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IB Math Studies Year 1 Exam Review

Mr. Howes

Date: Name........................................

1. (a) Calculate exactly [pic]

(1)

(b) Write the answer to part (a) correct to 2 significant figures.

(1)

(c) Calculate the percentage error when the answer to part (a) is written correct to 2 significant figures.

(2)

(d) Write your answer to part (c) in the form a × 10k where 1 ≤ a < 10 and k ∈ [pic].

(2)

[pic]

(Total 6 marks)

2. (a) Given x = 2.6 × 104 and y = 5.0 × 10–8, calculate the value of w = x × y. Give your answer in the form a × 10k where 1 ≤ a < 10 and k ∈ [pic].

(b) Which two of the following statements about the nature of x, y and w above are incorrect?

(i) x ∈ [pic]

(ii) y ∈ [pic]

(iii) y ∈ [pic]

(iv) w < y

(v) x + y ∈ [pic]

(vi) [pic] < x

[pic]

(Total 8 marks)

3. Let [pic] = {x : 1 ≤ x < 17, x ∈ [pic]}.

P , Q and R are the subsets of [pic] such that

P = {multiples of four};

Q = {factors of 36};

R = {square numbers}.

(a) List the elements of

(i) [pic]

(ii) P ∩ Q ∩ R.

(2)

(b) Describe in words the set P ∪ Q.

(1)

(c) (i) Draw a Venn diagram to show the relationship between sets P, Q and R.

(2)

(ii) Write the elements of [pic] in the appropriate places on the Venn diagram.

(3)

(d) Let p, q and r be the statements

p: x is a multiple of four;

q: x is a factor of 36;

r: x is a square number.

(i) Write a sentence, in words, for the statement

(p ∨ r) ∧ ¬ q

(2)

(ii) Shade the region on your Venn diagram in part (c)(i) that represents (p ∨ r) ∧ ¬ q

(1)

(iii) (a) Use a truth table to determine the values of (p ∨ r) ∧ ¬ q. Write the first three columns of your truth table in the following format.

|p |q |r |

|T |T |T |

|T |T |F |

|T |F |T |

|T |F |F |

|F |T |T |

|F |T |F |

|F |F |T |

|F |F |F |

(3)

(b) Write down one possible value of x for which (p ∨ r) ∧ ¬ q is true.

(1)

(Total 15 marks)

4. Vanessa wants to rent a place for her wedding reception. She obtains two quotations.

(a) The local council will charge her £30 for the use of the community hall plus £10 per guest.

(i) Copy and complete this table for charges made by the local council.

|Number of guests (N) |10 |30 |50 |70 |90 |

|Charges (C) in £ | | | | | |

(2)

(ii) On graph paper, using suitable scales, draw and label a graph showing the charges. Take the horizontal axis as the number of guests and the vertical axis as the charges.

(3)

(iii) Write a formula for C, in terms N, that can be used by the local council to calculate their charges.

(1)

(b) The local hotel calculates charges for their conference room using the formula:

C = [pic] + 500

where C is the charge in £ and N is the number of guests.

(i) Describe, in words only, what this formula means.

(2)

(ii) Copy and complete this table for the charges made by the hotel.

|Number of guests (N) |0 |20 |40 |80 |

|Charges (C) in £ | | | | |

(2)

(iii) On the same axes used in part (a)(ii), draw this graph of C. Label your graph clearly.

(2)

(c) Explain, briefly, what the two graphs tell you about the charges made.

(2)

(d) Using your graphs or otherwise, find

(i) the cost of renting the community hall if there are 87 guests;

(2)

(ii) the number of guests if the hotel charges £650;

(2)

(iii) the difference in charges between the council and the hotel if there are 82 guests at the reception.

(2)

(Total 20 marks)

5. The universal set U is the set of integers from 1 to 20 inclusive.

A and B are subsets of U where:

A is the set of even numbers between 7 and 17.

B is the set of multiples of 3.

List the elements of the following sets:

(a) A;

(1)

(b) B;

(1)

(c) A ∪ B;

(2)

(d) A ∩ B′.

(2)

[pic]

(Total 6 marks)

6. The perimeter of a rectangle is 24 metres.

(a) The table shows some of the possible dimensions of the rectangle.

Find the values of a, b, c, d and e.

|Length (m) |Width (m) |Area (m2) |

|1 |11 |11 |

|a |10 |b |

|3 |c |27 |

|4 |d |e |

(2)

(b) If the length of the rectangle is x m, and the area is A m2, express A in terms of x only.

