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## Question 1

A business has $$28$$ employees.
Their ages, in years, are given below.

\begin{align}32&&41&&57&&64&&19&&21&&35\\ 18&&43&&54&&63&&65&&33&&22\\ 39&&58&&18&&42&&20&&34&&21\\ 49&&33&&55&&34&&57&&43&&63\end{align}

(a) Complete the stem-and-leaf diagram, showing the ages of all $$28$$ employees.

Solution
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(b) Find the percentage of employees who are older than $$40$$ years of age.

$$50\%$$

Solution

\begin{align}\frac{14}{28}=50\%\end{align}

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(c) One employee is chosen at random on a day when all employees are present at work.

(i) Find the probability that the employee is a teenager ($$<20$$ years of age).

(ii) Find the probability that the employee chosen is a person in their thirties whose age is even or a person in their forties whose age is odd.

(i) $$\dfrac{3}{28}$$

(ii) $$\dfrac{7}{28}$$

Solution

(i)

\begin{align}\frac{3}{28}\end{align}

(ii)

\begin{align}\frac{3}{28}+\frac{4}{28}=\frac{7}{28}\end{align}

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## Question 2

The diagram shows the line $$PQ$$ and the line $$QR$$.
The co-ordinates of the points are $$P(4,2)$$, $$Q(8,5)$$ and $$R(2,11)$$.

(a) Find the slope of $$PQ$$.

$$\dfrac{3}{4}$$

Solution

\begin{align}m_{PQ}&=\frac{y_2-y_1}{x_2-x_1}\\&=\frac{5-2}{8-4}\\&=\frac{3}{4}\end{align}

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(b) Find the equation of the line $$PQ$$.
Give your answer in the form $$ax+by+c=0$$, where $$a,b,c\in\mathbb{Z}$$.

$$3x-4y-4=0$$

Solution

\begin{align}y-y_1=m(x-x_1)\end{align}

\begin{align}\downarrow\end{align}

\begin{align}y-2=\frac{3}{4}(x-4)\end{align}

\begin{align}\downarrow\end{align}

\begin{align}4y-8=3x-12\end{align}

\begin{align}\downarrow\end{align}

\begin{align}3x-4y-4=0\end{align}

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(c) Write down the slope of any line perpendicular to $$PQ$$.

$$-\dfrac{3}{4}$$

Solution

\begin{align}m_{\perp}=-\frac{3}{4}\end{align}

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(d) Find the area of the triangle $$PQR$$.

$$21\mbox{ units}^2$$

Solution

\begin{align}(4,2)&&(8,5)&&(2,11)\end{align}

\begin{align}\downarrow\end{align}

\begin{align}(4-4,2-2)&&(8-4,5-2)&&(2-4,11-2)\end{align}

\begin{align}\downarrow\end{align}

\begin{align}(0,0)&&(4,3)&&(-2,9)\end{align}

\begin{align}\downarrow\end{align}

\begin{align}A&=\frac{1}{2}|x_1y_2-x_2y_1|\\&=\frac{1}{2}|(4)(9)-(-2)(3)|\\&=21\mbox{ units}^2\end{align}

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## Question 3

In a population, the probability that a person has blue eyes is $$0.7$$.

(a) One person is chosen at random from the population.
What is the probability that this person does not have blue eyes?

$$0.3$$

Solution

\begin{align}P&=1-0.7\\&=0.3\end{align}

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(b) Two people are chosen at random.
What is the probability that both have blue eyes?

$$0.49$$

Solution

\begin{align}P&=0.7\times0.7\\&=0.49\end{align}

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(c) Three people are chosen at random.
What is the probability that exactly two of them have blue eyes?

$$0.441$$

Solution

\begin{align}P&={3\choose2}\times0.7^2\times0.3^1\\&=0.441\end{align}

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(d) Four people are chosen at random, one after another.
What is the probability that the fourth person of the four chosen is the only one to have blue
eyes?

$$0.0189$$

Solution

\begin{align}P&=0.3^3\times0.7\\&=0.0189\end{align}

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## Question 4

The circle $$c$$ is enclosed in the square $$PQRS$$ and touches all four sides, as shown in the diagram.
The co-ordinates of three of the vertices are $$P(2,3)$$, $$R(4,17)$$, and $$S(-4,11)$$.

