\[Maximum mark: 6\]

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]




\[Maximum mark: 4\]

The following table shows the number of overtime hours worked by employees in a company.

:::center
![AA928a](question:95466299-212d-4150-9bc4-af83348f7ce2)
:::

It is known that the mean number of overtime hours is $1$.

1. Find the value of $x$. :marks[2]

2. Find the standard deviation of the data. :marks[2]

\[Maximum mark: 6\]

A $3$D printer builds a set of $49$ Ei$\text{f}$fel Tower
Replicas in different sizes. The height of the largest tower in this set
is $64$ cm. The heights of successive smaller
towers are $95$ % of the preceding larger tower,
as shown in the diagram below.


:::center

![AA724a](question:e2ae24d8-0743-4825-b529-687c2f726965)

:::




1.  Find the height of the smallest tower in this set. :marks[3]

2.  Find the **total** height if all $49$ towers were placed one on top
    of another. :marks[3]






\[Maximum mark: 6\]

A farmer is going to fence two equal adjacent parcels of land. These
parcels share one side (which also requires fencing) as shown in the
following diagram. The farmer has only $80$ metres of fence.


:::center





![580f23d360211ad7dd2c45e79d30c7d9c7d24f56.svg](question:70cc4897-882c-489b-9551-58f6c2e26bb3)

:::




1.  Write down the equation for the total length of the fence,
    $80$ m, in terms of $x$ and $y$.
    :marks[1]

2.  Write down the total area of both parcels of land in terms of $x$.
    :marks[2]

3.  Find the maximum area, in m$^2$, of one parcel of land. :marks[3]




\[Maximum mark: 6\]

$A$ and $B$ are independent events such that $\mathrm{P}(A \cap {B\,} ^\prime) = 0.09$ and $\mathrm{P}({A\,}^\prime \cap B) = 0.49$. 

Let $x = \mathrm{P}(A\cap B)$.

1. 1. Express $\mathrm{P}(A)$ in terms of $x$.

    2. Express $\mathrm{P}(B)$ in terms of $x$. :marks[2]

2. Find the value of $x$. :marks[2]

3. Find $\mathrm{P}(B\,|\,A)$. :marks[2]

\[Maximum mark: 7\]

The velocity, $v\hspace{0.15em}\text{m}\hspace{0.15em}\text{s}^{-1}$, at time $t$ seconds, of a particle moving in a straight line is given by
:::center
$v=\dfrac{(t^2-1)\,\mathrm{sin}\hspace{0.15em}t}{2}\,$,
:::
for $0 \leq t \leq 2$.

1. Determine when the particle changes direction for the first time. :marks[2]

2. Find the times when the acceleration of the particle is $1.4\hspace{0.15em}\text{m}\hspace{0.15em}\text{s}^{-2}$. :marks[3]

3. Find the acceleration of the particle when its speed is at its greatest. :marks[2]

\[Maximum mark: 12\]

The following table shows the total revenue, $y$, in Australian dollars
(AUD), $\text{obtained}$ daily during the first week of January
$2020$, by Peppy's Pizza $\text{restaurant}$ and the number of
guests, $x$, served.


:::center




![9736e8eb9c9cfe8a065bfa8f5344721e7887b704.svg](question:bb9098c3-1fc2-42a0-8654-68ad60f8549d)

:::




1.  
    1.  Calculate the Pearson's product-moment correlation coefficient,
        $r$, for this data.

    2.  Comment on the result. :marks[3]

    

2.  Write down the equation of the regression line $y$ on $x$.
    :marks[1]

3.  Use the line of the regression to estimate the revenue of serving
    $70$ guests. Give your answer correct to **the nearest AUD**.
    :marks[2]

 

The daily maintenance cost for the restaurant is $300$ AUD. Additionally,
the cost of serving one guest is $5$ AUD. 



4.  Determine if the restaurant expects to make a profit when serving
    $50$ guests on a $\text{particular}$ day. :marks[2]

5.  
    1.  Find an expression for the profit of the restaurant when serving
        $x$ guests on a particular day.

