Mock Exam Set 1 - Paper 2

[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: 4]

The following table shows the number of overtime hours worked by employees in a company.

It is known that the mean number of overtime hours is $1$.

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

2. Find the standard deviation of the data. [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.

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

2. Find the total height if all $49$ towers were placed one on top of another. [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.

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

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

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

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

3. Find $\mathrm{P}(B\,|\,A)$. [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

$v=\dfrac{(t^2-1)\,\mathrm{sin}\hspace{0.15em}t}{2}\,$,

for $0 \leq t \leq 2$.

1. Determine when the particle changes direction for the first time. [2]
   

2. Find the times when the acceleration of the particle is $1.4\hspace{0.15em}\text{m}\hspace{0.15em}\text{s}^{-2}$. [3]
   

3. Find the acceleration of the particle when its speed is at its greatest. [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.

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

   2. Comment on the result. [3]
      

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

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

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. [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. [5]
      

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

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

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

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

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

2. Find the values of $t$ when $s_A(t)=s_B(t)$. [3]
   

3. For $t>6$, prove that particle $B$ was always further away from the fixed origin than particle $A$. [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 $B$. [6]