One-to-One Grinds
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2015 Ordinary Level - Paper One
Section A
Question 1
Padraic works in America and travels between Ireland and America.
(a) In Ireland, he exchanged €\(2000\) for US dollars when the exchange rate was €\(1= \)$\(1.29\).
Find how many US dollars he received.
\(\$2{,}580\)
\begin{align}2{,}000\times1.29=\$2{,}580\end{align}
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(b) Padraic returned to Ireland and exchanged $\(21{,}000\) for euro. He received €\(15{,}000\). Write the
exchange rate for this transaction in the form €\(1= \)$______.
€\(1=\$1.40\)
\begin{align}\frac{21{,}000}{15{,}000}=1.4\end{align}
\begin{align}\downarrow\end{align}
€\(1=\$1.40\)
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(c) Padraic wants to exchange some dollars for sterling. On a day when the euro to dollar exchange rate is €\(1= \)$\(1.24\) and the euro to sterling exchange rate is €\(1= \)£\(0.83\), find the dollar to sterling exchange rate. Write your answer in the form $\(1= \)£______.
\(\$1=0.67\mbox{ sterling}\)
\begin{align}\$1&=\frac{0.83}{1.24}\\&=0.67\mbox{ sterling}\end{align}
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Question 2
(a) John, Mary and Eileen bought a ticket in a draw. The ticket cost €\(50\). John paid €\(25\), Mary
paid €\(15\) and Eileen paid €\(10\). The ticket won a prize of €\(20{,}000\). The prize is divided in
proportion to how much each paid. How much prize money does each person receive?
John: \(10{,}000\mbox{ euro}\)
Mary: \(6{,}000\mbox{ euro}\)
Eileen: \(4{,}000\mbox{ euro}\)
John
\begin{align}\frac{25}{50}\times20{,}000=10{,}000\mbox{ euro}\end{align}
\[\,\]
Mary
\begin{align}\frac{15}{50}\times20{,}000=6{,}000\mbox{ euro}\end{align}
\[\,\]
Eileen
\begin{align}\frac{10}{50}\times20{,}000=4{,}000\mbox{ euro}\end{align}
\[\,\]
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(b) Assuming that the Earth is a sphere of radius \(6378\mbox{ km}\)\):
(i) Find the length of the equator, correct to the nearest km.
(ii) Find the volume of the Earth in the form , \(a\times10^n\), where \(1\leq a<10\) and \(n\in\mathbb{N}\).
Give the value of \(a\) correct to three decimal places.
(i) \(40{,}074\mbox{ km}\)
(ii) \(1.087\times10^{12}\mbox{ km}^3\)
(i)
\begin{align}C&=2\pi r\\&=2\pi(6{,}378)\\&\approx40{,}074\mbox{ km}\end{align}
(ii)
\begin{align}V&=\frac{4}{3}\pi r^3\\&=\frac{4}{3}\pi(6{,}378)^3\\&\approx1.087\times10^{12}\mbox{ km}^3\end{align}
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(c) The mass of the Earth is \(5.97\times10^{24}\mbox{ kg}\). The mass of the Sun is \(1.99\times10^{30\mbox{ kg}}\). How many times greater than the mass of the Earth is the mass of Sun? Give your answer correct to the nearest whole number.
\(333{,}333\)
\begin{align}\frac{1.99\times10^{30}}{5.97\times10^{24}}\approx333{,}333\end{align}
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Question 3
(a) Simplify \(3(4-5x)-2(5-6x)\).
\(2-3x\)
\begin{align}3(4-5x)-2(5-6x)&=12-15x-10+12x\\&=2-3x\end{align}
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(b) List all the values of \(x\) that satisfy the inequality \(2-3x\geq -6\), \(x\in\mathbb{N}\).
\(x\in\{1,2\}\)
\begin{align}2-3x\geq-6\end{align}
\begin{align}\downarrow\end{align}
\begin{align}-3x\geq-8\end{align}
\begin{align}\downarrow\end{align}
\begin{align}x\leq\frac{8}{3}\end{align}
\begin{align}\downarrow\end{align}
\(x\in\{1,2\}\) (as \(x\in\mathbb{N}\))
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(c) \(g(x)\) is a function and \((2-3x)\times g(x)=15x^2-22x+8\), for all \(x\in\mathbb{R}\). Find \(g(x)\).
