IB Studies Cumulative Exam – Part 1



IB Studies Cumulative Exam Question Pool – Part 1

1. Two students Ann and Ben play a game. Each time Ann passes GO she receives $15. Each time Ben passes GO he receives 8% of the amount he already has. Both students start with $100.

(a) How much money will Ann have after she has passed GO 10 times?

(b) How much money will Ben have after he passes GO 10 times?

(c) How many times will the students have to pass GO for Ben to have more money than Ann?

(Total 6 marks)

2. Clara visits Britain from the United States and exchanges 1000 US dollars (USD) into pounds (GBP). The exchange rate is 1 USD = 0.543 GBP. The bank charges 2% commission for each transaction.

(a) Calculate how many GBP she receives.

Next Clara wants to travel to France. She changes 150 GBP to euros (€) at a rate of 1 GBP = 1.35 €. The bank charges commission and then gives Clara 200 €.

(b) Find the amount of commission in GBP.

(Total 6 marks)

3. Bob invests 600 EUR in a bank that offers a rate of 2.75% compounded annually. The interest is added on at the end of each year.

(a) Calculate how much money Bob has in the bank after 4 years.

(b) Calculate the number of years it will take for the investment to double.

Ann invests 600 EUR in another bank that offers interest compounded annually. Her investment doubles in 20 years.

(c) Find the rate that the bank is offering.

(Total 6 marks)

4. William invests $1200 for 5 years at a rate of 3.75% compounded annually.

(a) Calculate the amount of money he has in total at the end of the 5 years.

(b) The interest rate then drops to 3.25%. If he continues to leave his money in the bank find how much it will be worth after a further 3 years.

(Total 6 marks)

5. Jacques can buy six CDs and three video cassettes for $163.17

or he can buy nine CDs and two video cassettes for $200.53.

(a) Express the above information using two equations relating the price of CDs and the price of video cassettes.

(b) Find the price of one video cassette.

(c) If Jacques has $180 to spend, find the exact amount of change he will receive if he buys nine CDs.

(Total 6 marks)

6. A group of 30 children are surveyed to find out which of the three sports cricket (C), basketball (B) or volleyball (V) they play. The results are as follows:

3 children do not play any of these sports

2 children play all three sports

6 play volleyball and basketball

3 play cricket and basketball

6 play cricket and volleyball

16 play basketball

12 play volleyball.

(a) Draw a Venn diagram to illustrate the relationship between the three sports played.

(1)

(b) On your Venn diagram indicate the number of children that belong to each region.

(3)

(c) How many children play only cricket?

(2)

(Total 6 marks)

7. Write down the values for a, b, c, d, e and f from the table below:

|p |q |¬p |p ∧ q |p ∨ q |p ∨ q |p ⇒ q |p ⇔ q |

|T |T |a | | |d | | |

|T |F | |b | | | |f |

|F |T | | |c | | | |

|F |F | | | | |e | |

(Total 6 marks)

8. The diagram shows a point P, 12.3 m from the base of a building of height h m. The angle measured to the top of the building from point P is 63°.

[pic]

(a) Calculate the height h of the building.

Consider the formula h = 4.9t2, where h is the height of the building and t is the time in seconds to fall to the ground from the top of the building.

(b) Calculate how long it would take for a stone to fall from the top of the building to the ground.

(Total 6 marks)

9. (a) Sketch the graph of the function y = 2x2 – 6x + 5.

(b) Write down the coordinates of the local maximum or minimum of the function.

(c) Find the equation of the axis of symmetry of the function.

(Total 6 marks)

10. In an experiment it is found that a culture of bacteria triples in number every four hours.

There are 200 bacteria at the start of the experiment.

|Hours |0 |4 |8 |12 |16 |

|No. of bacteria |200 |600 |a |5400 |16200 |

(a) Find the value of a.

(1)

(b) Calculate how many bacteria there will be after one day.

(2)

(c) Find how long it will take for there to be two million bacteria.

(3)

(Total 6 marks)

11. In triangle ABC, AB = 3.9 cm, BC = 4.8 cm and angle [pic] = 82°.

[pic]

(a) Calculate the length of AC.

