\[Maximum mark: 6\]

On the first day of September, $2019$, Gloria planted $5$ flowers in her
garden. The number of flowers, which she plants at every day of the
month, forms an arithmetic sequence. The number of flowers she is going
to plant in the last day of September is $63$. 



1.  Find the common difference of the sequence. :marks[2]

2.  Find the **total** number of flowers Gloria is going to plant during
    September.:marks[2]

3.  Gloria estimated she would plant $1000$ flowers in the month of
    September. Calculate the percentage error in Gloria's estimate.
    :marks[2]






\[Maximum mark: 6\]

**In this question give all answers correct to the nearest whole
number.** 

Benjamin spends € $32\hspace{0.15em}000$ buying
bitcoin mining hardware for his cryptocurrency $\text{mining}$
business. The hardware depreciates by $16$ %
each year. 



1.  Find the value of the hardware after two years. :marks[3]

2.  Find the number of years it will take for the hardware to be worth
    less than $\text{\euro\hspace{0.05em}$8000$}$. :marks[3]






\[Maximum mark: 6\]

**In this question give all answers correct to the nearest whole
number.** 

A population of goats on an island starts at $232$. The population is
expected\
to increase by $15$ % each year.




1.  Find the expected population size after: 
    1.  $10$ years;

    2.  $20$ years. :marks[4]

    

2.  Find the number of years it will take for the population to reach
    $15\hspace{0.15em}000$. :marks[2]




\[Maximum mark: 6\]

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]






\[Maximum mark: 6\]

Greg has saved $2000$ British pounds (GBP) over the last six months. He
decided to deposit his savings in a bank which offers a nominal annual
interest rate of $\text{$8$\hspace{0.05em}\%}$, **compounded
monthly**, for two years. 



1.  Calculate the total amount of interest Greg would earn over the two
    years. Give your answer correct to **two decimal places**.
    :marks[3]

 

Greg would earn the same amount of interest, **compounded
semi-annually**, for two years if he deposits his savings in a second
bank. 



2.  Calculate the nominal annual interest rate the second bank
    offers. :marks[3]




\[Maximum mark: 15\]

Charles has a New Years Resolution that he wants to be able to complete
$100$ pushups in one go without a break. He sets out a training regime
whereby he completes $20$ pushups on the first day, then adds $5$
pushups each day thereafter.



1.  Write down the number of pushups Charles completes
    1.  on the $4$th training day;

    2.  on the $n$th training day. :marks[3]

    

 

On the $k$th training day Charles completes $100$
pushups for the first time.




2.  Find the value of $k$. :marks[2]

3.  Calculate the total number of pushups Charles completes on
    the first $10$ training days. :marks[4]

 

Charles is also working on improving his long distance swimming in
preparation for an Iron Man event in $12$ weeks time. He swims a total
of $10\hspace{0.15em}000$ metres in the first week, and plans to
increase this by $10$ % each week up until the
event. 



4.  Find the distance Charles swims in the $6$th week of
    training. :marks[3]

5.  Calculate the total distance Charles swims until the event.
    :marks[3]




\[Maximum mark: 6\]

**In this question give all answers correct to the nearest whole
number.** 

Michelle takes out a loan of $$\$12\hspace{0.15em}000$$. The unpaid
balance on the loan has an interest rate of
$4.3$ % per year, **compounded annually**.




1.  The loan is to be repaid in payments of $$\$1500$$ made at the end of
    each year. 

    

    1.  Find the number of years it will take to repay the loan.
    2.  Calculate the total amount that has been paid in amortising
        the loan.:marks[3]

    

2.  The loan is to be amortised over $5$ years.
    1.  Find the annual payment made at the end of each year.

    2.  Calculate the total amount that has been paid in amortising
        the loan.:marks[3]

    




\[Maximum mark: 6\]

Melinda has $$\$300\hspace{0.15em}000$$ in a private foundation. Each year
she donates $10\hspace{0.05em}\%$ of the money remaining in her private
foundation to charity. 



