Mock Exam Set 1 - Paper 2

[Maximum mark: 13]

On September 1st, an orchard commences the process of harvesting $36$ hectares of apple trees. At the end of September 4th, there were $30$ hectares remaining to be harvested, and at the end of September 8th, there were $24$ hectares remaining. Assuming that the number of hectares harvested each day is constant, the total number of hectares remaining to be harvested can be described by an arithmetic sequence.

1. Find the number of hectares of apple trees that are harvested each day. [3]
   

2. Determine the number of hectares remaining to be harvested at the end of September 1st. [1]
   

3. Determine the date on which the harvest will be complete. [2]
   

In 2021 the orchard sold their apple crop for $\$220\,000$. It is expected that the selling price will then increase by $3.2\%$ annually for the next $7$ years.

4. Determine the amount of money the orchard will earn for their crop in 2026. Round your answer to the nearest dollar. [3]
   

5. 1. Find the value of $\displaystyle\sum_{n=1}^8 \big(220\hspace{0.15em}000 \times 1.032^{n-1}\big)$. Round your answer to the nearest integer.

   2. Describe, in context, what the value in part (e) (i) represents. [3]
      

6. Comment on whether it is appropriate to model this situation in terms of a geometric sequence. [1]
   

[Maximum mark: 15]

The lifespans of a new model of smart television are normally distributed with a mean of $8.3$ years and a standard deviation of $2.2$ years.

1. A customer buys a television of this model. Find the probability that the television lasts longer than $5$ years. [2]
   

2. $10\%$ of televisions of this model have a lifespan of less than $m$ years. Find the value of $m$. [2]
   

The manufacturer offers a five-year warranty for this television model. Eight smart televisions of this model are sold on a certain day.

3. Find the probability that at most one of them will be claimed for warranty. [4]
   

4. Find the probability that the eighth television sold will be the second one to be claimed for warranty. [3]
   

As company policy, televisions with a lifespan of less than $3$ years will be replaced with a new one of the same model without repairing.

5. Find the probability that a television will be replaced with a new one, given that it is claimed for warranty. [4]
   

[Maximum mark: 18]

The Voronoi diagram below shows four hotels in a small town represented by points with coordinates $\mathrm{A}(-4,4)$, $\mathrm{B}(3,5)$, $\mathrm{C}(3,-3)$, and $\mathrm{D}(-1,3)$. The vertices $\mathrm{V}_1$, $\mathrm{V}_2$ and $\mathrm{V}_3$ are also shown. Distances in the direction of the $x$ and $y$ axes are measured in increments of $100$ metres.

1. Find the midpoint of AD. [2]
   

2. Hence, find the equation of the line that passes through $\mathrm{V}_1$ and $\mathrm{V}_2$. [4]
   

The equation of line that passes through $\mathrm{V}_1$ and $\mathrm{V}_3$ is $y=-2x+6$.

3. Find the coordinates of $\mathrm{V}_1$. [3]
   

The coordinates of $\mathrm{V}_2$ are $(-5,-4)$ and the coordinates of $\mathrm{V}_3$ are $(2.5,1)$.

4. Find the distance from $\mathrm{V}_1$ to $\mathrm{V}_2$. Give your answer to the nearest metre. [2]
   

5. Given that the distance from $\mathrm{V}_1$ to $\mathrm{V}_3$ is $783$ metres, find the angle $\mathrm{V_2}\widehat{\mathrm{V}}_1\mathrm{V}_3$. Give your answer to the nearest degree. [4]
   

6. Hence, find the area of the Voronoi cell containing hotel $\mathrm{D}$, giving your answer in $\text{m}^2$, to three significant figures. [2]
   

The manager of hotel $\mathrm{D}$ believes that the larger the area of triangle $\mathrm{V}_1\mathrm{V}_2\mathrm{V}_3$, the more people will stay at hotel $\mathrm{D}$.

7. State one criticism of the manager's belief. [1]
   

[Maximum mark: 16]

A particle moves along the $x$-axis so that its velocity, $v$ m s$^{-1}$ at time $t$ seconds, is given by the equation

1. Find the time at which the particle changes direction. [3]
   

2. Find the magnitude of the particle's acceleration at time $t = 2$ seconds. [4]
   

The particle starts from the origin O.

3. Find an expression for the displacement of the particle from O at time $t$ seconds. [4]
   

4. Find the time at which the particle returns to the origin. [2]
   

5. Calculate the total distance travelled by the particle by the time it has returned to the origin. [3]
   

[Maximum mark: 12]

Lohan receives an antique vase as a gift from her grandparents. She decides to model the shape of the vase to calculate its volume.

She places the vase horizontally on a piece of paper and uses a pencil to mark five representative points $(0,5), (7.5,10), (17,7.5), (25,3)$ and $(35, 7.5)$, as shown below. These points are connected to form a symmetrical cross-section about the $x$-axis. All units are in centimetres.

Lohan initially uses a straight line to model the section from $(25,3)$ to $(35,7.5)$.

1. Determine the equation of the line that passes through these two points. [2]
   

Lohan thinks that a quadratic curve might be a good model for the section from $(0,5)$ to $(25,3)$. She carries out a least squares regression using this model for the points she has recorded.

2. 1. Determine the equation of the least squares regression quadratic curve found by Lohan.

   2. By considering the gradient of the curve at the point where $x = 10$, determine whether the quadratic regression curve is a good model or not. [3]
      

Lohan decides that a cubic curve for the entire section from $(0,5)$ to $(35,7.5)$ would be a better fit.

3. Find the equation of the cubic model. [2]
   

Using this model, Lohan estimates the volume of the vase by calculating the volume of revolution about the $x$-axis.

4. Find the volume of the vase estimated by Lohan. [3]
   

Lohan subsequently fills the vase with water and discovers that the true volume is $5\hspace{0.15em}500\, \text{cm}^3$.

5. Calculate the percentage error in Lohan's estimate of the volume. [2]
   

[Maximum mark: 18]

Adriano is riding a skateboard in a parking lot. His position vector from a fixed origin O at time $t$ seconds is modelled by

where $a$ and $b$ are non-zero constants to be determined. All distances are in metres.

1. Find the velocity vector at time $t$. [3]
   

2. Given that $a > 0$, show that the magnitude of the velocity vector at time $t$
   is given by $a\sqrt{\dfrac{1}{(t+b)^2}+(\ln{(t+b)})^2}$.

   [4]
   

At time $t = 0$, the velocity vector is $\bigg(\hspace{-0.1em}\begin{matrix} 2 \\[2pt] 2.773 \end{matrix}\hspace{0.1em}\bigg)$ .

3. Find the value of $a$ and the value of $b$. [3]
   

4. Find the magnitude of the velocity vector when $t = 3$. [2]
   

At point P, Adriano is riding parallel to the $y$-axis for the first time.

5. Find |OP|. [6]
   

[Maximum mark: 18]

Consider the following system of coupled dif$\text{f}$erential equations.

The system can be written in the form

where $A$ is a $2\times2$ matrix.

1. 1. Write down matrix $A$.

   2. Find the eigenvalues and corresponding eigenvectors of matrix $A$. [7]
      

2. Hence write down the general solution of the system. [2]
   

3. Determine whether the equilibrium point E$(0,0)$ is stable or unstable. Justify your answer. [2]
   

4. Find the value of $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ at point:

   1. P$(5,0)$;

   2. Q$(-5,0)$. [3]
      

5. Sketch a phase portrait for the general solution to the system of coupled dif$\text{f}$erential equations for $-8\leq x \leq 8$ and $-8\leq y \leq 8$. [4]