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2017 Ordinary Level - Paper Two
Section A
Question 1
According to the Central Statistics Office (CSO) there were \(65{,}909\) babies born in Ireland in \(2015\).
Of these \(32{,}290\) were girls.
(a)
(i) How many boys were born in Ireland in \(2015\)?
(ii) Find the probability that a baby picked at random from those born in Ireland in \(2015\) is a boy. Give your answer correct to \(2\) decimal places.
(i) \(33{,}619\)
(ii) \(0.51\)
(i)
\begin{align}65{,}909-32{,}290=33{,}619\end{align}
(ii)
\begin{align}P&=\frac{33{,}619}{65{,}909}\\&\approx0.51\end{align}
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(b) Eight babies were born in Limerick’s Maternity Hospital on \(1\) May \(2015\).
(i) Use your answer to part (a)(ii) to find the probability that the first three babies born were boys. Give your answer correct to \(4\) decimal places.
(ii) Find the probability that the third birth was the first girl born in the hospital that day.
Give your answer correct to \(4\) decimal places.
(i) \(0.1327\)
(ii) \(0.1274\)
(i)
\begin{align}P&=0.51^3\\&\approx0.1327\end{align}
(ii)
\begin{align}P&=0.51^2\times0.49\\&\approx0.1274\end{align}
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(c) The table below shows the probability of being born on a particular day of the week.
(i) Complete the table.
| Day | Mon | Tue | Wed | Thu | Fri | Sat | Sun |
|---|---|---|---|---|---|---|---|
|
Probability |
\(0.14\) |
|
\(0.15\) |
\(0.18\) |
\(0.15\) |
\(0.12\) |
\(0.1\) |
(ii) In a particular week \(1300\) babies were born.
Find the number of babies expected to be born on the Tuesday of that week.
(i)
| Day | Mon | Tue | Wed | Thu | Fri | Sat | Sun |
|---|---|---|---|---|---|---|---|
|
Probability |
\(0.14\) |
\(0.16\) |
\(0.15\) |
\(0.18\) |
\(0.15\) |
\(0.12\) |
\(0.1\) |
(ii) \(208\)
(i)
| Day | Mon | Tue | Wed | Thu | Fri | Sat | Sun |
|---|---|---|---|---|---|---|---|
|
Probability |
\(0.14\) |
\(0.16\) |
\(0.15\) |
\(0.18\) |
\(0.15\) |
\(0.12\) |
\(0.1\) |
(ii)
\begin{align}1{,}300\times0.16=208\end{align}
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Question 2
(a) The circle \(c\) has centre \((0,0)\) and radius \(5\) units. Write down the equation of \(c\).
\(x^2+y^2=25\)
\begin{align}x^2+y^2=25\end{align}
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(b) The diagram shows a semi-circle which is part of \(c\).
(i) The point \(P(-4,k)\), \(k>0\), is on the semi-circle. Find the value of \(k\).
(ii) Show that the triangle \(ABP\) is right-angled at \(P\).
(i) \(k=3\)
(ii) As \(AB\) is a diameter, the angle \(|\angle ABP|\) must be a right angle.
(i)
\begin{align}x^2+y^2=25\end{align}
\begin{align}\downarrow\end{align}
\begin{align}(-4)^2+k^2=25\end{align}
\begin{align}\downarrow\end{align}
\begin{align}16+k^2=25\end{align}
\begin{align}\downarrow\end{align}
\begin{align}k^2=9\end{align}
\begin{align}\downarrow\end{align}
\(k=3\) (as \(k>0\))
(ii) As \(AB\) is a diameter, the angle \(|\angle ABP|\) must be a right angle.
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(c) Find the area of the region which is inside the semi-circle but outside the triangle \(ABP\).
Give your answer, in square units, correct to \(2\) decimal places.
\(24.27\)
\begin{align}A&=\frac{1}{2}\pi r^2-\frac{1}{2}bh\\&=\frac{1}{2}\pi(5^2)-\frac{1}{2}(10)(3)\\&\approx24.27\end{align}
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Question 3
(a) The points \(A(2,1)\), \(B(6,3)\), \(C(5,5)\), and \(D(1,3)\) are the vertices of the rectangle \(ABCD\) as shown.
