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2018 Ordinary Level - Paper Two
Section A
Question 1
(a) An experiment consists of throwing two fair, standard, six-sided dice and recording the sum of the two numbers thrown. Some of the totals are shown in the table.
(a) An experiment consists of throwing two fair, standard, six-sided dice and recording the sum of the two numbers thrown. Some of the totals are shown in the table.
(i) Complete the table.
(ii) Find the probability of getting a total of \(7\) or \(11\).
(iii) Find the probability of getting a total which is a prime number.
(i)
(ii) \(\dfrac{2}{9}\)
(iii) \(\dfrac{5}{12}\)
(i)
(ii)
\begin{align}P&=\frac{6+2}{36}\\&=\frac{2}{9}\end{align}
(iii)
\begin{align}P&=\frac{1+2+4+6+2}{36}\\&=\frac{5}{12}\end{align}
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(b) A car distributor sells Ford cars and Renault cars.
It has \(30\) cars for sale on a particular day; \(18\) are Ford cars and \(12\) are Renault cars.
\(7\) of the Ford cars are red and \(4\) of the Renault cars are red. One of the \(30\) cars is chosen at random. What is the probability that the car chosen is a Ford car or a car which is not red?
\(\dfrac{13}{15}\)
\begin{align}P&=\frac{18}{30}+\frac{8}{30}\\&=\frac{13}{15}\end{align}
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Question 2
The points \(P(7,10)\), \(Q(1,2)\) and \(R(11,4)\) are the vertices of the triangle shown.
The point \(U(4,6)\) is the midpoint of \([PQ]\) and the point \(V\) is the midpoint of \([PR]\).
(a) Find the co-ordinates of \(V\).
\((9,7)\)
\begin{align}V&=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\&=\left(\frac{7+11}{2},\frac{10+4}{2}\right)\\&=(9,7)\end{align}
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(b) Show, by using slopes, that \(UV\) is parallel to \(QR\).
The answer is already in the question!
\begin{align}m_{UV}&=\frac{y_2-y_1}{x_2-x_1}\\&=\frac{7-6}{9-4}\\&=\frac{1}{5}\end{align}
and
\begin{align}m_{QR}&=\frac{y_2-y_1}{x_2-x_1}\\&=\frac{4-2}{11-1}\\&=\frac{1}{5}\end{align}
as required.
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(c) Find the area of the triangle \(PQR\).
\(34\mbox{ units}^2\)
\begin{align}(7,10)&&(1,2)&&(11,4)\end{align}
\begin{align}\downarrow\end{align}
\begin{align}(7-7,10-10)&&(1-7,2-10)&&(11-7,4-10)\end{align}
\begin{align}\downarrow\end{align}
\begin{align}(0,0)&&(-6,-8)&&(4,-6)\end{align}
\begin{align}\downarrow\end{align}
\begin{align}A&=\frac{1}{2}|x_1y_2-x_2y_1|\\&=\frac{1}{2}|(-6)(-6)-(4)(-8)|\\&=34\mbox{ units}^2\end{align}
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(d) The point \(S\) is the image of the point \(Q\) under the translation \(\vec{UV}\).
Find the coordinates of \(S\).
\((6,3)\)
\begin{align}S&=(1+5,2+1)\\&=(6,3)\end{align}
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Question 3
(a)
(i) Find the number of different arrangements that can be made using all the letters of the word RAINBOW. Each letter is used only once.
(ii) Find the number of different \(3\)-letter arrangements that can be made using the letters
of the word RAINBOW. Each letter is used at most once.
(i) \(5{,}040\)
(ii) \(210\)
(i)
\begin{align}7!=5{,}040\end{align}
(ii)
\begin{align}7\times6\times5=210\end{align}
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(b) A game, called Rainbow, uses an unbiased circular spinner.
The spinner has seven sectors coloured red (\(R\)), orange (\(O\)), yellow (\(Y\)), green (\(G\)), blue (\(B\)), indigo (\(I\)), and violet (\(V\)) as shown below.
The table below shows the angle of each sector.
