OL Mock Exam A
Question 1
(a) Find the value of each of the following:
(i) \(273+389\)
(ii) \(4.3\times6\)
(iii) \(8^2\div(6-2)\)
Answer
(i) \(662\)
(ii) \(25.8\)
(iii) \(16\)
Solution
(i) \(662\)
(ii) \(25.8\)
(iii) \(16\)
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(b) Consider the number \(e=2.71828…\).
Write this number:
(i) correct to two decimal places.
(ii) correct to the nearest whole number.
Answer
(i) \(2.72\)
(ii) \(3\)
Solution
(i) \(2.72\)
(ii) \(3\)
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Question 2
A school \(U\) of \(80\) students were surveyed and asked if they watched Netflix (\(N\)) or Disney+ (\(D\)).
- \(55\) said they watched Netflix.
- \(40\) said they watched Disney+.
- \(12\) said they watched Disney+ only.
(a) Using this information, complete the Venn diagram below.
Answer
Solution
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(b) How many of the students don’t watch either service?
Answer
\(14\)
Solution
\(14\)
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(c) Write how many of the students are in the following region:
\begin{align}\#(N’)\end{align}
Answer
\(26\)
Solution
\begin{align}\#(N’)&=12+14\\&=26\end{align}
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(d) One student from the survey is chosen at random. Write the probability that that student watches Disney+ as fraction in its simplest form.
Answer
\(\dfrac{1}{2}\)
Solution
\begin{align}P&=\frac{40}{80}\\&=\frac{1}{2}\end{align}
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Question 3
Clyde has \(180\mbox{ euro}\) saved in his bank account.
(a) \(1\mbox{ euro}=1.1\mbox{ US dollar}\).
Write Clyde’s savings in US dollars.
Answer
\(198\mbox{ US dollar}\)
Solution
\begin{align}180\times1.1=198\mbox{ US dollar}\end{align}
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Clyde will use his savings to buy a bicycle for \(135\mbox{ euro}\).
What percentage of Clyde’s saving will be used to buy a bicycle?
Answer
\(75\%\)
Solution
\begin{align}\frac{135}{180}\times100=75\%\end{align}
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(c) Clyde wish to use the remainder of his saving to buy a video game.
The cost of the video game is \(35\mbox{ euro}\) plus VAT.
The VAT rate is \(23\%\).
Can Clyde afford the video game? Explain your reasoning.
Answer
Yes
Solution
\begin{align}35\times1.23=43.05\mbox{ euro}\end{align}
As Clyde has \(45\mbox{ euro}\) remaining, he can afford the video game.
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(d) Clyde changes his mind and decides it would be better to invest his savings for three years.
The interest rate on his investment is \(4\%\).
Calculate how much money Clyde will have at the end of the three year period. Write your answer correct to the nearest cent.
Answer
\(202.48\mbox{ euro}\)
Solution
\begin{align}180\times1.04^3=202.48\mbox{ euro}\end{align}
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Question 4
To find unknown quantities in a non right-angled triangle, it is often simplest to split that triangle into two right-angled triangles.
Consider the non right-angled triangle below.
(a) Write the perpendicular height of this triangle.
Answer
\(2\mbox{ cm}\)
Solution
\(2\mbox{ cm}\)
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(b) Using trigonometry, find the angle \(A\) marked in the diagram above. Write your answer correct to one decimal place.
Answer
\(16.6^{\circ}\)
Solution
\begin{align}\sin A=\frac{2}{7}\end{align}
\begin{align}\downarrow\end{align}
\begin{align}A&=\sin^{-1}\left(\frac{2}{7}\right)\\&\approx16.6^{\circ}\end{align}
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Question 5
The shape below is composed of two squares of side length \(5\mbox{ m}\).
(a) Find the perimeter of this shape.
Answer
\(40\mbox{ m}\)
Solution
\begin{align}8\times5=40\mbox{ m}\end{align}
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(b) Using Pythagoras’ theorem, find \(|AB|\). Write your answer correct to the nearest metre.