(i) Show that \(|AD|=\sqrt{5}\) units.
(i) Show that \(|AD|=\sqrt{5}\) units.
(i) The answer is already in the question!
(ii) \(10\mbox{ units}^2\)
(i)
\begin{align}|AD|&=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\\&=\sqrt{(2-1)^2+(1-3)^2}\\&=\sqrt{1+4}\\&=\sqrt{5}\mbox{ units}\end{align}
as required.
(ii)
\begin{align}|AB|&=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\\&=\sqrt{(2-6)^2+(1-3)^2}\\&=\sqrt{16+4}\\&=\sqrt{20}\mbox{ units}\end{align}
\begin{align}\downarrow\end{align}
\begin{align}A&=|AB|\times|AD|\\&=\sqrt{20}\times\sqrt{5}\\&=\sqrt{100}\\&=10\mbox{ units}^2\end{align}
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(b) Find the equation of the line \(BC\).
Give your answer in the form \(ax+by+c=0\), where \(a,b,\) and \(c\in\mathbb{Z}\).
\(2x+y-15=0\)
\begin{align}m_{BC}&=\frac{y_2-y_1}{x_2-x_1}\\&=\frac{5-3}{5-6}\\&=-2\end{align}
\begin{align}\downarrow\end{align}
\begin{align}y-y_1=m(x-x_1)\end{align}
\begin{align}\downarrow\end{align}
\begin{align}y-3=-2(x-6)\end{align}
\begin{align}\downarrow\end{align}
\begin{align}y-3=-2x+12\end{align}
\begin{align}\downarrow\end{align}
\begin{align}2x+y-15=0\end{align}
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(c) Use trigonometry to find the measure of the angle \(ABD\).
Give your answer correct to the nearest degree.
\(27^{\circ}\)
\begin{align}\tan|\angle ABD|&=\frac{|AD|}{|AB|}\\&=\frac{\sqrt{5}}{\sqrt{20}}\\&=\frac{1}{2}\end{align}
\begin{align}\downarrow\end{align}
\begin{align}|\angle ABD|&=\tan^{-1}\left(\frac{1}{2}\right)\\&\approx27^{\circ}\end{align}
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Question 4
(a) Construct the triangle \(ABC\), where \(|AB|=8\mbox{ cm}\), \(|AC|=5\mbox{ cm}\), and \(|BC|=7\mbox{ cm}\).
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(b) The diagram shows a square of side length \(2k\mbox{ cm}\).
(i) Write down, in terms of \(k\), an expression for the area of the square.
(ii) An isosceles triangle with side lengths of \(20\mbox{ cm}\) and hypotenuse of length \(2k\mbox{ cm}\) is removed from the square, as shown.
Find the value of \(k\) (correct to \(2\) decimal places) and the area of the remaining (shaded) section.
(i) \(4k^2\mbox{ cm}^2\)
(ii) \(k=14.14\) and \(A=600\mbox{ cm}^2\)
(i)
\begin{align}A&=(2k)^2\\&=4k^2\mbox{ cm}^2\end{align}
(ii)
\begin{align}(2k)^2=20^2+20^2\end{align}
\begin{align}\downarrow\end{align}
\begin{align}4k^2=800\end{align}
\begin{align}\downarrow\end{align}
\begin{align}k&=\sqrt{200}\\&\approx14.14\end{align}
\begin{align}\downarrow\end{align}
\begin{align}A&=4k^2-\frac{1}{2}(20)(20)\\&=4(\sqrt{200}^2)-200\\&=600\mbox{ cm}^2\end{align}
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Question 5
(a) In a survey, the IQ scores of \(1200\) people were recorded. The mean score was \(100\) points
and the standard deviation was \(15\) points. Assuming that IQ scores are normally distributed, the data are shown on the diagram below.
(i) Fill in the missing numbers on the horizontal axis.
(ii) A person is chosen at random from those surveyed. Use the Empirical Rule to find the probability that this person has an IQ score between \(70\) and \(130\) points.
(iii) Use the Empirical Rule to find the approximate number of people surveyed with an IQ score of between \(85\) and \(115\) points.
