M12/5/MATSD/SP1/ENG/TZ1/XX - Brandon Academy · A clean copy of the Mathematical Studies SL...

M12/5/MATSD/SP1/ENG/TZ1/XX

22127403

MATHEMATICAL STUDIES Candidate session number

STANDARD LEVEL PAPER 1

Thursday 3 May 2012 (afternoon) Examination code

1 hour 30 minutes

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.

A clean copy of the Mathematical Studies SL information booklet is required for this paper.

Answer all questions

Write your answers in the boxes provided.

Unless otherwise stated in the question, all numerical answers should be given exactly or correct to

three significant figures.

The maximum mark for this examination paper is [90 marks].

0 0

2 2 1 2 – 7 4 0 3

Please do not write on this page.

Answers written on this page will

not be marked.

\ – 3 – M12/5/MATSD/SP1/ENG/TZ1/XX

Maximum marks will be given for correct answers. Where an answer is incorrect, some marks may be given

for a correct method, provided this is shown by written working. Write your answers in the answer boxes

provided. Solutions found from a graphic display calculator should be supported by suitable working, e.g. if

graphs are used to find a solution, you should sketch these as part of your answer.

1. The following six integers are arranged from smallest to largest

1 , x , 3 , y , 14 , z

The mode is 1 , the median is 5 and the mean is 7.

Find

(a) x ; [1 mark]

(b) y ; [2 marks]

(c) z . [3 marks]

Working:

Answers:

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

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

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

2. The coordinates of point A are (−4, p) and the coordinates of point B are (2, −3) .

The mid-point of the line segment AB, has coordinates (q, 1).

(a) Find the value of

(i) q ;

(ii) p . [4 marks]

(b) Calculate the distance AB. [2 marks]

Working:

Answers:

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

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

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


3. Ross is a star that is 82 414 080 000 000 km away from Earth. A spacecraft, launched

from Earth, travels at 48 000 kmh–1 towards Ross.

(a) Calculate the exact time, in hours, for the spacecraft to reach the star Ross. [2

marks]

(b) Give your answer to part (a) in years. (Assume 1 year = 365 days) [2

marks]

(c) Give your answer to part (b) in the form a×10k , where 1≤ a < 10 and k ∈. [2

marks]

Working:

Answers:

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

. .

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

. .

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

. .


(Question 4 continued)

Working:

Answers:

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

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

5. The daily rainfall for the town of St. Anna is collected over a 20-day period of time. The

collected data are represented in the box and whisker plot below.

Daily rainfall (mm)

(a) Write down

(i) the lowest daily rainfall;

(ii) the highest daily rainfall. [2 marks]

(b) State what the value of 12 mm represents on the given diagram. [1 mark]

(c) Find the interquartile range. [2 marks]

(d) Write down the percentage of the data which is less than the upper quartile. [1 mark]

4 6 8 10 12 14 16 18 20 22


6. Consider the statements

p : The numbers x and y are both even.

q : The sum of x and y is an even number.

(a) Write down, in words, the statement p ⇒ q . [2 marks]

(b) Write down, in words, the inverse of the statement p ⇒ q . [2 marks]

(c) State whether the inverse of the statement p ⇒ q is always true. Justify your

answer. [2 marks]

Working:

Answers:

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

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7. The temperature T of water in a lake, in °C, in a 24 hour period is given by the

trigonometric function T(t) =−2cos(bt)+ 20 , where b is a constant, and t is the time in

hours from midnight. The graph of the function is given below.

Temperature

T (°C)

Time t (hours)

(a) Write down the time at which the water reaches its maximum temperature. [1

mark]

(b) Write down the temperature at 06:00. [1

mark]

(c) Write down the time interval during which the temperature is higher than 20 C. [2

marks]

(d) Calculate the value of b .

[2

marks]

Working:

Answers:

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

. .

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

. .

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

. .

(d) . . . . . . . . . . . . . . . . . . . . . . . . .

. .


8. Sasha travelled from the USA to Mexico and converted 650 US dollars (USD) to Mexican

pesos (MXN). Her bank offered an exchange rate of 1 USD = 12.50 MXN.

(a) Find the number of MXN that Sasha received.