(3)

(b) Calculate the size of angle [pic]

(3)

(Total 6 marks)

12. P (4, 1) and Q (0, –5) are points on the coordinate plane.

(a) Determine the

(i) coordinates of M, the midpoint of P and Q;

(ii) gradient of the line drawn through P and Q;

(iii) gradient of the line drawn through M, perpendicular to PQ.

The perpendicular line drawn through M meets the y-axis at R (0, k).

(b) Find k.

(Total 6 marks)

13. The age in months at which a child first starts to walk is observed for a random group of children from a town in Brazil. The results are

14.3, 11.6, 12.2, 14.0, 20.4, 13.4, 12.9, 11.7, 13.1.

(a) (i) Find the mean of the ages of these children.

(ii) Find the standard deviation of the ages of these children.

(b) Find the median age.

(Total 6 marks)

14. Consider the function f (x) = 2x3 – 5x2 + 3x + 1.

(a) Find f ′ (x).

(3)

(b) Write down the value of f ′ (2).

(1)

(c) Find the equation of the tangent to the curve of y = f (x) at the point (2, 3).

(2)

(Total 6 marks)

15. (a) Differentiate the following function with respect to x:

f (x) = 2x – 9 – 25x–1

(b) Calculate the x-coordinates of the points on the curve where the gradient of the tangent to the curve is equal to 6.

(Total 6 marks)

16. The following stem and leaf diagram gives the heights in cm of 39 schoolchildren.

|Stem |Leaf |Key 13 |2 represents 132 cm. |

|13 |2, 3, 3, 5, 8, | | |

|14 |1, 1, 1, 4, 5, 5, 9, | | |

|15 |3, 4, 4, 6, 6, 7, 7, 7, 8, 9, 9, | | |

|16 |1, 2, 2, 5, 6, 6, 7, 8, 8, | | |

|17 |4, 4, 4, 5, 6, 6, | | |

|18 |0, | | |

(a) (i) State the lower quartile height.

(ii) State the median height.

(iii) State the upper quartile height.

(b) Draw a box-and-whisker plot of the data using the axis below.

[pic]

(Total 6 marks)

IB Studies Cumulative Exam Question Pool – Part 2

1. On Vera’s 18th birthday she was given an allowance from her parents. She was given the following choices.

Choice A $100 every month of the year.

Choice B A fixed amount of $1100 at the beginning of the year, to be invested at an interest

rate of 12% per annum, compounded monthly.

Choice C $75 the first month and an increase of $5 every month thereafter.

Choice D $80 the first month and an increase of 5% every month.

(a) Assuming that Vera does not spend any of her allowance during the year, calculate, for each of the choices, how much money she would have at the end of the year.

(8)

(b) Which of the choices do you think that Vera should choose? Give a reason for your answer.

(2)

(c) On her 19th birthday Vera invests $1200 in a bank that pays interest at r% per annum compounded annually. Vera would like to buy a scooter costing $1452 on her 21st birthday. What rate will the bank have to offer her to enable her to buy the scooter?

(4)

(Total 14 marks)

2. Note: For this question, it is important that you show your working and explain your method clearly.

A box contains 10 coloured light bulbs, 5 green, 3 red and 2 yellow. One light bulb is selected at random and put into the light fitting of room A.

(a) What is the probability that the light bulb selected is

(i) green?

(1)

(ii) not green?

(1)

A second light bulb is selected at random and put into the light fitting in room B.

(b) What is the probability that

(i) the second light bulb is green given the first light bulb was green?

(l)

(ii) both light bulbs are not green?

(2)

(iii) one room has a green light bulb and the other room does not have a green light bulb?

(3)

A third light bulb is selected at random and put in the light fitting of room C.

(c) What is the probability that

(i) all three rooms have green light bulbs?

(2)

(ii) only one room has a green light bulb?

(3)

(iii) at least one room has a green light bulb?

(2)

(Total 15 marks)

3. In a club with 60 members, everyone attends either on Tuesday for Drama (D) or on Thursday for Sports (S) or on both days for Drama and Sports.

One week it is found that 48 members attend for Drama and 44 members attend for Sports and x members attend for both Drama and Sports.

(a) (i) Draw and label fully a Venn diagram to illustrate this information.

(3)

(ii) Find the number of members who attend for both Drama and Sports.

