mathematics - m. selkirk confey college

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1065Confey College

Kildare

2017.1 L.17 1/20 of 19

L.17

NAME

SCHOOL

TEACHER

Pre-Leaving Certificate Examination, 2017

Mathematics

Paper 1

Higher Level

Time: 2 hours, 30 minutes

300 marks

For examiner

Question Mark

1

2

School stamp 3

4

5

6

7

8

9

Grade

Running total

Total

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Printed:

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SAMPLE© DEB Exams

1065Confey College

Kildare

Pre-Leaving Certificate, 2017 2017.1 L.17 2/20

of 19

MathematicsPaper 1 – Higher 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. There is space for extra work at the back of the booklet. 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 you do not show all necessary work.

You may lose marks if you do not include appropriate units of measurement, where relevant.

You may lose marks if you do not give your answers 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) Simplify

15112

92

2

+−−

xx

x ÷ 23

2

104

3

xx

xx

−+

.

(b) Find the range of values of x for which

5

23

−−

x

x ≤ 5, where x ∈ ℝ and x ≠ 5.

(c) Prove that the equation px2 − (2p + 1)x + 2 = 0 has real roots for all values of p ∈ ℝ and hence, or otherwise, write down the roots of the equation in terms of p.

Page Running

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(a) Given that 4z − 3z = i

i

−−2

181, express z in the form a + bi, where a, b ∈ ℝ and i2 = −1.

(b) The complex number w has modulus 38

3 and argument

3

2π.

(i) Use De Moivre’s Theorem to find, in polar form, the three complex cube roots of w. (That is, find the three values of v for which v3 = w.)

(ii) w is marked on the Argand diagram below. On the same diagram, show your answers to part (i) and hence, write down the equation of the curve on which all three roots lie.

Re

Im

2 3

1

2

3

�2

1�1��

�1

w

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(a) One root of the equation 4x3 − 8x2 + kx + 2 = 0 is 2

1.

Find the value of k ∈ ℝ and hence the other roots of the equation.

(b) (i) Express log9 xy in terms of log3

x and log3 y.

(ii) Hence, or otherwise, solve the simultaneous equations:

log9 xy =

2

5

log3 x .log3

y = −6.

Express your answers in their simplest form.

Previous Page Running

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(a) A curve is defined by the equation x2 + y2 − 6x − 2y + 1 = 0.

(i) Find dx

dy in terms of x and y.

(ii) Hence, find the equation of the tangent to the curve at the point (3, –2).

(b) (i) Show that the curve y = 3

2

−x, where x ≠ 3 and x ∈ ℝ, has no turning points and

no points of inflection.

(ii) Comment on the shape of the curve for all x ∈ ℝ.

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(a) The power supply to a space satellite is provided by means of a generator that converts heat released by the decay of a radioisotope into electricity. The power output, in watts, may be calculated using the function

w(t) = Aebt,

where t is the time, in days, from when the satellite is launched into space. The initial power output at the launch of the satellite is 60 watts.

(i) Given that after 14 days the power output falls to 56 watts, calculate the value of b, correct to three decimal places.

(ii) The satellite cannot function properly when the power output falls below 5 watts. After how many days will the satellite fail to function properly?

(b) Find the value of the constant k for which w(t + k) = 2

1w(t), for all t ≥ 0.

Give your answer in the form p lnq, where p, q ∈ ℕ.


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(a) Fiona arranged to pay €120 at the end of each week for 25 years into a pension fund that earns an annual equivalent rate (AER) of 3⋅9%.

(i) Show that the rate of interest, compounded weekly, which corresponds to an AER of 3⋅9% is 0⋅0736%, correct to four decimal places. [1 year = 52 weeks]

(ii) Calculate, correct to the nearest euro, the total value of Fiona’s pension fund when she retires.

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(b) On retirement, Fiona invests the total value of her pension fund in a scheme that earns an AER of 4⋅2%. Fiona will receive a fixed amount of money at the end of each month for twenty years, at which time the value of her investment will be zero. Calculate, correct to the nearest euro, the amount of each monthly payment.


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1065Confey College

Kildare


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Section B Contexts and Applications 150 marks

Answer all three questions from this section.


The diagram below shows the beginning of Pascal’s triangle.

2

1

1 1

3 31

1 1

1

The rows of Pascal’s triangle are conventionally enumerated, starting with row r = 0 at the top (row 0). The entries in each row are numbered from left to right, beginning with k = 0 (e.g. in row 3, k0 = 1, k1 = 3, k2 = 3 and k4 = 1).