Before her return to the USA, Sasha exchanged 2300 MXN back into USD. The bank charged

a commission of 1 %. The exchange rate was still 1 USD = 12.50 MXN.

[2 marks]

(b) Write down the commission charged by the bank in MXN.

(c) Calculate the amount of USD that Sasha received after commission. Give your

[1 mark]

answer correct to the nearest USD. [3 marks]

Working:

Answers:

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

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

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

9. Line L is given by the equation 3y + 2x = 9 and point P has coordinates (6 , –5).

(a) Explain why point P is not on the line L . [1 mark]

(b) Find the gradient of line L . [2 marks]

(c) (i) Write down the gradient of a line perpendicular to line L .

(ii) Find the equation of the line perpendicular to L and passing through

point P . [3 marks]

Working:

Answers:

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

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

(c) (i)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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


10. The diagram shows quadrilateral ABCD in which AB = 13 m , AD = 6 m and DC = 10

m. Angle ADC =120 and angle ABC = 40.

(a) Calculate the length of AC. [3

marks]

(b) Calculate the size of angle ACB.

[3

marks]

Working:

Answers:

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

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

13 m

6 m

10 m

A B

D

C

diagram not to scale

40

120

11. The number of calories a person burns during a walk depends on the time they spend

walking. The table below shows the number of calories, y, burned by a person in relation

to the time they spend walking, x, in minutes.


12. Merryn plans to travel to a concert tomorrow. Due to bad weather, there is a 60 % chance

that all flights will be cancelled tomorrow. If the flights are cancelled Merryn will travel

by car.

If she travels by plane the probability that she will be late for the concert is 10 %.

If she travels by car, the probability that she will not be late for the concert is 25 %.

(a) Complete the tree diagram below. [1 mark]

(b) Find the probability that Merryn will not be late for the concert.

Merryn was not late for the concert the next day.

[3

marks]

(c) Given that, find the probability that she travelled to the concert by car.

[2

marks]

Working:

Answers:

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

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

0.10

0.4 Travel

by plane

Travel

by car

Late

Late

Not late

Not late 0.25

0.6

13. In triangle ABC, BC = 8 m, angle ACB = 110, angle CAB = 40, and angle ABC = 30.

(a) Find the length of AC. [3 marks]

(b) Find the area of triangle ABC. [3 marks]

Working:

Answers:

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

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

B C

A

8 m

30 110

40

diagram not to scale


14. A function f(x) = p×2x +q is defined by the mapping diagram below.

15. Veronica wants to make an investment and accumulate 25 000 EUR over a period of 18

years. She finds two investment options.

Option 1 offers simple interest of 5 % per annum.

(a) Find out the exact amount she will have in her account after 18 years, if she

invests 12 500 EUR with this option. [3 marks]

Option 2 offers a nominal annual interest rate of 4 %, compounded monthly.

(b) Find the amount that Veronica has to invest with option 2 to have 25 000 EUR in her account after

18 years. Give your answer correct to two decimal places. [3 marks]

Working:

Answers:

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

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


Please do not write on this page.

Answers written on this page will

not be marked.

– 20 – M11/5/MATSD/SP2/ENG/TZ2/XX

2211-7406

M11/5/MATSD/SP2/ENG/TZ2/XX

22117406

MATHEMATICAL STUDIES

STANDARD LEVEL PAPER 2

Thursday 5 May 2011 (morning)

1 hour 30 minutes

INSTRUCTIONS TO CANDIDATES

Do not open this examination paper until instructed to do so.

Answer all the questions.

A graphic display calculator is required for this paper.

Unless otherwise stated in the question, all numerical answers must be given exactly or correct to

three significant figures.


Please start each question on a new page. You are advised to show all working, where possible. Where

an answer is wrong, some marks may be given for correct method, provided this is shown by written

working. Solutions found from a graphic display calculator should be supported by suitable working, e.g.

if graphs are used to find a solution, you should sketch these as part of your answer.

1. [Maximum mark: 23]

Part A

A university required all Science students to study one language for one year. A survey

was carried out at the university amongst the 150 Science students. These students all

studied one of either French, Spanish or Russian. The results of the survey are shown

below.

