2015 may exam paper 2 - mrs. profio-miller's math...

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2015 May Exam Paper 2

1a. [2 marks]

In a debate on voting, a survey was conducted. The survey asked people’s opinion on whether or not

the minimum voting age should be reduced to 16 years of age. The results are shown as follows.

A test at the 1% significance level was conducted. The critical value of the test is 9.21.

State

(i) , the null hypothesis for the test;

(ii) , the alternative hypothesis for the test.

Markscheme

(i) age and opinion (about the reduction) are independent. (A1)

Notes: Accept “not associated” instead of independent.

(ii) age and opinion are not independent. (A1)(ft)

Notes: Follow through from part (a)(i). Accept “associated” or “dependent”.

Award (A1)(ft) for their correct worded consistently with their part (a)(i).

1b. [1 mark]

Write down the number of degrees of freedom.

Markscheme

(A1)

1c. [2 marks]

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Show that the expected frequency of those between the ages of 26 and 40 who oppose the reduction in

the voting age is 21.5, correct to three significant figures.

Markscheme

OR (M1)

Note: Award (M1) for OR seen. The following (A1) cannot be awarded

without this statement.

(A1)

(AG)

Note: Both an unrounded answer that rounds to the given answer and rounded must be seen for the

(A1) to be awarded. Accept or as an unrounded answer.

1d. [3 marks]

Find

(i) the statistic;

(ii) the associated -value for the test.

Markscheme

(i) (G2)

Note: Accept as a correct 2 significant figure answer.

(ii) -value (G1)

1e. [2 marks]

Determine, giving a reason, whether should be accepted.

Markscheme

since -value should not be accepted (R1)(A1)(ft)

OR

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since should not be accepted (R1)(A1)(ft)

Note: Do not award (R0)(A1). Follow through from their answer to part (d). Award (R0)(A0) if part (d)

is unanswered.

Award (R1) for a correct comparison of either their -value to the test level or their statistic to the

critical value, award (A1) for the correct result from that comparison.

2a. [3 marks]

Consider the following statements.

: the land has been purchased

: the building permit has been obtained

: the land can be used for residential purposes

Write the following argument in symbolic form.

“If the land has been purchased and the building permit has been obtained, then the land can be used

for residential purposes.”

Markscheme

(A1)(A1)(A1)

Notes: Award (A1) for conjunction seen, award (A1) for implication seen, award (A1) for correct

simple propositions in correct order (the parentheses are required). Accept .

2b. [2 marks]

In your answer booklet, copy and complete a truth table for the argument in part (a).

Begin your truth table as follows.

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Markscheme

(A1)(ft)(A1)(ft)

Notes: Award (A1)(ft) for each correct column, follow through to the final column from their

column. For the second (A1)(ft) to be awarded there must be an implication in part (a).

Follow through from part (a).

2c. [2 marks]

Use your truth table to determine whether the argument in part (a) is valid.

Give a reason for your decision.

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Markscheme

The argument is not valid since not all entries in the final column are T. (A1)(ft)(R1)

Notes: Do not award (A1)(ft)(R0). Follow through from part (b).

Accept “The argument is not valid since is not a tautology”.

2d. [4 marks]

Write down the inverse of the argument in part (a)

(i) in symbolic form;

(ii) in words.

Markscheme

(i) (A1)(ft)(A1)(ft)

OR

(A1)(ft)(A1)(ft)

Notes:Award (A1)(ft) for the negation of their antecedent and the negation of their consequent,

(A1)(ft) for their fully correct answer.

Follow through from part (a). Accept or . Follow through from part

(a).

(ii) if it is not the case that the land has been purchased and the building permit has been obtained

then the land can not be used for residential purposes. (A1)(A1)(ft)

OR

if (either) the land has not been purchased or the building permit has not been obtained then the land

can not be used for residential purposes. (A1)(A1)(ft)

Notes: Award (A1) for “if… then…” seen, (A1)(ft) for correct statements in correct order. Follow

through from part (d)(i).

3a. [1 mark]

The cumulative frequency graph shows the speed, , in , of vehicles passing a hospital gate.

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Estimate the minimum possible speed of one of these vehicles passing the hospital gate.

Markscheme

(A1)

3b. [2 marks]

Find the median speed of the vehicles.

Markscheme

(G2)

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3c. [1 mark]

Write down the percentile.

Markscheme

(G1)

3d. [2 marks]

Calculate the interquartile range.

Markscheme

(M1)

(A1)(ft)(G2)

Notes: Award (M1) for quartiles seen. Follow through from part (c).

