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2015 May Exam

1a. [2 marks]

, where , and .

Calculate the exact value of .

Markscheme

(M1)

Note:Award (M1) for correct substitution into formula.

(A1) (C2)

Note:Using radians the answer is , award at most (M1)(A0).

1b. [2 marks]

Give your answer to correct to

(i) two significant figures;

(ii) three decimal places.

Markscheme

(i) (A1)(ft)

Note: Follow through from part (a).

(ii) (A1)(ft) (C2)

Note:Follow through from part (a).

1c. [2 marks]

Pyotr estimates the value of to be .

Calculate the percentage error in Pyotr’s estimate.

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Markscheme

(M1)

Notes: Award (M1) for their correct substitution into the percentage error formula. Absolute value

signs are not required.

Their unrounded answer from part (a) must be used.

Do not accept use of answers from part (b).

(A1)(ft) (C2)

Notes: The sign is not required.

The answer from radians is , award (M1)(A1)(ft).

Follow through from part (a).

2a. [2 marks]

The IB grades attained by a group of students are listed as follows.

Find the median grade.

Markscheme

(M1)

Note: Award (M1) for correct ordered set.

(A1) (C2)

2b. [2 marks]

Calculate the interquartile range.

Markscheme

(M1)

Note:Award (M1) for correct quartiles seen.

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(A1) (C2)

2c. [2 marks]

Find the probability that a student chosen at random from the group scored at least a grade .

Markscheme

(A2) (C2)

3a. [2 marks]

The diagram shows the straight line , which intersects the -axis at and the -axis at

.

Write down the coordinates of M, the midpoint of line segment AB.

Markscheme

(A1)(A1) (C2)

Note: Accept , . Award (A0)(A1) if parentheses are missing.

3b. [2 marks]

Calculate the gradient of .

Markscheme

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(M1)

Note: Award (M1) for correct substitution into gradient formula.

(A1) (C2)

Note: Accept .

3c. [2 marks]

The line is parallel to and passes through the point .

Find the equation of . Give your answer in the form .

Markscheme

(M1)

OR

(M1)

Note: Award (M1) for substitution of their gradient from part (b).

(A1)(ft) (C2)

Note: Follow through from part (b).

The answer must be an equation in the form for the (A1)(ft) to be awarded.

4a. [4 marks]

Identical mosquito traps are placed at different distances from a lake. On one day the number of

mosquitoes caught in 10 of the traps is recorded.

It is believed the number of mosquitoes caught varies linearly with the distance, in metres, of the trap

from the lake.

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Find

(i) Pearson’s product–moment correlation coefficient, ;

(ii) the equation of the regression line on .

Markscheme

(i) (A2)

Note: Award (A0)(A1) for .

Award (A1)(A0) for .

(ii) (A1)(A1) (C4)

Note: Award a maximum of (A0)(A1) if the answer is not an equation.

4b. [2 marks]

Use the equation of the regression line on to estimate the number of mosquitoes caught in a trap

that is m from the lake.

Markscheme

(M1)

Note: Award (M1) for correct substitution of into their equation of regression line.

(A1)(ft) (C2)

Note: Accept or or from use of sf values.

Follow through from part (a)(ii).

5a. [3 marks]

Assume the Earth is a perfect sphere with radius 6371 km.

Calculate the volume of the Earth in . Give your answer in the form , where

and .

Markscheme

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(M1)

Note: Award (M1) for correct substitution into volume formula.

(A2) (C3)

Notes: Award (A1)(A0) for correct mantissa between 1 and 10, with incorrect index.

Award (A1)(A0) for

Award (A0)(A0) for answers of the type: .

5b. [3 marks]

The volume of the Moon is .

Calculate how many times greater in volume the Earth is compared to the Moon.

Give your answer correct to the nearest integer.

Markscheme

(M1)

Note: Award (M1) for dividing their answer to part (a) by .

(A1)(ft)

Note: Accept from use of 3 sf answer to part (a).

(A1) (C3)

Notes: Follow through from part (a).

The final (A1) is awarded for their unrounded non-integer answer seen and given correct to the nearest

integer.

Do not award the final (A1) for a rounded answer of 0 or if it is incorrect by a large order of magnitude.

6a. [2 marks]

Pietro arrives in Singapore and, at the airport, changes 800 euros (EUR) to Singapore dollars (SGD).

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The bank rates quoted at the airport for exchanging EUR with SGD are given in the following table. Also

given are the rates for exchanging SGD with British pounds (GBP) and US dollars (USD). There is no

commission charged on exchanges.

Calculate the number of SGD Pietro receives.

Markscheme

(M1)

Note: Award (M1) for multiplication by .

(A1) (C2)

6b. [4 marks]

Pietro also has 100 GBP that he wishes to change to USD for a trip to Cambodia.

