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]

An airplane leaves Doha airport bound for Paris Charles de Gaulle airport. There are lights located at the end of the runway at points $A(0, 4)$ and $B(6, 8)$, relative to a terminal at the origin. The takeoff path of the airplane is the perpendicular bisector of line $AB$.

1. Find the equation of the takeoff path in the form $ax + by + d = 0$, where $a, b, d \in \mathbb{Z}$. [4]
   

The airplane travels at an average speed of $570\,\text{km} \hspace{0.15em} \text{hr}^{-1}$ in a straight line. Once the airplane has reached cruising altitutde, an air traffic controller at the top of a $200\hspace{0.15em}\text{m}$ high air traffic control tower at $C(7,0)$ observes that the angle of elevation to the airplane is $\ang{40}$. Five minutes later, the controller observes that the angle of elevation is $\ang{10}$.

2. Find the cruising altitude of the airplane in metres. [5]
   

As the airplane is about to land at the Paris Charles de Gaulle airport, the pilot is asked to delay the landing due to a traffic issue. The pilot is instructed to turn the airplane on a bearing of $\ang{045}$ for $10\,$km until reaching point P, then travel on a bearing of $\ang{165}$ for $30\,$km to point Q before flying back to the original point O for landing.

3. Find the angle OP̂Q. [2]
   

4. Find the shortest distance from Q back to O for landing. [3]
   

5. Find the bearing the airplane must travel on to get back to O from Q. [4]
   

[Maximum mark: 16]

The owner of a bakery has found that the profit obtained from selling $x$ cakes is given by the function

$P(x) = \dfrac{x}{20} \left(600 - \dfrac{x^2}{2k^2}\right)$

where $k$ is a positive constant and $x\geq 0$.

1. Find an expression for $P\,{'}(x)$ in terms of $k$ and $x$. [3]
   

2. Hence, find the maximum value of $P$ in terms of $k$. [4]
   

The owner knows that the bakery makes a profit of $\$873$ when they sell $30$ cakes.

3. Find the value of $k$. [3]
   

4. Determine how many cakes the bakery should sell to maximize their profit. [1]
   

5. Sketch the graph of $P$, labelling the maximum point and $x$-intercepts. [3]
   

6. Determine the maximum number of cakes the bakery can sell before they start losing money. [2]
   

[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]