Answer
\(14\mbox{ m}\)
Solution
\begin{align}|AB|&=2\sqrt{5^2+5^2}\\&=2\sqrt{50}\\&\approx14\mbox{ m}\end{align}
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Question 6
(a) Consider the following equation:
\begin{align}y=\frac{4x-3}{7}\end{align}
Find the value of \(y\) when \(x=6\).
Answer
\(y=3\)
Solution
\begin{align}y&=\frac{4(6)-3}{7}\\&=3\end{align}
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(b) Factorise the following quadratic expression:
\begin{align}x^2+3x-10=(x+5)(\mbox{_____})\end{align}
Answer
\((x+5)(x-2)\)
Solution
\begin{align}x^2+3x-10=(x+5)(x-2)\end{align}
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(c) Solve the following equation:
\begin{align}3x(x-5)=0\end{align}
Answer
\(x=0\) or \(x=5\)
Solution
\begin{align}3x(x-5)=0\end{align}
\begin{align}\downarrow\end{align}
\(3x=0\) or \(x-5=0\)
\begin{align}\downarrow\end{align}
\(x=0\) or \(x=5\)
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Question 7
Consider the following sequence of patterns made from sticks.
(a) Draw the fourth pattern.
Answer
Solution
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(b) How many sticks are present in the fifth pattern?
Answer
\(16\)
Solution
\(16\)
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(c) Is this pattern linear, quadratic or exponential?
Answer
Linear
Solution
Linear
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(d) The number of sticks in the \(n\)th pattern is \(T_n=3n+A\).
Find the value of \(A\).
Answer
\(A=1\)
Solution
\begin{align}T_1=4\end{align}
\begin{align}\downarrow\end{align}
\begin{align}3(1)+A=4\end{align}
\begin{align}\downarrow\end{align}
\begin{align}A&=4-3\\&=1\end{align}
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Question 8
(a) Draw the image of the following rhombus under central symmetry in the point \(P\).
Answer
Solution
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(b) Using construction methods, divide the line \(AB\) below into three equal parts. Show your construction lines and arcs.
Answer
Solution
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Question 9
The equation of the line below is given by:
\begin{align}2y=10x+5\end{align}
(a) Write the slope \(m\) of the line.
Answer
\(m=5\)
Solution
\begin{align}m&=\frac{10}{2}\\&=5\end{align}
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(b) Write the coordinates of any point that is on the line.
\begin{align}(x,y)=(\mbox{__},\mbox{__})\end{align}
Answer
\((0,2.5)\)
Solution
\((0,2.5)\)
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(c) Write one of the equations of the lines that pass through the origin.
Answer
\(y=x\)
Solution
\(y=x\)
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Question 10
A \(50\) cent coin is roughly in the shape of a cylinder of radius of \(12\mbox{ mm}\) and a thickness of \(2.4\mbox{ mm}\).
(a) Write down the diameter of a \(50\) cent coin in centimetres.
Answer
\(2.4\mbox{ cm}\)
Solution
\(2\times1.2=2.4\mbox{ cm}\)
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(b) Work out the volume of a \(50\) cent coin in \(\mbox{mm}^3\). Write your answer in terms of \(\pi\).
Answer
\(345.6\pi\mbox{ mm}^3\)
Solution
\begin{align}V&=\pi r^2h\\&=\pi(12^2)(2.4)\\&=345.6\pi\mbox{ mm}^3\end{align}
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(c) The composition of a \(50\) cent coin is \(89\%\) copper, \(5%\) aluminium, \(1%\) tin and the remainder zinc.
(i) Work out the percentage of the coin that is zinc.
(ii) The coin weighs \(7.8\mbox{ grams}\). Work out the how much the copper in the coin weighs.
Answer
(i) \(5\%\)
(ii) \(6.942\mbox{ g}\)
Solution
(i)
\begin{align}100\%-89\%-5\%-1\%=5\%\end{align}
(i)
\begin{align}7.8\times0.89=6.942\mbox{ g}\end{align}