The triangle may be constructed as follows: In row 0 (the topmost row), the entry is 1. Each entry in successive rows is found by adding the number above and to the left with the number above and to the right, treating blank entries as 0.

There are several patterns found within Pascal’s triangle. Consider the two sequences, A and B, shown below.

2

1

1

1

1 1

3

4

5 10 10 5

46

31

1

1

1 1

1

B

A

(a) Find an expression for Tn , the nth term, and Sn, the sum of the first n terms, of sequence A.

(b) Find an expression for Tn , the nth term of sequence B.

............................ row 0

....................... row 1

.................. row 2

.............. row 3

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(c) (i) Verify that the third entry of row 6 of Pascal’s triangle is found by adding T5 of sequence A and T4 of sequence B.

(ii) Find an expression, in n, for the third entry of the rth row and hence, verify your answer to part (i) above.

(iii) An entry in Pascal’s triangle is denoted

k

r and can be determined using the formula:

k

r =

−−

1

1

k

r +

−k

r 1,

where r is the row number (top row = 0) and k is the entry number in row r (first entry = 0).

Using the above formula, verify your expression, in n, for the third entry in the rth row.


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(d) Prove by induction that Sn, the sum of the first n terms of sequence B, is 6

)2)(1( ++ nnn

for all n ∈ ℕ.

(e) The coefficients of a binomial expansion can be found using Pascal’s triangle.

(i) Using Pascal’s triangle, or otherwise, expand (a + b)4 + (a − b)4 and simplify.

(ii) Hence, express (x + 12 +x )4 + (x − 12 +x )4 as a polynomial in terms of x.

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(a) A grain silo is a tank used for the bulk storage of grain after it is harvested. A particular grain silo is in the shape of an inverted right cone, as shown. The vertical height of the cone is 5 m and the diameter of the base of the cone is 4 m.

Grain is pumped into an empty silo at a uniform rate of 4 m3 per minute. Let h be the depth of the grain and r be the radius of the grain in the silo after t minutes.

(i) Using similar triangle, or otherwise, show that r = 5

2h.

(ii) Find, in terms of π and h, the volume of grain in the silo after t minutes.

(b) (i) Find, in terms of π, the rate at which the depth of grain is increasing when the depth of grain in the silo is 3 m.


5 m

h

r

4 m

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(ii) Find the rate at which the free surface of the grain is increasing when the radius is 1⋅5 m.

(c) The company which manufactures these grain silos wishes to minimise the amount of sheet metal required to produce each one while retaining the same capacity (volume) of the tank.

(i) Express the curved surface area of the silo in term of π and h.

(ii) Hence, find the value of the radius that minimises the curved surface area of the grain silo, correct to two decimal places.

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1065Confey College

Kildare


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(a) The acceleration of a particle, in m s–2, moving in a straight line during a particular time interval, is given by:

a = 2

1

t + 3t, for 1 ≤ t ≤ 5,

where t is the time, in seconds, from the instant the particle begins to move.

(i) Given that the speed of the particle is 2

1 after 1 s, find its speed after 5 s.

(ii) Find the average speed of the particle over the interval 1 ≤ t ≤ 5. Give your answer correct to two decimal places.


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(b) The proposed level of new roadways is achieved primarily through series of ‘cuts’ and ‘fills’, taking earth material from one area and using it somewhere else.

The diagram shows the vertical cross-section of a roadway through a particular terrain. The proposed elevation of the roadway is 14 m above sea-level. A cut is proposed between points A and B.

Using the co-ordinate plane with the y-axis as the initial point of the cut and the x-axis as sea-level, the elevation of the terrain can be described by the function

f (x) = 32 − 2(x − 3)2,

where both x and f (x) are measured in metres.

(i) Find the co-ordinates of A and B.

(ii) Use the trapezoidal rule and interval widths of 1 m to find the approximate area of the shaded cross-section of earth material to be excavated between the elevation of the terrain and the proposed elevation of the roadway.

f x( )

x

x

y = 14

y = 14

y

A B

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(iii) Use integration to find the actual area of the shaded cross-section.

(d) An alternative proposal is to construct the new roadway at an elevation of 24 m above sea-level. Find, correct to two decimal places, the percentage reduction in the cross-section of material to be excavated if this proposal was adopted.


f x( )

x

y = 14

y = 24

y

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Pre-Leaving Certificate, 2017 – Higher Level

Mathematics – Paper 1 Time: 2 hours, 30 minutes

mathematics - m. selkirk confey college

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