(i)
(ii) \(0.95\)
(iii) \(816\)
(i)
(ii) \(0.95\)
(iii)
\begin{align}0.68\times1{,}200=816\end{align}
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(b) A class of \(24\) pupils is made up of \(10\) boys and \(14\) girls. Chemistry is studied by \(6\) of the boys and \(9\) of the girls. A pupil is chosen at random from the class. Find the probability that the pupil chosen is a boy or is a pupil who does not study chemistry.
\(\dfrac{15}{24}\)
\begin{align}P&=\frac{10}{24}+\frac{5}{24}\\&=\frac{15}{24}\end{align}
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Question 6
(a) Find the distance \(x\) in the diagram below (not to scale).
Give your answer correct to \(2\) decimal places.
\(8.84\mbox{ cm}\)
\begin{align}\frac{x}{\sin(180^{\circ}-65^{\circ}-63^{\circ})}=\frac{10}{\sin63^{\circ}}\end{align}
\begin{align}\downarrow\end{align}
\begin{align}x&=\frac{10\sin52^{\circ}}{\sin63^{\circ}}\\&\approx8.84\mbox{ cm}\end{align}
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(b) Find the distance \(y\) in the diagram below (not to scale).
Give your answer correct to \(2\) decimal places.
\(8.60\mbox{ cm}\)
\begin{align}y&=\sqrt{8.5^2+10.2^2-2(8.5)(10.2)\cos53.8^{\circ}}\\&\approx8.60\mbox{ cm}\end{align}
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Section B
Question 7
Cones is a sculpture in the National Gallery of Australia. It consists of \(14\) identical steel cones arranged into pairs which are joined together.
(a)
(i) The height of each cone is equal to the diameter of its base. If the radius of the base is \(2.25\mbox{ m}\), write the height of a cone.
flickr.com @russellstreet
(ii) Show that, correct to \(2\) decimal places, the slant height, ݈\(l\), of a cone is \(5.03\mbox{ m}\).
(i) \(4.5\mbox{ m}\)
(ii) The answer is already in the question!
(i)
\begin{align}h&=2r\\&=2(2.25)\\&=4.5\mbox{ m}\end{align}
(ii)
\begin{align}l&=\sqrt{h^2+r^2}\\&=\sqrt{4.5^+2.25^2}\\&\approx5.03\mbox{ m}\end{align}
as required.
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(b) In order to maintain the steel’s reflective shine, the surface is polished regularly.
(i) Find the curved surface area of the entire sculpture (\(14\) cones).
Give your answer correct to \(2\) decimal places.
(ii) One litre of polish will cover \(12.25\mbox{ m}^2\). Find how many litres are needed to polish the entire sculpture. Give your answer correct to the nearest litre.
(iii) A container of polish contains 5 litres and costs A$\(110\). Find the number of containers of polish that must be purchased in order to polish the entire sculpture and hence find the cost of the polish in euro ( A$\(1=\) €\(0.68\)). Give your answer correct to the nearest euro.
(i) \(497.77\mbox{ m}\)
(ii) \(41\mbox{ litres}\)
(iii) The number of containers needed is \(9\) and the cost is \(673\mbox{ euro}\).
(i)
\begin{align}14\times\pi rl&=14\pi(2.25)(5.03)\\&\approx497.77\mbox{ m}\end{align}
(ii)
\begin{align}\frac{497.77}{12.25}\approx41\mbox{ litres}\end{align}
(iii)
\begin{align}\frac{41}{5}=8.2\end{align}
Therefore, the number of containers needed is \(9\).
\begin{align}\mbox{Cost}&=9\times110\times0.68\\&=673\mbox{ euro}\end{align}
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(c) The diagram (not to scale) shows the net of the outer surface of one of the cones of the sculpture. It is a sector of a circle of radius length \(5.03\mbox{ m}\) with arc length \(p\mbox{ m}\) and angle \(\theta\) at the centre, as shown below.
(i) Find \(p\), the length of the arc of the sector. Give your answer correct to \(2\) decimal places.
(ii) Find \(\theta\), the angle at the centre of the sector. Show all your working out.
Give your answer correct to the nearest degree.
