NOV 2008
1.
Given that h =
(a)
l2 
d2
,
4
Calculate the exact value of h when l = 0.03625 and d = 0.05.
(2)
(b)
Write down the answer to part (a) correct to three decimal places.
(1)
(c)
Write down the answer to part (a) correct to three significant figures.
(1)
(d)
Write down the answer to part (a) in the form a × 10k, where 1 ≤ a < 10, k 
.
(2)
(Total 6 marks)
2.
The grades obtained by a group of 20 IB students are listed below:
(a)
6
2
5
3
5
5
6
2
6
1
7
6
2
4
2
4
3
4
5
6
Complete the following table for the grades obtained by the students.
Grade
Frequency
1
2
3
2
4
5
4
6
7
1
(2)
(b)
Write down the modal grade obtained by the students.
(1)
IB Questionbank Mathematical Studies 3rd edition
1
NOV 2008
(c)
Calculate the median grade obtained by the students.
(2)
One student is chosen at random from the group.
(d)
Find the probability that this student obtained either grade 4 or grade 5.
(1)
(Total 6 marks)
IB Questionbank Mathematical Studies 3rd edition
2
NOV 2008
3.
The diagram below shows the cumulative frequency distribution of the heights in metres of 600
trees in a wood.
(a)
Write down the median height of the trees.
(1)
(b)
Calculate the interquartile range of the heights of the trees.
(2)
IB Questionbank Mathematical Studies 3rd edition
3
NOV 2008
(c)
Given that the smallest tree in the wood is 3 m high and the tallest tree is 28 m high, draw
the box and whisker plot on the grid below that shows the distribution of trees in the
wood.
(3)
(Total 6 marks)
4.
Let p and q represent the propositions
p: food may be taken into the cinema
q: drinks may be taken into the cinema
(a)
Complete the truth table below for the symbolic statement ¬(p  q).
p
q
T
T
T
F
F
T
F
F
p  q
¬(p  q)
(2)
(b)
Write down in words the meaning of the symbolic statement ¬(p  q).
(2)
IB Questionbank Mathematical Studies 3rd edition
4
NOV 2008
(c)
Write in symbolic form the compound statement:
“no food and no drinks may be taken into the cinema”.
(2)
(Total 6 marks)
5.
The exchange rate between Indian rupees (INR) and Singapore dollars (S$) is
100 INR = S$3.684
Kwai Fan changes S$500 to Indian rupees.
(a)
Calculate the number of Indian rupees she will receive using this exchange rate.
Give your answer correct to the nearest rupee.
(2)
On her return to Singapore, Kwai Fan has 2500 Indian rupees left from her trip.
She wishes to exchange these rupees back to Singapore dollars. There is a 3% commission
charge for this transaction and the exchange rate is 100 INR = S$3.672.
(b)
Calculate the commission in Indian rupees that she is charged for this exchange.
(2)
(c)
Calculate the amount of money she receives in Singapore dollars, correct to two decimal
places.
(2)
(Total 6 marks)
6.
The distribution of the weights, correct to the nearest kilogram, of the members of a football
club is shown in the following table.
Weight (kg)
40 – 49
50 – 59
60 – 69
70 – 79
Frequency
6
18
14
4
IB Questionbank Mathematical Studies 3rd edition
5
NOV 2008
(a)
On the grid below draw a histogram to show the above weight distribution.
(2)
(b)
Write down the mid-interval value for the 40 – 49 interval.
(1)
(c)
Find an estimate of the mean weight of the members of the club.
(2)
(d)
Write down an estimate of the standard deviation of their weights.
(1)
(Total 6 marks)
7.
A straight line, L1, has equation x + 4y + 34 = 0.
(a)
Find the gradient of L1.
(2)
The equation of line L2 is y = mx. L2 is perpendicular to L1.
(b)
Find the value of m.
(2)
(c)
Find the coordinates of the point of intersection of the lines L1 and L2.
(2)
(Total 6 marks)
IB Questionbank Mathematical Studies 3rd edition
6
NOV 2008
8.
Given the arithmetic sequence: u1 = 124, u2 = 117, u = 110, u4 = 103, …
(a)
Write down the common difference of the sequence.
(1)
(b)
Calculate the sum of the first 50 terms of the sequence.
(2)
uk is the first term in the sequence that is negative.
(c)
Find the value of k.
