DP IB Maths: AA SL

Topic Questions

6.1 Extended Questions

1a
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2 marks

A supermarket manager wishes to gather information about the spending habits of the store’s customers. During each of his lunchbreaks on Monday through Friday of a given week, he chooses 24 customers at random and notes the total cost, in dollars ($), of the items in their baskets when they check out at the tills.  The results of his survey are represented by the following cumulative frequency graph.

ib1a-ai-sl-6-1-ib-maths-veryhard

Find the median total cost of the items these customers had in their baskets.

1b
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3 marks

Find the interquartile range of the total cost of the items these customers had in their baskets.

1c
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3 marks

Given that two thirds of customers had a total of more than $   p of goods in their baskets, find the value of p.

1d
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4 marks

The same survey information is represented by the following table:

Total cost ($m) of goods in basket

0 less than m less or equal than 20 20 less than m less or equal than 40 40 less than m less or equal than 70 70 less than m less or equal than 90

Frequency

9 q r 20

Find the value of q and the value of r.

1e
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3 marks

In an average week, the manager estimates that the store has a total of 3600 customers.

Use the results of the manager’s survey to estimate the number of customers in a week who have goods totalling more than $50 in their baskets.

1f
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2 marks
(i)
Explain why the manager’s survey sample might not provide an accurate representation of the spending habits of all the shop’s customers. 

(ii)
Suggest a sampling method that might obtain a more representative sample. 

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2a
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1 mark

The diagram below shows a part of the graph of the function             

f left parenthesis x right parenthesis equals 1 third x cubed minus 4 x squared plus 9 x plus 12

ib2a-ai-sl-6-1-ib-maths-veryhard

Point A spaceis the point of intersection between the graph and the y-axis. Write down the coordinates of point A.

2b
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2 marks

Find f to the power of apostrophe left parenthesis x right parenthesis.

2c
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3 marks

Using the graph, explain why the equation f to the power of apostrophe left parenthesis x right parenthesis equals 0 must have exactly two distinct real solutions.

2d
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2 marks

Point B is the point on the graph with x-coordinate fraction numerator 8 minus square root of 26 over denominator 2 end fraction  .

Given that open parentheses fraction numerator 8 minus square root of 26 over denominator 2 end fraction close parentheses squared equals fraction numerator 45 minus 8 square root of 26 over denominator 2 end fraction ,  find the gradient of the tangent line to the graph at point B.

2e
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5 marks

Points C and D are the points on the graph at which the tangent lines are perpendicular to the tangent line at point .

By first determining the gradient of the tangents at points C and D, find the x-coordinates of points C and D.

2f
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4 marks

Given that point C lies between points A spaceand B on the graph, find the equation of the tangent line to the graph at point C.

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3a
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2 marks

After escaping from a research station, a small population of rabbits has become established on an island in the Southern Ocean.  Scientists have begun to study this rabbit population, and have determined that the number of rabbits, P, at a time t months after the beginning of the study can be modelled by the function

P open parentheses t close parentheses equals fraction numerator 3000 over denominator 1 plus 99 e to the power of negative k t end exponent end fraction

Where k is a positive constant.

Determine the number of rabbits on the island at the beginning of the study.

3b
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4 marks
(i)
Explain what happens to the values of e to the power of negative k t end exponent as  becomes large. 

(ii)
Hence determine the maximum number of rabbits that the model predicts the island can support.  Be sure to show clear mathematical reasoning to support your answer.      
3c
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3 marks

Show that

P apostrophe open parentheses t close parentheses equals fraction numerator 3000 cross times 99 k e to the power of negative k t end exponent over denominator open parentheses 1 plus 99 e to the power of negative k t end exponent close parentheses squared end fraction

3d
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4 marks
(i)
Use the result from part (c) to show that P left parenthesis t right parenthesis is an increasing function for all values of t greater or equal than 0

(ii)
Explain why this does not contradict the result of (b)(ii).    
3e
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7 marks

The model predicts that the population of rabbits will double in the first two months after the beginning of the study.

(i)
Use this information to show that  k equals 1 half ln open parentheses 99 over 49 close parentheses.
   
(ii)
Hence find the exact rate of change of the rabbit population at the beginning of the study, as predicted by the model.

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4a
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2 marks

The Strike A Light! matchstick company produces matchsticks with a length, X mm, that is normally distributed with mean 45 and variance sigma squared.

The probability that X is greater than 45.37 is 0.1714.