(i) \(4.14\mbox{ m}\)
(ii) \(161^{\circ}\)
(i)
\begin{align}p&=2\pi r_c\\&=2\pi(2.25)\\&\approx4.14\mbox{ m}\end{align}
(ii)
\begin{align}\frac{\theta}{360^{\circ}}\times2\pi r_s=14.14\end{align}
\begin{align}\downarrow\end{align}
\begin{align}\theta&=\frac{(14.14)(360^{\circ})}{2\pi(5.03)}\\&\approx161^{\circ}\end{align}
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Question 8
The diagram below shows the triangles \(CBA\) and \(CDE\).
(a) The coordinates of \(C\) are \((4.5,0)\).
From the diagram, write down the coordinates of the points \(A\), \(B\), \(D\) and \(E\).
\(A=(1,-2)\), \(B=(4,2)\), \(D=(6,-6)\) and \(E=(15,6)\)
\begin{align}A=(1,-2)&&B=(4,2)\end{align}
\begin{align}D=(6,-6)&&E=(15,6)\end{align}
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(b) Show, using slopes, that the line segments \([AB]\) and \([DE]\) are parallel.
The answer is already in the question!
\begin{align}m_{AB}&=\frac{y_2-y_1}{x_2-x_1}\\&=\frac{2-(-2)}{4-1}\\&=\frac{4}{3}\end{align}
and
\begin{align}m_{DE}&=\frac{y_2-y_1}{x_2-x_1}\\&=\frac{6-(-6)}{15-6}\\&=\frac{12}{9}\\&=\frac{4}{3}\end{align}
Therefore, the lines are indeed parallel.
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(c)
(i) Show that the area of the triangle \(CBA\) is \(4\) square units.
(ii) Find \(|AB|\), the distance from \(A\) to \(B\).
(iii) The triangle \(CDE\) is an enlargement of the triangle \(CBA\).
Given that \(|DE|=15\mbox{ units}\), find the scale factor of the enlargement.
(iv) Use this scale factor to find the area of the triangle \(CDE\).
(i) The answer is already in the question!
(ii) \(5\mbox{ units}\)
(iii) \(5\)
(iii) \(36\mbox{ square units}\)
(i)
\begin{align}(1,-2)&&(4,2)&&(4.5,0)\end{align}
\begin{align}\downarrow\end{align}
\begin{align}(1-4.5,-2)&&(4-4.5,2)&&(4.5-4.5,0)\end{align}
\begin{align}\downarrow\end{align}
\begin{align}(-3.5,-2)&&(-0.5,2)&&(0,0)\end{align}
\begin{align}\downarrow\end{align}
\begin{align}A&=\frac{1}{2}|x_1y_2-\_2y_1|\\&=\frac{1}{2}|(-3.5)(2)-(-2)(-0.5)|\\&=\frac{1}{2}|-8|\\&=4\mbox{ square units}\end{align}
as required.
(ii)
\begin{align}|AB|&=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\&=\sqrt{(4-1)^2+(2-(-2))^2}\\&=\sqrt{9+16}\\&=5\mbox{ units}\end{align}
(iii)
\begin{align}k&=\frac{15}{3}\\&=5\end{align}
(iii)
\begin{align}A&=4\times3^2\\&=36\mbox{ square units}\end{align}
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Question 9
In June \(2016\) the UK held a referendum on its membership of the EU. Table \(1\) below summarises
the results.
| Votes | |
|---|---|
|
Leave the EU |
\(17{,}410{,}742\) |
|
Remain a member of the EU |
\(16{,}141{,}241\) |
|
Valid votes |
\(33{,}551{,}983\) |
|
Invalid or blank votes |
|
|
Total votes |
\(33{,}577{,}342\) |
Source: The UK Electoral Commission
(a)
(i) Write the number of Invalid or blank votes into the table.
(ii) Write the number who voted to leave the EU as a percentage of the valid votes.
Give your answer correct to the nearest percent.