(a) Find the co-ordinates of Q.

$$(10,9)$$

Solution

\begin{align}Q&=(4+6,17-8)\\&=(10,9))\end{align}

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(b) Find the co-ordinates of the centre of $$c$$.

$$(3,10)$$

Solution

\begin{align}\mbox{Centre}&=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\&=\left(\frac{-4+10}{2},\frac{11+9}{2}\right)\\&=(3,10)\end{align}

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(c) Find the length of the radius of $$c$$.

$$5\mbox{ units}$$

Solution

\begin{align}r&=\frac{1}{2}|RS|\\&=\frac{1}{2}\sqrt{(-4-4)^2+(11-17)^2}\\&=5\mbox{ units}\end{align}

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(d) Find the equation of circle $$c$$.

$$(x-3)^2+(y-10)^2=25$$

Solution

\begin{align}(x-3)^2+(y-10)^2=25\end{align}

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## Question 5

(a) The crescent, shown in the shaded part of the diagram, was created by removing a disc of radius $$2.5\mbox{ cm}$$ from a disc of radius $$3\mbox{ cm}$$.
Find the area and the perimeter of the crescent.
Give each answer correct to two decimal places.

$$\mbox{Area}=8.64\mbox{ cm}^2$$ and $$\mbox{Perimeter}=34.56\mbox{ cm}$$

Solution

\begin{align}\mbox{Area}&=\pi(3^2)-\pi(2.5^2)\\&\approx8.64\mbox{ cm}^2\end{align}

and

\begin{align}\mbox{Perimeter}&=2\pi(3)+2\pi(2.5)\\&\approx34.56\mbox{ cm}\end{align}

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(b) An empty inverted cone of vertical height $$12\mbox{ cm}$$ and radius $$7\mbox{ cm}$$ is filled with water from a pipe.
The water flows from the pipe at a steady rate of $$0.5$$ litres per minute.
Find the time it takes to fill the cone.

$$74\mbox{ s}$$

Solution

\begin{align}V&=\frac{1}{3}\pi r^2h\\&=\frac{1}{3}\pi(7^2)(12)\\&=196\pi\mbox{ cm}^3\end{align}

\begin{align}\downarrow\end{align}

\begin{align}t&=\frac{196\pi}{0.5\times1{,}000}\\&=1.2315..\mbox{ min}\\&\approx74\mbox{ s}\end{align}

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## Question 6

(a)

(i) Construct the parallelogram $$PQRS$$, where $$|PQ|=9\mbox{ cm}$$, $$|Ps|=5\mbox{ cm}$$ and $$|SPQ|=65^{\circ}$$.
The point $$P$$ has been marked in for you.
Show all your construction lines, arcs and labels clearly.

(ii) Find the area of the parallelogram $$PQRS$$.
Give your answer correct to $$2$$ decimal places.

(i)

(ii) $$40.78\mbox{ cm}^2$$

Solution

(i)

(ii)

\begin{align}A&=ab\sin C\\&=(9)(5)\sin65^{\circ}\\&\approx40.78\mbox{ cm}^2\end{align}

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(b) In the diagram $$O$$ is the centre of the circle $$s$$.
Find the value of $$\alpha$$ and the value of $$\beta$$.

(ii) $$\alpha=52^{\circ}$$ and $$\beta=19^{\circ}$$

Solution

\begin{align}\alpha=52^{\circ}\end{align}

and

\begin{align}\beta&=\frac{38^{\circ}}{2}\\&=19^{\circ}\end{align}

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## Question 7

Table A below shows the price index for thirteen countries in four food categories for the year 2015. (Source: Central Statistics Office.)
It shows, for example, that if you paid €$$132$$ for meat in Austria, then the same quantity of similar meat would cost you €$$117$$ in Germany, €$$106$$ in Ireland, €$$63$$ in Lithuania and so on.
The description is the same for the other food categories.

