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OL Mock Exam A

Question 1

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

(i) \(273+389\)

(ii) \(4.3\times6\)

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

(i) \(662\)

(ii) \(25.8\)

(iii) \(16\)

(i) \(662\)

(ii) \(25.8\)

(iii) \(16\)

(b) Consider the number \(e=2.71828…\).

Write this number:

(i) correct to two decimal places.

(ii) correct to the nearest whole number.

(i) \(2.72\)

(ii) \(3\)

(i) \(2.72\)

(ii) \(3\)

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.

12U(80)N( )D( )
14122826U(80)N(54)D(40)
14122826U(80)N(54)D(40)

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

\(14\)

\(14\)

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

\begin{align}\#(N’)\end{align}

\(26\)

\begin{align}\#(N’)&=12+14\\&=26\end{align}

(d) One student from the survey is chosen at random. Write the probability that that student watches Disney+ as fraction in its simplest form.

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

\begin{align}P&=\frac{40}{80}\\&=\frac{1}{2}\end{align}

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.

\(198\mbox{ US dollar}\)

\begin{align}180\times1.1=198\mbox{ US dollar}\end{align}

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?

\(75\%\)

\begin{align}\frac{135}{180}\times100=75\%\end{align}

(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.

Yes

\begin{align}35\times1.23=43.05\mbox{ euro}\end{align}

As Clyde has \(45\mbox{ euro}\) remaining, he can afford the video game.

(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.

\(202.48\mbox{ euro}\)

\begin{align}180\times1.04^3=202.48\mbox{ euro}\end{align}

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.

2 cm7 cmA

(a) Write the perpendicular height of this triangle.

\(2\mbox{ cm}\)

\(2\mbox{ cm}\)

(b) Using trigonometry, find the angle \(A\) marked in the diagram above. Write your answer correct to one decimal place.

\(16.6^{\circ}\)

\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}

Question 5

The shape below is composed of two squares of side length \(5\mbox{ m}\).

5 mAB

(a) Find the perimeter of this shape.

\(40\mbox{ m}\)

\begin{align}8\times5=40\mbox{ m}\end{align}

(b) Using Pythagoras’ theorem, find \(|AB|\). Write your answer correct to the nearest metre.

\(14\mbox{ m}\)

\begin{align}|AB|&=2\sqrt{5^2+5^2}\\&=2\sqrt{50}\\&\approx14\mbox{ m}\end{align}

Question 6

(a) Consider the following equation:

\begin{align}y=\frac{4x-3}{7}\end{align}

Find the value of \(y\) when \(x=6\).

\(y=3\)

\begin{align}y&=\frac{4(6)-3}{7}\\&=3\end{align}

(b) Factorise the following quadratic expression:

\begin{align}x^2+3x-10=(x+5)(\mbox{_____})\end{align}

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

\begin{align}x^2+3x-10=(x+5)(x-2)\end{align}

(c) Solve the following equation:

\begin{align}3x(x-5)=0\end{align}

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

\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 \(x=5\)

Question 7

Consider the following sequence of patterns made from sticks.

Pattern 1Pattern 2Pattern 3

(a) Draw the fourth pattern.

Pattern 4
Pattern 4

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

\(16\)

\(16\)

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

Linear

Linear

(d) The number of sticks in the \(n\)th pattern is \(T_n=3n+A\).

Find the value of \(A\).

\(A=1\)

\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}

Question 8

(a) Draw the image of the following rhombus under central symmetry in the point \(P\).

P
P
P

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

AB
AXX'Y YZB
AXX'Y YZB

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.

\(m=5\)

\begin{align}m&=\frac{10}{2}\\&=5\end{align}

(b) Write the coordinates of any point that is on the line.

\begin{align}(x,y)=(\mbox{__},\mbox{__})\end{align}

\((0,2.5)\)

\((0,2.5)\)

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

\(y=x\)

\(y=x\)

Question 10

A \(50\) cent coin is roughly in the shape of a cylinder of radius of \(12\mbox{ mm}\) and a thickness of \(2.4\mbox{ mm}\).

12 mm2.4 mm

(a) Write down the diameter of a \(50\) cent coin in centimetres.

\(2.4\mbox{ cm}\)

\(2\times1.2=2.4\mbox{ cm}\)

(b) Work out the volume of a \(50\) cent coin in \(\mbox{mm}^3\). Write your answer in terms of \(\pi\).

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

\begin{align}V&=\pi r^2h\\&=\pi(12^2)(2.4)\\&=345.6\pi\mbox{ mm}^3\end{align}

(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.

(i) \(5\%\)

(ii) \(6.942\mbox{ g}\)

(i)

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

(i)

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

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