(3)
(Total 6 marks)
9.
The Venn diagram shows the numbers of pupils in a school according to whether they study the
sciences Physics (P), Chemistry (C), Biology (B).
(a)
Write down the number of pupils that study Chemistry only.
(1)
(b)
Write down the number of pupils that study exactly two sciences.
(1)
(c)
Write down the number of pupils that do not study Physics.
(2)
IB Questionbank Mathematical Studies 3rd edition
7
NOV 2008
(d)
Find n[(P  B)  C].
(2)
(Total 6 marks)
10.
Eva invests USD2000 at a nominal annual interest rate of 8 % compounded half-yearly.
(a)
Calculate the value of her investment after 5 years, correct to the nearest dollar.
(3)
Toni invests USD1500 at an annual interest rate of 7.8 % compounded yearly.
(b)
Find the number of complete years it will take for his investment to double in value.
(3)
(Total 6 marks)
11.
The marks obtained by 8 candidates in Physics and Chemistry tests are given below.
(a)
Physics (x)
6
8
10
11
10
5
4
12
Chemistry (y)
8
11
14
13
11
7
5
15
Write down the product moment correlation coefficient, r.
(1)
(b)
Write down, in the form y = mx + c, the equation of the regression line y on x for the 8
candidates.
(2)
A ninth candidate obtained a score of 7 in the Physics test but was absent for the Chemistry test.
(c)
Use your answer to (b) to estimate the score he would have obtained on the Chemistry
test.
(2)
(d)
Give a reason why it is valid to use this regression line to estimate the score on the
Chemistry test.
(1)
(Total 6 marks)
IB Questionbank Mathematical Studies 3rd edition
8
NOV 2008
12.
The diagram shows a pyramid VABCD which has a square base of length 10 cm and edges of
length 13 cm. M is the midpoint of the side BC.
diagram not to scale
(a)
Calculate the length of VM.
(2)
(b)
Calculate the vertical height of the pyramid.
(2)
(c)
Calculate the angle between a sloping face of the pyramid and its base.
(2)
(Total 6 marks)
13.
(a)
Factorise the expression x2 – kx.
(1)
(b)
Hence solve the equation x2 – kx = 0.
(1)
IB Questionbank Mathematical Studies 3rd edition
9
NOV 2008
The diagram below shows the graph of the function f(x) = x2 – kx for a particular value of k.
(c)
Write down the value of k for this function.
(1)
(d)
Find the minimum value of the function y = f(x).
(3)
(Total 6 marks)
14.
The function f(x) = ax + b is defined by the mapping diagram below.
(a)
Find the values of a and b.
(3)
IB Questionbank Mathematical Studies 3rd edition
10
NOV 2008
(b)
Write down the image of 2 under the function f.
(1)
(c)
Find the value of c.
(2)
(Total 6 marks)
15.
The diagram below shows the graphs of two sine functions, f(x) and g(x) , for –180° ≤ x ≤ 180°.
(a)
Write down
(i)
the equation of f(x);
(ii)
the equation of g(x).
(4)
(b)
Use your graphic display calculator to solve the equation f(x) = g(x) in the interval
–90° ≤ x ≤ 90°.
(2)
(Total 6 marks)
IB Questionbank Mathematical Studies 3rd edition
11
NOV 2008
16.
Throughout this question all the numerical answers must be given correct to the nearest
whole number.
Park School started in January 2000 with 100 students. Every full year, there is an increase of
6 % in the number of students.
(a)
Find the number of students attending Park School in
(i)
January 2001;
(ii)
January 2003.
(4)
(b)
Show that the number of students attending Park School in January 2007 is 150.
(2)
Grove School had 110 students in January 2000. Every full year, the number of students is 10
more than in the previous year.
(c)
Find the number of students attending Grove School in January 2003.
(2)
(d)
Find the year in which the number of students attending Grove School will be first 60 %
more than in January 2000.
(4)
Each January, one of these two schools, the one that has more students, is given extra money to
spend on sports equipment.
(e)
(i)
Decide which school gets the money in 2007. Justify your answer.
(ii)
Find the first year in which Park School will be given this extra money.
(5)
(Total 17 marks)
IB Questionbank Mathematical Studies 3rd edition
12
NOV 2008
17.
The quadrilateral ABCD shown below represents a sandbox. AB and BC have the same length.