3e. [2 marks]

The speed limit past the hospital gate is .

Find the number of these vehicles that exceed the speed limit.

Markscheme

(M1)

(A1)(G2)

Note: Award (M1) for seen.

3f. [2 marks]

The table shows the speeds of these vehicles travelling past the hospital gate.

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Find the value of and of .

Markscheme

(A1)(ft)(A1)(ft)

Note: Follow through from part (e).

3g. [2 marks]


(i) Write down the modal class.

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(ii) Write down the mid-interval value for this class.

Markscheme

(i) (A1)

(ii) (A1)(ft)

Note:Follow through from part (g)(i).

3h. [3 marks]


Use your graphic display calculator to calculate an estimate of

(i) the mean speed of these vehicles;

(ii) the standard deviation.

Markscheme

(i) (G2)(ft)

Notes:Follow through from part (f).

(ii) (G1)(ft)

Note: Follow through from part (f), irrespective of working seen.

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3i. [2 marks]

It is proposed that the speed limit past the hospital gate is reduced to from the current

.

Find the percentage of these vehicles passing the hospital gate that do not exceed the current speed

limit but would exceed the new speed limit.

Markscheme

(M1)

Note: Award (M1) for seen.

(A1)(G2)

4a. [4 marks]

A boat race takes place around a triangular course, , with , and angle

. The race starts and finishes at point .

Calculate the total length of the course.

Markscheme

(M1)(A1)

(A1)(G2)

length of course (A1)

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Notes: Award (M1) for substitution into cosine rule formula, (A1) for correct substitution, (A1) for

correct answer.

Award (G3) for seen without working.

The final (A1) is awarded for adding and to their irrespective of working seen.

4b. [3 marks]

It is estimated that the fastest boat in the race can travel at an average speed of .

Calculate an estimate of the winning time of the race. Give your answer to the nearest minute.

Markscheme

(M1)

Note: Award (M1) for their length of course divided by .


(A1)(ft)

(A1)(ft)(G2)

Notes:Award the final (A1) for correct conversion of their answer in seconds to minutes, correct to the

nearest minute.


4c. [3 marks]

It is estimated that the fastest boat in the race can travel at an average speed of .

Find the size of angle .

Markscheme

(M1)(A1)(ft)

OR

(M1)(A1)(ft)

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(A1)(ft)(G2)

Notes: Award (M1) for substitution into sine rule or cosine rule formula, (A1) for their correct

substitution, (A1) for correct answer.

Accept for sine rule and for cosine rule from use of correct three significant figure values.

Follow through from their answer to (a).

4d. [3 marks]

To comply with safety regulations, the area inside the triangular course must be kept clear of other

boats, and the shortest distance from to must be greater than metres.

Calculate the area that must be kept clear of boats.

Markscheme

(M1)(A1)

Note: Accept . Follow through from parts (a) and (c).

(A1)(G2)

Notes:Award (M1) for substitution into area of triangle formula, (A1) for correct substitution, (A1) for

correct answer.

Award (G1) if is seen without units or working.

4e. [3 marks]

To comply with safety regulations, the area inside the triangular course must be kept clear of other

boats, and the shortest distance from to must be greater than metres.

Determine, giving a reason, whether the course complies with the safety regulations.

Markscheme

(M1)

(A1)(ft)(G2)

Note: Follow through from part (c).

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OR

(M1)

(A1)(ft)(G2)

Note: Follow through from part (a) and part (d).

is greater than , thus the course complies with the safety regulations (R1)

Notes: A comparison of their area from (d) and the area resulting from the use of as the

perpendicular distance is a valid approach and should be given full credit. Similarly a comparison of

angle and should be given full credit.

Award (R0) for correct answer without any working seen. Award (R1)(ft) for a justified reason

consistent with their working.

Do not award (M0)(A0)(R1).

4f. [2 marks]

The race is filmed from a helicopter, , which is flying vertically above point .

The angle of elevation of from is .

Calculate the vertical height, , of the helicopter above .

Markscheme

(M1)

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

(A1)(ft)(G2)

4g. [3 marks]

The race is filmed from a helicopter, , which is flying vertically above point .

The angle of elevation of from is .

Calculate the maximum possible distance from the helicopter to a boat on the course.

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Markscheme

(M1)(A1)

Note: Award (M1) for substitution into Pythagoras, (A1) for their and their

correctly substituted in formula.

(A1)(ft)(G2)

Note: Follow through from their answer to parts (a) and (f).

5a. [3 marks]

Consider the function , where is a constant and .