\(g(x)=-5x+4\)
\[
\require{enclose}
\begin{array}{rll}
-5x+4\phantom{000000}\, \\[-3pt]
-3x+2 \enclose{longdiv}{\,15x^2-22x+8} \\[-3pt]
\underline{15x^2-10x\phantom{000}\,\,} \\[-3pt]
-12x+8\phantom{00}\,\\[-3pt]\phantom{0000}\underline{-12x+8\phantom{00}\,}\\0\phantom{00}
\end{array}
\]
\begin{align}\downarrow\end{align}
\begin{align}g(x)=-5x+4\end{align}
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Question 4
(a) Solve the equation \(-x^2+6x-4=0\). Give each solution correct to one decimal place.
\(x=0.8\) or \(x=5.2\)
\begin{align}a=-1&&b=6&&c=-4\end{align}
\begin{align}\downarrow\end{align}
\begin{align}x&=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\&=\frac{-6\pm\sqrt{6^2-4(-1)(-4)}}{2(-1)}\\&=\frac{6\pm\sqrt{20}}{2}\end{align}
\begin{align}\downarrow\end{align}
\(x\approx0.8\) or \(x\approx5.2\)
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(b) Find the co-ordinates of the turning point of the function \(f(x)=-x^2+6x-4\), \(x\in\mathbb{R}\).
\((3,5)\)
\begin{align}f'(x)=0\end{align}
\begin{align}\downarrow\end{align}
\begin{align}-2x+6=0\end{align}
\begin{align}\downarrow\end{align}
\begin{align}x=3\end{align}
and
\begin{align}f(3)&=-3^2+6(3)-4\\&=5\end{align}
Therefore, the turning point is \((3,5)\).
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(c) Use your answers to parts (a) and (b) above to sketch the curve \(y=f(x)\).
Show your scale on both axes.
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Question 5
The diagram shows the graph of the function \(f(x)=5x-x^2\) in the domain \(0\leq x\leq 6\), \(x\in\mathbb{R}\).
(a) The function \(g\) is \(g(x)=x+3\), \(x\in\mathbb{R}\).
The points \(A(1,k)\) and \(B\) are the points of intersection of \(f\) and \(g\).
Find the co-ordinates of \(A\) and of \(B\).
\(A(1,4)\) and \(B(3,6)\)
\begin{align}f(x)=g(x)\end{align}
\begin{align}\downarrow\end{align}
\begin{align}5x-x^2=x+3\end{align}
\begin{align}\downarrow\end{align}
\begin{align}x^2-4x+3=0\end{align}
\begin{align}\downarrow\end{align}
\begin{align}(x-1)(x-3)=0\end{align}
\begin{align}\downarrow\end{align}
\(x=1\) or \(x=3\)
\begin{align}\downarrow\end{align}
\begin{align}g(1)&=1+3\\&=4\end{align}
and
\begin{align}g(3)&=3+3\\&=6\end{align}
Therefore, the intersection points are \(A(1,4)\) and \(B(3,6)\).
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(b) The points \(O(0,0)\) and \(C(5,0)\) are on the graph of \(f\).
(i) Draw the quadrilateral \(OCBA\) on the diagram above.
(ii) Find the area of the quadrilateral \(OCBA\).
(i)
(ii) \(18\mbox{ units}^2\)
(i)
(ii)
\begin{align}A&=\frac{1}{2}(1)(4)+\frac{1}{2}(4+6)(2)+\frac{1}{2}(2)(6)\\&=18\mbox{ units}^2\end{align}
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Question 6
(a) The complex number \(z_1=a+bi\), where \(i^2=-1\), is shown on the Argand diagram below.
(i) Write down the value of a and the
value of b.
(ii) The image of \(z_1\) under reflection in the real axis is \(z_2=c+di\). Write down the value of \(c\) and the value of \(d\).
(i) \(a=3\) and \(b=1\)
(ii) \(c=3\) and \(d=-1\)
(i)
\begin{align}a=3&&b=1\end{align}
(ii)
\begin{align}c=3&&d=-1\end{align}
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(b)
(i) The angle \(\theta\) is formed by joining \(z_1\) to \(0+0i\) to \(z_2\).
Find \(\cos \theta\), correct to one decimal place.
(ii) Show that \(|z_1|\times|z_2|\times \cos \theta=ac+bd\), where \(a\), \(b\), \(c\) and \(d\) are the values from part (a) above.
(i) \(0.8\)
(ii) The answer is already in the question!
