Level 2

[Maximum mark: 6]

Given that $z = \dfrac{10\sin \alpha}{3x+y}$, where $\alpha = \ang{30}$, $x = 6$ and $y = 46$.

1. Find the exact value of $z$. [2]
   

2. Write your answer to part (a)

   1. correct to $2$ decimal places;

   2. correct to $3$ significant figures;

   3. in the form $a\times10^k$, where $1 \leq a < 10$ and $k\in \mathbb{Z}$.[4]
      

[Maximum mark: 6]

The column graph below shows the number of wet days per week recorded for a period of time in Melbourne, Australia.

1. Write down how many weeks were recorded. [1]
   

2. Write down the modal number of wet days per week. [1]
   

3. Calculate the mean number of wet days per week. [2]
   

4. Determine the percentage of weeks which had more than 2 wet days. [2]
   

[Maximum mark: 6]

Only one of the following four sequences is arithmetic and only one of them is geometric.

1. State which sequence is arithmetic and find the common difference of the sequence. [2]
   

2. State which sequence is geometric and find the common ratio of the sequence.[2]
   

3. For the geometric sequence find the exact value of the sixth term. Give your answer as a fraction. [2]
   

[Maximum mark: 6]

A ladder $3.7$ m long leans against a vertical wall. The distance from the top of the ladder to the ground is $3.1$ m.

1. Represent this information on a diagram in the space provided above. Show the ground, the ladder and the wall as the labelled line segments. [1]
   

2. Find the distance from the bottom of the ladder to the wall. [2]
   

3. Write down the acute angle made by the ladder with the wall. [3]
   

[Maximum mark: 6]

Maria invests $\$25\hspace{0.15em}000$ into a savings account that pays a nominal annual interest rate of $4.25$ %, compounded monthly.

1. Calculate the amount of money in the savings account after $3$ years. [3]
   

2. Calculate the number of years it takes for the account to reach $\$40\hspace{0.15em}000$. [3]
   

[Maximum mark: 6]

The surface area of a baseball is made up of two equal leather strips. The height of the baseball laying on the ground is $73$ mm. Assuming the surface of the baseball is a sphere:

1. Find the area of one leather strip used to make the baseball in mm$^2$. Give your answer correct to one decimal place. [4]
   

2. Find the circumference of the baseball. Give your answer in mm correct to three significant figures. [2]
   

[Maximum mark: 6]

A bag contains $6$ white and $4$ orange table tennis balls. Jack selects a ball at random from the bag and then, afterwards, John selects a ball at random from the bag.

1. Complete the tree diagram. [3]
   

   

2. Find the probability that John chooses a white ball. [3]
   

[Maximum mark: 6]

The diagram below shows a straight line $L_1$ which passes through A$(0,-2)$ and B$(8,0)$.

1. Write down the coordinates of the midpoint of line segment [AB]. [2]
   

Another line, $L_2$ , intersects the $y$-axis at C$(0,3)$ and is parallel to $L_1$.

2. Find the gradient of $L_2$. [2]
   

3. Find the equation of $L_2$, giving your answer in the form $y = mx+c$. [2]
   

[Maximum mark: 6]

The area $A$ is defined as the region bounded by the curve $y = -\dfrac{x^2}{6}(x - 6)$ and the $x$-axis for $0 \leq x \leq 6$.

1. Sketch the curve on the diagram below, shading the area $A$. [3]
   

   

2. Write down a definite integral that represents area $A$. [1]
   

3. Find the area of $A$. [2]
   

[Maximum mark: 6]

A population of $50$ hamsters was introduced to a new town. One month later, the number of hamsters was $62$. The number of hamsters, $P$, can be modelled by the function

where $t$ is the time, in months, since the hamsters were introduced to the town.

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

2. Calculate the number of hamsters in the town after $6$ months. [2]
   

A wildlife specialist estimates that the town has enough drink and food to support a maximum population of $2000$ hamsters.

3. Calculate the number of months it takes for the hamster population to reach this maximum. [2]