Country Meat Fish Milk, Cheese and Eggs Fruits, Vegetables and Potatoes

Austria

$$132$$

$$125$$

$$105$$

$$125$$

Germany

$$117$$

$$109$$

$$91$$

$$111$$

Ireland

$$106$$

$$108$$

$$128$$

$$136$$

Lithuania

$$63$$

$$73$$

$$85$$

$$77$$

Macedonia

$$56$$

$$59$$

$$71$$

$$48$$

Netherlands

$$111$$

$$99$$

$$93$$

$$104$$

Norway

$$157$$

$$117$$

$$175$$

$$150$$

Poland

$$54$$

$$64$$

$$65$$

$$62$$

Spain

$$85$$

$$89$$

$$96$$

$$95$$

Sweden

$$131$$

$$115$$

$$116$$

$$137$$

Switzerland

$$254$$

$$177$$

$$148$$

$$172$$

Turkey

$$78$$

$$105$$

$$122$$

$$86$$

United Kingdom

$$112$$

$$105$$

$$118$$

$$116$$

(a) Complete Table B below, to show the maximum value, the minimum value and the range of the data for the four food categories.

Meat Fish Milk, Cheese and Eggs Fruits, Vegetables and Potatoes

Maximum

$$254$$

Minimum

$$54$$

Range

$$200$$

Meat Fish Milk, Cheese and Eggs Fruits, Vegetables and Potatoes

Maximum

$$254$$

$$177$$

$$175$$

$$172$$

Minimum

$$54$$

$$59$$

$$65$$

$$48$$

Range

$$200$$

$$118$$

$$110$$

$$124$$

Solution
Meat Fish Milk, Cheese and Eggs Fruits, Vegetables and Potatoes

Maximum

$$254$$

$$177$$

$$175$$

$$172$$

Minimum

$$54$$

$$59$$

$$65$$

$$48$$

Range

$$200$$

$$118$$

$$110$$

$$124$$

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(b) Write the data for the Fruits, Vegetables and Potatoes category in increasing order and hence find the median for that food category.

\begin{align}48&&62&&77&&86&&95&&104&&111&&116&&125&&136&&137&&150&&172\end{align}

\begin{align}\mbox{Median}=111\mbox{ euro}\end{align}

Solution

\begin{align}48&&62&&77&&86&&95&&104&&111&&116&&125&&136&&137&&150&&172\end{align}

\begin{align}\downarrow\end{align}

\begin{align}\mbox{Median}=111\mbox{ euro}\end{align}

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(c)

(i) Find the mean of the data in the Meat category.

(ii) Use your calculator to find the standard deviation for the data in the Meat category.

(iii) List all the countries in the Meat category that are not within one standard deviation of the mean.

(i) $$112\mbox{ euro}$$

(ii) $$51\mbox{ euro}$$

(iii) Macedonia, Poland and Switzerland.

Solution

(i)

\begin{align}\mbox{Mean}&=\frac{132+117+106+63+56+111+157+54+85+131+254+78+112}{13}\\&=112\mbox{ euro}\end{align}

(ii)

\begin{align}\mbox{Standard Deviation}=51\mbox{ euro}\end{align}

(iii) Macedonia, Poland and Switzerland.

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(d) A customer in Ireland buys some meat, fish, milk, cheese, eggs, fruits, vegetables and potatoes as detailed in Table C.
A customer in Poland buys the same items in the same quantities.
Complete Table C.

