Question 1
Below is a 3D diagram for an office building. The dimensions are in feet (ft).
Question 1(a)
Calculate the volume of the office building in cubic feet.
▶️Answer/Explanation
Ans:
$\begin{aligned}
& 50 \times 20 \times 30(=30000) \\
& 20 \times 15 \times 30(=9000)==39000\left(\mathrm{ft}^3\right)
\end{aligned}$
Question 1(b)
The number of employees in the office building each day is given in the table below.
Find the mean number of employees in the office building during the working days.
▶️Answer/Explanation
Ans:
\[
\text{{Mean}} = \frac{{\sum \text{{Number of employees}}}}{{\text{{Number of working days}}}}
\]
In this case, the sum of the number of employees is $105 + 70 + 90 + 75 + 60 = 400$, and the number of working days is 5.
Plugging these values into the formula, we get:
\[
\text{{Mean}} = \frac{{400}}{{5}} = 80
\]
Hence, the mean number of employees in the office building during the working days is 80.
Question 1(c)
To control the temperature in the office building, a central air-conditioning unit is needed. The power $(\mathrm{P})$ of the air-conditioning unit is measured in horsepower (hp) and can be found using the following formula:
$$
P=\frac{(\rm{6 V+500} \mathrm{~N})}{9000}
$$
Where:
$\mathrm{V}$ is the volume in cubic feet.
$\mathrm{N}$ is the mean number of employees during the working days.
Using your answers from part (a) and part (b), determine the value of $\mathrm{P}$ needed for controlling the temperature in this office building.
▶️Answer/Explanation
Ans:
To determine the value of \( P \) (power) needed for controlling the temperature in the office building, we can substitute the given values into the formula:
\[ P = \frac{{6V + 500N}}{{9000}} \]
where \( V = 39000 \) ft\(^3\) (volume) and \( N = 80 \) (mean number of employees).
Substituting the values:
\[ P = \frac{{6 \times 39000 + 500 \times 80}}{{9000}} \]
\[ P = \frac{{234000 + 40000}}{{9000}} \]
\[ P = \frac{{274000}}{{9000}} \]
\[ P \approx 30.44 \]
Therefore, the value of \( P \) needed for controlling the temperature in this office building is approximately 30.44 horsepower (hp).
Question 2(a)
Robot 1 is at point $S$. It moves following the vector $3 a-c-2 b$. Label the grid to show the path of Robot 1.
▶️Answer/Explanation
Ans:
Question 2(b)
Robot 2 is at point $\mathrm{M}$. It must collect books from point $\mathrm{P}$ and then deliver them to the conveyor belt. Determine the vector of path of Robot 2, in terms of $\boldsymbol{a}, \boldsymbol{b}$ and $\boldsymbol{c}$, in its simplest form.
▶️Answer/Explanation
Ans:
$–2b + 3.5a – 4b$ or $4a – 2b – 0.5a – 4b$
Question 2(c)
4 robots can prepare 15 orders in 3 minutes. Calculate how many minutes it would take 10 robots to prepare 300 orders.
▶️Answer/Explanation
Ans:
Given that 4 robot can prepare 15 orders in 3 minutes
So, 1 robot can prepare 3.75 orders $\frac{14}{4}$
and 10 robots can prepare 37.5 orders, the ratio of orders to robots is the same. Therefore:
\[\frac{300}{37.5} = 8\]
This means that with 10 robots, it would take 8 units of time to prepare 300 orders. Since each unit of time corresponds to 3 minutes (as given in the problem), the total time would be:
\[8 \times 3 = 24\]
So, it would take 24 minutes for 10 robots to prepare 300 orders.
Question 3 (8 Marks)
Izumi is a volunteer at a pet rescue centre which has cats, dogs and rabbits for adoption. At the next school festival, she will try to convince students to adopt a pet from the pet rescue centre.
Izumi decides to run a survey in her school before the festival.
She asked the following questions:
The image and Venn diagram show the survey results for the girls.
Event $\mathrm{C}$ represents: Would like to adopt a cat
Event $\mathrm{D}$ represents: Would like to adopt a dog
Event $\mathrm{R}$ represents: Would like to adopt a rabbit
No girl selected more than one pet.
