loan in fixed monthly instalments.

1. Determine the amount Edward should pay each month. Give your answer to the nearest dollar.[3]
   

2. Calculate the amount Edward will still owe the bank at the end of the third year. [3]
   

[Maximum mark: 7]

A small town is planning the construction of a new road. The new road will pass through the point A$(0,5)$ and will be perpendicular to the road connecting the points B$(3,0)$ and C$(6,6)$. This information is shown in the following diagram.

1. Find the gradient of the line through points B and C. [2]
   

2. Hence, state the gradient of the line through points A and D. [1]
   

3. Find the equation of the line through A and D. Give your answer in the form $y=mx+c$.[2]
   

4. Point D lies on the $x$-axis. Find the coordinates of point D.[2]
   

[Maximum mark: 6]

The following table shows the total revenue, $y$, in Australian dollars (AUD), obtained daily during the first week of January $2020$, by Peppy's Pizza restaurant and the number of guests, $x$, served.

1. 1. Calculate the Pearson's product-moment correlation coefficient, $r$, for this data.

   2. Hence comment on the result. [3]
      

2. Write down the equation of the regression line $y$ on $x$. [1]
   

3. Use the equation of the regression line to estimate the revenue of serving $70$ guests. Give your answer correct to $\textbf{the nearest AUD}$. [2]
   

[Maximum mark: 5]

Joe creates a small baseball field in his backyard to play with his children. The field is covered by grass in the shape of a sector of a circle with radius $8$ metres, as shown in the following diagram. The angle at the centre of the sector is $\theta^{\circ}$.

A total of $48$ m$^2$ of grass was used to cover the field.

1. Find the value of $\theta$. [2]
   

Joe asks his son, Lancer, to run around the field twice to warm up before playing a game.

1. Find the total distance that Lancer runs. [3]
   

[Maximum mark: 6]

On the Persian Gulf, there is a small peninsula that can be modelled by the equation $y = -0.2x^3+1.2x^2-x$ where $x$ and $y$ are measured in kilometres, as shown in the following diagram. Points P and Q are on the coastline.

1. Find the coordinates of P and Q. [2]
   

2. Find the furthest distance that the land protrudes from the coastline. [2]
   

3. Find the area of the peninsula. [2]
   

[Maximum mark: 4]

The following table shows the number of overtime hours worked by employees in a company.

It is known that the mean number of overtime hours is $1$.

1. Find the value of $x$. [2]
   

2. Find the standard deviation of the data. [2]
   

[Maximum mark: 5]

The following table shows the rate, $R$, at which rubbish is produced by a town, in tonnes per day. $t$ represents the time in days since the rate of rubbish production was first recorded.

The total amount of rubbish produced by the town over the 3-day period is equal to the area underneath the graph of the curve $R(t)$.

1. Use the trapezoidal rule with 6 intervals to estimate the total amount of rubbish produced by the town over the 3-day period.[3]
   

The actual amount of rubbish produced over the 3 days was $191\,$tonnes.

2. Find the percentage error between the estimate in part (a) and the actual amount of rubbish produced. [2]
   

[Maximum mark: 7]

The radius of a cylindrical container with height $x$ cm is

1. Find the range of $r$.[3]
   

The function $r^{-1}$ is the inverse function of $r$.

2. 1. Find $r^{-1}(8)$.

   2. Interpret the answer to part (b) (i) in context.

   3. Find the range of $r^{-1}$.[4]
      

[Maximum mark: 8]

Melody is camping with her friends in a forest. After setting up the tent, she goes for a walk to see the nearby surroundings. She walks $200$ m on a bearing of $\ang{280}$, changes direction, then walks another $150$ m on a bearing of $\ang{035}$.

1. Find the distance Melody is from the tent at this point. [4]
   

2. Determine the bearing Melody should walk on to return to the tent. [4]
   

[Maximum mark: 8]

BioNature Inc. is considering the monthly cost of producing straws made from paper. It is estimated that the rate of change in monthly cost with respect to the number of paper straws produced is modelled by

$C'(q) = 0.006q^2 -0.48q - 12, \text{ for } q > 0$

Where the monthly cost, $$\,C$, is in thousands and the number of straws, $q$, is also in thousands.

1. Find the number of paper straws that the company should produce in a month to minimize the cost. [4]
   

The company produces $110\hspace{0.15em}000$ paper straws with a cost of $\$88\hspace{0.15em}000$ in a certain month.

2. Find an expression for $C$ in terms of $q$. [4]
   

[Maximum mark: 8]

At a university, newly admitted first-year students can start their degrees at any of its seven campuses. The admissions director for the university is interested in finding out whether first-year students show any specific preference in their choice of campus. She forms a model in which she expects the same number of first-year students will select each campus.

The director records the number of first-year students registering at each campus this year to test the model's suitability. The results are shown in the table below.

A goodness of fit test at a $5\%$ significance level is carried out to test the accuracy of her model.

1. State the null and alternative hypotheses.[2]
   

2. Determine the expected number of registrations at each campus.[1]
   

3. Write down the number of degrees of freedom for this test. [1]
   

4. State the conclusion to the test, justifying your answer. [4]
   

[Maximum mark: 5]

A bouncy ball is dropped out of a second story classroom window, $5\,$m off the ground. Every time the ball hits the ground it bounces $89\,$% of its previous height.

1. Find the height the ball reaches after the $2$nd bounce. [2]
   

2. Find the total distance the ball has travelled when it hits the ground for the $5$th time. [3]