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## Question 1

Emma works part time after school at the local takeaway. She is paid a rate per hour and also receives €$$2$$ for each delivery she makes.

(a) One day, she works for $$2$$ hours, makes $$5$$ deliveries and is paid a total of €$$28$$.
Find her hourly rate of pay.

$$9\mbox{ euro}$$

Solution

\begin{align}\frac{28-5\times2}{2}=9\mbox{ euro}\end{align}

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(b) One week, she works for $$h$$ hours and makes $$d$$ deliveries.
Write a formula in $$h$$ and $$d$$ for the wage ($$w$$) she receives.

$$w=9h+2d$$

Solution

\begin{align}w=9h+2d\end{align}

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(c) Another week, she works for $$6$$ hours and makes $$12$$ deliveries. She also works $$5$$ hours on a Sunday, at time and a half, and makes some deliveries. In total, she receives €$$161.50$$ for that week. Find how many deliveries she makes on the Sunday.

$$8$$

Solution

\begin{align}9\times6+12\times2+5\times1.5\times9+2d=161.50\end{align}

\begin{align}\downarrow\end{align}

\begin{align}145.5+2d=161.50\end{align}

\begin{align}\downarrow\end{align}

\begin{align}d&=\frac{161.50-145.5}{2}\\&=8\end{align}

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## Question 2

$$z_1=1+3i$$ and $$z_2=2-i$$, where $$i^2=-1$$, are two complex numbers.

(a) Let $$z_3=z_1+2z_2$$. Find $$z_3$$ in the form ܽ$$a+bi$$ where $$a,b\in\mathbb{Z}$$.

$$z_3=5+i$$

Solution

\begin{align}z_3&=z_1+2z_2\\&=(1+3i)+2(2-i)\\&=1+3i+4-2i\\&=5+i\end{align}

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(b) Plot $$z_1$$, $$z_2$$ and $$z_3$$ on the given Argand diagram and label each point clearly.

Solution
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(c) Investigate if $$|z_2-z_3|=|z_1+z_2|$$.

$$|z_2-z_3|=|z_1+z_2|$$

Solution

\begin{align}|z_2-z_3|&=|(2-i)-(5+i)|\\&=|-3-2i|\\&=\sqrt{(-3)^2+(-2)^2}\\&=\sqrt{13}\end{align}

and

\begin{align}|z_1+z_2|&=|(1+3i)+(2-i)|\\&=|3+2i|\\&=\sqrt{3^2+2^2}\\&=\sqrt{13}\end{align}

\begin{align}\downarrow\end{align}

\begin{align}|z_2-z_3|=|z_1+z_2|\end{align}

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(d) Find the complex number $$w$$, such that $$w=\frac{z_1}{z_2}$$.
Give your answer in the form $$a+bi$$ where $$a,b\in\mathbb{R}$$.

$$w=-\dfrac{1}{5}+\dfrac{7}{5}i$$

Solution

\begin{align}w&=\frac{z_1}{z_2}\\&=\frac{1+3i}{2-i}\\&=\frac{1+3i}{2-i}\times\frac{2+i}{2+i}\\&=\frac{2+i+6i-3}{4+2i-2i+1}\\&=\frac{7i-1}{5}\\&=-\frac{1}{5}+\frac{7}{5}i\end{align}

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## Question 3

(a) Solve for $$x$$:

$$3(x-7)+5(x-4)=15$$, where $$x\in\mathbb{R}$$

$$x=7$$

Solution

\begin{align}3(x-7)+5(x-4)=15\end{align}

\begin{align}\downarrow\end{align}

\begin{align}3x-21+5x-20=15\end{align}

\begin{align}\downarrow\end{align}

\begin{align}8x=56\end{align}

\begin{align}\downarrow\end{align}

\begin{align}x&=\frac{56}{8}\\&=7\end{align}

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(b) Solve the equations below to find the value of $$a$$ and the value of $$b$$:

\begin{align}4a+3b&=-3\\5a&=25+2b\end{align}

$$a=3$$ and $$b=-5$$

Solution

\begin{align}4a+3b=-3\end{align}

\begin{align}5a-2b=25\end{align}

\begin{align}\downarrow\end{align}

\begin{align}20a+15b=-15\end{align}

\begin{align}20a-8b=100\end{align}

\begin{align}\downarrow\end{align}

\begin{align}23b=-115\end{align}

\begin{align}\downarrow\end{align}

\begin{align}b&=-\frac{115}{23}\\&=-5\end{align}

and

\begin{align}a&=\frac{-3-3b}{4}\\&=\frac{-3-3(-5)}{4}\\&=\frac{12}{4}\\&=3\end{align}

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(c) List all the values of $$x$$ that satisfy the inequality $$2(2x-3)+6x<25$$, where $$x\in\mathbb{N}$$.

$$x\in\{1,2,3\}$$

Solution

\begin{align}2(2x-3)+6x<25\end{align}

\begin{align}\downarrow\end{align}

\begin{align}4x-6+6x<25\end{align}

\begin{align}\downarrow\end{align}

\begin{align}10x<31\end{align}

\begin{align}\downarrow\end{align}

\begin{align}x<3.1\end{align}

\begin{align}\downarrow\end{align}

$$x\in\{1,2,3\}$$ (as $$x\in\mathbb{N}$$)

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## Question 4

The function $$f:x \rightarrow x^3+x^2-2x+7$$ is defined for $$x\in\mathbb{R}$$.

(a) Find the coordinates of the point at which the graph of $$f$$ cuts the $$y$$-axis.

$$(0,7)$$

Solution

\begin{align}f(0)&=0^3+0.^2-2(0)+7\\&=7\end{align}

\begin{align}\downarrow\end{align}

\begin{align}(x,y)=(0,7)\end{align}

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(b) Verify, using algebra, that the point $$A(1,7)$$ is on the graph of $$f$$.

Solution

\begin{align}f(1)&=1^3+1^2-2(1)+7\\&=7\end{align}

as required.

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(c)

(i) Find $$f'(x)$$, the derivative of $$f(x)$$.
Hence find the slope of the tangent to the graph of $$f$$ when $$x=1$$.

(ii) Hence, find the equation of the tangent to the graph of f at the point $$A(1,7)$$.

(i) $$f'(x)=3x^2+2x-2$$ and $$m=3$$

(ii) $$y=3x+4$$

Solution

(i)

\begin{align}f'(x)=3x^2+2x-2\end{align}

\begin{align}\downarrow\end{align}

\begin{align}f'(1)&=3(1^2)+2(1)-2\\&=3\end{align}

(ii)

\begin{align}y-7=3(x-1)\end{align}

\begin{align}\downarrow\end{align}

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

\begin{align}\downarrow\end{align}

\begin{align}y=3x+4\end{align}

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## Question 5

The first five numbers in a pattern of numbers are given in the table below.

Term Number

$$U_1$$

$$13$$

$$U_2$$

$$15$$

$$U_3$$

$$19$$

$$U_4$$

$$25$$

$$U_5$$

$$33$$

$$U_6$$

$$U_7$$

$$U_8$$

(a)

(i) Follow the pattern in the table above to write the next three numbers into the table.

(ii) Use the data in the table to show that the pattern is quadratic.

(i)

Term Number

$$U_1$$

$$13$$

$$U_2$$

$$15$$

$$U_3$$

$$19$$

$$U_4$$

$$25$$

$$U_5$$

$$33$$

$$U_6$$

$$43$$

$$U_7$$

$$55$$

$$U_8$$

$$69$$

(ii) The second difference is a constant value of $$2$$.

Solution

(i)

Term Number

$$U_1$$

$$13$$

$$U_2$$

$$15$$

$$U_3$$

$$19$$

$$U_4$$

$$25$$

$$U_5$$

$$33$$

$$U_6$$

$$43$$

$$U_7$$

$$55$$

$$U_8$$

$$69$$

(ii)

First Difference

$$U_2-U_1$$

$$2$$

$$U_3-U_2$$

$$4$$

$$U_4-U_3$$

$$6$$

$$U_5-U_4$$

$$8$$

$$U_6-U_5$$

$$10$$

$$U_7-U_6$$

$$12$$

$$U_8-U_7$$

$$14$$

As there is therefore a constant second difference of $$2$$, the pattern is quadratic.

