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Exemplar für Prüfer/innen

Supplementary Examination for the

Standardised Competence-Oriented

Written School-Leaving Examination

AHS

Main Examination Session 2021

Mathematics

Supplementary Examination 6

Examiner’s Version

Supplementary Examination 6 / Main Examination Session 2021 / MAT / Examiner p. 2/

Instructions for the standardised implementation of the

supplementary examination

The following supplementary examination booklet contains five tasks that can be completed inde- pendently of one another. Each task comprises two sub-tasks: the “task” and the “guiding ques- tion”.

The preparation time is to be at least 30 minutes; the examination time is at most 25 minutes.

The use of the official formula booklet that has been approved by the relevant government authority for use in the standardised school-leaving examination in mathematics is allowed. Furthermore, the use of electronic devices (e. g. graphic display calculators or other appropriate technology) is allowed provided there is no possibility to communicate (e. g. via the internet, intranet, Bluetooth, mobile networks etc.) and there is no access to an individual’s data stored on the device.

After the examination, all materials (tasks, extra sheets of paper etc.) from all candidates are to be collected in. The examination materials (tasks, extra sheets of paper, digital materials etc.) may only be made public after the time period allocated for the examination has passed.

Evaluation grid for the supplementary examination

The evaluation grid below may be used to assist in assessing the candidates’ performances.

Candidate 1 Candidate 2 Candidate 3 Candidate 4 Candidate 5

Task 1

Task 2

Task 3

Task 4

Task 5

Total

Supplementary Examination 6 / Main Examination Session 2021 / MAT / Examiner p. 4/

Task 1

Arrow

Task:

The diagram below shows a model of an arrow in 2-dimensional space. The dotted line is the axis of symmetry of this arrow.

f

d

c

• Using c, d and f, write down a formula that can be used to calculate the area A of the area shaded in grey.

A =

Guiding question:

The diagram below shows a model of the tip of an arrow. The dotted line is the axis of symmetry of the tip of the arrow.

h

a

a

b

The isosceles triangle shown in the diagram above has a base b = 6 cm, side length a cm and height h = 7 cm. The length of the base is to be kept the same, but the area of the triangle is to be increased by 20 %.

• Determine the side lengths of the triangle after the area has been increased.

Supplementary Examination 6 / Main Examination Session 2021 / MAT / Examiner p. 5/

Solution to Task 1

Arrow

Expected solution to the statement of the task:

A = 12 ∙ (c ∙ f – c ∙ d)

Answer key:

The point for the core competency is to be awarded if the formula has been given correctly.

Expected solution to the guiding question:

Abefore = 6 ∙ 7 2 = 21

Aafter = 21 ∙ 1.2 = 25.

Aafter = hafter 2 ∙ 6 ⇒ hafter = 8.4 cm

aafter =

 8.4^2 + 3^2 = 8.91...

The sides of the triangle have a length of around 8.9 cm after increasing the area.

Answer key:

The point for the guiding question is to be awarded if the length has been calculated correctly.

Supplementary Examination 6 / Main Examination Session 2021 / MAT / Examiner p. 7/

Solution to Task 2

Mountain Railway

Expected solution to the statement of the task:

α ... angle of elevation of the mountain railway x ... difference between the altitude of the station at the top of the mountain and the altitude of the station at the bottom of the mountain

tan( α) = 0.41 ⇒ α = 22.29...°

0.41 = 2 500 x ⇒ x = 1 025

The station at the top of the mountain is at an altitude of 2 025 m.

Answer key:

The point for the core competency is to be awarded if the angle of elevation and the altitude of the station at the top of the mountain have been calculated correctly.

Expected solution to the guiding question:

p(2 025) – p(1 000) 5 = –21.3... The average absolute change in the air pressure is around 21 mbar/min.

Answer key:

The point for the guiding question is to be awarded if the average absolute change in air pressure has been calculated correctly. The unit “mbar/min” does not need to be given.

Supplementary Examination 6 / Main Examination Session 2021 / MAT / Examiner p. 8/

Task 3

Trigonometric Functions

The diagram below shows the graphs of the functions f and g with f(x) = a ∙ sin(b ∙ x) and g(x) = c ∙ sin(d ∙ x) with a, b, c, d ∈ ℝ+.

f(x), g(x)

f

g

H

x 0

0

Task:

• Complete each of the gaps below with the appropriate symbol “<”, “>” or “=” and justify your answers.

a c

b d

Guiding question:

The maximum point of the graph of f labelled H in the diagram above has coordinates H = (^) (π 4 , 3).

• Determine a and b.

Supplementary Examination 6 / Main Examination Session 2021 / MAT / Examiner p. 10/

Task 4

Drag Race

Jan and Tom are participating in a drag race. They set off at the same time when t = 0. The velocities of their vehicles in the first few seconds can be described by the two functions vJ and vT.

t ... time in s vJ(t) ... velocity of Jan’s vehicle at time t in m/s vT(t) ... velocity of Tom’s vehicle at time t in m/s

Task:

For the time-velocity function vJ, the following relationship holds:

vJ(t) = 0.6 ∙ t^2 ∙ ℯ–0.09^ ∙^ t

• Determine the acceleration of Jan’s vehicle when t = 10.