(1)

(c) What are the length and width of the rectangle if the area is to be a maximum?

(3)

(Total 6 marks)

7. Events A and B have probabilities P(A) = 0.4, P (B) = 0.65, and P(A ∪ B) = 0.85.

(a) Calculate P(A ∩ B).

(b) State with a reason whether events A and B are independent.

(c) State with a reason whether events A and B are mutually exclusive.

[pic]

(Total 6 marks)

8. The truth table below shows the truth-values for the proposition

p ∨ q ⇒ ¬ p ∨ ¬ q

|p |q |¬ p |¬ q |p ∨ q |¬ p ∨ ¬ q |p ∨ q ⇒ ¬ p ∨ ¬ q |

|T |T |F |F | |F | |

|T |F |F | |T |T |T |

|F |T |T |F |T |T |T |

|F |F |T |T |F | |T |

(a) Explain the distinction between the compound propositions, p ∨ q and ¬ p ∨ ¬ q.

(b) Fill in the four missing truth-values on the table.

(c) State whether the proposition p ∨ q ⇒ ¬ p ∨ ¬ q is a tautology, a contradiction or neither.

9. The following diagrams show the graphs of five functions.

[pic]

Each of the following sets represents the range of one of the functions of the graphs.

(a) {y ⎜ y ∈ [pic]}

(b) {y ⎜ y ≥ 2}

(c) {y ⎜ y > 0}

(d) {y ⎜1 ≤ y ≤ 2}

Write down which diagram is linked to each set.

|Working: | |

| |Answers: |

| |(a) ………………………………………….. |

| |(b) ………………………………………….. |

| |(c) ………………………………………….. |

| |(d) ………………………………………….. |

(Total 4 marks)

10. (a) Factorize the expression 2x2 – 3x – 5.

(b) Hence, or otherwise, solve the equation 2x2 – 3x = 5.

|Working: | |

| |Answers: |

| |(a) |

| |................................................................|

| |.. |

| |(b) |

| |................................................................|

| |.. |

(Total 4 marks)

11. A swimming pool is to be built in the shape of a letter L. The shape is formed from two squares with side dimensions x and [pic]as shown.

[pic]

(a) Write down an expression for the area A of the swimming pool surface.

(b) The area A is to be 30 m2. Write a quadratic equation that expresses this information.

(c) Find both the solutions of your equation in part (b).

(d) Which of the solutions in part (c) is the correct value of x for the pool? State briefly why you made this choice.

[pic]

(Total 8 marks)

12. Amos travels to school either by car or by bicycle. The probability of being late for school is [pic] if he travels by car and [pic] if he travels by bicycle. On any particular day he is equally likely to travel by car or by bicycle.

(a) Draw a probability tree diagram to illustrate this information.

(4)

(b) Find the probability that

(i) Amos will travel by car and be late.

(2)

(ii) Amos will be late for school.

(3)

(c) Given that Amos is late for school, find the probability that he travelled by bicycle.

(3)

(Total 12 marks)

13. Claire and Kate both wish to go to the cinema but one of them has to stay at home to baby-sit.

The probability that Kate goes to the cinema is 0.2. If Kate does not go Claire goes.

If Kate goes to the cinema the probability that she is late home is 0.3.

If Claire goes to the cinema the probability that she is late home is 0.6.

(a) Copy and complete the probability tree diagram below.

[pic]

(3)

(b) Calculate the probability that

(i) Kate goes to the cinema and is not late;

(2)

(ii) the person who goes to the cinema arrives home late.

(3)

(Total 8 marks)

14. The diagram shows the graph of y = x2 – 2x – 8. The graph crosses the x-axis at the point A, and has a vertex at B.

[pic]

(a) Factorize x2 – 2x – 8.

(b) Write down the coordinates of each of these points

(i) A;

(ii) B.

|Working: | |

| |Answers: |

| |(a) ………………………………………….. |

| |(b) (i) …………………………………….. |

| |(ii) …………………………………….. |

(Total 4 marks)

15. Consider the following logic propositions:

p: Sean is at school

q: Sean is playing a game on his computer.

(a) Write in words, p ∨ q.

(2)

(b) Write in words, the converse of p ⇒ ¬q.