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(b) $$U_n=n^2+6n+c$$ where $$b,c\in\mathbb{Z}$$. Find the value of $$b$$ and the value of $$c$$.

$$b=-1$$ and $$c=13$$

Solution

\begin{align}1^2+b(1)+c=13\end{align}

\begin{align}2^2+b(2)+c=15\end{align}

\begin{align}\downarrow\end{align}

\begin{align}b+c=12\end{align}

\begin{align}2b+c=11\end{align}

\begin{align}\downarrow\end{align}

\begin{align}b=-1\end{align}

and

\begin{align}c&=12-b\\&=12-(-1)\\&=13\end{align}

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(c) The table below shows the first five terms of an arithmetic sequence.
Find an expression for $$T_n$$, the $$n$$th term of the sequence.
Hence, or otherwise, find the value of $$T_{30}$$, the $$30$$th term.

Term Number

$$T_1$$

$$12$$

$$T_2$$

$$14$$

$$T_3$$

$$16$$

$$T_4$$

$$18$$

$$T_5$$

$$20$$

$$T_n=2n+10$$ and $$T_{30}=70$$

Solution

\begin{align}a=12&&d=2\end{align}

\begin{align}\downarrow\end{align}

\begin{align}T_n&=a+(n-1)d\\&=12+(n-1)(2)\\&=12+2n-2\\&=2n+10\end{align}

\begin{align}\downarrow\end{align}

\begin{align}T_{30}&=2(30)+10\\&=70\end{align}

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## Question 6

Fiona earns a gross wage of €$$1550$$ every fortnight. She pays income tax, a Universal Social Charge (USC), and Pay Related Social Insurance (PRSI) on this wage.

(a) Each fortnight, Fiona pays income tax at the rate of $$20\%$$ on the first €$$1300$$ she earns and $$40\%$$ on the remainder. She has tax credits of €$$126$$ per fortnight.
Find how much income tax she pays per fortnight

$$234\mbox{ euro}$$

Solution

\begin{align}1{,}300\times0.2+(1{,}550-1{,}300)\times0.4-126=234\mbox{ euro}\end{align}

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(b) Each fortnight, Fiona also pays USC on her gross wage. The rates are:
$$1\%$$ on the first €$$462$$ she earns, $$3\%$$ on the next €$$214$$, and $$5.5\%$$ on the balance.
Find the total amount of USC she pays each fortnight.

$$59.11\mbox{ euro}$$

Solution

\begin{align}462\times0.01+214\times0.03+(1{,}550-62-214)\times0.055=59.11\mbox{ euro}\end{align}

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(c)

(i) Fiona pays PRSI amounting to €$$18$$ each fortnight.
Find the sum of her fortnightly deductions.

(ii) Write the sum of her fortnightly deductions as a percentage of her gross wage.

(i) $$311.11\mbox{ euro}$$

(ii) $$20.1\%$$

Solution

(i)

\begin{align}234+59.11+18=311.11\mbox{ euro}\end{align}

(ii)

\begin{align}\frac{311.11}{1{,}550}\times100\approx20.1\%\end{align}

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## Question 7

Paul has €$$8000$$ that he wants to invest for a maximum of $$3$$ years. His local bank is offering him two
options, Option $$1$$ and Option $$2$$, as shown in the table below.

Option 1

$$2\%$$ interest in Year $$1$$

$$3\%$$ interest in Year $$2$$

$$5\%$$ interest in Year $$3$$

Money can be taken out at the end of Year $$1$$ or Year $$2$$ without penalty

Option 2

$$3.7\%$$ compound interest per
year, for $$3$$ years.

Money may not be taken out until the end of Year $$3$$.

