Mathematics: applications and interpretation Standard ...ionma.org/share/maths/mathsai/mockexam/mockexamfullpaper.pdf · Standard level Mock paper 1 i i 11 pages Tuesday 2 June 2020

NCC1701

Candidate session number

Mathematics: applications and interpretationStandard levelMock paper 1

i i

11 pages

Tuesday 2 June 2020 (morning)

1 hour

Instructions to candidates

Write your session number in the boxes above.Do not open this examination paper until instructed to do so.A graphic display calculator is required for this paper.Answer all questions.Answers must be written within the answer boxes provided.Unless otherwise stated in the question, all numerical answers should be given exactly or correct to three signif cant f gures.A clean copy of the mathematics: applications and interpretation formula booklet is required for this paper.The maximum mark for this examination paper is

M20/5/MATAI/SP1/ENG/TZ0/XX

[45 marks]

© Jesús García Rodríguez 2020

Please do not write on this page.

Answers written on this page will not be marked.

– 2 – M20/5/MATAI/SP1/ENG/TZ0/XX

NCC1701

1. A calculator f ts into a cuboid case with height 29 mm, width 98 mm and length 186 mm.

(a) Find the volume, in mm3, of this calculator case. Give your answer to two signif cant f gures. [2]

(b) Write down your answer to part (a) in the form a × 10k where 1 ≤ a<10 and k∈ . [2]

(c) Find the volume, in cm3, of this calculator case. [2]

Working:

Answers:

(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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– 3 –

Answers must be written within the answer boxes provided. Full marks are not necessarily awardedfor a correct answer with no working. Answers must be supported by working and/or explanations.Solutions found from a graphic display calculator should be supported by suitable working. For example,if graphs are used to f nd a solution, you should sketch these as part of your answer. Where an answeris incorrect, some marks may be given for a correct method, provided this is shown by written working.You are therefore advised to show all working.

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NCC1701


3. In this question, give all answers to two decimal places.

Velina travels from New York to Copenhagen with 1200 US dollars (USD). She exchanges her money to Danish kroner (DKK). The exchange rate is 1 USD = 7.0208 DKK .

(a) Calculate the amount that Velina receives in DKK. [2]

At the end of her trip Velina has 3450 DKK left that she exchanges to USD. The bank charges a 5 % commission. The exchange rate is still 1 USD = 7.0208 DKK .

(b) (i) Calculate the amount, in DKK, that will be left to exchange after commission.

(ii) Hence, calculate the amount of USD she receives. [4]

Working:

Answers:

(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) (i) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(ii) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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NCC1701

3. [Maximum mark: 6]

The Osaka Tigers basketball team play in a multilevel stadium.

The most expensive tickets are in the f rst row. The ticket price, in Yen (¥), for each row forms an arithmetic sequence. Prices for the f rst three rows are shown in the following table.

Ticket pricing per game

1st row 6800 Yen

2nd row 6550 Yen

3rd row 6300 Yen

(a) Write down the value of the common dif erence, d [1]

(b) Calculate the price of a ticket in the 16th row. [2]

(c) Find the total cost of buying 2 tickets in each of the f rst 16 rows. [3]

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NCC1701

[Maximum mark: 6]

Professor Vinculum investigated the migration season of the Bulbul bird from their natural wetlands to a warmer climate.

He found that during the migration season their population, P could be modelled by 1350 400(1.25) tP -= + , t ≥ 0 , where t is the number of days since the start of the

migration season.

(a) Find the population of the Bulbul birds,

(i) at the start of the migration season.

(ii) in the wetlands after 5 days. [3]

(b) Calculate the time taken for the population to decrease below 1400. [2]

(c) According to this model, f nd the smallest possible population of Bulbul birds during the migration season. [1]

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4.


NCC1701

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[Maximum mark: 6]

Points A (3 , 1) , B (3 , 5) , C (11 , 7) , D (9 , 1) and E (7 , 3) represent snow shelters in the Blackburn National Forest. These snow shelters are illustrated in the following coordinate axes.

Horizontal scale: 1 unit represents 1 km.

Vertical scale: 1 unit represents 1 km.