AD is 9 m long and CD is 4.2 m long. Angles AD̂C and AB̂C are 95° and 130° respectively.
diagram not to scale
(a)
Find the length of AC.
(3)
(b)
(i)
Write down the size of angle BĈA.
(ii)
Calculate the length of AB.
(4)
(c)
Show that the area of the sandbox is 31.1 m2 correct to 3 s.f.
(4)
The sandbox is a prism. Its edges are 40 cm high. The sand occupies one third of the volume of
the sandbox.
(d)
Calculate the volume of sand in the sandbox.
(3)
(Total 14 marks)
IB Questionbank Mathematical Studies 3rd edition
13
NOV 2008
18.
Jorge conducted a survey of 200 drivers. He asked two questions:
How long have you had your driving licence?
Do you wear a seat belt when driving?
The replies are summarized in the table below.
(a)
Wear a seat belt
Do not wear a
seat belt
Licence less than 2
years
38
42
Licence between 2
and 15 years
30
45
Licence more than
15 years
30
15
Jorge applies a χ2 test at the 5 % level to investigate whether wearing a seat belt is
associated with the time a driver has had their licence.
(i)
Write down the null hypothesis, H0.
(ii)
Write down the number of degrees of freedom.
(iii)
Show that the expected number of drivers that wear a seat belt and have had their
driving licence for more than 15 years is 22, correct to the nearest whole number.
(iv)
Write down the χ2 test statistic for this data.
(v)
Does Jorge accept H0? Give a reason for your answer.
(8)
(b)
Consider the 200 drivers surveyed. One driver is chosen at random.
Calculate the probability that
(i)
this driver wears a seat belt;
(ii)
the driver does not wear a seat belt, given that the driver has held a licence for
more than 15 years.
(4)
IB Questionbank Mathematical Studies 3rd edition
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NOV 2008
(c)
Two drivers are chosen at random. Calculate the probability that
(i)
both wear a seat belt.
(ii)
at least one wears a seat belt.
(6)
(Total 18 marks)
19.
The temperature in °C of a pot of water removed from the cooker is given by
T(m) = 20 + 70 × 2.72–0.4m, where m is the number of minutes after the pot is removed from the
cooker.
(a)
Show that the temperature of the water when it is removed from the cooker is 90 °C.
(2)
The following table shows values for m and T(m).
(b)
m
1
2
4
6
8
10
T(m)
66.9
51.4
34.1
26.3
22.8
s
(i)
Write down the value of s.
(ii)
Draw the graph of T(m) for 0 ≤ m ≤ 10. Use a scale of 1 cm to represent 1 minute
on the horizontal axis and a scale of 1 cm to represent 10 °C on the vertical axis.
(iii)
Use your graph to find how long it takes for the temperature to reach 56 °C. Show
your method clearly.
(iv)
Write down the temperature approached by the water after a long time.
Justify your answer.
(9)
Consider the function S(m) = 20m – 40 for 2 ≤ m ≤ 6.
The function S(m) represents the temperature of soup in a pot placed on the cooker two minutes
after the water has been removed. The soup is then heated.
(c)
Draw the graph of S(m) on the same set of axes used for part (b).
(2)
IB Questionbank Mathematical Studies 3rd edition
15
NOV 2008
(d)
Comment on the meaning of the constant 20 in the formula for S(m) in relation to the
temperature of the soup.
(1)
(e)
(i)
Use your graph to solve the equation S(m) = T(m). Show your method clearly.
(ii)
Hence describe by using inequalities the set of values of m for which
S(m) > T(m).
(4)
(Total 18 marks)
20.
Consider the curve y = x3 
(a)
3 2
x – 6x – 2
2
(i) Write down the value of y when x is 2.
(ii)
Write down the coordinates of the point where the curve intercepts the y-axis.
(3)
(b)
Sketch the curve for –4 ≤ x ≤ 3 and –10 ≤ y ≤ 10. Indicate clearly the information found
in (a).
(4)
(c)
Find
dy
.
dx
(3)
(d)
Let L1 be the tangent to the curve at x = 2.
Let L2 be a tangent to the curve, parallel to L1.
(i)
Show that the gradient of L1 is 12.
(ii)
Find the x-coordinate of the point at which L2 and the curve meet.
(iii)
Sketch and label L1 and L2 on the diagram drawn in (b).