Country Meat Fish Milk, Cheese and Eggs Fruits, Vegetables and Potatoes Total Cost

Ireland

$$36.04$$

$$31.32$$

$$24.32$$

$$31.28$$

$$122.96$$

Poland

$$18.36$$

Country Meat Fish Milk, Cheese and Eggs Fruits, Vegetables and Potatoes Total Cost

Ireland

$$36.04$$

$$31.32$$

$$24.32$$

$$31.28$$

$$122.96$$

Poland

$$18.36$$

$$18.56$$

$$12.35$$

$$14.26$$

$$63.53$$

Solution
Country Meat Fish Milk, Cheese and Eggs Fruits, Vegetables and Potatoes Total Cost

Ireland

$$36.04$$

$$31.32$$

$$24.32$$

$$31.28$$

$$122.96$$

Poland

$$18.36$$

$$18.56$$

$$12.35$$

$$14.26$$

$$63.53$$

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## Question 8

(a) A solid sphere of radius $$3\mbox{ cm}$$ is placed inside a cylinder and then water is poured into the cylinder until it is full, as shown in the diagram.

(i) Find the volume of the sphere, in terms of $$\pi$$.

(ii) The sphere is now removed. The internal radius of the cylinder is $$5\mbox{ cm}$$.
Find the drop, in $$\mbox{cm}$$, in the height of the water.

(iii) The cylinder has a height of $$18\mbox{ cm}$$. The curved surface of the cylinder is cut from a rectangular piece of metal measuring $$35\mbox{ cm}$$ by $$20\mbox{ cm}$$, as shown. Find how much metal will be left over when the curved surface of the cylinder is cut out.
Give your answer correct to $$1$$ decimal place.

(i) $$36\pi\mbox{ cm}^3$$

(ii) $$\dfrac{36}{25}\mbox{ cm}$$

(iii) $$34.5\mbox{ cm}^2$$

Solution

(i)

\begin{align}V&=\frac{4}{3}\pi r^3\\&=\frac{4}{3}\pi(3^3)\\&=36\pi\mbox{ cm}^3\end{align}

(ii)

\begin{align}36\pi=\pi(5^2)h\end{align}

\begin{align}\downarrow\end{align}

\begin{align}h&=\frac{36}{5^2}\\&=\frac{36}{25}\mbox{ cm}\end{align}

(iii)

\begin{align}35\times20-2\pi(5)(18)\approx134.5\mbox{ cm}^2\end{align}

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(b) In September $$2018$$ a European Commission report stated that it is considering a proposal to abolish daylight saving time. This would mean that the clocks would not go forward one hour in March and then not go back one hour the following October. A Transition Year class carried out a survey to find out local people’s views on the proposal.

(i) The class surveyed a random sample of $$800$$ people in the local area.
Find the margin of error of the survey.
Give your answer as a percentage, correct to $$2$$ decimal places.

(ii) In the survey $$350$$ people said they supported the EU proposal to abolish daylight saving time. Use your answer to part (b)(i) above to create a $$95\%$$ confidence interval for the percentage of the population who supported the EU proposal.

(iii) A local newspaper had reported that $$50\%$$ of people in the area supported the EU proposal. Use your answer to part (b)(ii) above to conduct a hypothesis test, at the $$5\%$$ level of significance, to test the newspaper’s claim. Clearly state your conclusion in the context of the question and give a reason for your conclusion.

(i) $$3.54\%$$

(ii) $$40.21\%<\hat{p}<47.29\%$$

(iii)

Null hypothesis: The percentage of people in the area that supported the EU proposal is $$50\%$$.

Alternative hypothesis: The percentage of people in the area that supported the EU proposal is not $$50\%$$.

Calculations: $$50\%$$ is not within the confidence interval derived above.

Conclusion: The percentage of people in the area that supported the EU proposal is not $$50\%$$.

Solution

(i)

\begin{align}\mbox{Margin of error}&=\frac{1}{\sqrt{n}}\\&=\frac{1}{\sqrt{800}}\\&=0.035355…\\&\approx3.54\%\end{align}

(ii)

\begin{align}\frac{350}{800}\times 100-3.54<\hat{p}<\frac{350}{800}\times 100+3.54\end{align}

\begin{align}\downarrow\end{align}

\begin{align}40.21\%<\hat{p}<47.29\%\end{align}

(iii)

Null hypothesis: The percentage of people in the area that supported the EU proposal is $$50\%$$.

