IB Questionbank Mathematical Studies 3rd edition 1

1. Five pipes labelled, “6 metres in length”, were delivered to a building site. The contractor

measured each pipe to check its length (in metres) and recorded the following;

5.96, 5.95, 6.02, 5.95, 5.99.

(a) (i) Find the mean of the contractor’s measurements.

(ii) Calculate the percentage error between the mean and the stated, approximate

length of 6 metres.

(b) Calculate 5.05 73.889.3 , giving your answer

(i) correct to the nearest integer;

(ii) in the form a ×10k, where 1 a 10, k .

(Total 6 marks)

2. The Venn diagram below shows the universal set of real numbers and some of its important

subsets:

: the rational numbers,

: the integers,

: the natural numbers.

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

–1, 1, , ,3333.3,16

7 3 .

(Total 6 marks)

IB Questionbank Mathematical Studies 3rd edition 2

3. The mean of the ten numbers listed below is 6.8.

8, 5, 5, 10, 8, 4, 9, 7, p, q

(a) Write down an equation in terms of p and q. (2)

The mode of these ten numbers is five and p is less than q.

(b) Write down the value of

(i) p;

(ii) q. (2)

(c) Find the median of the ten numbers. (2)

(Total 6 marks)

IB Questionbank Mathematical Studies 3rd edition 3

4. There are 120 teachers in a school. Their ages are represented by the cumulative frequency

graph below.

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75

Age

130

120

110

100

90

80

70

60

50

40

30

20

10

0

Cu

mu

lati

ve

freq

uen

cy

(a) Write down the median age. (1)

(b) Find the interquartile range for the ages. (2)

(c) Given that the youngest teacher is 21 years old and the oldest is 72 years old, represent

the information on a box and whisker plot using the scale below.

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75

Age (3)

(Total 6 marks)

IB Questionbank Mathematical Studies 3rd edition 4

5. 10 000 people attended a sports match. Let x be the number of adults attending the sports match

and y be the number of children attending the sports match.

(a) Write down an equation in x and y. (1)

The cost of an adult ticket was 12 AUD. The cost of a child ticket was 5 AUD.

(b) Find the total cost for a family of 2 adults and 3 children. (2)

The total cost of tickets sold for the sports match was 108 800 AUD.

(c) Write down a second equation in x and y. (1)

(d) Write down the value of x and the value of y. (2)

(Total 6 marks)

6. Consider the function f (x) = 2x3 – 5x

2 + 3x + 1.

(a) Find f ′ (x). (3)

(b) Find the values for x when f’(x) = 2 (3)

(Total 6 marks)

IB Questionbank Mathematical Studies 3rd edition 5

7. The graph of y = 2x2 – rx + q is shown for –5 ≤ x ≤ 7.

The graph cuts the y-axis at (0, 4).

(a) Write down the value of q. (1)

The axis of symmetry is x = 2.5.

(b) Find the value of r. (2)

(c) Write down the minimum value of y. (1)

(d) Write down the range of y. (2)

(Total 6 marks)

IB Questionbank Mathematical Studies 3rd edition 6

8. A line joins the points A(2, 1) and B(4, 5).

(a) Find the gradient of the line AB. (2)

Let M be the midpoint of the line segment AB.

(b) Write down the coordinates of M. (1)

(c) Find the equation of the line perpendicular to AB and passing through M. (3)

(Total 6 marks)

9. 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°.

12.3

63°P

h m

(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)

10. Triangle ABC is such that AC is 7 cm, angle CBA is 65 and angle BCA is 30.

(a) Sketch the triangle writing in the side length and angles. (1)

(b) Calculate the length of AB. (2)

(c) Find the area of triangle ABC. (3)

(Total 6 marks)

IB Questionbank Mathematical Studies 3rd edition 7

11. Tony wants to carry out a χ2 test to determine whether or not a person’s choice of one of the

three professions – engineering, medicine or law – is influenced by the person’s sex (gender).

(a) State the null hypothesis, H0, for this test.

(1)

(b) Write down the number of degrees of freedom. (1)

Of the 400 people Tony interviewed, 220 were male and 180 were female.

80 of the people had chosen engineering as a profession.

(c) Calculate the expected number of female engineers. (2)

Tony used a 5 % level of significance for his test and obtained a p-value of 0.0634 correct to 3

significant figures.

(d) State Tony’s conclusion to the test. Give a reason for this conclusion. (2)

(Total 6 marks)

12. A shop keeper recorded daily sales s of ice cream along with the daily maximum temperature

t°C. The results for one week are shown below.

t 29 31 34 23 19 20 27

s 104 92 112 48 56 72 66

(a) Write down the equation of the regression line for s on t. (3)

(b) Use your equation to predict the ice cream sales on a day when the maximum temperature

is 24 °C. Give your answer correct to the nearest whole number. (3)

(Total 6 marks)

IB Questionbank Mathematical Studies 3rd edition 8

13. The weights of a group of children are normally distributed with a mean of 22.5 kg and a

standard deviation of 2.2 kg.

(a) Write down the probability that a child selected at random has a weight more than 25.8

kg.

(b) Of the group 95 weigh less than k kilograms. Find the value of k.

(c) The diagram below shows a normal curve.

On the diagram, shade the region that represents the following information:

87 of the children weigh less than 25 kg (Total 6 marks)

IB Questionbank Mathematical Studies 3rd edition 9

14. The following figures show the graphs of y = f (x) with f (x) chosen to be various cubic

functions. The value of a is positive.

A B C

D E

y

x

a

a

a

y

xa

a

a

y

x

y y

aa

aa

xx

(a) In the table below, write the letter corresponding to the graph of y = f (x) in the space next

to the cubic function.

(Note: one of the graphs is not represented in this table)

cubic function f (x) graph label

f (x) = x3 + a

f (x) = (x – a)3 + a

f (x) = x3

f (x) = (x – a)3

(b) State which one of the graphs represents a function that has a positive gradient for every

value of x.

(c) State how many of the graphs have the x-axis as a tangent at some point. (Total 6 marks)

IB Questionbank Mathematical Studies 3rd edition 10

15. The length of a square garden is (x + 1) m. In one of the corners a square of 1 m length is used

only for grass. The rest of the garden is only for planting roses and is shaded in the diagram

below.

diagram not to scale

The area of the shaded region is A.

(a) Write down an expression for A in terms of x. (1)

(b) Find the value of x given that A = 109.25 m2.

(3)

(c) The owner of the garden puts a fence around the shaded region. Find the length of this

fence. (2)

(Total 6 marks)