French Spanish Russian

Female 9 29 12

Male 31 40 29

Ludmila decides to use the χ2 test at the 5 % level of significance to determine whether the

choice of language is independent of gender.

(a) State Ludmila’s null hypothesis. [1 mark]

(b) Write down the number of degrees of freedom. [1 mark]

(c) Find the expected frequency for the females studying Spanish. [2 marks]

(d) Use your graphic display calculator to find the χ2 test statistic for this data. [2 marks]

(e) State whether Ludmila accepts the null hypothesis. Give a reason for your

answer.

[2 marks]

(This question continues on the following page)


Give all your numerical answers correct to two decimal places.

On 1 January 2005, Daniel invested 30 000 AUD at an annual simple interest

rate in a Regular Saver account. On 1 January 2007, Daniel had 31 650 AUD

in the account.

(a) Calculate the rate of interest.

On 1 January 2005, Rebecca invested 30 000 AUD in a Supersaver account at a

nominal annual rate of 2.5 % compounded annually.

[3 marks]

(b) Calculate the amount in the Supersaver account after two years.

(c) Find the number of complete years since 1 January 2005 it would take for

the

[3 marks]

amount in Rebecca’s account to exceed the amount in Daniel’s account.

On 1 January 2007, Daniel reinvested 80 % of the money from the Regular

Saver account in an Extra Saver account at a nominal annual rate of 3 %

compounded quarterly.

(d) (i) Calculate the amount of money reinvested by Daniel on the 1

January 2007.

(ii) Find the number of complete years it will take for the amount in Daniel’s

[3 marks]

Extra Saver account to exceed 30 000 AUD. [5 marks]

3. [Maximum mark: 18] Part A

A geometric sequence has 1024 as its first term and 128 as its fourth term.

(a) Show that the common ratio is . [2 marks]

(b) Find the value of the eleventh term. [2 marks]

(c) Find the sum of the first eight terms. [3 marks]

(d) Find the number of terms in the sequence for which the sum first

exceeds 2047.968.

[3 marks]

Part B

Consider the arithmetic sequence 1, 4, 7, 10, 13, …

(a) Find the value of the eleventh term. [2 marks]

(b) The sum of the first n terms of this sequence is 𝑛

2(3n−1).

(i) Find the sum of the first 100 terms in this arithmetic sequence.

(ii) The sum of the first n terms is 477.

(a) Show that 3n2 −n−954 = 0.

(b) Using your graphic display calculator or otherwise, find the

number of terms, n. [6 marks]


The diagram represents a small, triangular field, ABC, with BC = 25 m, angle BAC

= 55 and angle ACB = 75.

(a) Write down the size of angle ABC. [1 mark]

(b) Calculate the length of AC. [3 marks]

(c) Calculate the area of the field ABC.

N is the point on AB such that CN is perpendicular to AB. M is the midpoint

of CN.

[3 marks]

(d) Calculate the length of NM.

A goat is attached to one end of a rope of length 7 m. The other end of the

rope is attached to the point M.

(e)Decide whether the goat can reach point P, the midpoint of CB. Justify

your

[3 marks]

answer. [5 marks]

55

75

A

C

B

M

N diagram not to scale


The function f(x) is defined by f(x) =1.5x + 4 + 6

𝑥, x ≠ 0.

(a) Write down the equation of the vertical asymptote. [2 marks]

(b) Find f ′(x). [3 marks]

(c) Find the gradient of the graph of the function at x = −1. [2 marks]

(d) Using your answer to part (c), decide whether the function f(x) is increasing

or decreasing at x = −1. Justify your answer. [2 marks]

(e) Sketch the graph of f(x) for −10 ≤ x ≤10 and −20 ≤ y ≤ 20. [4 marks]

P1 is the local maximum point and P2 is the local minimum point on the graph of f(x).

(f) Using your graphic display calculator, write down the coordinates of

(i) P1;

(ii) P2. [4 marks]

(g) Using your sketch from (e), determine the range of the function

f(x) for −10 ≤ x ≤ 10.

[3 marks]

M12/5/MATSD/SP1/ENG/TZ1/XX - Brandon Academy · A clean copy of the Mathematical Studies SL...

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