Level 1

[Maximum mark: 6]

The distance between two points with coordinates $(x_1,y_1)$ and $(x_2,y_2)$ is equal to $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.

1. Calculate the distance between points A$(40,-100)$ and B$(1,-2)$. Give your answer correct to three significant figures. [3]
   

2. Give your answer from part (a) correct to one decimal place. [1]
   

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

[Maximum mark: 6]

The number of days rain per month in London varies depending on the time of year. The data shows the number of wet days per month.

$17 \hspace{1em} 13 \hspace{1em} 11 \hspace{1em} 14 \hspace{1em} 13 \hspace{1em} 11 \hspace{1em} 13 \hspace{1em} 12 \hspace{1em} 13 \hspace{1em} 14 \hspace{1em} 16 \hspace{1em} 16$

1. For this data, find $\,$

   1. the median;

   2. the minimum and maximum values. [3]
      

The lower quartile of the data is $12.5$ and the upper quartile of the data is $15$.

2. Draw a box-and-whisker diagram to represent the data. [3]
   

[Maximum mark: 6]

Mary (M) sits on the top of the Eastern side of the Grand Canyon. Below her stands Peter (P), at the bottom of the canyon. It is known that the depth of the Grand Canyon is $1\hspace{0.15em}850$ meters at this point. Nathan (N) stands on the Western side of the bottom of the Canyon, with a distance of $2.5$ km to Peter.

1. Write down the depth of the Grand Canyon, MP, in kilometers. [1]
   

2. On the diagram, label the angle of elevation from N to M with an $x$. [1]
   

3. Find the size of the angle of elevation from N to M. [2]
   

4. Find MN, the distance in kilometers between Mary and Nathan. [2]
   

[Maximum mark: 6]

Jeremy invests $\$8000$ into a savings account that pays an annual interest rate of $5.5$ %, compounded annually.

1. Write down a formula which calculates that total value of the investment after $n$ years. [2]
   

2. Calculate the amount of money in the savings account after:

   1. $1$ year;

   2. $3$ years. [2]
      

3. Jeremy wants to use the money to put down a $\$10\hspace{0.15em}000$ deposit on an apartment. Determine if Jeremy will be able to do this within a $5$-year timeframe.[2]
   

[Maximum mark: 6]

A water storage tank has a cylindrical shape. The diameter of the base of the tank is $0.568$ m. The height of the tank is $0.855$ m. This is shown in the following diagram.

1. Write down the radius, in m, of the base of the tank. [1]
   

2. Calculate the area of the base of the tank. [2]
   

George is going to paint the curved surface and the base of the water storage tank.

3. Calculate the area to be painted. [3]
   

[Maximum mark: 6]

The number of daily visitors to the famous 'Bondi Beach' in Sydney, Australia, and the daily temperature on those days, were recorded for eight days in February. The table below shows this data and has been ordered in ascending order by temperature.

The number of visitors to Bondi Beach varies linearly with the temperature.

1. Find

   1. Pearson's product-moment correlation coefficient, $r$ ;

   2. the equation of the regression line $y$ on $x$. [4]
      

2. Use the equation of the regression line $y$ on $x$ to estimate the number of visitors to Bondi Beach during a day the temperature is $26^\circ$C. [2]
   

[Maximum mark: 6]

An arithmetic sequence has $u_1 = 12$, $u_2 = 21$, $u_3 = 30$.

1. Find the common difference, $d$. [2]
   

2. Find $u_{10}$. [2]
   

3. Find $S_{10}$. [2]
   

[Maximum mark: 6]

The probability that Amanda successfully passes her maths exam depends on the exam topic. The probability that the topic is statistics is $0.35$ and the probability she passes this topic is only $0.3$. The probability that Amanda passes any other topic during the exam is $0.8$.

1. Complete the following tree diagram. [3]
   

   

2. Find the probability that Amanda does not pass the exam. [3]
   

[Maximum mark: 6]

In an ecology experiment, a number of mosquitoes are placed in a container with water and vegetation. The population of mosquitoes, $P$, increases and can be modelled by the function

where $t$ is the time, in days, since the mosquitoes were places in the container.

1. Find the number of mosquitoes:

   1. initially placed in the container;

   2. in the container after $5$ days. [4]
      

The maximum capacity of the container is $5000$ mosquitoes.

2. Find the number of days until the container reaches its maximum capacity. [2]
   

[Maximum mark: 6]

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

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

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