(2)

(c) Complete the following truth table for p ⇒ ¬q.

|p |q |¬q |p ⇒ ¬q |

|T |T | | |

|T |F | | |

|F |T | | |

|F |F | | |

(2)

[pic]

(Total 6 marks)

16. Children in a class of 30 students are asked whether they can swim (S) or ride a bicycle (B).

There are 12 girls in the class. 8 girls can swim, 6 girls can ride a bicycle and 4 girls can do both.

16 boys can swim, 13 boys can ride a bicycle and 12 boys can do both. This information is represented in a Venn diagram.

[pic]

(a) Find the values of a and b.

(2)

(b) Calculate the number of students who can do neither.

(2)

(c) Write down the probability that a student chosen at random can swim.

(2)

(d) Given that the student can ride a bicycle, write down the probability that the student is a girl.

(2)

(Total 8 marks)

17. The following histogram shows the weights of a number of frozen chickens in a supermarket. The weights are grouped such that 1 ≤ weight < 2, 2 ≤ weight, < 3 and so on.

[pic]

(a) On the graph above, draw in the frequency polygon.

(2)

(b) Find the total number of chickens.

(1)

(c) Write down the modal group.

(1)

Gabriel chooses a chicken at random.

(d) Find the probability that this chicken weighs less than 4 kg.

(2)

[pic]

(Total 6 marks)

18. 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)

19. (a) A function f is represented by the following mapping diagram.

[pic]

Write down the function f in the form

f : x [pic]y, x ∈ {the domain of f}.

(b) The function g is defined as follows

g : x [pic]sin 15x°, {x ∈ [pic] and 0 < x ≤ 4}.

Complete the following mapping diagram to represent the function g.

[pic]

|Working: | |

| |Answer: |

| |(a) ………………………………………….. |

(Total 4 marks)

20. The four diagrams below show the graphs of four different straight lines, all drawn to the same scale. Each diagram is numbered and c is a positive constant.

[pic]

In the table below, write the number of the diagram whose straight line corresponds to the equation in the table.

|Equation |Diagram number |

|y = c | |

|y = – x + c | |

|y = 3 x + c | |

|y = [pic] x + c | |

(Total 8 marks)

21. Consider the graphs of the following functions.

(i) y = 7x + x2;

(ii) y = (x – 2)(x + 3);

(iii) y = 3x2 – 2x + 5;

(iv) y = 5 – 3x – 2x2.

Which of these graphs

(a) has a y-intercept below the x-axis?

(b) passes through the origin?

(c) does not cross the x-axis?

(d) could be represented by the following diagram?

[pic]

|Working: | |

| |Answers: |

| |(a) ..............................................|

| |(b) ..............................................|

| |(c) ..............................................|

| |(d) |

| |................................................ |

(Total 8 marks)

22. The height of a vertical cliff is 450 m. The angle of elevation from a ship to the top of the cliff is 23°. The ship is x metres from the bottom of the cliff.

(a) Draw a diagram to show this information.

Diagram:

(b) Calculate the value of x.

|Working: | |

| |Answer: |

| |(b) ………………………………………….. |

(Total 4 marks)

23. The following diagram shows the points P, Q and M. M is the midpoint of [PQ].

[pic]

(a) Write down the equation of the line (PQ).

(b) Write down the equation of the line through M which is perpendicular to the line (PQ).

|Working: | |

| |Answers: |

| |(a) ………………………………………….. |

| |(b) ………………………………………….. |

(Total 4 marks)

24. In the diagram, the lines L1 and L2 are parallel.

[pic]

(a) What is the gradient of L1?

(b) Write down the equation of L1.

(c) Write down the equation of L2 in the form ax + by + c = 0.

|Working: | |

| |Answers: |

| |(a) ………………………………………….. |

| |(b) ………………………………………….. |

| |(c) ………………………………………….. |

(Total 4 marks)

25. The perimeter of this rectangular field is 220 m. One side is x m as shown.

[pic]

(a) Express the width (W) in terms of x.

(b) Write an expression, in terms of x only, for the area of the field.

(c) If the length (x) is 70 m, find the area.

|Working: | |

| |Answers: |

| |(a) ………………………………………….. |

| |(b) ………………………………………….. |

| |(c) ………………………………………….. |

(Total 4 marks

26. The diagram shows a cuboid 22.5 cm by 40 cm by 30 cm.

[pic]

(a) Calculate the length of [AC].