It also shows the cash prize that a player wins if the spinner stops in that sector.
| Colour | Angle | Probability | Prize |
|---|---|---|---|
|
Red |
\(72^{\circ}\) |
|
\(20\mbox{ euro}\) |
|
Orange |
\(30^{\circ}\) |
|
\(60\mbox{ euro}\) |
|
Yellow |
\(45^{\circ}\) |
\(\dfrac{1}{8}\) |
\(24\mbox{ euro}\) |
|
Green |
\(90^{\circ}\) |
|
\(8\mbox{ euro}\) |
|
Blue |
\(60^{\circ}\) |
|
\(42\mbox{ euro}\) |
|
Indigo |
\(18^{\circ}\) |
|
\(90\mbox{ euro}\) |
|
Violet |
\(45^{\circ}\) |
|
\(48\mbox{ euro}\) |
(i) Complete the “Probability” column of the table which shows the probability of the spinner coming to rest in each sector after one spin.
(ii) Find the expected value of the prize that a player wins if they play Rainbow.
(i)
| \(t\) | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) |
|---|---|---|---|---|---|---|---|---|---|
|
\(\mathbf{h(t)}\) |
|
\(42\) |
|
|
|
|
|
|
|
(ii) \(31.50\mbox{ euro}\)
(i)
| \(t\) | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) |
|---|---|---|---|---|---|---|---|---|---|
|
\(\mathbf{h(t)}\) |
|
\(42\) |
|
|
|
|
|
|
|
(ii)
\begin{align}E(X)&=20\left(\frac{1}{5}\right)+60\left(\frac{1}{12}\right)+24\left(\frac{1}{8}\right)+8\left(\frac{1}{4}\right)+42\left(\frac{1}{6}\right)+90\left(\frac{1}{20}\right)+48\left(\frac{1}{8}\right)\\&=31.50\mbox{ euro}\end{align}
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Question 4
The points \(A(1,8)\) and \(B(9,0)\) are the end-points of a diameter of the circle \(w\), as shown in the diagram.
(a) Find the co-ordinates of the centre of \(w\).
\((5,4)\)
\begin{align}\mbox{Centre}&=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\&=\left(\frac{1+9}{2},\frac{8+0}{2}\right)\\&=(5,4)\end{align}
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(b) Find the length of the radius of \(w\). Give your answer in the form \(p\sqrt{q}\), where \(p,q\in\mathbb{N}\).
\(4\sqrt{2}\mbox{ units}\)
\begin{align}r&=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\&=\sqrt{(5-1)^2+(4-8)^2}\\&=\sqrt{32}\\&=\sqrt{16\times2}\\&=\sqrt{16}\times\sqrt{2}\\&=4\sqrt{2}\mbox{ units}\end{align}
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(c) Hence write down the equation of the circle \(w\).
\((x-5)^2+(y-4)^2=32\)
\begin{align}(x-5)^2+(y-4)^2=32\end{align}
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(d) Find the equation of the line that is a tangent to the circle \(w\) at \(A\).
Give your answer in the form \(ax+by+c=0\), where \(a,b\) and \(c\in\mathbb{Z}\).
\(x-y+7=0\)
\begin{align}m_{AB}&=\frac{y_2-y_1}{x_2-x_1}\\&=\frac{0-8}{9-1}\\&=-1\end{align}
\begin{align}\downarrow\end{align}
\begin{align}m_{\perp}=1\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-8=1(x-1)\end{align}
\begin{align}\downarrow\end{align}
\begin{align}y-8=x-1\end{align}
\begin{align}\downarrow\end{align}
\begin{align}x-y+7=0\end{align}
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Question 5
(a) The square \(ABCD\) has sides of length \(7\mbox{ cm}\).
The vertices of the square \(PQRS\) lie on the perimeter of \(ABCD\), as shown in the diagram, with \(|AQ|=5\mbox{ cm}\).
Find the area of the square \(PQRS\).