    2.  Find the least number of guests required to be served to result
        in a profit for the day. :marks[4]

    






\[Maximum mark: 18\]

A cannonball is fired from the top of a tower. The rate of change of the height, $h$, of the cannonball above the ground is modelled by

$$
\begin{aligned}
h'(t) = -4t + 20, \hspace{0.5em} t \geq 0,\end{aligned}
$$
where $h$ is in metres and $t$ is the time, in seconds, since the moment the cannonball was fired.




1.  Determine the time $t$ at which the cannonball reached its maximum height. :marks[2]

After one second, the cannonball is 26 metres above the ground.

2.  1. Find an expression for $h(t)$, the height of the cannonball above the ground at time $t$.

    2. Hence, determine the maximum height reached by the cannonball. :marks[5]

3.  Write down the height of the tower. :marks[1]

4.  Calculate the height of the cannonball $4$ seconds after it was fired. :marks[2]

The cannonball hits its target on the ground $n$ seconds after it was fired.

5.  Find the value of $n$. :marks[2]

6.  Determine the total time the cannonball was above the height of the tower.:marks[3]

A second cannonball is fired from exactly halfway up the tower, with the same projectile motion as the first cannonball.

7. Given that both cannonballs land at the same time, determine the length of time between the first cannonball and the second cannonball being fired. :marks[3]




\[Maximum mark: 15\]

A physicist is studying the motion of two separate particles moving in a straight line. She measures the displacement of each particle from a fixed origin over the course of 10 seconds.

The physicist found that the displacement of particle $A$, $s_A$ cm, at time $t$ seconds can be modelled by the function $s_A(t)=7t+9$, where $0 \leq t \leq 10$.

The physicist found that the displacement of particle $B$, $s_B$ cm, at time $t$ seconds can be modelled by the function $s_B(t)=\mathrm{cos}(3t+5)+8t+4$.

1. Use the physicist's models to find the initial displacement of

    1. Particle $A$;

    2. Particle $B$ correct to three significant figures. :marks[3]

2. Find the values of $t$ when $s_A(t)=s_B(t)$. :marks[3]

3. For $t>6$, prove that particle $B$ was always further away from the fixed origin than particle $A$. :marks[3]

4. For $0 \leq t \leq 10$, find the total amount of time that the velocity of particle $A$ was greater than the velocity of particle $B$. :marks[6]

Trial Examinations for IB Mathematics AA SL

AA728

AA777

AA725

AA921

AA922

AA920

AA942

AA923

AA943

Paper 1

AA705

AA928

AA724

AA729

AA929

AA936

AA726

AA727

AA939

Paper 2

Analysis & Approaches SL

Number & Algebra

Topic 1 All

AA004

AA121

AA062

AA007

AA059

AA111

AA052

AA010

AA108

AA017

AA041

AA012

AA018

AA071

AA020

AA072

AA117

AA019

AA021

AA123

Sequences & Series

AA008

AA006

AA016

AA011

AA013

AA119

AA026

Compound Interest

AA704

AA707

AA109

AA840

AA110

Exponents & Logs

AA047

AA034

AA751

AA042

AA056

AA143

AA820

The Binomial Theorem

AA058

AA057

AA060

AA067

AA120

Proofs

AA950

AA834

AA835

AA836

Arithmetic/Geometric, Sigma Notation, Applications, Compound Interest…

AA001

AA002

AA623

AA624

AA003

AA636

AA638

AA702

AA709

AA625

AA626

AA627

AA628

AA629

AA005

AA668

AA832

AA703

AA706

AA710

AA630

AA631

AA632

AA657

AA1023

AA009

AA014

AA635

AA748

AA633

AA634

AA639

AA024

AA122

AA199

AA839

AA637

AA640

AA023

AA022

AA986

AA1044

AA1037

Exponent & Log Laws, Solving Exponential & Logarithmic Equations…

AA046

AA048

AA050

AA031

AA032

AA037

AA038

AA053

AA029

AA035

AA036

AA055

AA193

AA284

AA039

AA040

AA045

AA299

AA646

AA984

Binomial Expansion & Theorem, Pascal’s Triangle & The Binomial Coefficient nCr…

IB Mathematics AA SL - Mock Exams

Mock Exam Set 1 - Paper 2

--- Section A ---

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

--- Section B ---

Question 7

Question 8

Question 9

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