(i)
\begin{align}|z_1|&=\sqrt{3^2+1^2}\\&=\sqrt{10}\end{align}
\begin{align}\downarrow\end{align}
\begin{align}2^2=(\sqrt{10})^2+(\sqrt{10})^2-2(\sqrt{10})(\sqrt{10})\cos\theta\end{align}
\begin{align}\downarrow\end{align}
\begin{align}\cos\theta&=\frac{10+10-4}{2(10)}\\&=0.8\end{align}
(ii)
\begin{align}|z_1|\times|z_2|\times\cos\theta&=\sqrt{10}\times\sqrt{10}\times0.8\\&=8\end{align}
and
\begin{align}ac+bd&=(3)(3)+(1)(-1)\\&=8\end{align}
as required.
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Section B
Question 7
The first three patterns in a sequence of patterns are shown below.
(a) Draw the fourth pattern in the sequence.
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(b) Complete the table below.
| \(t\) | Number of Black Triangles | Number of White Triangles | Total Number of Small Triangles |
|---|---|---|---|
|
Pattern 1 |
\(3\) |
\(1\) |
\(T_1=4\) |
|
Pattern 2 |
|
|
\(T_2=...\) |
|
Pattern 3 |
|
|
\(T_3=...\) |
|
Pattern 4 |
|
|
\(T_4=...\) |
|
Pattern 5 |
|
|
\(T_5=...\) |
| \(t\) | Number of Black Triangles | Number of White Triangles | Total Number of Small Triangles |
|---|---|---|---|
|
Pattern 1 |
\(3\) |
\(1\) |
\(T_1=4\) |
|
Pattern 2 |
\(6\) |
\(3\) |
\(T_2=9\) |
|
Pattern 3 |
\(10\) |
\(6\) |
\(T_3=16\) |
|
Pattern 4 |
\(15\) |
\(10\) |
\(T_4=25\) |
|
Pattern 5 |
\(21\) |
\(15\) |
\(T_5=36\) |
| \(t\) | Number of Black Triangles | Number of White Triangles | Total Number of Small Triangles |
|---|---|---|---|
|
Pattern 1 |
\(3\) |
\(1\) |
\(T_1=4\) |
|
Pattern 2 |
\(6\) |
\(3\) |
\(T_2=9\) |
|
Pattern 3 |
\(10\) |
\(6\) |
\(T_3=16\) |
|
Pattern 4 |
\(15\) |
\(10\) |
\(T_4=25\) |
|
Pattern 5 |
\(21\) |
\(15\) |
\(T_5=36\) |
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(c) Show that the numbers of black triangles form a quadratic sequence.
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(d)
(i) How many black triangles are in the \(9\)th pattern?
(ii) How many white triangles are in the \(9\)th pattern?
(iii) How many small triangles, in total, are in the \(9\)th pattern?
(i) \(55\)
(ii) \(45\)
(iii) \(100\)
(i) \(55\)
(ii) \(45\)
(iii) \(T_9=100\)
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(e) Write an expression in \(n\) for the total number of triangles in the \(n\)th pattern.
\(T_n=(n+1)^2\)
\begin{align}T_n=(n+1)^2\end{align}
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(f) The number of black triangles in the \(n\)th pattern is given by the formula \(B_n=\dfrac{1}{2}n^2+\dfrac{3}{2}n+c\).
Find the value of \(c\).
\(c=1\)
\begin{align}B_1=3\end{align}
\begin{align}\downarrow\end{align}
\begin{align}\frac{1}{2}(1^2)+\frac{3}{2}(1)+c=3\end{align}
\begin{align}\downarrow\end{align}
\begin{align}2+c=3\end{align}
\begin{align}\downarrow\end{align}
\begin{align}c=1\end{align}
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(g) Use your answers to parts (e) and (f) above to find a formula for the number of white triangles in the \(n\)th pattern.
\(W_n=\dfrac{1}{2}n^2+\dfrac{1}{2}n\)
\begin{align}W_n&=(n+1)^2-B_n\\&=(n+1)^2-\left(\frac{1}{2}n^2+\frac{3}{2}n+1\right)\\&=n^2+2n+1-\frac{1}{2}n^2-\frac{3}{2}n-1\\&=\frac{1}{2}n^2+\frac{1}{2}n\end{align}
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(h) One particular pattern has a total of \(625\) triangles. Find the number of black triangles and the number of white triangles in that pattern.
\(B_{24}=325\) and \(W_{24}=300\)
\begin{align}T_n=625\end{align}
\begin{align}\downarrow\end{align}
\begin{align}(n+1)^2=625\end{align}
\begin{align}\downarrow\end{align}
\begin{align}n&=\sqrt{625}-1\\&=24\end{align}
\begin{align}\downarrow\end{align}
\begin{align}B_{24}&=\frac{1}{2}(24^2)+\frac{3}{2}(24)+1\\&=325\end{align}
and
\begin{align}W_{24}&=\frac{1}{2}(24^2)+\frac{1}{2}(24)\\&=300\end{align}
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Question 8
The daily profit of an oil trader is given by the profit function \(p=96x-0.03x^2\), where \(p\) is the daily profit, in euro, and \(x\) is the number of barrels of oil traded in a day.