To perform this transaction, the GBP must first be converted to SGD and then to USD.

Calculate the number of USD Pietro receives.

Markscheme

(A1)(M1)(M1)

Notes: Award (M1) for multiplication by a GBP rate ( or ), (M1) for division by a USD rate (

or ), (A1) for two correct rates used.

(A1) (C4)

Note: Award a maximum of (A1)(ft)(M1)(M1)(A1)(ft) for , if in part (a) a rate of is used.

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Award a maximum of (A1)(ft)(M1)(M1)(A1)(ft) if division by an EUR rate is seen in part (a) and

multiplication by and division by is seen in (b).

7a. [3 marks]

The second term of an arithmetic sequence is 30. The fifth term is 90.

Calculate

(i) the common difference of the sequence;

(ii) the first term of the sequence.

Markscheme

(i) (or equivalent) (M1)

Note: Award (M1) for one correct equation. Accept a list of at least 5 correct terms.

(A1)

(ii) (A1)(ft) (C3)

Note: Follow through from (a)(i), irrespective of working shown if OR

7b. [3 marks]

The first, second and fifth terms of this arithmetic sequence are the first three terms of a geometric

sequence.

Calculate the seventh term of the geometric sequence.

Markscheme

OR (M1)(A1)(ft)

Note: Award (M1) for substituted geometric sequence formula, (A1)(ft) for their correct substitutions.

OR

(M1)(A1)(ft)

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Note: Award (M1) for a list of at least 5 consecutive terms of a geometric sequence, (A1)(ft) for terms

corresponding to their answers in part (a).

(A1)(ft) (C3)

Note: Follow through from part (a).

8a. [2 marks]

Aleph has an unbiased cubical (six faced) die on which are written the numbers

1 , 2 , 3 , 4 , 5 and 6.

Beth has an unbiased tetrahedral (four faced) die on which are written the numbers

2 , 3 , 5 and 7.

Complete the Venn diagram with the numbers written on Aleph’s die ( ) and Beth’s die ( ).

Markscheme

10

(A1)(A1) (C2)

Note: Award (A1) for 2, 3, 5 in intersection, (A1) for 1, 4, 6, 7 correctly placed.

8b. [2 marks]

Find .

Markscheme

(M1)(A1)(ft) (C2)

Notes:Award (M1)(A0) for listing the elements of their set ;shading the correct region on

diagram; or an answer of with a correct Venn diagram. Follow through from part (a).

8c. [2 marks]

Aleph and Beth are each going to roll their die once only. Shin says the probability that each die will

show the same number is .

Determine whether Shin is correct. Give a reason.

Markscheme

Correct, from and on sample space

OR

Correct, from a labelled tree diagram

OR

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Correct, from a sample space diagram

OR

Correct, from (or equivalent) (A1)(ft)(R1) (C2)

Notes: Do not award (A1)(ft)(R0). Award (R1) for a consistent reason with their part (a). Follow

through from part (a).

9a. [4 marks]

A right pyramid has apex and rectangular base , with , and

. The vertical height of the pyramid is .

Calculate .

Markscheme

(M1)

Note: Award (M1) for correct substitution into Pythagoras, or recognition of Pythagorean triple.

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(A1)

Note: Award (A2) for OR with no working seen.

(M1)

Note: Award (M1) for correct second use of Pythagoras, using the result from the first use of

Pythagoras.

(A1) (C4)

Notes: Accept alternative methods and apply the markscheme as follows: Award (M1)(A1) for first

correct use of Pythagoras with lengths from the question, (M1) for a correct second use of Pythagoras,

consistent with the method chosen, (A1) for correct height.

9b. [2 marks]

Calculate the volume of the pyramid.

Markscheme

(M1)

Note: Award (M1) for their correct substitutions into volume formula.

(A1)(ft) (C2)

Notes: Follow through from part (a), only if working seen.

10a. [3 marks]

Pierre invests 5000 euros in a fixed deposit that pays a nominal annual interest rate of 4.5%,

compounded monthly, for seven years.

Calculate the value of Pierre’s investment at the end of this time. Give your answer correct to two

decimal places.

Markscheme

(M1)(A1)

Note: Award (M1) for substitution into compound interest formula, (A1) for correct substitutions.

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OR

(A1)(M1)

Note: Award (A1) for seen, (M1) for all other correct entries.

OR

(A1)(M1)

Note: Award (A1) for seen, (M1) for all other correct entries.

(A1) (C3)

Note: Answer must be correct to 2 decimal places for the final (A1) to be awarded.

10b. [3 marks]

Carla has 7000 dollars to invest in a fixed deposit which is compounded annually.

She aims to double her money after 10 years.

Calculate the minimum annual interest rate needed for Carla to achieve her aim.

