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2016 Ordinary Level - Paper Two
Section A
Question 1
(a) A survey of \(168\) people was carried out. Participants were asked whether they owned a cat or a dog. Some of the results are recorded in the Venn diagram below.
(i) Of those surveyed, \(19\) people did not own either a cat or a dog. Complete the diagram.
(i) Of those surveyed, \(19\) people did not own either a cat or a dog. Complete the diagram.
(i)
(ii) \(\dfrac{64}{168}\)
(iii) \(50.6\%\)
(i)
(ii)
\begin{align}\frac{64}{168}\end{align}
(iii)
\begin{align}\frac{37+48}{168}&=50.59\\&=50.6\%\end{align}
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(b) Mandy wants to buy a ticket to the theatre. She can choose a ticket in the Balcony (\(B\)) or the Stalls (\(S\)). Mandy can go on Monday (\(M\)), Wednesday (\(W\)) or Friday (\(F\)). For an extra charge she can choose a VIP ticket (\(V\)) where she will meet the band or she can choose the show only (\(O\)).
Complete the tree diagram below, and hence or otherwise, find the probability that Mandy chooses a VIP ticket on a Wednesday.
You may assume that all choices are equally likely.
\(\dfrac{1}{6}\)
\begin{align}P&=\frac{1}{12}+\frac{1}{12}\\&=\frac{1}{6}\end{align}
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Question 2
(a) Find the area of the given triangle.
\(24\mbox{ cm}^2\)
\begin{align}A&=\frac{1}{2}ab\sin C\\&=\frac{1}{2}(8)(12)\sin30^{\circ}\\&=24\mbox{ cm}^2\end{align}
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(b) A triangle has sides of length \(3\mbox{ cm}\), \(5\mbox{ cm}\), and \(7\mbox{ cm}\).
Find the size of the largest angle in the triangle.
\(120^{\circ}\)
\begin{align}7^2=5^2+3^2-2(5)(3)\cos A\end{align}
\begin{align}\downarrow\end{align}
\begin{align}A&=\cos^{-1}\left(\frac{5^2+3^2-7^2}{2(5)(3)}\right)\\&=120^{\circ}\end{align}
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Question 3
(a) The circle \(c\) has centre \((2,-3)\) and a radius of \(4\mbox{ cm}\).
Write down the equation of \(c\).
\((x-2)^2+(y+3)^2=16\)
\begin{align}(x-2)^2+(y+3)^2=16\end{align}
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(b) Draw the circle \(c\) on the grid. Each unit on the co-ordinate grid is \(1\mbox{ cm}\).
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(c) Verify, using algebra, that the point \((3,1)\) is outside of \(c\).
The answer is already in the question!
\begin{align}(x-2)^2+(y+3)^2&=(3-2)^2+(1+3)^2\\&=1+16\\&=17\\&>16\end{align}
as required.
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(d) Find the area of the smallest four-sided figure that will fit around the circle \(c\).
\(64\mbox{ cm}^2\)
\begin{align}A&=2r\times2r\\&=4r^2\\&=4(4^2)\\&=64\mbox{ cm}^2\end{align}
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Question 4
(a) The line \(l\) contains the points \(A(4,5)\) and \(B(2,0)\). Find the equation of \(l\).
Give your answer in the form \(ax+by+c=0\) where \(a\), \(b\), and \(c\in\mathbb{Z}\).
\(5x-2y-10=0\)
\begin{align}m&=\frac{y_2-y_1}{x_2-x_1}\\&=\frac{0-5}{2-4}\\&=\frac{5}{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-5=\frac{5}{2}(x-4)\end{align}
\begin{align}\downarrow\end{align}
\begin{align}2(y-5)=5(x-4)\end{align}
\begin{align}\downarrow\end{align}
\begin{align}2y-10=5x-20\end{align}
\begin{align}\downarrow\end{align}
\begin{align}5x-2y-10=0\end{align}
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(b) Draw the line \(k:x+2y=8\) on the axes below.
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(c) Use a graphic, numeric or algebraic method to find the co-ordinates of \(l\cap k\).
\begin{align}l\cap k=(3,2.5)\end{align}
\begin{align}l\cap k=(3,2.5)\end{align}
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Question 5
The waiting times, in minutes, for \(16\) patients at a dentist’s surgery are recorded for a particular week (Week 1) on the following stem and leaf plot.
(a) Find the mode and the median of the data.
\(\mbox{Mode}=12\mbox{ min}\) and \(\mbox{Median}=24.5\mbox{ min}\)
\begin{align}\mbox{Mode}=12\mbox{ min}\end{align}
and
\begin{align}\mbox{Median}&=\frac{24+25}{2}\\&=24.5\mbox{ min}\end{align}
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(b) Find the mean waiting time for Week 1, correct to \(1\) decimal place.