(8)
IB Questionbank Mathematical Studies 3rd edition
16
NOV 2008
(e)
It is known that
dy
> 0 for x < –2 and x > b where b is positive.
dx
(i)
Using your graphic display calculator, or otherwise, find the value of b.
(ii)
Describe the behaviour of the curve in the interval –2 < x < b.
(iii)
Write down the equation of the tangent to the curve at x = –2.
(5)
(Total 23 marks)
IB Questionbank Mathematical Studies 3rd edition
17
NOV 2008
SOLUTIONS:
21.
(a)
h=
0.036252 
0.052
4
(M1)
= 0.02625
(A1) (C2)
Note: Award (A1) only for 0.0263 seen without working
(b)
0.026
(A1)(ft) (C1)
(c)
0.0263
(A1)(ft) (C1)
(d)
2.625 × 10–2
for 2.625 (ft) from unrounded (a) only
for × 10–2
(A1)(ft)
(A1)(ft) (C2)
[6]
22.
(a)
Grade
Frequency
1
1
2
4
3
(2)
4
3
5
(4)
6
5
7
(1)
(A2) (C2)
Note: Award (A1) for three correct.
Award (A0) for two or fewer correct.
(b)
Mode = 6
(c)
Median = 4.5
(M1)
Note: (M1) for attempt to order raw data (if frequency table
not used)
(A1)(ft)
th
th
or (M1) halfway between 10 and 11 result.
(C2)
(d)
7
(0.35, 35 %)
20
(A1)(ft) (C1)
(A1)(ft) (C1)
[6]
IB Questionbank Mathematical Studies 3rd edition
18
NOV 2008
23.
Unit penalty (UP) applies in part (a) in this question
(a)
Median = 11m
(A1) (C1)
(b)
Interquartile range = 14 – 10
(A1)
=4
(A1)(ft) (C2)
Note: (M1) for taking a sensible difference or for both correct
quartile values seen.
(c)
correct median
(A1)(ft)
correct quartiles and box
(A1)(ft)
endpoints at 3 and 28, joined to box by straight lines
(A1) (C3)
Note: Award (A0) if the lines go right through the box.
Award final (A1) if the whisker goes to 20 with an outlier at 28
[6]
24.
(a)
p
q
p  q
¬(p  q)
T
T
T
F
T
F
T
F
F
T
T
F
(A1)
F
F
F
T
(A1)(ft)
Note: (A1) for each correct column
(b)
(C2)
It is not true that food or drinks may be taken into the cinema.
Note: (A1) for “it is not true”. (A1) for “food or drinks”.
OR
Neither food nor drinks may be taken into the cinema.
Note: (A1) for “neither”. (A1) for “nor”.
OR
No food and no drinks may be taken into the cinema.
Note: (A1) for “no food”, “no drinks”. (A1) for “and”.
OR
No food or drink may be brought into the cinema.
IB Questionbank Mathematical Studies 3rd edition
(A2) (C2)
19
NOV 2008
Note: (A1) for “no”, (A1) for “food or drink”
Do not penalize for use of plural / singular
Note: the following answers are incorrect:
No food and drink may be brought into the cinema. Award
(A1)(A0)
Food and drink may not be brought into the cinema. Award
(A1)(A0)
No food or no drink may be brought into the cinema. Award
(A1)(A0)
(c)
¬p  ¬q
Note: (A1) for both negations, (A1) for conjunction.
OR
¬(p  q)
(A1)(A1) (C2)
Note: (A1) for negation, (A1) for p  q in parentheses.
[6]
25.
Financial penalty (FP) applies in parts (a) and (c) in this question.
(a)
FP
500 ×
100
3.684
(M1)
= 13572
(A1) (C2)
Note:(M1) for multiplication by
(b)
2500 × 0.03
= 75 (75.0, 75.00)
If 2500 × 0.03 ×
100
3.684
(M1)
(A1)
3.672
100
= 2.75
(C2)
Note: Award (M1)(A0)
(c)
FP
2425 ×
3.672
100
= 89.05
(M1)(ft)
(A1)(ft)
OR
FP
3.672
× 0.97 × 2500
100
= 89.05
(M1)(ft)
(A1)(ft)
OR
FP
3% of 91.8 = 2.754
91.8 – 2.754
(M1)(ft)
= 89.05
(A1)(ft) (C2)
Note: (ft) in (c) if the conversion process is reversed
consistently through the question. i.e. multiplication in (a)
followed by division in (c)
IB Questionbank Mathematical Studies 3rd edition
20
NOV 2008
[6]
26.