Alternative hypothesis: The percentage of people in the area that supported the EU proposal is not $$50\%$$.

Calculations: $$50\%$$ is not within the confidence interval derived above.

Conclusion: The percentage of people in the area that supported the EU proposal is not $$50\%$$.

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## Question 9

$$R$$ is a radar station located $$120\mbox{ km}$$ north of a port $$P$$.
The circle $$c$$, centred at $$R$$ and with radius $$100\mbox{ km}$$ shows the detection range of the radar. When a ship enters the circle it will be detected by the radar station at $$R$$.
Figure $$1$$ shows a ship leaving port $$P$$ and sailing in the direction north $$30^{\circ}$$ east.
The ship enters the circle $$c$$ at $$S$$ and exits at $$T$$.
At $$Q$$, the ship is closest to $$R$$ and $$|\angle PQR=90^{\circ}$$.

(a) The triangle $$PQR$$ taken from Figure 1 is shown in Figure 2.
Find $$|QR|$$, the length of $$[QR]$$.

$$60\mbox{ km}$$

Solution

\begin{align}\sin30^{\circ}=\frac{|QR|}{120}\end{align}

\begin{align}\downarrow\end{align}

\begin{align}|QR|&=120\sin30^{\circ}\\&=60\mbox{ km}\end{align}

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(b) The triangle $$QRS$$ taken from Figure 1 is shown in Figure 3.
Use your answer from part (a) to find $$|QS|$$.

$$80\mbox{ km}$$

Solution

\begin{align}|QS|&=\sqrt{|RS|^2-|RQ|^2}\\&=\sqrt{100^2-60^2}\\&=80\mbox{ km}\end{align}

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(c) Find $$|PS|$$. Give your answer correct to the nearest $$\mbox{km}$$.

$$24\mbox{ km}$$

Solution

\begin{align}|PS|&=|PQ|-|QS|\\&=\sqrt{|PR|^2-|QR|^2}-|QS|\\&=\sqrt{120^2-60^2}-80\\&\approx24\mbox{ km}\end{align}

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(d)

(i) Consider the triangle $$RST$$.
Use the Cosine Rule to find an expression for $$\cos \theta$$,
where $$\theta$$ is the measure of the angle $$TRS$$.
Hence show that $$\theta=106^{\circ}$$, correct to the nearest degree.

(ii) John sails directly from $$S$$ to $$T$$. Mary sails from $$S$$ to $$T$$ along the minor arc $$ST$$.
Find the difference between the distance that John sails and the distance that Mary sails. Give your answer correct to the nearest $$\mbox{ km}$$.

(iii) The sea in this region is estimated to have an average of $$1$$ ship per $$25$$ square kilometres at any time.
Use this estimate to find the number of ships in the sector $$RST$$.

(i) $$\cos\theta=-\dfrac{7}{25}$$

(ii) $$25\mbox{ km}$$

(iii) $$370$$

Solution

(i)

\begin{align}|TS|&=2|QS|\\&=2(80)\\&=160\mbox{ km}\end{align}

\begin{align}\downarrow\end{align}

\begin{align}160^2=100^2+100^2-2(100)(100)\cos\theta\end{align}

\begin{align}\downarrow\end{align}

\begin{align}\cos\theta&=\frac{160^2-100^2-100^2}{2(100)(100)}\\&=-\frac{7}{25}\end{align}

\begin{align}\downarrow\end{align}

\begin{align}\theta&=\cos^{-1}\left(\frac{7}{25}\right)\\&\approx106^{\circ}\end{align}

as required.

(ii)

\begin{align}2\pi(100)\times\frac{106^{\circ}}{360^{\circ}}-160\approx25\mbox{ km}\end{align}

(iii)

\begin{align}\mbox{Area}&=\pi(100)^2\times\frac{106^{\circ}}{360^{\circ}}\\&=9{,}250.245…\mbox{ km}^2\end{align}

\begin{align}\downarrow\end{align}

\begin{align}N&=\frac{9{,}250.245…}{25}\\&\approx370\end{align}

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