\(22.7\mbox{ min}\)
\begin{align}\mbox{Mean}&=\frac{5+8+12+12+12+12+13+20+24+25+27+30+31+32+34+34+44}{16}\\&\approx22.7\mbox{ min}\end{align}
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(c) The waiting times were recorded again the following week. The results were:
\(27, 23, 6, 15, 18, 29, 16, 17, 15, 18, 40, 32, 16, 12, 28, 9\)
Show these results on the plot above (under Week 2), creating a back-to-back stem and leaf plot to display the data.
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Question 6
(a)
(i) Construct a triangle \(ABC\), where \(|AB|=7\mbox{ cm}\), \(|\angle BAC|=50^{\circ}\), and \(|AC|=4.5\mbox{ cm}\).
(ii) Measure the length of \([BC]\) and hence find the sum of the lengths of the
sides \([AC]\) and \([BC]\), correct to one decimal place.
(i)
(ii) \(|AC|=4.5\mbox{ cm}\), \(|BC|=5.4\mbox{ cm}\) and \(\mbox{sum}=9.9\mbox{ cm}\)
(i)
(ii) \(|AC|=4.5\mbox{ cm}\), \(|BC|=5.4\mbox{ cm}\) and \(\mbox{sum}=9.9\mbox{ cm}\)
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(b) State which one of the following triangles can not be constructed.
Give a reason to support your answer.
- Triangle 1: Sides of length (cm) \(3.9\), \(2.9\), \(5.4\)
- Triangle 2: Sides of length (cm) \(6\), \(7\), \(15\)
Triangle \(2\) as the two smaller sides cannot reach the longer side.
Triangle \(2\) as the two smaller sides cannot reach the longer side.
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(c) The lengths of the sides of a right-angled triangle are \(5\), \(x\) and \(x+1\) as shown.
Use the Theorem of Pythagoras to find the value of \(x\).
\(x=12\)
\begin{align}(x+1)^2=5^2+x^2\end{align}
\begin{align}\downarrow\end{align}
\begin{align}x^2+2x+1=25+x^2\end{align}
\begin{align}\downarrow\end{align}
\begin{align}2x=24\end{align}
\begin{align}\downarrow\end{align}
\begin{align}x=12\end{align}
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Section B
Question 7
The Atomium in Brussels is one of Belgium’s most famous landmarks.
It consists of \(9\) identical spheres joined by two types of cylindrical pipes.
(a) The Atomium is modelled on an iron atom that has been magnified \(165\) billion times.
Given that a billion is a thousand million, write \(165\) billion in the form \(a\times 10^n\), where \(n\in\mathbb{Z}\), and \(1\leq a<10\).
\(1.65\times10^{11}\)
\begin{align}165\mbox{ billion}&=165\times10^3\times10^6\\&=165\times10^9\\&=1.65\times10^{11}\end{align}
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(b) The diameter of each sphere in the Atomium is \(18\) metres.
(i) Find the radius of each sphere.
(ii) Find the volume of each sphere, correct to 2 decimal places.
(i) \(9\mbox{ m}\)
(ii) \(3{,}053.63\mbox{ m}^3\)
(i)
\begin{align}r&=\frac{d}{2}\\&=\frac{18}{2}\\&=9\mbox{ m}\end{align}
(ii)
\begin{align}V&=\frac{4}{3}\pi r^3\\&=\frac{4}{3}\pi(9^3)\\&\approx3{,}053.63\mbox{ m}^3\end{align}
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(c) Find the combined surface area of all 9 spheres in the Atomium, correct to the nearest \(\mbox{m}^2\).
\(9{,}161\mbox{ m}^2\)
\begin{align}A&=9\times4\pi r^2\\&=36\pi r^2\\&=36\pi(9^2)\\&\approx9{,}161\mbox{ m}^2\end{align}
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(d) Each of the \(8\) cylindrical pipes extending from the centre sphere has a radius of \(1.65\mbox{ m}\) and a length of \(23\mbox{ m}\).
(i) Find the sum of the curved surface areas of all \(8\) pipes, correct to the nearest \(\mbox{m}^2\).
(ii) The other \(12\) cylindrical pipes connect the outer spheres to each other. Each pipe has a radius of \(1.45\mbox{ m}\). All \(12\) pipes are equal in length. The sum of the curved surface areas of the \(12\) pipes is \(3170\mbox{ m}^2\). Find the length of one pipe.
Give your answer correct to the nearest metre.
(iii) The curved surfaces of the \(20\) pipes and \(9\) spheres are covered in stainless steel.
Stainless steel costs €\(70\) per square metre. Use the areas you have calculated or have been given above to find the approximate cost of the stainless steel required to resurface the Atomium.
