2016.1 L.16 -16 sk-final - WordPress.com · 2016-03-14 · Pre-Leaving Certificate, 2016 2016.1 L.16 5/16 Page 5 of 15 Mathematics Paper 1 – Ordinary Level Question 3 (25 marks)

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L.16

NAME

SCHOOL

TEACHER

Pre-Leaving Certificate Examination, 2016

Mathematics

Paper 1

Ordinary Level

Time: 2 hours, 30 minutes

300 marks

For examiner

Question Mark

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School stamp 3

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Pre-Leaving Certificate, 2016 2016.1 L.16 2/16

Page 2 of 15

MathematicsPaper 1 – Ordinary Level

Instructions There are two sections in this examination paper:

Section A Concepts and Skills 150 marks 6 questions

Section B Contexts and Applications 150 marks 3 questions

Answer all nine questions.

Write your answers in the spaces provided in this booklet. You may lose marks if you do not do so. You may also ask the superintendent for more paper. Label any extra work clearly with the question number and part.

The superintendent will give you a copy of the Formulae and Tables booklet. You must return it at the end of the examination. You are not allowed to bring your own copy into the examination.

You will lose marks if all necessary work is not clearly shown.

You may lose marks if the appropriate units of measurement are not included, where relevant.

You may lose marks if your answers are not given in simplest form, where relevant.

Write the make and model of your calculator(s) here:


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Section A Concepts and Skills 150 marks

Answer all six questions from this section.

Question 1 (25 marks)

(a) (i) Solve for x:

5 − 7(2 − x) = −2x − 3(1 − 4x).

(ii) Verify your answer to part (i) above.

(b) Solve the inequality:

7 − 4x ≥ −2, x ∈ ℤ,

and show the solution set on the number line below.

(c) Write down the inequality that is shown on the number line below:

0 1�1 2�2 3�3

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0 1�1 2�2 3�3


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The Summer Olympic Games will take place in Rio de Janeiro, Brazil, in 2016.

(a) According to the 2010 Census, the population of the host city was 5 940 224. Write this number in the form a × 10n, where 1 ≤ a ≤ 10 and n ∈ ℕ, correct to two significant figures.

(b) According to the most recent estimate, the population of the city is now 6 453 682.

(i) Use your answer to part (a) above to calculate the population growth rate (percentage change in population) of the city, correct to two decimal places.

(ii) Find the percentage error in using your answer to part (a) above instead of the actual figure to calculate the population growth rate, correct to two decimal places.

(c) Teresa plans to travel to Rio de Janeiro for the Olympic Games this summer. She wishes to bring $2500 real (Brazilian currency) with her to spend. Given that the rate of exchange is €1 = R$3·4962 and the bank charges 2·5% commission on the exchange, calculate how much Teresa will have to pay, correct to the nearest cent.


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(a) A teacher needs to replace 7 jerseys and 3 balls in the sports kit for the Under-16 team in a school. The total cost of replacing this kit is €495. Let €x be the cost of a jersey and €y be the cost of a ball.

(i) Write down an equation in x and y to represent the cost of this kit.

(ii) The teacher later finds out he needs to replace 3 of the same-sized jerseys and 2 balls for the Under-14 team at a cost of €255. Write down an equation in x and y to represent the cost of this kit.

(iii) Solve these simultaneous equations to find the cost of a jersey and the cost of a ball.

(b) If the teacher orders everything at the same time, he will receive a discount of 12·5% on all the jerseys. Find the overall percentage discount he will receive on his entire order.

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(a) Solve the equation x2 + 2x − 4 = 0 and give your answers correct to two decimal places.

(b) Show that the co-ordinates of the turning point of the function f (x) = x2 + 2x − 4, x ∈ ℝ, are (−1, −5).


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(c) Use your answers to parts (a) and (b) above to sketch the graph of f (x). Show your scale on both axes.

(d) On the same axes above, sketch the graph of each of the functions:

g(x) = f (x) − 2, h(x) = f (x − 2).

Label each graph clearly.

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2 3 41�1�2�3�4�5

1

�1

2

�2

3

�3

4

�4

5

�5

6

�6

7

�7

x

y


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z = 2 + 5i is a complex number, where i2 = −1.

(a) Plot z and iz on the Argand diagram.

(b) (i) Verify algebraically that | z | = | iz |.

(ii) Give a reason why | z | = | iz | will always be true for any complex number z.

(c) Show that z is a root of the equation z2 − 4z + 29 = 0 and find the other root of the equation.

2

2

�2

4

6

�6

�4

4 6

Re( )z

Im( )z

��4�6


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The first three terms of an arithmetic sequence are 2, 8, 14.