(a) Find the value of the investment at the end of $$3$$ years if Paul chooses Option $$1$$ and does not
take any money out.

$$8{,}825.04\mbox{ euro}$$

Solution

\begin{align}8{,}000\times1.02\times1.03\times1.05=8{,}825.04\mbox{ euro}\end{align}

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(b) Find the value of the investment at the end of $$3$$ years if Paul chooses Option $$2$$.

$$8{,}921.26\mbox{ euro}$$

Solution

\begin{align}8{,}000\times(1.037^3)=8{,}921.26\mbox{ euro}\end{align}

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(c) Give one issue, other than the rate of interest earned, that Paul might take into account when
deciding between Option $$1$$ and Option $$2$$.

It’s not possible to withdraw money early with option $$2$$.

Solution

It’s not possible to withdraw money early with option $$2$$.

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(d) Paul would like his investment of €$$8000$$ to amount to €$$9000$$ after $$3$$ years.
What annual rate of compound interest would be required for this to happen?

$$4\%$$

Solution

\begin{align}8{,}000(1+i)^3=9{,}000\end{align}

\begin{align}\downarrow\end{align}

\begin{align}i&=\sqrt{\frac{9{,}000}{8{,}000}}-1\\&=0.04\\&=4\%\end{align}

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(e) Another bank offers to invest his €$$8000$$ in a special fund. The bank has found that, in the short term, the amount of money in this fund follows the formula

\begin{align}v=8000+36t-1.2t^2\end{align}

where $$v$$ is the value of the fund, in euro, and $$t$$ is the time, in months, since the €$$8000$$ was invested.

(i) Find the value of the money in the fund after $$12$$ months.

(ii) Mary invests €$$8000$$ for $$1$$ year at $$r\%$$ per annum. After $$1$$ year her investment is worth the same amount as the answer to part (e)(i) above.
Find the value of $$r$$, correct to $$2$$ decimal places.

(i) $$8{,}259.20\mbox{ euro}$$

(ii) $$3.24\%$$

Solution

(i)

\begin{align}v(12)&=8{,}000+36(12)-1.2(12^2)\\&=8{,}259.20\mbox{ euro}\end{align}

(ii)

\begin{align}r&=\frac{8{,}259.20-8{,}000}{8{,}000}\times100\\&=3.24\%\end{align}

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## Question 8

Kieran has $$21$$ metres of fencing. He wants to enclose a vegetable garden in a rectangular shape as shown.

(a) By writing an expression for the perimeter of the vegetable garden in terms of $$x$$ (length in metres) and $$y$$ (width in metres), show that $$y=10.5-x$$.

Solution

\begin{align}2x+2y=21\end{align}

\begin{align}\downarrow\end{align}

\begin{align}2y=21-2x\end{align}

\begin{align}\downarrow\end{align}

\begin{align}y=10.5-x\end{align}

as required.

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(b)

(i) Complete the table below to show the values of $$y$$ and $$A$$ (the area of the garden) for each given value of $$x$$.

$$x$$ (m) $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$

$$y$$ (m)

$$6.5$$

$$A$$ (m$$^2$$)

$$26$$

(ii) Use the values of $$x$$ and $$A$$ from the table to plot the graph of $$A$$ on the grid below.

(i)

$$x$$ (m) $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$

$$y$$ (m)

$$10.5$$

$$9.5$$

$$8.5$$

$$7.5$$

$$6.5$$

$$5.5$$

$$4.5$$

$$3.5$$

$$2.5$$

$$1.5$$

$$0.5$$

$$A$$ (m$$^2$$)

$$0$$

$$9.5$$

$$17$$

$$22.5$$

$$26$$

$$27.5$$

$$27$$

$$24.5$$

$$20$$

$$13.5$$

$$5$$

(ii)

Solution

(i)

$$x$$ (m) $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$

$$y$$ (m)

$$10.5$$

$$9.5$$

$$8.5$$

$$7.5$$

$$6.5$$

$$5.5$$

$$4.5$$

$$3.5$$

$$2.5$$

$$1.5$$

$$0.5$$

$$A$$ (m$$^2$$)

$$0$$

$$9.5$$

$$17$$

$$22.5$$

$$26$$

$$27.5$$

$$27$$

$$24.5$$

$$20$$

$$13.5$$

$$5$$

(ii)

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(c) Use your graph to estimate the maximum value of $$A$$ as well as the corresponding length and width.