Markscheme

(M1)(A1)

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Notes: Award (M1) for substitution into compound interest formula equated to 14000 or equivalent.

Award (A1) for correct substitutions.

OR

(A1)(M1)

Note: Award (A1) for seen, (M1) for other correct entries. and must have opposite

signs.

(A1) (C3)

Note: Do not penalize if sign is missing. Do not accept .

11a. [1 mark]

Peter either walks or cycles to work. The probability that he walks is 0.25. If Peter walks to work, the

probability that he is late is 0.1. If he cycles to work, the probability that he is late is 0.05. The tree

diagram for this information is shown.

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On a day chosen at random, Peter walked to work.

Write down the probability that he was on time.

Markscheme

(A1) (C1)

11b. [2 marks]

For a different day, also chosen at random,

find the probability that Peter cycled to work and was late.

Markscheme

(M1)

(A1) (C2)

11c. [3 marks]

For a different day, also chosen at random,

find the probability that, given Peter was late, he cycled to work.

Markscheme

(M1)(M1)

Note: Award (M1) for their correct numerator, (M1) for their correct denominator, ie,

.

Do not award (M1) for their or if not a correct part of a fraction.

(A1)(ft) (C3)

Note: Follow through from part (b).

12a. [1 mark]

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An iron bar is heated. Its length, , in millimetres can be modelled by a linear function, ,

where is the temperature measured in degrees Celsius (°C).

At 150°C the length of the iron bar is 180 mm.

Write down an equation that shows this information.

Markscheme

(or equivalent) (A1) (C1)

12b. [1 mark]

At 210°C the length of the iron bar is 181.5 mm.

Write down an equation that shows this second piece of information.

Markscheme

(or equivalent) (A1) (C1)

12c. [4 marks]

At 210°C the length of the iron bar is 181.5 mm.

Hence, find the length of the iron bar at 40°C.

Markscheme

(A1)(A1)(ft)

Note: Follow through from part (a) and part (b), irrespective of working shown.

(M1)

Note:Award (M1) for substitution of their , their and 40 into equation.

(A1)(ft) (C4)

Note: Follow through, within part (c), from their and only if working shown.

13a. [2 marks]

The weight, , of bags of rice follows a normal distribution with mean 1000 g and standard deviation 4

g.

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Find the probability that a bag of rice chosen at random weighs between 990 g and 1004 g.

Markscheme

(M1)

Note: Award (M1) for approximate curve with 990 and 1004 in correct place.

(A1) (C2)

13b. [2 marks]

95% of the bags of rice weigh less than grams.

Find the value of .

Markscheme

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(M1)

Note: Award (M1) for approximate curve with placed to the right of the mean.

(A1) (C2)

Note: Award full marks only for , or an answer with more than sf resulting from correct

rounding of .

13c. [2 marks]

For a bag of rice chosen at random, .

Find the value of .

Markscheme

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(M1)

Note: Award (M1) for some indication of symmetry on diagram.

OR

OR (M1)

Note: Award (M1) for probability with single inequality resulting from symmetry of diagram.

(A1) (C2)

14a. [2 marks]

Consider the quadratic function, , where and are positive integers.

The graph of passes through the point .

Calculate the value of .

Markscheme

(M1)

(A1)

OR

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(M1)

(A1)

OR

(M1)

(A1) (C2)

14b. [2 marks]

The vertex of the function is .

Find the value of .

Markscheme

(M1)

Note: Award (M1) for correct substitution of the vertex and their into or equivalent

or equivalent.

(A1)(ft) (C2)

Note: Follow through from part (a).

14c. [2 marks]

The vertex of the function is .

Write down the range of .

Markscheme

(A1)(A1) (C2)

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Notes: Award (A1) for , (A1) for as part of an inequality.

Accept alternative notation: .

Award (A0)(A1) for .

Award (A0)(A0) for .

15a. [3 marks]

A cuboid has a rectangular base of width cm and length 2 cm . The height of the cuboid is cm . The

total length of the edges of the cuboid is cm.

The volume, , of the cuboid can be expressed as .

Find the value of .

Markscheme

(or equivalent) (M1)

Note: Award (M1) for a correct equation obtained from the total length of the edges.

(A1)

(A1) (C3)

15b. [3 marks]

Find the value of that makes the volume a maximum.

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Markscheme

(A1)

OR (M1)

Notes:Award (A1) for seen. Award (M1) for equating derivative to zero.

(A1)(ft) (C3)

Note: Follow through from part (a).

OR

Sketch of with visible maximum (M1)

Sketch with and indication of maximum (e.g. coordinates) (A1)(ft)

(A1)(ft) (C3)

Notes: Follow through from part (a).

Award (M1)(A1)(A0) for .

Award (C3) for .

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