Topic 1 All

[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. [2]
   

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

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

2. Find the number of years it will take for the hardware to be worth less than $\text{\euro\hspace{0.05em}\(8000\)}$. [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. [4]
      

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

[Maximum mark: 6]

Alex invests an amount of USD into a savings account which pays 3.3% (p.a.) interest, compounded monthly. After 5 years Alex has $8\hspace{0.15em}000$ USD in the account.

1. Find the amount of USD initially invested, rounding your answer to two decimal places.[3]
   

With this money, Alex purchases a used car for $5\hspace{0.15em}000$ dollars, and sells it 3 years later for $4\hspace{0.15em}200$.

2. Find the rate at which the car depreciates per year over the 3 year period.[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. [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. [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. [3]
      

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

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

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

5. Calculate the total distance Charles swims until the event. [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.[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.[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. [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. [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.

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

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

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

[Maximum mark: 6]

Data scientists, web designers and developers are paid according to an $\text{industry}$ standard. The total annual salary spend for three tech startups paying to the industry standard are summarised in the table below.

Let $x$, $y$ and $z$ represent the salaries, in thousand dollars, for data scientists, web designers and web developers respectively.

1. Write down a system of three linear equations in terms of $x$, $y$ and $z$
   that represent the information in the table above. [2]
   

2. Using matrices, solve the system of linear equations from part (a)
   to determine the salaries for the three roles. [2]
   

Data Quant is a tech startup that also pays to the industry standard and employs $10$ data scientists, $4$ web designers and $6$ web developers.

3. Calculate the exact value of the total salary bill for Data Quant. [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. [1]
   

2. Calculate the total cost of the loan under this financing option. [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 $y$. [1]
   

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

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

[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. [3]
   

Eddy cycles his last lap in $55.4$ seconds.

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

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

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

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

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

[Maximum mark: 15]

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

The sequence continues in the same manner.

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

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

Now consider the sequence $v_1,\, v_2,\, v_3,\, \dots,\, v_n,\, \dots$ where

This sequence continues in the same manner.

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

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

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

5. Calculate the value of $k$. [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 $w = \vv{\mathrm{OA}}$.

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

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

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

[Maximum mark: 6]

Points A and B represent the complex numbers $z_1 = \sqrt{3} - {\mathrm{\hspace{0.05em}i}\mkern 1mu}$ and $z_2 = -3 - 3{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ as shown on an Argand diagram below.

1. Find the angle AOB. [2]
   

2. Find the argument of $z_1z_2$. [1]
   

3. Given that the real powers of $pz_1z_2$, for $p > 0$, all lie on a unit circle centred at the origin, find the exact value of $p$. [3]
   

[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

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

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

3. Find the total distance travelled by the ball. [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. [4]
      

      

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 $r$. [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 $A\cos\hspace{0.05em}(\omega t+\varphi)$. [4]
   

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

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

   2. Hence write down matrices $\bm{P}$ and $\bm{D}$ such that $\bm{T} = \bm{PDP}^{-1}$. [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 $n\in\mathbb{Z}$. [5]
   

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