Question 3(a) : 1 marks
Determine the percentage of girls who would not like to adopt a pet. Write your answer on the Venn diagram.
▶️Answer/Explanation
Ans:
\( \begin{aligned} & =[100-(41+29+26)] \% \\ & =4 \%\end{aligned} \)
Question 3(b) : 1 marks
Events $\mathrm{C}, \mathrm{D}$ and $\mathrm{R}$ are mutually exclusive. State how this is represented in the Venn diagram.
▶️Answer/Explanation
Ans:
$A \cap B \cap C=0$ or $\varnothing$
In the context of your scenario with the Venn diagram representing the survey results for the girls, the mutually exclusive nature of events $\mathrm{C}$ (Would like to adopt a cat), $\mathrm{D}$ (Would like to adopt a dog), and $\mathrm{R}$ (Would like to adopt a rabbit) means that no girl selected more than one pet. Each girl’s response falls into one and only one of these categories.
So, in the Venn diagram, you would have three separate circles representing events $\mathrm{C}$, $\mathrm{D}$, and $\mathrm{R}$ with no overlapping regions between them. This visually demonstrates that the events are mutually exclusive, and each girl’s preference falls into one specific category.
Question 3(c) : 3 marks
The image shows the survey results for the boys. Some boys indicated that they would like to adopt more than one type of pet. Izumi draws the following Venn diagram to summarize the survey results for the boys.
Find the missing values and complete the Venn diagram.
▶️Answer/Explanation
Ans:
sum of percentages add up to more than $100\%$
Question 3(d) : 3 marks
On the festival day, $60 \%$ of the students are girls and $40 \%$ are boys. Calculate the probability that a student at the festival will adopt a pet from the pet rescue centre.
▶️Answer/Explanation
Ans:
$0.6 \times 0.96=0.58$ and $0.4 \times 0.84=0.34$
Question 4 (8 Marks)
The image shows a jar containing 50 cent coins and $\$ 1$ coins.
The ratio of the 50 cent coins to the $\$ 1$ coins is $5: 7$.
Question 4(a) : 1 marks
The number of $\$ 1$ coins in the jar is 140 . Show that the number of 50 cent coins is 100 .
▶️Answer/Explanation
Ans:
The ratio of the 50 cent coins to the $\$ 1$ coins is given as $5:7$.
Let’s denote the number of 50 cent coins as $x$.
We are given that the number of $\$ 1$ coins in the jar is 140.
Using the given ratio, we can set up the equation:
$\frac{x}{140} = \frac{5}{7}$
To find the number of 50 cent coins, we can solve for $x$:
$x = \frac{140 \times 5}{7}$
Simplifying this equation, we have:
$x = \frac{700}{7}$
$x = 100$
Therefore, the number of 50 cent coins in the jar is indeed 100.
Question 4(b) : 2 marks
Hence, determine the total value of the coins in the jar.
▶️Answer/Explanation
Ans:
To determine the total value of the coins in the jar, we need to consider the value of each type of coin and multiply it by its respective quantity.
Given that there are 100 50 cent coins and 140 $1 coins in the jar, we can calculate the total value as follows:
Value of 50 cent coins $= \$ 0.50 \times 100 = \$ 50$
Value of $\$ 1$ coins $= 1 \times 140 = \$ 140$
To find the total value of the coins in the jar, we add up the values of the 50 cent coins and $1 coins:
Total value = Value of 50 cent coins + Value of $\$ 1$ coins
$= \$ 50 + \$ 140$
$=\$ 190$
Therefore, the total value of the coins in the jar is $\$ 190$ .
Question 4(c) : 5 marks
Another jar contains 50 cent coins, $\$ 1$ coins and $\$ 2$ coins.
The number of 50 cent coins, $\$ 1$ coins and $\$ 2$ coins are in the ratio $2: 4: 3$.
The number of the three types of coins are in the ratio $2: 4: 3$, respectively. The total value of the coins is $\$ 1760$. Find the total number of coins in the jar.
▶️Answer/Explanation
Ans:
Let’s solve this problem step by step.
