\[Maximum mark: 6\]

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



1.  Find the **exact** value of $z$. :marks[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
        $k\in \mathbb{Z}$.:marks[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.


:::center




![74734100a77b8396208ce74aaaa2d09f5d724a3d.svg](question:15069aa9-91ac-4511-85b1-3d0f144ab0da)

:::



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

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

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

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






\[Maximum mark: 6\]

Only one of the following four sequences is arithmetic and only one of
them is geometric. 
$$
\begin{array}{rcccccl}
a_n &=& \dfrac{1}{3},\,\dfrac{1}{4},\,\dfrac{1}{5},\,\dfrac{1}{6},\,\dots &\,\hspace{4em}\,& c_n &=& 3,\,1,\,\dfrac{1}{3},\,\dfrac{1}{9},\,\dots \\[12pt]
b_n &=& 2.5,\,5,\,7.5,\,10,\,\dots &\,\hspace{4em}\,& d_n &=& 1,\,3,\,6,\,10,\,\dots
\end{array}
$$




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

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

3.  For the **geometric** sequence find the **exact** value of the sixth
    term. Give your answer as a fraction. :marks[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. 

:::center




![b2d163545a6a77affd33b3cb66aa0089157bff91.svg](question:090057e3-0cf3-4d24-b69a-7ca7f8dd2484)

:::




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. :marks[1]

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

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

2.  Calculate the number of years it takes for the account to reach
    $$\$40\hspace{0.15em}000$$. :marks[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. :marks[4]

2.  Find the circumference of the baseball. Give your answer in mm
    correct to three significant figures. :marks[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. :marks[3]
    :::center




    ![4833dbdc60abfa0bd07aaf9560193c577886f077.svg](question:6a805f04-ea86-422b-98ea-3e98a6e073e9)
    :::

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




\[Maximum mark: 6\]

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


:::center




![dd225af0bca38352f0853ac9a9eb9c1cd3829f36.svg](question:e476ece8-ad04-4e3b-8740-1be6a35203d6)

:::




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

 

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



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

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

    
    :::center




    ![cfa89c15088451c629d2cfbdfb87757bdf7bf68f.svg](question:86fa8cfb-66a4-4334-ba99-ef5952eb71f5)

    :::

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

3.  Find the area of $A$. :marks[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

$$
P(t) = 50\times b^t,\hspace{0.5em} t\geq0, \\[3pt]
$$
where $t$ is the
time, in months, since the hamsters were introduced to the town.




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

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






IB Math AI SL - Revision Ladder

Level 2

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

#1 IB Math Resource

34% Grade Increase

70% of IB Students

More IB Math AI SL Resources

Questionbank

Practice Exams

Key Concepts

Past Papers