Write down .

Markscheme

(A1)(A1)(A1)

Note: Award (A1) for , (A1) for , (A1) for (only).

5b. [2 marks]

The graph of has a local minimum point at .

Show that .

Markscheme

at local minimum (M1)

Note: Award (M1) for seeing (may be implicit in their working).

(A1)

(AG)

Note: Award (A1) for substituting in their , provided it leads to . The conclusion

must be seen for the (A1) to be awarded.

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5c. [2 marks]


Find .

Markscheme

(M1)

Note:Award (M1) for substituting and in .

(A1)(G2)

5d. [2 marks]


Find

Markscheme

(M1)

Note: Award (M1) for substituting and in their .

(A1)(ft)(G2)

Note: Follow through from part (a).

5e. [3 marks]


Find the equation of the normal to the graph of at the point where .

Give your answer in the form where .

Markscheme

(A1)(ft)(M1)

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Notes:Award (A1)(ft) for their seen, (M1) for the correct substitution of their point and their

normal gradient in equation of a line.

Follow through from part (c) and part (d).

OR

gradient of normal (A1)(ft)

(M1)

(A1)(ft)(G2)

Notes: Accept equivalent answers.

5f. [4 marks]


Sketch the graph of , for and .

Markscheme

(A1)(A1)(A1)(A1)

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Notes: Award (A1) for correct window (at least one value, other than zero, labelled on each axis), the

axes must also be labelled; (A1) for a smooth curve with the correct shape (graph should not touch -

axis and should not curve away from the -axis), on the given domain; (A1) for axis intercept in

approximately the correct position (nearer than zero); (A1) for local minimum in approximately the

correct position (first quadrant, nearer the -axis than ).

If there is no scale, award a maximum of (A0)(A1)(A0)(A1) – the final (A1) being awarded for the zero

and local minimum in approximately correct positions relative to each other.

5g. [2 marks]


Write down the coordinates of the point where the graph of intersects the -axis.

Markscheme

(G1)(G1)

Notes: If parentheses are omitted award (G0)(G1)(ft).

Accept . Award (G1) for seen.

5h. [2 marks]


State the values of for which is decreasing.

Markscheme

(A1)(A1)

Notes: Award (A1) for correct end points of interval, (A1) for correct notation (note: lower inequality

must be strict).

Award a maximum of (A1)(A0) if or used in place of .

6a. [2 marks]

A cup of boiling water is placed in a room to cool. The temperature of the room is 20°C . This situation

can be modelled by the exponential function , where is the temperature of the

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water, in °C , and is the number of minutes for which the cup has been placed in the room. A sketch

of the situation is given as follows.

Explain why .

Markscheme

the temperature of the water cannot fall below room temperature (R1)

an (informal) explanation that as R1)

OR

recognition that there is a horizontal asymptote at (R1)

Note: Award (R1) for a contextual reason involving room temperature.

Award (R1) for a mathematical reason similar to one of the two alternatives.

6b. [2 marks]

Initially, at , the temperature of the water is 100°C.

Find the value of .

Markscheme

(M1)

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Note: Award (M1) for substituting , and .

(A1)(G2)

Note: The (A1) is awarded only if all working seen is consistent with the final answer of .

6c. [2 marks]

After being placed in the room for one minute, the temperature of the water is 84°C.

Show that .

Markscheme

(M1)

Note: Substituting at any stage is an invalid method and is awarded (M0)(M0). Award (M1)

for correctly substituting , and their .

(M1)

(AG)

Note:Award (M1) for correct rearrangement that isolates ; must be consistent with their

working and the conclusion must be seen.

6d. [2 marks]


Find the temperature of the water three minutes after it has been placed in the room.

Markscheme

(M1)

Note: Award (M1) for their correct substitutions into . Follow through from part (b) and .

(A1)(ft)(G2)

6e. [2 marks]


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Find the total time needed for the water to reach a temperature of 35°C. Give your answer in minutes

and seconds, correct to the nearest second.

Markscheme

(M1)

Note: Award (M1) for their correct substitutions into . Follow through from part (b). Accept graphical

solutions. Award (M1) for sketch of function.

(A1)(ft)(G2)

7 minutes and 30 seconds (A1)

Note:Award the final (A1) for correct conversion of their in minutes to minutes and seconds, but

only if answer in minutes is explicitly shown.

Printed for Millbrook High School

© International Baccalaureate Organization 2017

International Baccalaureate® - Baccalauréat International® - Bachillerato Internacional®

2015 may exam paper 2 - mrs. profio-miller's math...

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