1.  Find the maximum number of years Melinda can donate to charity while
    keeping at least $$\$100\hspace{0.15em}000$$ in the private
    foundation. :marks[3]

 

Bill invests $$\$400\hspace{0.15em}000$$ in a bank account that pays a
nominal interest rate of $4$ %, **compounded
quarterly**, for ten years. 



2.  Calculate the value of Bill's investment at the end of this
    time. Give your answer correct to **the nearest dollar**.
    :marks[3]






\[Maximum mark: 6\]

Taste of Home Magazine recommends using a combination of Cheddar, Brie
and Swiss when putting together cheese boards for parties. The
recommended total cheese board size for a party of
$10$ - $15$ people is
$1$ kilogram. The table below shows the weight, in hundred of grams, of
each kind of cheese required to make one kilogram of cheese combination,
and the cost of making each combination.


:::center




![1f95e531ff9a99fac83619dc0ed06ffca82c640c.svg](question:ce8d9594-1464-4833-a91b-37c05aab6a9c)

:::




1.  By setting up a system of linear equations and using matrices, find\
    the price per kilogram of each type of cheese. :marks[4]

 

John prepares a cheese board with proportion of each cheese type, in
hundred grams, as shown in the table below.


:::center




![585de61d3f014a0443069387501b6401e1d28016.svg](question:6da9718c-0e50-4f4d-aaf2-c86f790a573e)

:::




2.  Calculate the amount of money John spent on cheese.
    :marks[2]






\[Maximum mark: 6\]

Let $u_n = 5n-1$, for $n \in \mathbb{Z}^+$. 



1.  
    1.  Using sigma notation, write down an expression for
        $u_1 + u_2 + u_3 + \dots + u_{10}$.

    2.  Find the value of the sum from **part (a) (i)**. :marks[4]

    

 

A geometric sequence is defined by $v_n = 5\times 2^{n-1}$, for
$n \in \mathbb{Z}^+$. 



2.  Find the value of the sum of the geometric series
    $\medmath{\displaystyle\sum_{k = 1}^6}\hspace{0.1em}v_k$.:marks[2]




\[Maximum mark: 14\]

Bruce goes into a car dealership to purchase a new vehicle. The one he
wants to buy costs $$\$16\hspace{0.15em}000$$, however he doesn't have
that much money in his bank. The salesman offers him a financing option
of a $30$ % deposit followed by $12$ monthly
payments of $$\$1150$$.



1.  Find the amount of the deposit. :marks[1]

2.  Calculate the total cost of the loan under this financing option.
    :marks[2]


Bruce's father generously offers him an interest free loan of
$$\$16\hspace{0.15em}000$$ to buy the car to avoid the expensive loan
repayments. They agree that Bruce will repay the loan by paying his
father $$\$\hspace{0.05em}x$$ in the first month and $$\$\hspace{0.05em}y$$
every following month until the $$\$16\hspace{0.15em}000$$ is repaid.

The total amount Bruce's father receives after $12$ months is $$\$5200$$.
This can be expressed by the equation $x + 11y = 5200$. The total amount
that Bruce's father receives after $24$ months is
$$\$10\hspace{0.15em}600$$.



3.  Write down a second equation involving $x$ and $y$.
    :marks[1]

4.  Determine the value of $x$ and the value of $y$.
    :marks[2]

5.  Calculate the number of months it will take Bruce's father
    to receive\
    the $$\$16\hspace{0.15em}000$$. :marks[3]


Bruce decides to buy a cheaper car for $$\$12\hspace{0.15em}000$$ and
invest the remaining $$\$4000$$. He is considering two investment options
over four years.

Option A: Compound interest at an annual rate of
$6.5$ %.

Option B: Compound interest at a nominal annual rate of
$6$ %, **compounded monthly**.