Unit penalty (UP) applies in part (c) in this question.
(a)
(A1)(A1) (C2)
Note: (A1) for all correct heights, (A1) for all correct end
points (39.5, 49.5 etc.).
Histogram must be drawn with a ruler (straight edge) and
endpoints must be clear.
Award (A1) only if both correct histogram and correct
frequency polygon drawn.
(b)
44.5
IB Questionbank Mathematical Studies 3rd edition
(A1) (C1)
21
NOV 2008
(c)
Mean =
UP
44.5  6  54.5 18  ...
(M1)
42
Note: (M1) for a sum of frequencies multiplied by midpoint
values divided by 42.
= 58.3 kg
Note: Award (A1)(A0)(AP) for 58.
(d)
Standard deviation = 8.44
Note: If (b) is given as 45 then award
(b) 45
(c) 58.8 kg
(d) 8.44
(A1)(ft) (C2)
(A1) (C1)
(A0)
(M1)(A1)(ft) or (C2)(ft) if no working seen.
(C1)
[6]
27.
(a)
(b)
4y = –x – 34 or similar rearrangement
1
Gradient = 
4
m=4
Note: (A1) Change of sign
(A1) Use of reciprocal
(c)
Reasonable attempt to solve equations simultaneously
(–2, –8)
(M1)
(A1) (C2)
(A1)(ft)
(A1)(ft) (C2)
(M1)
(A1)(ft) (C2)
Note: Accept x = –2 y = –8
Award (A0) if brackets not included.
[6]
28.
(a)
d = –7
(b)
S50 =
(A1) (C1)
50
(2(124) + 49(–7))
2
Note: (M1) for correct substitution.
= –2375
IB Questionbank Mathematical Studies 3rd edition
(M1)
(A1)(ft) (C2)
22
NOV 2008
(c)
124 – 7(k – 1) < 0
(M1)
k > 18.7 or 18.7 seen
(A1)(ft)
k = 19
(A1)(ft) (C3)
Note: (M1) for correct inequality or equation seen or for list of
values seen or for use of trial and error.
[6]
29.
(a)
9
(A1) (C1)
(b)
12
(A1) (C1)
(c)
8+3+9+6
= 26
Note: Award (A1) for 20 seen if answer is incorrect.
(M1)
(A1) (C2)
(d)
5+2+3
= 10
(M1)
(A1) (C2)
Note: Award (A1) for 29 or 19 seen if answer is incorrect.
[6]
30.
Financial penalty (FP) applies in part (a) in this question.
(a)
2000(1.04)10
(M1)(A1)
Note: (M1) for substitution into CI formula.
(A1) for correct substitution.
FP
2960
(A1)
OR
10
8 

2000 1 
 – 2000
 200 
Note: (M1) for substitution into CI formula
(A1) for correct substitution
2960
IB Questionbank Mathematical Studies 3rd edition
(M1)(A1)
(A1) (C3)
23
NOV 2008
(b)
1500(1.078)n = 3000
(M1)(M1)
Note: (M1) for correct substitution in CI formula, (M1) for
3000 seen.
n = 10 years (n = 9.23 years not accepted)
(A1)
OR
1500(1.078)n – 1500 = 1500
(M1)(M1)
Note: (M1) for correct substitution in CI formula, (M1) for
1500 seen.
n = 10 years (n = 9.23 years not accepted)
(A1)
OR
Note: (M2) for list or graph.
n = 10 years (n = 9.23 years not accepted)
(M2)
(A1)
Note: If simple interest formula is used in both parts (a) and (b) then award
(M0)(M0)(A0) in (a) and
1500(7.8)n
(b) 1500 =
(M1)(A1)
100
(M1) for substitution in SI formula or lists, (A1) for
correct substitution
n = 13
Correct answer only.
(A1) (C3)
If 9.23 seen without working award (A2).
If calculator notation is used in either part with correct
unrounded answer award (A1)(d) only if (FP) is applied in (a)
or (AP) in (b).
Otherwise (A2)(d) if penalty has already been applied in a
previous question.
[6]
31.