Given that the number of 50 cent coins, $\$ 1$ coins, and $\$ 2$ coins are in the ratio $2: 4: 3$ respectively.
Let’s denote the number of 50 cent coins as $2x$, the number of $\$ 1$ coins as $4x$, and the number of $\$ 2$ coins as $3x$. Here, $x$ represents a common multiplier for the ratio.
The total value of the coins in the jar is given as $\$1760$.
Now, we need to determine the value of each type of coin and set up an equation.
The value of 50 cent coins is $0.50 \times 2x = 1x$ (since there are 2, 50 cent coins per $\$ 1$).
The value of $\$ 1$ coins is $1 \times 4x = 4x$.
The value of $\$ 2$ coins is $2 \times 3x = 6x$.
The total value of the coins is $\$1760$, so we can set up the equation:
$1x + 4x + 6x = 1760$
Combining like terms:
$11x = 1760$
To find the value of $x$, we divide both sides of the equation by 11:
$x = \frac{1760}{11}$
$x = 160$
Now that we have the value of $x$, we can find the number of each type of coin:
Number of 50 cent coins $= 2x = 2 \times 160 = 320$
Number of $\$ 1$ coins $= 4x = 4 \times 160 = 640$
Number of $\$ 2$ coins $= 3x = 3 \times 160 = 480$
Finally, to find the total number of coins in the jar, we add up the number of each type of coin:
Total number of coins = Number of 50 cent coins + Number of $\$ 1$ coins + Number of $\$ 2$ coins
$= 320 + 640 + 480$
$= 1440$
Therefore, the total number of coins in the jar is 1440.
Question 4 (6 Marks)
The following diagram shows part of the graph of a quadratic function $f(x)=3 x^2-5 x-2$
Question 4(a) : 1 marks
Write down the coordinates of point $\rm C$.
▶️Answer/Explanation
Ans:
To find the coordinates of point C on the y-axis, we need to determine the x-coordinate of C and then substitute it into the equation of the quadratic function to find the corresponding y-coordinate.
Since point C is on the y-axis, its x-coordinate is $0.$
Now, let’s substitute $x = 0$ into the equation of the quadratic function to find the y-coordinate:
$f(x) =3 x^2 – 5x – 2$
$f(0) = (3)0^2 – 5(0) -2$
$f(0) = 0 – 0 -2$
$f(0) = -2$
Therefore, the coordinates of point C are $(0, -2).$
Question 4(b) : 4 marks
Find the coordinates of points $\mathrm{A}$ and $\mathrm{B}$.
▶️Answer/Explanation
Ans:
To find the coordinates of points A and B on the x-axis, we need to determine the y-coordinate of each point. Since points A and B lie on the x-axis, their y-coordinates are both zero.
Let’s first find the x-coordinate of point A. To do this, we set the equation of the quadratic function equal to zero and solve for x:
$3x^2 – 5x -2 = 0$
We can factor this quadratic equation as follows:
$3x^2-6x+x-2$
$(3x + 1)(x – 2) = 0$
Setting each factor equal to zero:
$x – 2 = 0 \quad \text{or} \quad 3x +1 = 0$
Solving these equations:
$x = 2 \quad \text{or} \quad x = -\frac{1}{3}$
Therefore, the x-coordinate of point A is $-\frac{1}{3}$.
Similarly, we can find the x-coordinate of point B: $x = 2$
So, the coordinates of point A are (1/3, 0) and the coordinates of point B are (3, 0).
Question 4(c) : 1 marks
The function $f$ is reflected on the $y$-axis Write down the coordinates of point $B$ after the reflection.
▶️Answer/Explanation
Ans:
To find the coordinates of point B after reflecting the function on the y-axis, we simply change the sign of the x-coordinate while keeping the y-coordinate unchanged.
The original coordinates of point B are $(2, 0)$.
After reflecting on the y-axis, the x-coordinate becomes its opposite:
New x-coordinate of point B $= -2$
Therefore, the coordinates of point B after the reflection on the y-axis are $(-2, 0).$
Question 5(a) : 1 marks
Given that $\log 2 x=\log 12 y-\log 3$, show that $x=2 y$.