2 4 6 8 10 12 14 16−2

−4

0

2

4

6

8

10

12

B

C

E

DA

x

y

(a) Calculate the gradient of the line segment AE. [2]

(This question continues on the following page)

5.


NCC1701

(Question 5 continued)

The Park Ranger draws three straight lines to form an incomplete Voronoi diagram.

2 4 6 8 10 12 14 16−2

−4

0

2

4

6

8

10

12

B

C

E

DA

x

y

(b) Find the equation of the line which would complete the Voronoi cell containing site E. Give your answer in the form ax + by + d = 0 where a ,b ,d ∈ . [3]

(c) In the context of the question, explain the signif cance of the Voronoi cell containing site E. [1]

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NCC1701

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Line L has a y-intercept at (0, 3) and an x-intercept at (4, 0) , as shown on the following diagram.

−4−5 −3 −2 −10 1 2 3 4 5

1

2

3

4

5

y

x

L

(a) (i) Find the gradient of L .

(ii) Write down the equation of L in the form y =mx +c . [3]

Line N is perpendicular to L , and passes through point P (2, 1).

(b) (i) Write down the gradient of N .

(ii) Find the equation of N in the form y =mx +c . [3]

Working:

Answers:

(a) (i) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(ii) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) (i) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(ii) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.


NCC1701

A potter sells x vases per month.

His monthly prof t in Australian dollars (AUD) can be modelled by

3 21( ) 7 120 0

5,=- + - ≥P x x x x .

(a) Find the value of P if no vases are sold. [1]

(b) Dif erentiate 3 21( ) 7 120 0

5,=- + - ≥P x x x x. [2]

(c) Hence, f nd the number of vases that will maximize the prof t. [3]

Working:

Answers:

(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.


NCC1701

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The following diagram shows part of the graph of f (x) = (6 - 3x) (4 + x) , x∈ . The shaded region R is bounded by the x-axis, y-axis and the graph of f .

y

x

R

(a) Write down an integral for the area of region R . [2]

(b) Find the area of region R . [1]

(c) Knowing that the coordinates of A are (-4, 0), use the trapezoid method to estimatethe area of region S

S

A

[3]

8.


NCC1701



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NCC1701

Mathematics: applications and interpretationStandard levelMock paper 2

Tuesday 2 June 2020 (morning)

7 pages

1 hour

Instructions to candidates

Do not open this examination paper until instructed to do so.A graphic display calculator is required for this paper.Answer all the questions in the answer booklet provided.Unless otherwise stated in the question, all numerical answers should be given exactly or correct to three signif cant f gures.A clean copy of the mathematics: applications and interpretation formula booklet is required for this paper.The maximum mark for this examination paper is [60 marks].


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© Jesús García Rodríguez 2020

Please do not write on this page.

Answers written on this page will not be marked.


NCC1701

[Maximum mark: 11]

Tommaso plans to compete in a regional bicycle race after he graduates, however he needs to buy a racing bicycle. He f nds a bicycle that costs 1100 euro (EUR). Tommaso has 950 EURand invests this money in an account that pays 5 % interest per year, compounded monthly.

(a) Determine the amount that he will have in his account after 3 years. Give your answer correct to two decimal places. [3]

The cost of the bicycle, C , can be modelled by C =20x+1100 , where x is the number of years since Tommaso invested his money.

(b) Find the dif erence between the cost of the bicycle and the amount of money in Tommaso’s account after 3 years. Give your answer correct to two decimal places. [3]

After m complete months Tommaso will, for the f rst time, have enough money in his account to buy the bicycle.

(c) Find the value of m . [5]

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Answer all questions in the answer booklet provided. Please start each question on a new page.Full marks are not necessarily awarded for a correct answer with no working. Answers must besupported by working and/or explanations. Solutions found from a graphic display calculator shouldbe supported by suitable working. For example, if graphs are used to f nd a solution, you should sketch these as part of your answer. Where an answer is incorrect, some marks may be given for a correctmethod, provided this is shown by written working. You are therefore advised to show all working.