\(29\mbox{ cm}^2\)
\begin{align}|PQ|&=\sqrt{|AP|^2+|AQ|^2}\\&=\sqrt{2^2+5^2}\\&=\sqrt{29}\end{align}
\begin{align}\downarrow\end{align}
\begin{align}A&=|PQ|^2\\&=(\sqrt{29})^2\\&=29\mbox{ cm}^2\end{align}
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(b) The circles \(u\) and \(v\) represent two wheels that are free to rotate about their centres, as shown.
The radius of \(u\) is \(4\mbox{ cm}\) and the radius of \(v\) is \(6\mbox{ cm}\).
(i) Find the length of the circumference of each circle.
Give your answers in \(\mbox{cm}\) in terms of \(\pi\).
(ii) The wheels \(u\) and \(v\) are in non-slip contact and therefore the rotation of one causes the other to rotate. Find the number of complete rotations wheel \(u\) makes if wheel \(v\) completes \(100\) rotations.
(i) \(C_u=8\pi\) and \(C_v=12\pi\)
(ii) \(150\mbox{ rotations}\)
(i)
\begin{align}C_u&=2\pi r_u\\&=2\pi(4)\\&=8\pi\mbox{ cm}\end{align}
and
\begin{align}C_v&=2\pi r_v\\&=2\pi(6)\\&=12\pi\mbox{ cm}\end{align}
(ii)
\begin{align}\frac{12\pi}{8\pi}\times100=150\mbox{ rotations}\end{align}
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Question 6
(a)
(i) Construct the triangle \(ABC\). where \(|AB|=10\mbox{ cm}\), \(|\angle CAB|=60^{\circ}\) and \(|\angle ABC|=40^{\circ}\).
Label each vertex clearly.
(ii) Measure \(|BC|\) and write your answer in \(\mbox{cm}\), correct to \(1\) decimal place.
(i)
(ii) \(8.8\mbox{ cm}\)
(i)
(ii)
\begin{align}|BC|=8.8\mbox{ cm}\end{align}
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(b) The diagram shows a parallelogram with vertices \(P\), \(Q\), \(R\) and \(S\).
\(|\angle SPQ|=115^{\circ}\), \(|\angle QRS|=\alpha^{\circ}\) and \(|\angle RSP|=\beta^{\circ}\).
(i) Write down the value of \(\alpha\) and the value of \(\beta\).
(ii) Explain why the triangle \(PQR\) is congruent to triangle \(RSP\).
Give a reason for any statement you make in your explanation.
(i) \(\alpha=115^{\circ}\) and \(\beta=65^{\circ}\)
(ii) The answer is already in the question!
(i)
\begin{align}\alpha=115^{\circ}\end{align}
and
\begin{align}\beta&=\frac{1}{2}(360^{\circ}-115^{\circ}-115^{\circ})\\&=65^{\circ}\end{align}
(ii)
\(|SR|=|PQ|\) (opposite sides of parallelogram)
and
\(|SP|=|RQ|\) (opposite sides of parallelogram)
and third side is shared. Therefore, according to SSS rule, they are congruent.
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Section B
Question 7
The table below shows the total rainfall, in millimetres, and the total sunshine, in hours, at Valentia, County Kerry, during the month of June from \(2001\) to \(2010\).
| Year | \(2001\) | \(2002\) | \(2003\) | \(2004\) | \(2005\) | \(2006\) | \(2007\) | \(2008\) | \(2009\) | \(2010\) |
|---|---|---|---|---|---|---|---|---|---|---|
|
Total Rainfall (mm) |
\(72\) |
\(133\) |
\(155\) |
\(101\) |
\(94\) |
\(47\) |
\(149\) |
\(134\) |
\(94\) |
\(84\) |
|
Total Sunshine (hours) |
\(169\) |
\(124\) |
\(180\) |
\(173\) |
\(173\) |
\(239\) |
\(159\) |
\(168\) |
\(228\) |
\(205\) |
Source: Met Éireann
(a) Based on the data in the table above write down:
(i) the range of the rainfall data
(ii) the year with the highest June rainfall
(iii) the year with the least June sunshine
(i) \(108\mbox{ mm}\)
(ii) \(2003\)
(ii) \(2002\)
(i) \(155-47=108\mbox{ mm}\)
(ii) \(2003\)
(ii) \(2002\)
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(b) Based on the data in the table, write down the year with the best June weather and give a reason for your answer.