Find P left parenthesis 44.63 less than X less than 45.37 right parenthesis.

4b
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5 marks
(i)
Find sigma , the standard deviation of X.

(ii)
Hence, find the probability that a randomly selected matchstick has a length less than 44.5 mm.
4c
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3 marks

Andrew has a box of Strike A Light! matches with fifteen matchsticks remaining in it.  Those matchsticks may be assumed to be a random sample.  Let Y represent the number of matchsticks in Andrew’s box with lengths less than 44.5 mm.

Find E left parenthesis Y right parenthesis.

4d
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2 marks

Find the probability that exactly one of the matchsticks in Andrew’s box has a length less than 44.5 mm.

4e
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3 marks

A Strike A Light! matchstick is selected at random and is found to have a length greater than 44.5 mm.

Find the probability that the length of the matchstick is between 44.63 mm and 45.37 mm.

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5a
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3 marks

K.C. Jones & Company produces tunnels for model railroad layouts. Each tunnel has the form of a right prism, and the cross-section of one of the tunnels the company produces is shown in the diagram below. The upper and right-hand borders of the shaded area are parallel to the x-axis and y-axis respectively, and all units are in centimetres.

ib5a-ai-sl-6-1-ib-maths-veryhard

The shape of the opening of the tunnel may be modelled by the function

f left parenthesis x right parenthesis equals negative k left parenthesis x squared minus 14 x plus 24 right parenthesis

where k is a positive constant.

Points A and B are the points where the tunnel opening meets the x-axis in the diagram.

Find the coordinates of points A and B.

5b
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3 marks

The maximum height of the tunnel opening above the x-axis is 8 cm.

Use this information to determine the value of k.

5c
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4 marks

By setting up and solving an appropriate definite integral, show that the area of the tunnel opening is 160 over 3cm squared  .  You must use calculus and show the steps of your working.

5d
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5 marks

The material from which the tunnel is made has a density of 1060 kg divided by straight m cubed.

Given that the mass of the tunnel is 2067 g, find the length of the tunnel.

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6a
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2 marks

Badon Iron Works is building a new ship called the Gargantuan, which will be a full-sized replica of the original RMS Titanic.  Eleanor is an engineer at the company, and is involved with construction and testing of the ship’s screws (commonly known as ‘propellers’).  The diagram below depicts one of the ship’s screws mounted in the testing facility.

ib6a-ai-sl-6-1-ib-maths-veryhard

ib6aa-ai-sl-6-1-ib-maths-veryhard

Point C is the centre of the screw, which is fixed in place so that the screw is able to rotate about it.  Point straight A is the marked tip of one of the three identical blades of the screw.  Point straight O is the point on the horizontal floor of the testing facility that lies directly below point straight C.  Points straight O,straight A , straight C, straight P and straight Q lie at all times in the same plane.

The height, h m, of point  above the testing facility floor once the screw begins to rotate may be modelled by the function

h left parenthesis t right parenthesis equals 5.59 plus 3.6 space cos left parenthesis k pi t right parenthesis

where t is the time in seconds since the screw began rotating, and k is a constant.

Use the above information to determine:

(i)
The distance of point straight A from point straight C.

(ii)
The height of point straight C above point straight O.
6b
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3 marks

Given that the tips of the three blades of the screw are located at equal distances from each other around the circumference of a circle with centre straight C, determine the exact distance of point straight A from the tip of one of the other blades of the screw.

6c
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3 marks

When it is rotating, the screw makes 75 complete revolutions every minute.

Given that the argument of the cosine in the equation for h left parenthesis t right parenthesis spaceis measured in radians, use this information to determine the value of the constant k.

6d
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2 marks

Paul, a mathematician, has been hired as a consultant on the Gargantuan project.  Because of his height of 1.96 m, Eleanor is concerned about whether he will be able to walk safely beneath the screw while it is rotating.

Determine whether Eleanor is right to be concerned, giving a mathematical reason for your answer.

6e
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3 marks

The screw has been locked in place so that point straight A is at its highest possible position above the floor.  Paul is standing at point straight P, which is at a distance of 9.69 m from point straight O.  He walks towards point straight O until he arrives at point straight Q, which is located such that

tan O Q with hat on top C equals fraction numerator 3 space over denominator 2 end fraction space tan O P with hat on top C

Determine the distance of point straight Q from point straight P.

6f
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4 marks

Given that point straight A remains fixed at its highest possible position above the floor, determine the area of triangle APQ.

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