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AI067

AI254

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Paper 1

AI031

AI316

AI251

AI156

AI431

Paper 2

\[Maximum mark: 6\]

The volume of a hemisphere, $V$, is given by the formula

$$
V = \sqrt{\dfrac{4S^3}{243\pi}}\hspace{0.05em},
$$
where $S$ is the
total surface area. 

The total surface area of a given hemisphere is
$529$ cm$^2$. 



1.  Calculate the volume of this hemisphere in cm$^3$. Give your answer
    correct to **one decimal place**. :marks[3]

2.  Write down your answer to part (a) correct to **the nearest
    integer**. :marks[1]

3.  Write down your answer to **part (b)** in the form $a\times10^k$,
    where $1 \leq a < 10$ and $k \in \mathbb{Z}$.:marks[2]






\[Maximum mark: 6\]

Consider the function $f(x) = x^3 - 6x^2 + 5x + 18$. Part of the graph
of $f$ is shown below.


:::center




![cb43ccf36b7c8052efb237974883947bb0183950.svg](question:156a798f-6c8a-420e-968a-e9eda11390b9)

:::




1.  Find $f'(x)$. :marks[3]

2.  There are two points at which the gradient of the graph of $f$ is
    $20$. Find the $x$-coordinates of these points. :marks[3]






\[Maximum mark: 6\]

The Voronoi diagram below shows the fitness studios A, B and C, located
in a particular district.


:::center




![b3aaeee21fcd45406c5c929ca6b8501265c0c1b5.svg](question:b2ace22e-907b-485b-99c2-75d4a49ceac4)

:::




1.  On the diagram, shade the cell with the missing fitness studio.
    :marks[2]

2.  Determine the coordinates of the missing fitness studio, D.
    :marks[2]

3.  Jennifer lives an equal distance from fitness studios A and D. Find
    theshortest possible distance Jennifer's house
    could be from fitness studios A and D.:marks[2]






\[Maximum mark: 6\]

Oliver throws a coin from the open window onto flat horizontal ground in
front of his office building. 

The path of the coin is modelled by the quadratic curve
$y = 8 + 3x - 0.5x^2$ , where $x$ represents the
horizontal distance the coin is thrown and $y$ represents the height of
the coin above the ground. All distances are measured in metres. The
wall of the building lies along the $y$-axis. The coin starts its flight
at point A, reaches its maximum height at point B, and lands on the
ground at point C, as shown on the diagram below.


:::center




![3194f4c38ebe70fcc80ad86fb61b4fd630cd08ef.svg](question:be672629-0ece-49eb-b0ba-8f5c76b93870)

:::




1.  Write down the height in metres from which the coin was thrown.
    :marks[1]

2.  Calculate the maximum height, above the ground, reached by the coin.
    :marks[3]

3.  Find the horizontal distance from the base of the wall to the point
    at which the coin hits the ground. :marks[2]



\[Maximum mark: 6\]

The runtime, $R$, of lithium batteries made by a battery manufacturing
company follows a normal distribution. $R$ is measured in hours of
continuous use in a camping head lamp.

The normal distribution curve of $R$ is shown below.


:::center




![ea19a30bee9041db718ef8c59822d9d75879bbb1.svg](question:694c3d96-5d73-4f69-b25c-39cd0510a967)

:::



1.  Write down the mean runtime of the batteries. :marks[1]

 

The standard deviation of the runtime of the batteries is $12$
hours.



2.  Find the probability that $85 \leq R \leq 100$, for a
    randomly chosen battery. :marks[2]



The company states that $95\hspace{0.1em}\%$ of the batteries have a
runtime of at least $n$ hours.



3.  Find the value of $n$. Give your answer correct to the
    nearest integer. :marks[3]




\[Maximum mark: 6\]

Charles plans to invest in a retirement plan for $30$ years. In this
plan, he will deposit $1000$ British pounds (GBP) at the end of every
month and receive a $6.5\hspace{0.05em}\%$ interest rate per annum,
**compounded monthly**. 



1.  Find the future value of the investment at the end of the $30$
    years. Give your answer correct to **the nearest pound**.:marks[3]

 

After the $30$-year period, Charles will start receiving regular monthly
payments of $1500$ GBP. 



2.  Calculate the number of years it will take Charles's
    monthly retirement $\text{payments}$ to match the total
    amount originally invested. :marks[3]






\[Maximum mark: 4\]

Cylindrical logs of length $2.2$ m and radius
$7.6$ cm are used to construct a fence line.
Part of the logs are removed, as illustrated in the diagram below.


:::center




![7ab1702add176182b785d942e7d7230ac6752635.svg](question:7843e469-ca0e-42de-a9cb-5dede00680ce)

:::


Find the volume of each log.





\[Maximum mark: 6\]

Susan has read a scientific paper which states that there is a
significant difference in body temperature between non-pregnant women
and pregnant women. She is interested in verifying this result and
decides to measures the body temperature of two groups: a group of $7$
non-pregnant women and a group of $8$ pregnant women at weeks $8$ to
$12$ of their pregnancy. The results are shown below.