Maximum area: $$28\mbox{ m}^2$$

x value: $$5\mbox{ m}$$

y value: $$5\mbox{ m}$$

Solution

Maximum area: $$28\mbox{ m}^2$$

x value: $$5\mbox{ m}$$

y value: $$5\mbox{ m}$$

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(d)

(i) Show that the area of the rectangle can be written as $$A=10.5x-x^2$$.

(ii) Find $$\dfrac{dA}{dx}$$.

(iii) Hence, find the value of x which will give the maximum area.

(iv) Find this maximum area.

(ii) $$\dfrac{dA}{dx}=10.5-2x$$

(iii) $$5.25\mbox{ m}$$

(iv) $$27.5625\mbox{ m}^2$$

Solution

(i)

\begin{align}A&=xy\\&=x(10.5-x)\\&=10.5x-x^2\end{align}

as required.

(ii)

\begin{align}\frac{dA}{dx}=10.5-2x\end{align}

(iii)

\begin{align}A'(x)=0\end{align}

\begin{align}\downarrow\end{align}

\begin{align}10.5-2x=0\end{align}

\begin{align}\downarrow\end{align}

\begin{align}x&=\frac{10.5}{2}\\&=5.25\mbox{ m}\end{align}

(iv)

\begin{align}A(5.25)&=10.5(5.25)-5.25^2\\&=27.5625\mbox{ m}^2\end{align}

Video Walkthrough

## Question 9

Company $$A$$ uses the following formula to charge a customer for a job:

\begin{align}A(h)=30+9.5h\end{align}

where $$A(h)$$ is the cost of the job, in euro, and $$h$$ is the length of time that the job takes in hours.

Company $$B$$ uses the following formula to charge a customer for the same job:

\begin{align}B(h)=10(1.74)^h\end{align}

where $$B(h)$$ is the cost of the job, in euro, and $$h$$ is again the length of time that the job takes in hours.

(a)

(i) Complete the table below to show what Company $$A$$ charges and what Company $$B$$ charges for jobs that take up to $$5$$ hours. Where necessary give the charge correct to the nearest cent.

Time (hours) $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$

$$A(h)$$ (euro)

$$68$$

$$B(h)$$ (euro)

$$91.66$$

(ii) On the grid below draw separate graphs to show the charge for Company $$A$$ and the charge for Company $$B$$. Label each graph clearly.

(i)

Time (hours) $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$

$$A(h)$$ (euro)

$$30$$

$$39.5$$

$$49$$

$$58.5$$

$$68$$

$$77.5$$

$$B(h)$$ (euro)

$$10$$

$$17.4$$

$$30.28$$

$$52.68$$

$$91.66$$

$$159.49$$

(ii)

Solution

(i)

Time (hours) $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$

$$A(h)$$ (euro)

$$30$$

$$39.5$$

$$49$$

$$58.5$$

$$68$$

$$77.5$$

$$B(h)$$ (euro)

$$10$$

$$17.4$$

$$30.28$$

$$52.68$$

$$91.66$$

$$159.49$$

(ii)

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(b) Which company would charge least for a job that takes $$2\dfrac{1}{2}$$ hours to complete?

Company $$B$$ as they charge $$40\mbox{ euro}$$ compared with Company $$A$$ who would instead charge $$52\mbox{ euro}$$.

Solution

Company $$B$$ as they charge $$40\mbox{ euro}$$ compared with Company $$A$$ who would instead charge $$52\mbox{ euro}$$.

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(c) Use your graphs to estimate the value of $$h$$ for which the charge is the same for both companies.

\begin{align}h\approx3.2\end{align}

Solution

\begin{align}h\approx3.2\end{align}

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(d) Find the difference in cost for a job that takes 6 hours to complete.

$$190.52\mbox{ euro}$$