IBMathsDPAnalysis & Approaches (AA)HLExam QuestionsExtended QuestionsExtended Questions (Section B SL Only)

Extended Questions (Section B SL Only) (DP IB Analysis & Approaches (AA): HL): Exam Questions

3 hours12 questions
Medium
Hard
1a
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.

A cumulative frequency graph on a grid. The horizontal axis shows the total cost of items in basket ($) from 0 to 90 and the vertical axis shows cumulative frequency from 0 to 120. The curve rises from (0, 0) to (90, 120), steepest between about $40 and $60.

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

How did you do?

1b
3 marks

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

How did you do?

1c
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.

How did you do?

1d
4 marks

The same survey information is represented by the following table:

Total cost, m dollars, of goods in basket

0 < m \leq 20

20 < m \leq 40

40 < m \leq 70

70 < m \leq 90

Frequency

9

q

r

20

Find the value of q and the value of r.

How did you do?

1e
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.

How did you do?

1f
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. 

How did you do?

2a
1 mark

The following diagram shows part of the graph of the function

f\left(x\right) = \frac{1}{3}x^{3} - 4x^{2} + 9x + 12

Part of the graph of y = f(x), a cubic curve. It crosses the y-axis at point A, rises to a local maximum, falls to a local minimum and then rises again. Point B is marked on the curve close to the local maximum.

Point A is the point where the graph intersects the y-axis.

Write down the coordinates of A.

How did you do?

2b
2 marks

Find f'\left(x\right).

How did you do?

2c
2 marks

Using the graph, explain why the equation f'\left(x\right) = 0 must have exactly two distinct real solutions.

How did you do?

2d
2 marks

Point B is the point on the graph with x-coordinate \frac{8 - \sqrt{26}}{2}.

Given that \left(\frac{8 - \sqrt{26}}{2}\right)^{2} = \frac{45 - 8\sqrt{26}}{2}, find the gradient of the tangent to the graph at B.

How did you do?

2e
4 marks

Points C and D are the points on the graph at which the tangents are perpendicular to the tangent at B.

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

How did you do?

2f
4 marks

Given that C lies between A and B on the graph, find the equation of the tangent to the graph at C. Give your answer in the form y = mx + c.

How did you do?

3a
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.

How did you do?

3b
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.

How did you do?

3c
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.

How did you do?

3d
2 marks

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

How did you do?

3e
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.

How did you do?

4a
3 marks

K.C. Jones & Company produces tunnels for model railroad layouts. Each tunnel has the form of a right prism, and the following diagram shows the cross-section of one of the tunnels the company produces. 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.

The shaded cross-section of the tunnel on x- and y-axes: a rectangle from x = 0 to x = 14 and from y = 0 to y = 10, with an arch-shaped opening cut out of its base. The opening meets the x-axis at points A and B, between 0 and 14.

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

f\left(x\right) = -k\left(x^{2} - 14x + 24\right)

where k is a positive constant.

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

Find the coordinates of A and B.

How did you do?

4b
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.

How did you do?

4c
4 marks

By setting up and evaluating an appropriate definite integral, show that the area of the tunnel opening is \frac{160}{3} \; \text{cm}^{2}.

How did you do?

4d
5 marks

The material from which the tunnel is made has a density of 1060 \text{kg m}^{-3}.

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

How did you do?

5a
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 following diagrams show one of the ship’s screws mounted in the testing facility.

A ship's screw with three identical blades, each shaded, joined at the centre C. Point A is marked at the tip of the blade that points straight up from C, and a dashed line runs from C along each blade to its tip.
A horizontal line representing the floor of the testing facility, with points O, Q and P marked on it from left to right.

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

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

h\left(t\right) = 5.59 + 3.6\cos\left(k\pi t\right)

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 A from point C

(ii) the height of point C above point O.

How did you do?

5b
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 C, determine the exact distance of point A from the tip of one of the other blades of the screw.

How did you do?

5c
3 marks

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

Find the value of k.

How did you do?

5d
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. Justify your answer.

How did you do?

5e
3 marks

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

\tan \text{O}\hat{\text{Q}}\text{C} = \frac{3}{2} \tan \text{O}\hat{\text{P}}\text{C}

Determine the distance of point Q from point P.

How did you do?

5f
3 marks

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

How did you do?

6a
2 marks

As part of a study, a group of 80 people from a particular city were randomly selected and tested to find out the number of hours they spend on social media per week. The results are represented on the following cumulative frequency graph.

A cumulative frequency graph on a grid. The horizontal axis shows the number of hours from 0 to 24 and the vertical axis shows cumulative frequency from 0 to 80. The curve starts at 2 hours, rises slowly, climbs very steeply between about 12 and 16 hours, then levels off to reach 80 at 24 hours.

Find the median number of hours these people spend on social media per week.

How did you do?

6b
3 marks

Given that 87.5% of these people spend less than k hours per week on social media, find the value of k.

How did you do?

6c
4 marks

The same information is represented by the following table.

Number of hours, h

2 < h \leq 7

7 < h \leq 9

9 < h \leq 15

15 < h \leq 21

21 < h \leq 24

Frequency

4

a

b

16

c

Find the values of a, b and c.

How did you do?

6d
3 marks

There are 2 200 000 people living in the city.

Use the results of the study to estimate the number of people that spend more than 15 hours per week on social media.

How did you do?