\(2006\) as it had both the least rainfall and the most sunshine.
\(2006\) as it had both the least rainfall and the most sunshine.
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(c) Write the rainfall data in increasing order and hence find the median of the rainfall.
\(47,72,84,94,94,101,133,134,149,155\) and \(\mbox{median}=97.5\mbox{ mm}\)
\begin{align}47&&72&&84&&94&&94&&101&&133&&134&&149&&155\end{align}
\begin{align}\downarrow\end{align}
\begin{align}\mbox{Median}&=\frac{94+101}{2}\\&=97.5\mbox{ mm}\end{align}
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(d)
(i) Find the mean number of sunshine hours for June in Valentia between \(2001\) and \(2010\).
(ii) For what years was the sunshine data within \(5\%\) of the mean number of sunshine hours in Valentia?
(i) \(181.8\mbox{ hours}\)
(ii) \(2003\), \(2004\) and \(2005\)
(i)
\begin{align}\mbox{Mean}&=\frac{169+124+180+173+173+239+159+168+228+205}{10}\\&=181.8\mbox{ hours}\end{align}
(ii)
\begin{align}0.95\times181<x<1.05\times181\end{align}
\begin{align}\downarrow\end{align}
\begin{align}172.71<x<190.89\end{align}
The years within this range are \(2003\), \(2004\) and \(2005\).
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(e) Find the standard deviation of the rainfall data, in \(\mbox{mm}\), correct to \(1\) decimal place.
\(33.5\)
\begin{align}\sigma\approx33.5\end{align}
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(f) Part of a scatterplot of the data in the table is shown below.
The first four data points are plotted.
(i) Complete the scatterplot.
(ii) One of the numbers in the table is the correlation coefficient for the data above, correct to \(1\) decimal place.
Based on the scatterplot, select the number that you think most accurately reflects this data. Explain your choice.
| Tick one box | |
|---|---|
|
\(0.6\) |
|
|
\(0.1\) |
|
|
\(-0.1\) |
|
|
\(-0.6\) |
|
(i)
(ii) \(-0.6\) as the correlation is negative and moderate.
(i)
(ii) \(-0.6\) as the correlation is negative and moderate.
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Question 8
The diagram shows a section of a garden divided into three parts.
In the diagram: \(|Pr|=3.3\mbox{ m}\), \(|PQ|=6.5\mbox{ m}\), \(|QT|=|QS|=8\mbox{ m}\), \(|\angle QRP|=90^{\circ}\), \(|\angle PQR|=\alpha^{\circ}\), and \(|\angle RQS|=\beta^{\circ}\).
(a) Use the theorem of Pythagoras to find \(|RQ|\).
\(5.6\mbox{ m}\)
\begin{align}|RQ|&=\sqrt{|PQ|^2-|PR|^2}\\&=\sqrt{6.5^2-3.3^2}\\&=5.6\mbox{ m}\end{align}
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(b) Show that \(\alpha=31^{\circ}\), correct to the nearest degree.
The answer is already in the question!
\begin{align}\alpha&=\sin^{-1}\left(\frac{3.3}{6.5}\right)\\&\approx31^{\circ}\end{align}
as required.
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(c) Use the value of \(\alpha\) given in part (b) to find the value of \(\beta\).
\(149^{\circ}\)
\begin{align}\beta&=180^{\circ}-31^{\circ}\\&=149^{\circ}\end{align}
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(d) Use the Cosine Rule to find the length of \([RS]\).
Give your answer correct to the nearest metre.
\(13\mbox{ m}\)
\begin{align}|RS|&=\sqrt{|RQ|^2+|QS|^2-2|RQ||QS|\cos\beta}\\&=\sqrt{5.6^2+8^2-2(5.6)(8)\cos(149^{\circ})}\\&\approx13\mbox{ m}\end{align}
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(e) \(SQT\) is a sector of a circle whose centre is \(Q\).
Find the length of the arc \(TS\).