(a) Complete the table below.
| Number of barrels traded in a day | \(x\) | \(500\) | \(1000\) | \(1500\) | \(2000\) | \(2500\) |
|---|---|---|---|---|---|---|
|
Daily profit (euro) |
\(p\) |
\(40{,}500\) |
|
|
|
|
| Number of barrels traded in a day | \(x\) | \(500\) | \(1000\) | \(1500\) | \(2000\) | \(2500\) |
|---|---|---|---|---|---|---|
|
Daily profit (euro) |
\(p\) |
\(40{,}500\) |
\(66{,}000\) |
\(76{,}500\) |
\(72{,}000\) |
\(52{,}500\) |
| Number of barrels traded in a day | \(x\) | \(500\) | \(1000\) | \(1500\) | \(2000\) | \(2500\) |
|---|---|---|---|---|---|---|
|
Daily profit (euro) |
\(p\) |
\(40{,}500\) |
\(66{,}000\) |
\(76{,}500\) |
\(72{,}000\) |
\(52{,}500\) |
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(b) Draw the graph of the trader’s profit function on the axes below for \(500\leq x\leq 2500\), \(x\in\mathbb{R}\).
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(c) Use your graph to estimate:
(i) The daily profit when \(1750\) barrels are traded.
(ii) The numbers of barrels traded when the daily profit is €\(60{,}000\).
(i) \(76{,}125\mbox{ euro}\)
(ii) \(850\) or \(2{,}350\)
(i) \(76{,}125\mbox{ euro}\)
(ii) \(850\) or \(2{,}350\)
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(d)
(i) Use calculus to find the number of barrels of oil traded that will earn the maximum daily profit.
(ii) Find this maximum profit.
(i) \(1{,}600\mbox{ barrels}\)
(ii) \(76{,}800\mbox{ euro}\)
(i)
\begin{align}p'(x)=0\end{align}
\begin{align}\downarrow\end{align}
\begin{align}96-0.03x=0\end{align}
\begin{align}\downarrow\end{align}
\begin{align}x&=\frac{96}{0.06}\\&=1{,}600\mbox{ barrels}\end{align}
(ii)
\begin{align}p(1{,}600)&=96(1{,}600)-0.03(1{,}600)^2\\&=76{,}800\mbox{ euro}\end{align}
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(e) The trader will not make a profit if he trades more than \(k\) barrels of oil in a day.
Calculate the value of \(k\).
\(k=3{,}200\)
\begin{align}p(x)=0\end{align}
\begin{align}\downarrow\end{align}
\begin{align}96x-0.03x^2=0\end{align}
\begin{align}\downarrow\end{align}
\begin{align}x(96-0.0×3)\end{align}
\begin{align}\downarrow\end{align}
\(x=0\) or \(x=3{,}200\)
\begin{align}\downarrow\end{align}
\(k=3{,}200\)
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Question 9
Amanda buys a new car for €\(25{,}000\). She knows that the value of cars depreciates by a fixed percentage each year. She draws the graph below to represent the value of her car for the next six years.
(a)
(i) Use Amanda’s graph to estimate the length of time it will take for the car to depreciate from €\(15{,}000\) to €\(10{,}000\).
(ii) Continue Amanda’s graph to find the approximate value of the car after \(7.5\) years.
(iii) What name is given to the type of curve Amanda has drawn above?
(i) \(1.8\mbox{ years}\)
(ii) \(4{,}700\mbox{ euro}\)
(iii) Exponential
(i) \(4.1-2.3=1.8\mbox{ years}\)
(ii) \(4{,}700\mbox{ euro}\)
(iii) Exponential
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(b) On the day that Amanda bought her car Cathal also bought a new car. His car cost €\(22{,}500\).
Amanda’s car will travel \(15\mbox{ km}\) on a litre of fuel. Cathal’s car will travel \(10\mbox{ km}\) on a litre of fuel. How many kilometres will the cars have to travel until the cost price of the car plus the cost of fuel are equal, for the two cars? Assume that fuel costs €\(1.50\) per litre over the period.
\(50{,}000\mbox{ euro}\)
\begin{align}\frac{2{,}500\times100}{\frac{150}{10}-\frac{150}{15}}=50{,}000\mbox{ euro}\end{align}