**Express each answer in part (f) to the nearest dollar**.



6.  Calculate the value of each investment option after four
    years.

    

    1.  Option A.
    2.  Option B. :marks[5]

    



\[Maximum mark: 7\]

The half-life, $T$, in days, of a radioactive isotope can be modelled by
the function 
$$
\begin{aligned}
T(k) &= \dfrac{\ln 0.5}{\ln\left(1 - \frac{k}{100}\right)}, \hspace{0.5em} 0 < k < 100,\end{aligned}
$$

where $k$ is the decay rate, in percent, per day of the isotope.




1.  The decay rate of Gold-$196$ is $6.2$ % per
    day. Find its half-life.:marks[2]

 

The half-life of Phosphorus-$32$ (P-$32$) is $14.3$ days. A sample
containing $120$ grams of P-$32$ is produced and stored in a
biochemistry laboratory. 



2.  Find the decay rate per day of P-$32$. :marks[2]

3.  Find the amount of P-$32$ left in the sample after:
    1.  $14.3$ days;

    2.  $50$ days. :marks[3]

    






\[Maximum mark: 19\]

On Wednesday Eddy goes to a velodrome to train. He cycles the first lap
of the track in $25$ seconds. Each lap Eddy cycles takes him $1.6$
seconds longer than the previous lap. 



1.  Find the time, in seconds, Eddy takes to cycle his tenth lap.
    :marks[3]

 

Eddy cycles his last lap in $55.4$ seconds. 



2.  Find how many laps he has cycled on Wednesday. :marks[3]

3.  Find the **total** time, in **minutes**, cycled by Eddy on
    Wednesday. :marks[4]

 

On Friday Eddy brings his friend Mario to train. They both cycled the
first lap of the track in $25$ seconds. Each lap Mario cycles takes him
$1.05$ times as long as his previous lap. 



4.  Find the time, in seconds, Mario takes to cycle his fifth
    lap. :marks[3]

5.  Find the **total** time, in **minutes**, Mario takes to
    cycle his first ten laps. :marks[3]

 

Each lap Eddy cycles again takes him $1.6$ seconds longer that his
previous lap.\
After a certain number of laps Eddy takes less time per lap than Mario.




6.  Find the number of the lap when this happens. :marks[3]




\[Maximum mark: 6\]

When a coin is thrown from the top of a skyscraper, its height above the
ground after $t$ seconds is given by $s(t) = at^2 + bt + c$, where
$a,b,c \in \mathbb{R}$ and $s(t)$ is measured in metres. After $1$
second, the coin is $179.3$ m above the ground;
after $2$ seconds, $188.2$ m; after $6$ seconds,
$159.8$ m. 



1.  
    1.  Write down a system of three linear equations in terms of $a$,
        $b$ and $c$.

    2.  Hence find the values of $a$, $b$ and $c$. :marks[3]

    

2.  Find the height of the skyscraper. :marks[1]

3.  Find the time it takes for the coin to hit the ground. :marks[2]




\[Maximum mark: 15\]

Consider the sequence $u_1,\, u_2,\, u_3,\, \dots,\, u_n,\, \dots$ where

$$
\begin{aligned}
u_1 = 860,\hspace{0.3em} u_2 = 980,\hspace{0.3em} u_3 = 1100,\hspace{0.3em} u_4 = 1220.\end{aligned}
$$

The sequence continues in the same manner. 



1.  Find the value of $u_{50}$. :marks[3]

2.  Find the sum of the first $10$ terms of the sequence. :marks[3]

 

Now consider the sequence $v_1,\, v_2,\, v_3,\, \dots,\, v_n,\, \dots$
where 
$$
\begin{aligned}
v_1 = 2,\hspace{0.3em} v_2 = 4,\hspace{0.3em} v_3 = 8,\hspace{0.3em} v_4 = 16.\end{aligned}
$$

This sequence continues in the same manner. 