(i)
| Votes | |
|---|---|
|
Leave the EU |
\(17{,}410{,}742\) |
|
Remain a member of the EU |
\(16{,}141{,}241\) |
|
Valid votes |
\(33{,}551{,}983\) |
|
Invalid or blank votes |
\(25{,}359\) |
|
Total votes |
\(33{,}577{,}342\) |
(ii) \(52\%\)
(i)
| Votes | |
|---|---|
|
Leave the EU |
\(17{,}410{,}742\) |
|
Remain a member of the EU |
\(16{,}141{,}241\) |
|
Valid votes |
\(33{,}551{,}983\) |
|
Invalid or blank votes |
\(25{,}359\) |
|
Total votes |
\(33{,}577{,}342\) |
(ii)
\begin{align}\frac{17{,}410{,}742}{33{,}551{,}983}\times100\approx52\%\end{align}
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(b) Table \(2\) shows the percentages of “Remain” and “Leave” voters in various age groups of those who voted in the referendum.
| Age Group | Remain (%) | Leave (%) |
|---|---|---|
|
18 - 24 |
\(73\) |
\(27\) |
|
25 - 34 |
\(62\) |
\(38\) |
|
35 - 44 |
\(52\) |
\(48\) |
|
45 - 54 |
\(44\) |
\(56\) |
|
55 - 64 |
\(43\) |
\(57\) |
|
65+ |
\(40\) |
\(60\) |
(i) Draw a suitable chart or charts to represent the data in Table \(2\).
(ii) Find the mean of the “Remain” values given in Table \(2\) and find the mean of the “Leave” values given in Table \(2\). Give your answers as percentages, correct to \(2\) decimal places.
(iii) Explain why the answers to part b(ii) do not accurately reflect the actual outcome of the referendum.
(i)
(ii) Remain: \(52.33\%\). Leave: \(47.67\%\).
(iii) There is not an equal number of people in each age group.
(i)
(ii)
Remain
\begin{align}\frac{73+62+52+44+43+40}{6}\approx52.33\%\end{align}
\[\,\]
Leave
\begin{align}\frac{27+38+48+56+57+60}{6}\approx47.67\%\end{align}
(iii) There is not an equal number of people in each age group.
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(c) In the days following the UK referendum, a survey was conducted in Ireland on attitudes
towards a British exit from the European Union.
(i) \(1200\) people were surveyed. Find the margin of error of this survey.
Write your answer as a percentage. Give your answer correct to the nearest percent.
(ii) In the Irish survey \(578\) of the \(1200\) surveyed agreed that a UK exit would have a negative effect on the Irish economy. Use your answer to part c(i) above to create a \(95\%\) confidence interval for the proportion of the Irish population who agreed that a UK exit would have a negative effect on the Irish economy.
(iii) After the survey, a political party claimed that \(53\%\) of the Irish population believed that the decision of the UK to leave the EU would have a negative effect on the Irish economy.
Use your answer to part c(ii) above to conduct a hypothesis test, at the \(5\%\) level of significance, to test the party’s claim.
Give your conclusion in the context of the question.
(i) \(3\%\)
(ii) \(45\%<\hat{p}<51\%\)
(iii)
Null hypothesis: The percentage of the Irish population that believe that the decision of the UK to leave the EU would have a negative effect on the Irish economy is \(53\%\).
Alternative hypothesis: The percentage of the Irish population that believe that the decision of the UK to leave the EU would have a negative effect on the Irish economy is not \(53\%\).
Calculations: \(53%\) is not within the confidence interval calculated above.
Conclusion: The percentage of the Irish population that believe that the decision of the UK to leave the EU would have a negative effect on the Irish economy is not \(53\%\).
(i)
\begin{align}\mbox{Margin of error}&=\frac{1}{\sqrt{n}}\\&=\frac{1}{\sqrt{1200}}\\&=0.0288…\\&\approx3\%\end{align}
(ii)
\begin{align}\frac{578}{1200}\times100-3<\hat{p}<\frac{578}{1200}\times100+3\end{align}
\begin{align}\downarrow\end{align}
\begin{align}45\%<\hat{p}<51\%\end{align}
(iii)
Null hypothesis: The percentage of the Irish population that believe that the decision of the UK to leave the EU would have a negative effect on the Irish economy is \(53\%\).
Alternative hypothesis: The percentage of the Irish population that believe that the decision of the UK to leave the EU would have a negative effect on the Irish economy is not \(53\%\).
Calculations: \(53%\) is not within the confidence interval calculated above.
Conclusion: The percentage of the Irish population that believe that the decision of the UK to leave the EU would have a negative effect on the Irish economy is not \(53\%\).