6e
2 marks

Explain why the sampling method used might not provide an accurate representation of the amount of time all people in the city spend on social media per week, and suggest a more appropriate sampling method.

How did you do?

1a
3 marks

The following diagram shows the graph of the function f defined by f\left(x\right) = 4x^{2} + 4x - 4x^{3} as well as the graph of y = g\left(x\right) for a function g, with both functions defined for all x \in \mathbb{R}.

The graphs of y = f(x), a cubic, and y = g(x), a downward parabola, on the same axes. Points C and D are where the graph of f meets the non-negative x-axis, E is the local maximum point of f, and F is where the two graphs cross below the x-axis. Region R, between the parabola and the x-axis, is shaded dark; region S, above the x-axis between the two graphs, is shaded lighter.

Points C and D are the non-negative x-axis intercepts of the graph of f, and E is the local maximum point on the graph of f. Point F is the point of intersection of the graphs of f and g. The shaded region R is enclosed by the graph of g and the x-axis. The shaded region S lies above the positive x-axis, between the graphs of f and g.

Find the coordinates of C, D and E, giving your answers correct to three significant figures.

How did you do?

1b
3 marks

The graph of g is a parabola with x-intercepts \left(\frac{1}{2}, \; 0\right) and \left(\frac{3}{2}, \; 0\right), and vertex \left(1, \; 2\right).

Find an expression for g\left(x\right).

How did you do?

1c
2 marks

Hence find the area of region R.

How did you do?

1d
6 marks

The following diagram shows a cylindrical log with a wedge cut from it. The area of the major sector AOB is equal to the area of the shaded region S. The length l is equal to the distance between points E and F on the first diagram.

A cylindrical log with a wedge cut from it. The circular end face has centre O and points A and B on its circumference, with the major sector AOB shaded and a dashed arc from A to B where the wedge is missing. The length of the log is marked l.

Find the volume of the log.

How did you do?

2a
5 marks

The average interest rates on accounts for all the different banks in a country can be modelled by a normal distribution with mean \mu% and standard deviation \sigma%.

Given that 10% of banks have interest rates lower than 1.2% and 8% have interest rates higher than 2%, find two linear equations satisfied by \mu and \sigma, and hence find the values of \mu and \sigma, giving your answers correct to three significant figures.

How did you do?

2b
2 marks

Find the probability that a particular bank’s average interest rate is over 1.8%.

How did you do?

2c
4 marks

Given that a bank’s average interest rate is less than 1.8%, find the probability that its average interest rate is higher than 1.2%.

How did you do?

2d
3 marks

For a city in this country with 54 banks, find the expected number of banks that have average interest rates over 2.2%.

How did you do?

2e
3 marks

For a child’s second birthday, parents decide to put $10 000 into a bank account paying a fixed rate of 2.2% annual interest, and also to put $10 000 into an investment portfolio. The annual returns for the investment portfolio can be modelled by a normal distribution with mean 12% and standard deviation 5.5%.

Assume that the investment portfolio grows by its mean annual return every year.

Find the combined value of the bank account and the investment portfolio when the child turns 18.

How did you do?

3a
3 marks

The first two terms of an infinite geometric sequence are u_{1} = 12 and u_{2} = 9\cos^{2}\theta, where 0 < \theta < 2\pi, \theta \neq \frac{\pi}{2} and \theta \neq \frac{3\pi}{2}.

Find the range of possible values for the common ratio, r, of the geometric sequence.

How did you do?

3b
5 marks

Show that the sum of the infinite sequence can be expressed as fraction numerator 96 over denominator 5 minus 3 space cos space 2 straight theta end fraction.

How did you do?

3c
4 marks

Find the value of \theta that maximises the sum of the infinite sequence.

How did you do?

4a
3 marks

Consider a function f where the line l_{1} with equation y = 2x - 8 is the normal to the graph of f at the point where x = 5.

Find the values of f'\left(5\right) and f\left(5\right).

How did you do?

4b
5 marks

Let g be the function defined by g\left(x\right) = f\left(2x^{2} - 3\right), and let A be the point on the graph of g where x = 2. The line l_{2} is the tangent to the graph of g at A.

Find the equation of l_{2}.

How did you do?

4c
4 marks

Find the coordinates of the point of intersection of l_{1} and l_{2}. Give your answers for the coordinates as exact values.

How did you do?

5a
3 marks

Let  f left parenthesis x right parenthesis equals k x ln left parenthesis 3 x to the power of 4 right parenthesis  for  x greater than 0,  where  k greater than 0  is a constant.

Given that f left parenthesis a right parenthesis equals 0,  find the value of a.

How did you do?

5b
5 marks

Find

(i) f'\left(x\right)

(ii) f''\left(x\right).

How did you do?

5c
5 marks

Show that the graph of f has exactly one minimum point and determine its x-coordinate.

How did you do?

5d
3 marks

Given that the y-coordinate of the minimum point is -4, find the value of k.

How did you do?

6a
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\left(t\right) = \frac{3000}{1 + 99\mathrm{e}^{-kt}}

where k is a positive constant.

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

How did you do?

6b
3 marks

(i) Explain what happens to the values of \mathrm{e}^{-kt} as t 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.

How did you do?

6c
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

How did you do?

6d
3 marks

(i) Use the result from part (c) to show that P\left(t\right) is an increasing function for all values of t \geq 0.

(ii) Explain why this does not contradict the result of part (b)(ii).

How did you do?

6e
6 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.

How did you do?