(b) Calculate the size of [pic].

|Working: | |

| |Answers: |

| |(a) |

| |................................................................|

| |.. |

| |(b) |

| |................................................................|

| |.. |

(Total 4 marks)

27. In the diagram, triangle ABC is isosceles. AB = AC, CB = 15 cm and angle ACB is 23°.

Diagram not to scale

[pic]

Find

(a) the size of angle CAB;

(b) the length of AB.

|Working: | |

| |Answers: |

| |(a) ………………………………………….. |

| |(b) ………………………………………….. |

(Total 4 marks)

28. The mean of the ten numbers listed below is 5.5.

4, 3, a, 8, 7, 3, 9, 5, 8, 3

(a) Find the value of a.

(b) Find the median of these numbers.

|Working: | |

| |Answers: |

| |(a) ………………………………………….. |

| |(b) ………………………………………….. |

(Total 4 marks)

29. A marine biologist records as a frequency distribution the lengths (L), measured to the nearest centimetre, of 100 mackerel. The results are given in the table below.

|Length of mackerel |Number of |

|(L cm) |mackerel |

|27 < L ≤ 29 |2 |

|29 < L ≤ 31 |4 |

|31 < L ≤ 33 |8 |

|33 < L ≤ 35 |21 |

|35 < L ≤ 37 |30 |

|37 < L ≤ 39 |18 |

|39 < L ≤ 41 |12 |

|41 < L ≤ 43 |5 |

| |100 |

(a) Construct a cumulative frequency table for the data in the table.

(2)

(b) Draw a cumulative frequency curve.

Hint: Plot your cumulative frequencies at the top of each interval.

(3)

(c) Use the cumulative frequency curve to find an estimate, to the nearest cm for

(i) the median length of mackerel;

(2)

(ii) the interquartile range of mackerel length.

(2)

(Total 9 marks)

30. A group of 25 females were asked how many children they each had. The results are shown in the histogram below.

[pic]

(a) Show that the mean number of children per female is 1.4.

(2)

(b) Show clearly that the standard deviation for this data is approximately 1.06.

(3)

(c) Another group of 25 females was surveyed and it was found that the mean number of children per female was 2.4 and the standard deviation was 2. Use the results from parts (a) and (b) to describe the differences between the number of children the two groups of females have.

(2)

(d) A female is selected at random from the first group. What is the probability that she has more than two children?

(2)

(e) Two females are selected at random from the first group. What is the probability that

(i) both females have more than two children?

(2)

(ii) only one of the females has more than two children?

(3)

(iii) the second female selected has two children given that the first female selected had no children?

(1)

(Total 15 marks)

31. Twenty students are asked how many detentions they received during the previous week at school. The results are summarised in the frequency distribution table below.

|Number of |Number of |fx |

|detentions |students | |

|x |f | |

|0 |6 | |

|1 |3 | |

|2 |10 | |

|3 |1 | |

|Total |20 | |

(a) What is the modal number of detentions received?

(b) (i) Complete the table.

(ii) Find the mean number of detentions received.

|Working: | |

| |Answers: |

| |(a) |

| |................................................................|

| |.. |

| |(b) (ii) ……………………………………... |

(Total 4 marks)

32. The graph below shows the cumulative frequency for the yearly incomes of 200 people.

[pic]

Use the graph to estimate

(a) the number of people who earn less than 5000 British pounds per year;

(b) the median salary of the group of 200 people;

(c) the lowest income of the richest 20% of this group.

|Working: | |

| |Answers: |

| |(a) |

| |................................................................|

| |.. |

| |(b) |

| |................................................................|

| |.. |

| |(c) |

| |................................................................|

| |.. |

(Total 4 marks)

33. The table shows the number of children in 50 families.

|Number of |Frequency |Cumulative |

|children | |frequency |

|1 |3 |3 |

|2 |m |22 |

|3 |12 |34 |

|4 |p |q |

|5 |5 |48 |

|6 |2 |50 |

| |T | |

(a) Write down the value of T.

(b) Find the values of m, p and q.

|Working: | |

| |Answers: |

| |(a) ………………………………………….. |

| |(b) ………………………………………….. |

(Total 4 marks)

34. The Venn diagram below shows the universal set of real numbers [pic] and some of its important subsets:

[pic]: the rational numbers,

[pic]: the integers,

[pic]: the natural numbers.

Write the following numbers in the correct position in the diagram.

–1, 1, π, [pic] [pic].

[pic]

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