3.  Find the exact value of $v_{13}$. :marks[3]

4.  Find the sum of the first $10$ terms of this sequence.
    :marks[3]

 

$k$ is the smallest value of $n$ for which $v_n$ is greater than $u_n$.




5.  Calculate the value of $k$. :marks[3]




\[Maximum mark: 6\]

A circle is drawn on an Argand diagram as shown below. The tangent to
the circle from the point B$(0,9)$ meets the circle at the point A as
shown. Let $w = \vv{\mathrm{OA}}$.


:::center





![756bc10d49856365edc8188e18631d432452e02d.svg](question:a7e2f81c-7176-46a6-9367-c80340035fc4)

:::




1.  Show that $|w| = 3\sqrt{3}$. :marks[2]

2.  Find $\arg w$. :marks[2]

3.  Hence write $w$ in the form
    $a+b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where
    $a, b \in \mathbb{R}$. :marks[2]




\[Maximum mark: 6\]

A bouncy ball is dropped from a height of $2$ metres onto a concrete
floor. After hitting the floor, the ball rebounds back up to
$80$ % of it's previous height, and this pattern
continues on repeatedly, until coming to rest. 



1.  Show that the total distance travelled by the ball until coming to
    rest can be expressed by 

    $2 + 4(0.8) + 4(0.8)^2 + 4(0.8)^3 + \cdots$:marks[2]

2.  Find an expression for the total distance travelled by the ball, in
    terms of the number of bounces, $n$. :marks[2]

3.  Find the total distance travelled by the ball. :marks[2]




\[Maximum mark: 6\]

Let $z = re^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\frac{\pi}{3}}$ where
$r \in \mathbb{R}^+$. 



1.  For $r = \sqrt{2}$, 
    1.  express $z^2$ and $z^3$ in the form
        $a+b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where
        $a, b \in \mathbb{R}$;

    2.  draw $z^2$ and $z^3$ on the following Argand diagram.
        :marks[4]

        
        :::center




        ![7e711daec273ec9c0f630b852aae229a5a09d558.svg](question:f2f17a3e-671d-4ce2-8659-1788668037b1)
        :::

        

    

2.  Given that the integer powers of
    $w = (3-3{\mathrm{\hspace{0.05em}i}\mkern 1mu})\hspace{0.05em}z$ lie
    on a unit circle centred at the origin, find the value of $r$.
    :marks[2]




\[Maximum mark: 6\]

In an unbalanced three-phase electrical circuit, the current at time $t$
ms is given by

$I(t) = 2\sin\hspace{0.05em}(5t) + 5\sin\hspace{-0.1em}\Big(5t-\dfrac{3\pi}{4}\Big) + 10\sin\hspace{-0.1em}\Big(5t-\dfrac{5\pi}{4}\Big)$,

where $I(t)$ is measured in milliamperes (mA). 



1.  Write $I(t)$ in the form $A\cos\hspace{0.05em}(\omega t+\varphi)$.
    :marks[4]

2.  Hence find the highest current flowing through the circuit, and the
    time it first occurs. :marks[2]




\[Maximum mark: 14\]

A city has two major security guard companies, company A and company B.
Each year, $15$ % of customers using company A
move to company B and $5$ % of the customers
using company B move to company A. All additional losses and gains of
customers by the companies can be ignored. 



1.  Write down a transition matrix $\bm{T}$ representing the movements
    between the two companies in a particular year. :marks[2]

2.  
    1.  Find the eigenvalues and corresponding eigenvectors of $\bm{T}$.

    2.  Hence write down matrices $\bm{P}$ and $\bm{D}$ such that
        $\bm{T} = \bm{PDP}^{-1}$. :marks[6]

    

 

Initially company A and company B both have $3600$ customers.




3.  Find an expression for the number of customers company A
    has after $n$ years, where $n\in\mathbb{Z}$. :marks[5]

4.  Hence write down the number of customers that company A\
    can expect to have in the long term. :marks[1]




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