(a)
0.965
(b)
y = 1.15x + 0.976
(A1) for 1.15x (A1) for +0.976
(c)
(A1) (C1)
(A1)(A1) (C2)
y = 1.15 (7) + 0.976
(M1)
Chemistry = 9.03
(accept 9)
(A1)(ft) (C2)
Note: Follow through from candidate’s answer to (b) even if no
working is seen. Award (A2)(ft).
IB Questionbank Mathematical Studies 3rd edition
24
NOV 2008
(d)
the correlation coefficient is close to 1
OR strongly correlated variables
OR 7 lies within the range of physics marks.
(R1) (C1)
[6]
32.
Unit penalty (UP) applies in parts (a) and (b) in this question.
(a)
VM2 = 132 – 52
= 12 cm
(b)
h2 = 122– 52 (or equivalent)
= 10.9 cm
(c)
cos θ =
UP
UP
5
(or equivalent)
12
θ = 65.4°
Note: Accept θ = 65.3° (use of 10.9 with sine ratio).
(M1)
(A1) (C2)
(M1)
(A1)(ft) (C2)
(M1)
(A1)(ft) (C2)
[6]
33.
(a)
x(x – k)
(A1) (C1)
(b)
x = 0 or x = k
Note: Both correct answers only
(A1) (C1)
(c)
k=3
(A1) (C1)
IB Questionbank Mathematical Studies 3rd edition
25
NOV 2008
(d)
Vertex at x =
 (3)
2(1)
(M1)
Note: (M1) for correct substitution in formula
x = 1.5
y = –2.25
(A1)(ft)
(A1)(ft)
OR
f′(x) = 2x – 3
(M1)
Note: (M1) for correct differentiation
x = 1.5
y = –2.25
(A1)(ft)
(A1)(ft)
OR
for finding the midpoint of their 0 and 3
(M1)
x = 1.5
(A1)(ft)
y = –2.25
(A1)(ft) (C3)
Note: If final answer is given as (1.5, –2.25) award a maximum
of (M1)(A1)(A0)
[6]
34.
(a)
f(0) = a0 + b = 6
(M1)
b=5
(A1)
1
f(1) = a + 5 = 9  a = 4
(A1) (C3)
Note: (M1) for attempt at solving any appropriate equations
(simultaneously).
(A1)(A1) for each correct answer.
(b)
f(2) = 21
(A1)(ft) (C1)
Note: Follow through from their f(x)
(c)
4c + 5 = 5.5
(M1)
Note: Correct substitution in their f(x)
c= 
1
2
(A1)(ft) (C2)
[6]
35.
(a)
(i)
f(x) = sin(2x)
(A1)(A1) (C2)
Note: (A1) for amplitude of '1' implied in sin function, if the
function is reasonable. (A1) for 2x.
IB Questionbank Mathematical Studies 3rd edition
26
NOV 2008
(ii)
(b)
g(x) = 1.5 sin (x) + 0.5
Note: (A1) for 1.5 sin (x). (A1) for +0.5
(A1)(A1) (C2)
x = –62.1° (Answer must be reasonable from graph)
(A2)(ft) (C2)
nd
Note: Award (A1) if 2 answer given (–172°) in addition.
Award (A1) if correct coordinate pair is given.
[6]
36.
(a)
(i)
100 × 1.06 = 106
(M1)(A1)(G2)
Note: (M1) for multiplying by 1.06 or equivalent. (A1) for
correct answer.
(ii)
100 × 1.063 = 119
(M1)(A1)(G2)
3
Note: (M1) for multiplying by 1.06 or equivalent or for list of
values.
(A1) for correct answer.
(b)
100 × 1.067 = 150.36... = 150 correct to the nearest whole
(M1)(A1)(AG)
Note: (M1) for correct formula or for list of values. (A1) for
correct substitution or for 150 in the correct position in the list.
Unrounded answer must be seen for the (A1).
(c)
110 + 3 × 10 = 140
(M1)(A1)(G2)
Note: (M1) for adding 30 or for list of values. (A1) for correct
answer.
In (d) and (e) follow through from (c) if consistent wrong use of correct AP formula.
(d)
110 + ( n – 1) × 10 > 176
(A1)(M1)
n = 8  year 2007
(A1)(A1)(ft)(G2)
Note: (A1) for 176 or 66 seen. (M1) for showing list of values
and comparing them to 176 or for equating formula to 176 or
for writing the inequality.