:::center




![5088af5c79c2275ae9de7cb47f2e302966e3356c.svg](question:b79ca5e6-966a-42b6-9d81-36628ba2b275)

:::


At the $10\%$ level of significance, a $t$-test was used to compare the
means of two groups. The data is assumed to be normally distributed and
the standard deviations are equal between the two groups.




1.  
    1.  State the null hypothesis.

    2.  State the alternative hypothesis. :marks[2]

    

2.  Calculate the $p$-value for this test. :marks[2]

3.  State, giving a reason, whether Susan's conclusion agrees with the
    finding of :marks[2] the scientific paper.






\[Maximum mark: 7\]

The following cumulative frequency diagram shows the distance students
need to travel to get to school.


:::center




![03439bda8db314adc9c2b710fcb3499973f1049f.svg](question:9f9d93fa-4d91-4bb3-80a1-45960bc5d42d)



:::



1.  Find the median distance a student travels to school. :marks[2]

2.  Find the number of students that travel between $2$ km and $4$ km to
    get to school. :marks[3]

3.  Find the percentage of students that travel more than $4.5$ km to
    get to school. :marks[2]






\[Maximum mark: 5\]

The following diagram shows part of the graph of
$f(x) = (16-x^2)(x+3), \, x \in \mathbb{R}$. The shaded region $R$ is
bounded by the $x$-axis, $y$-axis and the graph of $f$.


:::center




![3751391507994630eeac54a189163b7a9bf8ff4b.svg](question:97461ef8-68e4-438c-b987-847de2c2a26b)

:::




1.  Write down an integral for the area of region $R$. :marks[2]

2.  Find the area of region $R$. :marks[1]

 

A trapezoid ABCD has four vertices with the coordinates shown as below.


:::center




![7483ada6476d727e0afb4f73aa114d5cd94fa693.svg](question:fa9cf112-d517-4181-8376-33dd5b464916)

:::



3.  Find the value of $a$, the $x$-coordinate of point D, such
    that the area of the trapezoid is equal to the area of region $R$.
    :marks[2]

 

\[Maximum mark: 5\]

The data below shows the price, $y$, in Kazakhstani tenge (KZT), that
each bag of wheat flour can be purchased for at a wholesale market in
Almaty if $x$ bags are ordered. 


:::center




![5a253ce60e250108cf649fd1f8dc000abcca0657.svg](question:6fdb89eb-7ab4-4f60-a277-9ebc4bccb019)

:::




1.  Find an expression for the price, $y$ KZT, of each bag purchased in
    terms of the number of bags, $x$, ordered. :marks[2]

2.  Hence find the exact total cost of purchasing $250$ bags of wheat
    flour. :marks[3]






\[Maximum mark: 6\]

Julia wants to buy a house that requires a deposit of
$74\hspace{0.15em}000$ Australian dollars (AUD).


Julia is going to invest an amount of AUD in an account that pays a
nominal annual interest rate of $5.5$ %,
**compounded monthly**. 



1.  Find the amount of AUD Julia needs to invest to reach
    $74\hspace{0.15em}000$ AUD after $8$ years. Give your answer correct
    to **the nearest dollar**. :marks[3]

 

Julia's parents offer to add $5000$ AUD to her initial investment from
part (a), however, only if she invests her money in a more reliable bank
that pays a nominal annual interest rate only of
$3.5$ %, **compounded quarterly**.




2.  Find the number of years it would take Julia to save the
    $74\hspace{0.15em}000$ AUD if she accepts her parents money and
    follows their advice. Give your answer correct to **the nearest
    year**. :marks[3]






\[Maximum mark: 6\]

In a college lottery, there are two types of tickets: 'standard' tickets
and 'lucky' tickets. Every student is given one ticket at random. The
total number of the standard tickets is four times greater than the
number of lucky tickets.

The probability of winning the lottery with a standard ticket is
$\dfrac{1}{100}$ and $\dfrac{1}{40}$ for a lucky ticket. The tree
diagram below illustrates this. 

:::center




![31cce4a2200474104de8ffe3e34c6deb706fdcd1.svg](question:063063f7-e756-48a2-8021-9d4e9aef5f71)

:::



1.  Calculate the probability that a randomly chosen student will win
    the lottery. :marks[2]

2.  Calculate the probability that a randomly chosen student will get a
    lucky ticket and still lose the lottery. :marks[2]

3.  Calculate the probability that a randomly chosen student, given they
    won the lottery, had a standard ticket. :marks[2]






\[Maximum mark: 6\]

Two unbiased, standard, 6 sided dice are thrown and the scores are
recorded. Let the random variable $X$ be the minimum of these two
scores. The probability distribution of $X$ is given in the following
table.


:::center




![7b8039a8716eab46adfe0a4927410dcb7e953a07.svg](question:2c06687a-4617-4534-aeb0-efda8da95e1a)



:::




1.  Find the value of $p$ and the value of $q$. :marks[3]

2.  Find the expected value of $X$. :marks[3]




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