(i) \(1{,}908\mbox{ m}^2\)
(ii) \(9\mbox{ m}\)
(iii) \(996{,}730\mbox{ euro}\)
(i)
\begin{align}CSA&=8\times(2\pi rh)\\&=16\pi rh\\&=16\pi(1.65)(23)\\&\approx1{,}908\mbox{ m}^2\end{align}
(ii)
\begin{align}12\times[2\pi (1.45)h]=3{,}170\end{align}
\begin{align}\downarrow\end{align}
\begin{align}h&=\frac{3{,}170}{(12)(2)\pi(1.45)}\\&\approx29\mbox{ m}\end{align}
(iii)
\begin{align}(9{,}161+1{,}908+3{,}170)\times70=996{,}730\mbox{ euro}\end{align}
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Question 8
The following table shows some of the data gathered by a company on average water usage in homes.
| Activity | Percentage Used | Angle (Degrees) |
|---|---|---|
|
WC flushing |
\(30\%\) |
|
|
Personal Washing - Baths and Taps |
\(21\%\) |
|
|
Personal Washing - Showers |
\(12\%\) |
|
|
Clothes Washing |
\(13\%\) |
\(46.8^{\circ}\) |
|
Other |
|
|
(a) Find the percentage used for the ‘Other’ activity, and write it in the table.
\(\mbox{Other}=24\%\)
| Activity | Percentage Used | Angle (Degrees) |
|---|---|---|
|
WC flushing |
\(30\%\) |
|
|
Personal Washing - Baths and Taps |
\(21\%\) |
|
|
Personal Washing - Showers |
\(12\%\) |
|
|
Clothes Washing |
\(13\%\) |
\(46.8^{\circ}\) |
|
Other |
\(24\%\) |
|
\begin{align}\mbox{Other}&=100\%-30\%-21\%-12\%-13\%\\&=24\%\end{align}
| Activity | Percentage Used | Angle (Degrees) |
|---|---|---|
|
WC flushing |
\(30\%\) |
|
|
Personal Washing - Baths and Taps |
\(21\%\) |
|
|
Personal Washing - Showers |
\(12\%\) |
|
|
Clothes Washing |
\(13\%\) |
\(46.8^{\circ}\) |
|
Other |
\(24\%\) |
|
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(b) A pie chart representing the data is to be drawn.
(i) The size of the angle representing ‘Clothes Washing’ on the pie chart is \(46.8^{\circ}\).
Complete the table above to show the remaining angles in the pie chart.
(ii) Draw a pie chart to represent the data. Label the sector corresponding to each activity and write the size of the angle in each sector.
(i)
| Activity | Percentage Used | Angle (Degrees) |
|---|---|---|
|
WC flushing |
\(30\%\) |
\(108^{\circ}\) |
|
Personal Washing - Baths and Taps |
\(21\%\) |
\(75.6^{\circ}\) |
|
Personal Washing - Showers |
\(12\%\) |
\(43.2^{\circ}\) |
|
Clothes Washing |
\(13\%\) |
\(46.8^{\circ}\) |
|
Other |
\(24\%\) |
\(86.4^{\circ}\) |
(ii)
(i)
| Activity | Percentage Used | Angle (Degrees) |
|---|---|---|
|
WC flushing |
\(30\%\) |
\(108^{\circ}\) |
|
Personal Washing - Baths and Taps |
\(21\%\) |
\(75.6^{\circ}\) |
|
Personal Washing - Showers |
\(12\%\) |
\(43.2^{\circ}\) |
|
Clothes Washing |
\(13\%\) |
\(46.8^{\circ}\) |
|
Other |
\(24\%\) |
\(86.4^{\circ}\) |
(ii)
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(c) John compiled a table showing the amount of water used in his own household over the course of a week. He recorded the number of litres required for each activity and the number of times each activity was undertaken.
The data are shown in the table below.
| Activity | Water Required per Activity | Frequency | Number of litres Used |
|---|---|---|---|
|
One bath |
\(80\) litres |
\(3\) baths |
\(240\) |
|
One Shower |
\(125\) litres |
\(20\) showers |
|
|
Brushing Teeth with Tap Running |
\(6\) litres per minute |
\(32\) minutes |
|
|
One WC Flush |
\(6\) litres |
\(60\) flushes |
|
|
One Use of Washing Machine |
\(45\) litres |
\(8\) uses |
|
|
One Use of Dishwasher |
\(20\) litres |
\(7\) uses |
|
|
Washing One Car with a Bucket |
\(10\) litres |
\(1\) wash |
|
|
Hosepipe |
\(9\) litres per minute |
\(15\) minutes |
|
|
|
|
|
Total = |
(i) Complete the table to show the number of litres used for the various activities and the total number of litres used in the week.
(ii) If water were charged at €\(1.85\) per \(1000\) litres, find what John’s household would pay for the water used in one year if the household uses the same amount of water each week.