(2)

(iii) Describe, in words, the set represented by (D ∩ S)'.

(2)

(iv) What is the probability that a member selected at random attends for Drama only or Sports only?

(3)

The club has 28 female members, 8 of whom attend for both Drama and Sports.

(b) What is the probability that a member of the club selected at random

(i) is female and attends for Drama only or Sports only?

(2)

(ii) is male and attends for both Drama and Sports?

(2)

(Total 14 marks)

4. Consider the function f (x) = 2x3 – 3x2 – 12x + 5.

(a) (i) Find f ' (x).

(ii) Find the gradient of the curve f (x) when x = 3.

(4)

(b) Find the x-coordinates of the points on the curve where the gradient is equal to –12.

(3)

(c) (i) Calculate the x-coordinates of the local maximum and minimum points.

(ii) Hence find the coordinates of the local minimum.

(6)

(d) For what values of x is the value of f (x) increasing?

(2)

(Total 15 marks)

5. The line L1 shown on the set of axes below has equation 3x + 4y = 24. L1 cuts the x-axis at A and cuts the y-axis at B.

Diagram not drawn to scale

[pic]

(a) Write down the coordinates of A and B.

(2)

M is the midpoint of the line segment [AB].

(b) Write down the coordinates of M.

(2)

The line L2 passes through the point M and the point C (0, –2).

(c) Write down the equation of L2.

(2)

(d) Find the length of

(i) MC;

(2)

(ii) AC.

(2)

(e) The length of AM is 5. Find

(i) the size of angle CMA;

(3)

(ii) the area of the triangle with vertices C, M and A.

(2)

(Total 15 marks)

6. An office tower is in the shape of a cuboid with a square base. The roof of the tower is in the shape of a square based right pyramid.

The diagram shows the tower and its roof with dimensions indicated. The diagram is not drawn to scale.

[pic]

(a) Calculate, correct to three significant figures,

(i) the size of the angle between OF and FG;

(3)

(ii) the shortest distance from O to FG;

(2)

(iii) the total surface area of the four triangular sections of the roof;

(3)

(iv) the size of the angle between the slant height of the roof and the plane EFGH;

(2)

(v) the height of the tower from the base to O.

(2)

A parrot’s nest is perched at a point, P, on the edge, BF, of the tower. A person at the point A, outside the building, measures the angle of elevation to point P to be 79°.

(b) Find, correct to three significant figures, the height of the nest from the base of the tower.

(2)

(Total 14 marks)

7. A survey of 400 people is carried out by a market research organization in two different cities, Buenos Aires and Montevideo. The people are asked which brand of cereal they prefer out of Chocos, Zucos or Fruti. The table below summarizes their responses.

| |Chocos |Zucos |Fruti |Total |

|Buenos Aires |43 |85 |62 |190 |

|Montevideo |57 |35 |118 |210 |

|Total |100 |120 |180 |400 |

(a) One person is chosen at random from those surveyed. Find the probability that this person

(i) does not prefer Zucos;

(ii) prefers Chocos, given that they live in Montevideo.

(4)

(b) Two people are chosen at random from those surveyed. Find the probability that they both prefer Fruti.

(3)

The market research organization tests the survey data to determine whether the brand of cereal preferred is associated with a city. A chi-squared test at the 5% level of significance is performed.

(c) State the null hypothesis.

(1)

(d) State the number of degrees of freedom.

(1)

(e) Show that the expected frequency for the number of people who live in Montevideo and prefer Zucos is 63.

(2)

(f) Write down the chi-squared statistic for this data.

(2)

(g) State whether the market research organization would accept the null hypothesis. Clearly justify your answer.

(2)

(Total 15 marks)

8. A closed box has a square base of side x and height h.

(a) Write down an expression for the volume, V, of the box.

(1)

(b) Write down an expression for the total surface area, A, of the box.

(1)

The volume of the box is 1000 cm3

(c) Express h in terms of x.

(2)

(d) Hence show that A = 4000x–l + 2x2.

(2)

(e) Find [pic].

(2)

(f) Calculate the value of x that gives a minimum surface area.

(4)

(g) Find the surface area for this value of x.

(3)

(Total 15 marks)

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