Guiding question:

At time t 1 , Tom’s vehicle is ahead of Jan’s vehicle. The distance between the vehicles at time t 1 is d metres.

• Using vJ and vT, write down a formula that can be used to calculate d.

d =

Supplementary Examination 6 / Main Examination Session 2021 / MAT / Examiner p. 11/

Solution to Task 4

Drag Race

Expected solution to the statement of the task:

vJ′(10) = 2.68...

The acceleration is around 2.7 m/s^2.

Answer key:

The point for the core competency is to be awarded if the acceleration has been determined correctly.

Expected solution to the guiding question:

d = ∫

t 1

0 vT(t)^ dt^ –^ ∫

t 1 0 vJ(t)^ dt

Answer key:

The point for the guiding question is to be awarded if the formula has been given correctly.

Supplementary Examination 6 / Main Examination Session 2021 / MAT / Examiner p. 13/

Solution to Task 5

Balls

Expected solution to the statement of the task:

14 30 ∙^

13 29 +^

16 30 ∙^

15 29 =^

211 435 = 0.4850... ≈ 48.5 %

Answer key:

The point for the core competency is to be awarded if the probability has been determined correctly.

Expected solution to the guiding question:

expectation value: 1 ∙ P(X = 1) + 2 ∙ P(X = 2) + 3 ∙ P(X = 3) + 4 ∙ P(X = 4)

= 1 ∙ 14 + 2 ∙ 34 ∙ 13 + 3 ∙ 43 ∙ 23 ∙ 12 + 4 ∙ 34 ∙ 23 ∙ 12 = 2.

Answer key:

The point for the guiding question is to be awarded if the expectation value has been calculated correctly.

Exemplar für Prüfer/innen

Supplementary Examination for the

Standardised Competence-Oriented

Written School-Leaving Examination

AHS

May 2020

Mathematics

Supplementary Examination 6

Examiner’s Version

Supplementary Examination 6 / May 2020 / MAT / Examiner p. 3/

Evaluation grid for the supplementary examination

This evaluation grid may be used to assist the examiner in assessing the candidate’s performance.

Point for core competencies reached

Point for the guiding question reached

Task 1

Task 2

Task 3

Task 4

Task 5

Supplementary Examination 6 / May 2020 / MAT / Examiner p. 4/

Task 1

Angle of a Slope

In order to determine the danger of avalanches, it is important to know the angle of a slope.

Task:

A particular slope has an angle of 30° to the horizontal.

• Determine the gradient of the slope as a percentage.

Guiding question:

The diagram below shows a method used to estimate the angle of a slope using ski poles. The angle of the slope, α, is determined using two ski poles of equal length, AB and CD.

The ski pole CD is held horizontally to the slope; the ski pole AB is held vertically at the end of the pole CD (as in the diagram).

B

C D

A

α

• Write down the angle of the slope if using this method it is found that the points B and C have the same position as each other.
• Determine the angle of the slope, α, when the length of the line segment BC is one third of the length of the ski pole AB.

Supplementary Examination 6 / May 2020 / MAT / Examiner p. 6/

Task 2

Ideal Gas Equation

The equation p ∙ V = n ∙ R ∙ T models the relationship between the pressure p, the volume V, the amount of the substance n, and the absolute temperature T of an ideal gas. In the equation, R is a constant.

Task:

• Justify why the relationship between how the pressure p changes with respect to the temperature T can be modelled by a linear function of the form p(T) = k ∙ T + d (where k, d ∈ ℝ) if the other values are constant.
• Write down the parameters k and d of this linear function (in terms of the values given above).

Guiding question:

The pressure p of an ideal gas can be given as a function of the volume V if the values of n, R and T are constant.

• Complete the table of values shown below, sketch the graph of the function p in the coordinate system and write down which type of function p is.

V in cm^3 50 100 150 200

p(V ) in hPa 100

p(V ) in hPa

V in cm^3 0 50 100 150 200 250 300 350 400

350 300 250 200 150 100 50 0

400

Supplementary Examination 6 / May 2020 / MAT / Examiner p. 7/

Solution to Task 2

Ideal Gas Equation

Expected solution to the statement of the task:

If n, R and V are constant, then p(T ) = n^ V∙^ R∙ T. This equation corresponds to a linear function

with parameters k = n^ V∙^ R and d = 0.

Answer key:

The point for the statement of the task is to be awarded if a correct justification has been given and the parameters of the corresponding linear function k and d have been given correctly.

Expected solution to the guiding question:

p(V) = n^ ∙^ RV^ ∙^ T ⇒ p(V ) ∙ V = constant ⇒ p(V) ∙ V = 15 000

V in cm^3 50 100 150 200 p(V ) in hPa 300 150 100 75 50

p(V ) in hPa

V in cm 3 0 50 100 150 200 250 300 350 400

350 300 250 200 150 100 50 0

400

The function p is a power function (or reciprocal function or indirectly proportional function).

Answer key:

The point for the guiding question is to be awarded if the table of values has been completed correctly, a correct graph has been sketched and a correct function type has been given.