(iii) Find John’s total bill for this water if VAT were to be included at a rate of \(13.5\%\).
(i)
| Activity | Water Required per Activity | Frequency | Number of litres Used |
|---|---|---|---|
|
One bath |
\(80\) litres |
\(3\) baths |
\(240\) |
|
One Shower |
\(125\) litres |
\(20\) showers |
\(2{,}500\) |
|
Brushing Teeth with Tap Running |
\(6\) litres per minute |
\(32\) minutes |
\(192\) |
|
One WC Flush |
\(6\) litres |
\(60\) flushes |
\(360\) |
|
One Use of Washing Machine |
\(45\) litres |
\(8\) uses |
\(360\) |
|
One Use of Dishwasher |
\(20\) litres |
\(7\) uses |
\(140\) |
|
Washing One Car with a Bucket |
\(10\) litres |
\(1\) wash |
\(10\) |
|
Hosepipe |
\(9\) litres per minute |
\(15\) minutes |
\(135\) |
|
|
|
|
Total = 3,937 |
(ii) \(378.74\mbox{ euro}\)
(iii) \(429.87\mbox{ euro}\)
(i)
| Activity | Water Required per Activity | Frequency | Number of litres Used |
|---|---|---|---|
|
One bath |
\(80\) litres |
\(3\) baths |
\(240\) |
|
One Shower |
\(125\) litres |
\(20\) showers |
\(2{,}500\) |
|
Brushing Teeth with Tap Running |
\(6\) litres per minute |
\(32\) minutes |
\(192\) |
|
One WC Flush |
\(6\) litres |
\(60\) flushes |
\(360\) |
|
One Use of Washing Machine |
\(45\) litres |
\(8\) uses |
\(360\) |
|
One Use of Dishwasher |
\(20\) litres |
\(7\) uses |
\(140\) |
|
Washing One Car with a Bucket |
\(10\) litres |
\(1\) wash |
\(10\) |
|
Hosepipe |
\(9\) litres per minute |
\(15\) minutes |
\(135\) |
|
|
|
|
Total = 3,937 |
(ii)
\begin{align}\frac{3{,}937\times52}{1{,}000}\times1.85\approx378.74\mbox{ euro}\end{align}
(iii)
\begin{align}378.74\times1.135=429.87\mbox{ euro}\end{align}
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(d) John would like to reduce his water bill to €\(260\) per year.
If water were charged at €\(1.85\) plus VAT at \(13.5\%\) per \(1000\) litres, find the number of litres of water he could purchase for €\(260\).
\(123{,}825\mbox{ litres}\)
\begin{align}\frac{260}{1.85\times1.135}\times1{,}000\approx123{,}825\mbox{ litres}\end{align}
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Question 9
Joe wants to draw a diagram of his farm. He uses axes and co-ordinates to plot his farmhouse at the point \(F\) on the diagram below.
(a)
(i) Write down the co-ordinates of the point \(F\).
(ii) A barn is \(5\) units directly North of the farmhouse. Plot the point representing the
position of the barn on the diagram. Label this point \(B\).
(i) \(F=(4,1)\)
(ii)
(i)
\begin{align}F=(4,1)\end{align}
(ii)
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(b) Joe’s quad bike is marked with the point \(Q\) on the diagram.
Find the distance from the barn (\(B\)) to the quad (\(Q\)).
Give your answer correct to \(2\) decimal places.
\(6.08\mbox{ units}\)
\begin{align}|BQ|&=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\&=\sqrt{(-2-4)^2+(7-6)^2}\\&=\sqrt{36+1}\\&=\sqrt{37}\\&\approx6.08\mbox{ units}\end{align}
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(c) Joe’s tractor is at the point \(T\), where \(FBQT\) is a parallelogram.
Plot \(T\) on the diagram and state the co-ordinates of \(T\).
\(T=(-2,2)\)
\begin{align}T=(-2,2)\end{align}
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(d) Joe wants to plough the land enclosed by the parallelogram \(FBQT\).
Find the area of this parallelogram in square units.
\(30\mbox{ units}^2\)
\begin{align}A&=a\times h_{\perp}\\&=5\times6\\&=30\mbox{ units}^2\end{align}
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(e) Given that \(|\angle QFB|=45^{\circ}\), use trigonometric methods to find \(|\angle BQF|\).
Give your answer in degrees correct to one decimal place.
\(35.6^{\circ}\)
\begin{align}\frac{\sin|\angle BQF|}{5}=\frac{\sin45^{\circ}}{6.08}\end{align}
\begin{align}\downarrow\end{align}
\begin{align}|\angle BQF|&=\sin^{-1}\left(\frac{5\sin45^{\circ}}{6.08}\right)\\&\approx35.6^{\circ}\end{align}