▶️Answer/Explanation
Ans:
We are given the equation:
$$\log 2x = \log 12y – \log 3.$$
Let’s work on simplifying this equation step by step:
1. Start by using the properties of logarithms:
$\log 2x = \log \frac{12y}{3}.$
2. Simplify the right side of the equation:
$\log 2x = \log 4y.$
3. Apply the logarithm property $\log a + \log b = \log(ab)$:
$\log 2x = \log (2 \cdot 2y).$
4. Use the property $\log a = \log b$ if and only if $a = b$:
$2x = 2 \cdot 2y.$
5. Divide both sides by 2:
$x = 2y.$
Thus, we have shown that if $\log 2x = \log 12y – \log 3$, then it follows that $x = 2y$.
Question 5(b) : 1 marks
Solve the simultaneous equations
$$
\begin{aligned}
& \log _4(3 x-2 y)=2 \\
& \log 2 x=\log 12 y-\log 3
\end{aligned}
$$
▶️Answer/Explanation
Ans:
Let’s solve this system step by step:
**Equation 1:**
$\log_4(3x – 2y) = 2$
Since $\log_4(3x – 2y) = 2$, we can rewrite this in exponential form:
$4^2 = 3x – 2y$
$16 = 3x – 2y$
**Equation 2:**
$\log 2x = \log 12y – \log 3$
Apply logarithm properties:
$\log 2x = \log \frac{12y}{3}$
$\log 2x = \log 4y$
Again, using exponential form:
$2x = 4y$
$x = 2y$
Now, we can substitute the value of $x$ in terms of $y$ into the first equation (equation 1):
$16 = 3(2y) – 2y$
$16 = 6y – 2y$
$16 = 4y$
Now, solve for $y$:
$y = \frac{16}{4}$
$y = 4$
Finally, substitute the value of $y$ back into the equation $x = 2y$:
$x = 2(4)$
$x = 8$
So, the solution to the simultaneous equations is $x = 8$ and $y = 4$.
Question 6 (15 marks)
The following question introduces how we are able to observe the Earth from the International Space Station.
Question 6(a) : 1 marks
The radius of the Earth is $r=6372 \mathrm{~km}$ Given that MS is $400 \mathrm{~km}$, show that OS is $6772 \mathrm{~km}$.
▶️Answer/Explanation
Ans:
$OS= r+MS$
$OS=( 6372+400) \mathrm{~km}$
$OS=6772 \mathrm{~km}$
Question 6(b) : 2 marks
OS is perpendicular to the chord $\mathrm{XY}$, show that triangle NOX is similar to triangle XOS.
▶️Answer/Explanation
Ans:
Question 6(c) : 5 marks
$\rm OS$ is perpendicular to the chord $\rm X Y$. Find the length of $\mathrm{ON}$ to the nearest km.
▶️Answer/Explanation
Ans:
In $\triangle \rm{ONX}$
$\cos 20=\frac{\mathrm{ON}}{6372}$
To find ON, we can isolate it by multiplying both sides of the equation by 6372:
$\cos 20 \times 6372 = \text{ON}$
Using a calculator to evaluate $\cos 20$, we have:
$\text{ON} \approx 0.9397 \times 6372$
$\text{ON} \approx 5989.2$
Rounding to the nearest kilometer, the length of $\text{ON}$ is approximately 5989 km.
Therefore, the length of $\text{ON}$ , to the nearest kilometer, is $\mathrm{5989 ~km}$.
Question 6(d) : 3 marks
The surface area $(A)$ of the spherical cap is $A=2 \pi r h$ where
$r$ is the radius of the Earth, $h$ is the height of the spherical cap $(\rm{MN})$
Calculate the surface area $(A)$ of the spherical cap. Give your answer in standard form correct to two significant figures.
▶️Answer/Explanation
Ans:
To calculate the surface area (A) of the spherical cap, we can use the formula $A = 2 \pi r h$, where $r$ is the radius of the Earth and $h$ is the height of the spherical cap (MN).
We were previously given that the radius of the Earth, $r$, is 6372 km, and the height of the spherical cap, $h$, is 383 km.
