OL Mock Exam A

Question 1

**(a) **Find the value of each of the following:

**(i)** \(273+389\)

**(ii)** \(4.3\times6\)

**(iii)** \(8^2\div(6-2)\)

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Answer

**(i)** \(662\)

**(ii)** \(25.8\)

**(iii)** \(16\)

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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.

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Answer

**(i)** \(2.72\)

**(ii)** \(3\)

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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.

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Answer

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Solution

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**(b) **How many of the students don’t watch either service?

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Answer

\(14\)

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Solution

\(14\)

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**(c) **Write how many of the students are in the following region:

\begin{align}\#(N’)\end{align}

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Answer

\(26\)

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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.

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Answer

\(\dfrac{1}{2}\)

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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.

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Answer

\(198\mbox{ US dollar}\)

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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?

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Answer

\(75\%\)

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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.

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Answer

Yes

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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.

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Answer

\(202.48\mbox{ euro}\)

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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.

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Answer

\(2\mbox{ cm}\)

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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.

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Answer

\(16.6^{\circ}\)

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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.

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Answer

\(40\mbox{ m}\)

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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.

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Answer

\(14\mbox{ m}\)

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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 \(x=6\).

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Answer

\(y=3\)

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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}

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Answer

\((x+5)(x-2)\)

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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}

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Answer

\(x=0\) or \(x=5\)

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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.

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Answer

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**(b)** How many sticks are present in the fifth pattern?

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Answer

\(16\)

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Solution

\(16\)

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**(c)** Is this pattern linear, quadratic or exponential?

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Answer

Linear

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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 \(A\).

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Answer

\(A=1\)

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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 \(P\).

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Answer

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**(b)** Using construction methods, divide the line \(AB\) below into three equal parts. Show your construction lines and arcs.

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Answer

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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.

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Answer

\(m=5\)

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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}

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Answer

\((0,2.5)\)

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Solution

\((0,2.5)\)

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**(c)** Write one of the equations of the lines that pass through the origin.

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Answer

\(y=x\)

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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.

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Answer

\(2.4\mbox{ cm}\)

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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 \(\pi\).

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Answer

\(345.6\pi\mbox{ mm}^3\)

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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.

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Answer

**(i) **\(5\%\)

**(ii) **\(6.942\mbox{ g}\)

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Solution

**(i)**

\begin{align}100\%-89\%-5\%-1\%=5\%\end{align}

**(i)**

\begin{align}7.8\times0.89=6.942\mbox{ g}\end{align}