(a) Find Tn , the nth term of the sequence.

(b) How many terms of the sequence are less than 150?

(c) Find S18, the sum of the first eighteen terms of the arithmetic series 2 + 8 + 14 …

(d) Find the value of n for which the sum of the first n terms of the series is 420.

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Section B Contexts and Applications 150 marks

Answer all three questions from this section.


The pilot of a stunt plane wishes to perform a particular aerobatic manoeuvre at an air show. The flight plan for a section of the manoeuvre was found to fit the following model:

h = 3

1t3 −

2

15t2 + 50t + 200, where 0 ≤ t ≤ 12

where h is the altitude of the stunt plane (its height above ground level) in metres and t is the time in seconds after the manoeuvre begins.

(a) Find the altitude at which the pilot must begin the manoeuvre.

(b) (i) Find dt

dh.

(ii) Use your answer to part (i) to find the time it would take the stunt plane to reach the top of its initial climb before the plane begins to dive.

(iii) Hence, find the altitude of the stunt plane at this time, correct to one decimal place.


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(c) The rate of climb (RoC) is the speed at which a plane can gain altitude.

(i) Find the RoC of the stunt plane after 3 seconds.

(ii) The pilot is using a Zivko Edge 540 stunt plane, which is commonly used in the Red Bull Air Race World Series. The plane has a maximum RoC of 20 metres per second.

Given that part of the initial climb requires an RoC greater than this speed, the pilot must join the climb at a later point.

Find the altitude at which the pilot must join the climb in order to attempt the manoeuvre. Give your answer correct to the nearest metre. [Hint: Let dh/dt = 20.]

(d) The greatest downward speed in the manoeuvre occurs when the acceleration of the stunt plane is zero.

(i) Find 2

2

dt

hd.

(ii) Use your answer to part (i) to find the value of t at which the acceleration of the stunt plane is zero.

(iii) Hence, find the greatest downward speed of the stunt plane.

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Jim plans to go to college this year. His parents invested a sum of money for his future education when he was born, 18 years ago, in a government bond which pays interest, compounded monthly, that corresponds to an annual equivalent rate (AER) of 3%.

(a) Given that (1 + r)12 = 1 + i, where r is the monthly rate and i is the annual rate of interest, find the rate of interest which, compounded monthly, corresponds to an AER of 3%, correct to four decimal places.

(b) Find the original sum of money that Jim’s parents invested if the value of the investment fund today is €12 768·25.

(c) Jim must pay €3000 in registration fees at the start of each year he attends college. Assuming that he makes no other withdrawals and the investment fund remains accruing interest at the same rate, calculate how much money will be left in the fund at the end of his four-year course, correct to the nearest cent.


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(d) Jim is considering whether to buy a scooter to go to and from college every day. He lives 16 km away from the college. A typical scooter uses 2·5 litres of fuel to travel 100 km.

(i) Assuming that fuel costs €1·40 per litre over the entire period, how much per day does it cost Jim to travel to and from college?

(ii) Another option would be to travel to and from college by bus. A weekly student ticket costs €21·50. Given that he attends college five days per week, how much will Jim save per week by using a scooter?

(iii) There are two semesters in each academic year and fourteen weeks in each semester. Jim does not wish to spend any more on buying the scooter and the fuel than buying bus tickets over four years. How much money does Jim have to purchase the scooter?

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The first three patterns in a sequence of patterns formed by arranging square tiles are shown below.

(a) Draw the fourth pattern in the sequence.

(b) (i) Complete the table below.

Pattern Number 1 2 3 4 5 6 7

Number of Tiles 4 8 14

(ii) Show that the number of tiles in each pattern forms a quadratic sequence.

(iii) Write an expression in n for the number of tiles needed to turn the nth pattern into the (n + 1)th pattern.


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(c) Draw a graph to represent the number of tiles needed for each of the first seven patterns on the axes below.

(d) The number of tiles in the nth pattern is given by the formula

Tn = n2 + bn + c, where b, c ∈ ℚ and n ∈ ℕ.

(i) Find the value of b and the value of c.

(ii) How many tiles are needed for the 30th pattern in the sequence?

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x

Nu

mb

ero

fT

iles

Pattern NumberPattern Number


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Pre-Leaving Certificate, 2016 – Ordinary Level

Mathematics – Paper 1 Time: 2 hours, 30 minutes

Documents

2016.1 L.16 -16 sk-final - WordPress.com · 2016-03-14 · Pre-Leaving Certificate, 2016 2016.1 L.16 5/16 Page 5 of 15 Mathematics Paper 1 – Ordinary Level Question 3 (25 marks)