Substituting these values into the formula, we have:
$A = 2 \pi (6372 \, \mathrm{km}) (383 \, \mathrm{km})$
$A \approx 15,326,189.28 \, \mathrm{km^2}$
Rounding to two significant figures, the surface area (A) of the spherical cap is approximately $1.5 \times 10^7 \, \mathrm{km^2}$.
Therefore, the surface area of the spherical cap, to two significant figures, is $1.5 \times 10^7 \, \mathrm{km^2}$.
Question 6(e) : 3 marks
Hence, find the percentage of Earth the International Space Station can see at any one time.
▶️Answer/Explanation
Ans:
To find the percentage of the Earth that the International Space Station (ISS) can see at any one time, we can use the given hint and formula.
The total surface area of the Earth is $4\pi \times 6372^2 = 510224605 \, \mathrm{km^2}$.
The area visible to the ISS at any one time is given as $1.5 \times 10^7 \, \mathrm{km^2}$.
To find the percentage, we divide the visible area by the total surface area of the Earth and multiply by 100:
Percentage = $\left(\frac{1.5 \times 10^7}{510224605}\right) \times 100$
Percentage = $\left(\frac{1.5 \times 10^7}{510224605}\right) \times 100$
Evaluating this expression, we find:
Percentage $≈ 2.94$
Therefore, the International Space Station can see approximately $2.94\%$ of the Earth’s surface at any one time.
Question 7 (19 marks)
The following video describes how different fuels for vehicles can impact emissions on communities and environments.
Question 7(a) : 3 marks
Vehicle A is advertised to buy for $£ 31250$. As part of a government incentive the vehicle cost is reduced by $20 \%$.
Calculate the vehicle cost after the government incentive for vehicle $A$.
▶️Answer/Explanation
Ans:
To calculate the vehicle cost after the government incentive, we need to subtract the discount amount from the original price.
The discount amount can be found by multiplying the original price by the discount percentage, which is $20\%$ or $0.20$.
\[
\text{{Discount amount}} = 0.20 \times £31250
\]
\[
\text{{Discount amount}} = £6250
\]
The vehicle cost after the government incentive is the original price minus the discount amount.
\[
\text{{Vehicle cost after incentive}} = £31250 – £6250
\]
\[
\text{{Vehicle cost after incentive}} = £25000
\]
Therefore, the vehicle cost after the government incentive for vehicle A is £25000.
Question 7(b) : 3 marks
A person drives 14000 miles on average per year.
Determine the annual fuel cost of vehicle $A$.
▶️Answer/Explanation
Ans:
For Vehicle A (electric-powered):Annual fuel cost for Vehicle A $= \mathrm{14000~ miles }\times \mathrm{£0.035~ per ~mile} = £490$
Question 7(c) : 2 marks
The annual fuel cost of vehicle $B$ is $£ 1190$. Find the percentage savings of the annual fuel cost when purchasing vehicle A instead of vehicle B.
▶️Answer/Explanation
Ans:
\( \begin{aligned} \text { Percentage } & =\left|\frac{1190-490}{1190}\right| \times 100 \% \\ & =\frac{700}{1190} \times 100 \% \\ & =58.8 \%\end{aligned} \)Question 7(d) : 2 marks
The following graph shows the linear relationship for the total cost, $T$, of owning Vehicle $\mathrm{A}$ for $n$ years, driving 14000 miles per year.
Write down a linear equation for the total cost, $T$, of owning vehicle A for $n$ years.
▶️Answer/Explanation
Ans:
the total cost ($T$) of owning vehicle A for $n$ years, driving 14000 miles per year, is:
$y=(14000 \times 0.035) n+25000$
$T = 490n + 25000$
In this equation, $n$ represents the number of years for which the vehicle is owned. The term $(14000 \times 0.035) \cdot n$ represents the annual fuel cost for $n$ years, and $25000$ is the initial vehicle cost after the government incentive. By adding these two terms, we obtain the total cost, $T$, of owning vehicle A for $n$ years.