Give your answer in metres, correct to one decimal place
\(4.3\mbox{ m}\)
\begin{align}l&=2\pi r\left(\frac{\theta}{360^{\circ}}\right)\\&=2\pi (8)\left(\frac{31^{\circ}}{360^{\circ}}\right)\\&\approx4.3\mbox{ m}\end{align}
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(f) Find the area of the sector \(SQT\).
Give your answer in square metres, correct to one decimal place.
\(17.3\mbox{ m}^2\)
\begin{align}A&=\pi r^2\left(\frac{\theta}{360^{\circ}}\right)\\&2\pi (8)^2\left(\frac{31^{\circ}}{360^{\circ}}\right)\\&\approx17.3\mbox{ m}^2\end{align}
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Question 9
(a)
(i) Find the volume of a solid sphere of radius \(0.3\mbox{ cm}\).
Give your answer in \(\mbox{cm}^3\), correct to \(3\) decimal places.
(ii) The sphere is made of pure gold. Each \(\mbox{cm}^3\) of pure gold weighs \(19.3\) grams.
Find the number of grams of pure gold in the sphere.
Give your answer correct to \(2\) decimal places.
(iii) It is known that there are approximately \(6.02\times 10^{23}\) atoms in \(197\) grams of pure gold.
Find the number of atoms of pure gold in the sphere.
Give your answer in the form \(a\times 10^n\), where \(1\leq a<10\), \(n\in\mathbb{N}\), and where \(\alpha\) is correct to \(2\) significant figures.
(i) \(0.113\mbox{ cm}^3\)
(ii) \(2.18\mbox{ g}\)
(iii) \(6.7\times10^{21}\mbox{ atoms}\)
(i)
\begin{align}V&=\frac{4}{3}\pi r^3\\&=\frac{4}{3}\pi(0.3)^3\\&\approx0.113\mbox{ cm}^3\end{align}
(ii)
\begin{align}0.113\times19.3\approx2.18\mbox{ g}\end{align}
(iii)
\begin{align}2.18\times\frac{6.02\times10^{23}}{197}\approx6.7\times10^{21}\mbox{ atoms}\end{align}
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(b) A survey was carried out on behalf of a television station to investigate the popularity of a certain show.
(i) A random sample of \(1560\) television viewers was surveyed. Find the margin of error of the survey. Give your answer as a percentage, correct to \(1\) decimal place.
(ii) In the survey, \(546\) of the \(1560\) viewers surveyed said that they liked the show. Use your answer to part b(i) above to create a \(95%\) confidence interval for the percentage of viewers who liked the show.
(iii) An executive for the television station had claimed that \(40\%\) of viewers liked the show.
Use your answer to part b(ii) above to conduct a hypothesis test, at the \(5\%\) level of significance, to test the executive’s claim. State your null hypothesis, your alternative hypothesis and give your conclusion in the context of the question.
(i) \(2.5\%\)
(ii) \(32.5\%<\hat{p}<37.5\%\)
(iii)
Null hypothesis: The percentage of viewers that liked the show was \(40\%\).
Alternative hypothesis: The percentage of viewers that liked the show was not \(40\%\).
Calculations: \(40%\) is not within the confidence interval derived above.
Reasoning: The percentage of viewers that liked the show was not \(40\%\).
(i)
\begin{align}\mbox{Margin of error}&=\frac{1}{\sqrt{n}}\\&=\frac{1}{\sqrt{1{,}560}}\\&=0.0253…\\&\approx2.5\%\end{align}
(ii)
\begin{align}\frac{546}{1{,}560}\times100-2.5<\hat{p}<\frac{546}{1{,}560}\times100+2.5\end{align}
\begin{align}\downarrow\end{align}
\begin{align}32.5\%<\hat{p}<37.5\%\end{align}
(iii)
Null hypothesis: The percentage of viewers that liked the show was \(40\%\).
Alternative hypothesis: The percentage of viewers that liked the show was not \(40\%\).
Calculations: \(40%\) is not within the confidence interval derived above.
Reasoning: The percentage of viewers that liked the show was not \(40\%\).