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[Maximum mark: 16]

On her f rst day in a hospital, Kiri receives u1 milligrams (mg) of a therapeutic drug. The amount of the drug Kiri receives increases by the same amount, d , each day. On the seventh day, she receives 21mg of the drug, and on the eleventh day she receives 29 mg.

(a) Write down an equation, in terms of u1 and d , for the amount of the drug that she receives

(i) on the seventh day;

(ii) on the eleventh day. [2]

(b) Write down the value of d and the value of u1 . [2]

Kiri receives the drug for 30 days.

(c) Calculate the total amount of the drug, in mg, that she receives. [3]

Ted is also in a hospital and on his f rst day he receives a 20 mg antibiotic injection. The amount of the antibiotic Ted receives decreases by 50 % each day. On the second day, Ted receives a 10mg antibiotic injection, on the third day he receives 5mg, and so on.

(d) (i) Find the amount of antibiotic, in mg, that Ted receives on the f fth day.

(ii) The daily amount of antibiotic Ted receives will f rst be less than 0.06 mg on the k th day. Find the value of k .

(iii) Hence f nd the total amount of antibiotic, in mg, that Ted receives during the f rst k days. [9]

Turn over

2.


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[Maximum mark: 17]

The braking distance of a vehicle is def ned as the distance travelled from where the brakes are applied to the point where the vehicle comes to a complete stop.

The speed, s m s- 1, and braking distance, d m, of a truck were recorded. This information is summarized in the following table.

Speed, s m s- 1 0 6 10

Braking distance, d m 0 12 60

This information was used to create Model A, where d is a function of s , s ≥ 0 .

Model A: d (s) = ps2 + qs , where p , q∈

At a speed of 6 m s- 1 , Model A can be represented by the equation 6 p + q = 2 .

(a) (i) Write down a second equation to represent Model A, when the speed is 10m s- 1 .

(ii) Find the values of p and q [4]

(b) Find the coordinates of the vertex of the graph of y = d (s) . [2]

(c) Using the values in the table and your answer to part (b), sketch the graph of y = d (s) for 0 ≤ s ≤ 10 and - 10 ≤ d ≤ 60 , clearly showing the vertex. [3]

(d) Hence, identify why Model A may not be appropriate at lower speeds. [1]

Additional data was used to create Model B, a revised model for the braking distance of a truck.

Model B: d (s) = 0.95 s2 - 3.92 s

(e) Use Model B to calculate an estimate for the braking distance at a speed of 20 m s- 1 . [2]

The actual braking distance at 20 m s- 1 is 320m .

(f) Calculate the percentage error in the estimate in part (e). [2]


3.


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It is found that once a driver realizes the need to stop their vehicle, 1.6 seconds will elapse, on average, before the brakes are engaged. During this reaction time, the vehicle will continue to travel at its original speed.

A truck approaches an intersection with speed s m s- 1 . The driver notices the intersection’s traf c lights are red and they must stop the vehicle within a distance of 330 m.

diagram not to scale

330

(g) Using model B and taking reaction time into account, calculate the maximum possible speed of the truck if it is to stop before the intersection. [3]


fi

[Maximum mark: 15]

The Happy Straw Company manufactures drinking straws.

The straws are packaged in small closed rectangular boxes, each with length 8 cm, width 4 cmand height 3 cm. The information is shown in the diagram.

H G

C

BA

E

D

F

84

3

(a) Calculate the surface area of the box in cm2. [2]

(b) Calculate the length AG. [2]

Each week, the Happy Straw Company sells x boxes of straws. It is known that d

2 220d

Px

x= - + ,

x ≥ 0 , where P is the weekly prof t, in dollars, from the sale of x thousand boxes.

(c) Find the number of boxes that should be sold each week to maximize the prof t. [3]

The prof t from the sale of 20 000 boxes is $1700.

(d) Find P (x) . [5]

(e) Find the least number of boxes which must be sold each week in order to make a prof t. [3]

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Documents

Mathematics: applications and interpretation Standard ...ionma.org/share/maths/mathsai/mockexam/mockexamfullpaper.pdf · Standard level Mock paper 1 i i 11 pages Tuesday 2 June 2020