Question 7(e) : 10 marks
Discuss whether vehicle $A$ or vehicle $B$ is a better buy. Use the information provided in the table and your answers from parts (a) to (d). In your answer, you should:
$\bullet$ identify three relevant factors to consider when deciding whether to buy vehicle $A$ or vehicle $B$
$\bullet$ draw a graph that describes the linear relationship for the total cost $(T)$ of owning vehicle $A$ for $n$ years
$\bullet$ determine after how many years the total cost of owning vehicle $A$ is equal to vehicle $B$
$\bullet$ justify whether vehicle $A$ or vehicle $B$ is a better buy and how this may impact communities and the environment
$\bullet$ comment on the accuracy of the total cost for owning the different vehicles.
▶️Answer/Explanation
Ans:
To determine whether vehicle A or vehicle B is a better buy, we need to consider several factors and analyze the information provided. Here are three relevant factors to consider:
1. Initial Vehicle Cost: Vehicle A’s cost after the government incentive is £25,000, while vehicle B’s cost is £18,000. The initial vehicle cost is an important factor to consider when making a purchase decision.
2. Annual Fuel Cost: Vehicle A, being electric-powered, has an annual fuel cost of £490, while vehicle B, being petrol-powered, has an annual fuel cost of £1,190. The annual fuel cost over the years of ownership can significantly impact the total cost of owning the vehicle.
3. Duration of Ownership: The number of years the vehicle will be owned affects the total cost. We can analyze the linear relationship for the total cost of owning vehicle A to determine after how many years the total cost of owning vehicle A is equal to vehicle B.
Let’s draw a graph that describes the linear relationship for the total cost (T) of owning vehicle A for n years, using the equation we derived earlier:
\[T = (14000 \times 0.035) \cdot n + 25000\]
Here’s the graph:
To determine after how many years the total cost of owning vehicle A is equal to vehicle B, we can set the total costs equal to each other and solve for n:
\[(14000 \times 0.035) \cdot n + 25000 = 18000 + 1190 \cdot n\]
Simplifying the equation, we find:
\[700n = 7000\]
\[n = 10\]
So, after approximately 10 years, the total cost of owning vehicle A will be equal to that of vehicle B.
To justify which vehicle is a better buy, we need to consider the factors mentioned earlier.
$\bullet$ Initial Vehicle Cost: Vehicle B has a lower initial cost (£18,000) compared to vehicle A (£25,000), making it more affordable upfront.
$\bullet$ Annual Fuel Cost: Vehicle A has a significantly lower annual fuel cost (£490) compared to vehicle B (£1,190) due to its electric power source. This can result in long-term savings.
$\bullet$ Duration of Ownership: If the intended ownership period is less than 10 years, vehicle B might be a better buy. However, if the ownership period exceeds 10 years, vehicle A could be a better choice due to its lower total cost in the long run.
From an environmental perspective, vehicle A (electric-powered) produces zero tailpipe emissions, making it more environmentally friendly compared to vehicle B (petrol-powered), which contributes to air pollution. Choosing vehicle A can have a positive impact on local air quality and reduce greenhouse gas emissions.
Regarding the accuracy of the total cost for owning the different vehicles, the calculations are based on the given information, assuming the variables remain constant over time. However, it’s essential to note that factors such as fuel prices, maintenance costs, and resale value can vary, impacting the accuracy of the total cost estimation. It’s advisable to consider these factors and update the calculations with current data when making a purchasing decision.
Question 8 (31 marks)
Question 8(a) : 3 marks
In this task you will investigate sides and perimeters of trapeziums.
Show that the diagonal of the trapezium is $\sqrt{2}$ units.
▶️Answer/Explanation
Ans:
$\begin{gathered}1^2+1^2 \\ \sqrt{2}\end{gathered}$
 |  |
 |  |
Question 8(b) : 1 marks
Write down the missing values in the table up to row 6 .
▶️Answer/Explanation
Ans:
To fill in the missing values in the table:
Question 8(c) : 1 marks
Describe in words two patterns you see in the table for $D$.
▶️Answer/Explanation
Ans:
Exponential Growth Pattern: The first pattern is that the diagonal values $(D)$ are increasing exponentially as the stage $(n)$ increases. Specifically, each diagonal value is double the previous one. This can be observed by looking at the values in the table: $2 \sqrt{2}$ is twice the value of $\sqrt{2}$, $4 \sqrt{2}$ is twice the value of $2 \sqrt{2}$, $8 \sqrt{2}$ is twice the value of $4 \sqrt{2}$, and so on. This exponential growth signifies a geometric progression where each term is obtained by multiplying the previous term by a constant factor.
Multiplicative Pattern:Each diagonal value $(D)$ is obtained by multiplying the initial value $\sqrt{2}$ by increasing powers of $2$ based on the stage $(n)$. In other words, the diagonal values follow the formula:
\[ D = \sqrt{2} \times 2^{(n-1)} \]
Where $n$ is the stage number. This explains why each diagonal value in the table is double the value of the previous stage’s diagonal, and this pattern continues as the stage number increases.
Question 8(d) : 3 marks
Determine a general rule for $D$ in terms of $n$.
▶️Answer/Explanation
Ans:
The general rule for the diagonal values $(D)$ in terms of the stage number $(n)$ is:
\[ D = \sqrt{2} \times 2^{(n-1)} \]
Where:
$\bullet$ \( D \) represents the diagonal value at stage \( n \).
$\bullet$ \( \sqrt{2} \) is the initial value of the diagonal at stage 1.
$\bullet$ \( 2^{(n-1)} \) represents the exponentiation of 2 to the power of \( (n-1) \), which accounts for the exponential growth of the diagonal values with each increasing stage.
Question 8(e) : 3 marks
Verify your general rule for $D$.
▶️Answer/Explanation
Ans:
let’s verify the general rule \(D = \sqrt{2} \times 2^{(n-1)}\) for a few stages using the given table:
As we can see, the calculated values using the general rule match the diagonal values given in the table for each stage. This confirms that the general rule \(D = \sqrt{2} \times 2^{(n-1)}\) is correct and accurately describes the relationship between the stage number \(n\) and the corresponding diagonal value \(D\).
Question 8(f) : 20 marks
The diagram below shows the trapeziums formed in each stage.
Investigate the values in the table to find a relationship for the perimeter $(P)$ of each trapezium in terms of $n$. In your answer, you should:
$\bullet$ predict more values and record these in the table
$\bullet$ describe in words a pattern for column $P$
$\bullet$ find a general rule for $P$ in terms of $n$
$\bullet$ test your general rule for $P$
$\bullet$ verify and justify your general rule for $P$
$\bullet$ ensure that you communicate all your working appropriately.
▶️Answer/Explanation
Ans:
Let’s investigate the relationship between the stage number \(n\) and the perimeter \(P\) of each trapezium using the provided table. We’ll follow the steps you mentioned:
Step 1: Predict More Values
Let’s predict more values for \(n = 5\) and \(n = 6\) and calculate their corresponding perimeters.
Step 2: Describe the Pattern
Upon examining the table, we can observe that the perimeter \(P\) consists of two parts: a fixed value and a term involving the diagonal \(D\). The fixed value is twice the side \(S\) of the trapezium. The term involving the diagonal \(D\) is always double the diagonal value, i.e., \(2D\).
Step 3: General Rule for \(P\) in Terms of \(n\)
Based on the pattern, we can write the general rule for the perimeter \(P\) in terms of the stage number \(n\) as follows:
\[ P = 2S + 2D = 2S + 2 \times 2^{(n-1)} \sqrt{2} \]
Step 4: Test the General Rule
Let’s test the general rule for \(n = 5\) and \(n = 6\):
$\bullet$ For \(n = 5\): \(P = 2 \times 16 + 2 \times 2^{(5-1)} \sqrt{2} = 32 + 32 \sqrt{2}\)
$\bullet$ For \(n = 6\): \(P = 2 \times 32 + 2 \times 2^{(6-1)} \sqrt{2} = 64 + 64 \sqrt{2}\)
These values match the predicted values in the extended table.
Step 5: Verify and Justify the General Rule
The general rule \(P = 2S + 2 \times 2^{(n-1)} \sqrt{2}\) holds true for each stage \(n\), as demonstrated by the consistent match between the calculated values and the values in the table. The rule captures the doubling of the diagonal term and the inclusion of twice the side length, which collectively define the perimeter of the trapezium at each stage.
By following these steps, we’ve established a general rule for the perimeter \(P\) of each trapezium in terms of the stage number \(n\).