If n = 8 not seen can still get (A2) for 2007. Answer n = 8 with
no working gets (G1).
OR
110 + n × 10 > 176
n = 7  year 2007
IB Questionbank Mathematical Studies 3rd edition
(A1)(M1)
(A1)(A1)(ft)(G2)
27
NOV 2008
(e)
(i)
180
Grove School gets the money.
Note: (A1) for 180 seen. (A1) for correct answer.
(A1)(ft)
(A1)(ft)
(ii)
100 × 1.06n–1 > 110 + (n – 1) × 10
(M1)
n = 20  year 2019
(A1)(A1)(ft)(G2)
Note: (M1) for showing lists of values for each school and
comparing them or for equating both formulae or writing the
correct inequality. If n = 20 not seen can still get (A2) for 2019.
Follow through with ratio used in (b) and/or formula used in
(d).
OR
100 × 1.06n > 110 + n × 10
n = 19  year 2019
(M1)
(A1)(A1)(ft)(G2)
OR
graphically
Note: (M1) for sketch of both functions on the same graph, (A1)
for the intersection point, (A1) for correct answer.
[17]
37.
Unit penalty (UP) is applicable in (a), (b)(ii) and (d)
(a)
AC2 = 92 + 4.22 – 2 × 9 × 4.2 × cos 95°
(M1)(A1)
AC = 10.3 m
(A1)(G2)
Note: (M1) for correct substituted formula and (A1) for correct
substitution If radians used answer is 6.59. Award at most
(M1)(A1)(A0)
(b)
(i)
UP
BĈA = 25°
IB Questionbank Mathematical Studies 3rd edition
(A1)
28
NOV 2008
(ii)
AB
10.258...

(M1)(A1)
sin25 sin 130
AB = 5.66 m
(A1)(ft)(G2)
Note: (M1) for correct substituted formula and (A1) for correct
substitution. (A1) for correct answer.
Follow through with angle BCˆ A and their AC.
Allow AB = 5.68 if AC = 10.3 used.
If radians used answer is 0.938 (unreasonable answer). Award
at most (M1)(A1)(A0)(ft)
OR
Using that ABC is isosceles
1
 10.258...
2
cos 25° =
(or equivalent)
AB
UP
AB = 5.66 m
(A1)(M1)(ft)
(A1)(ft)(G2)
1
of their AB seen, (M1) for correct
2
trigonometric ratio and correct substitution, (A1) for correct
answer.
1
If
AB seen and correct answer is given award (A1)(G1).
2
Allow AB = 5.68 if AC = 10.3 used.
If radians used answer is 3.32. Award (A1)(M1)(A1)(ft).
If sin65 and radians used answer is 3.99. Award
(A1)(M1)(A1)(ft)
Note: (A1) for
(c)
UP
Area =
1
1
× 9 × 4.2 × sin 95° +
× (5.6592...)2 × sin 130°(M1)(M1)(ft)(M1)
2
2
= 31.095... = 31.1 m2 (correct to 3 s.f.)
(A1)(AG)
Note: (M1)(M1) each for correct substitution in the formula of
the area of each triangle, (M1) for adding both areas. (A1) for
unrounded answer.
Follow through with their length of AB but last mark is lost if
they do not reach the correct answer.
IB Questionbank Mathematical Studies 3rd edition
29
NOV 2008
(d)
1
(31.09... × 0.4)
(M1)(M1)
3
= 4.15 m3
(A1)(G2)
Note: (M1) for correct formula of volume of prism and for
1
correct substitution, (M1) for multiplying by
and last (A1)
3
for correct answer only.
Volume of sand =
[14]
38.
(a)
(i)
H0 = wearing of a seat belt and the time a driver has held
a licence are independent.
(A1)
Note: For independent accept “not associated” but do not
accept “not related” or “not correlated”
(ii)
2
(iii)
(A1)
98 45
= 22.05 = 22 (correct to the nearest whole number)(M1)(A1)(AG)
200
Note: (M1) for correct formula and (A1) for correct substitution.
Unrounded answer must be seen for the (A1) to be awarded.
(iv)
χ2 = 8.12
(G2)
Note: For unrounded answer award (G1)(G0)(AP)
If formula used award (M1) for correct substituted formula with
correct substitution (6 terms) (A1) for correct answer.
(v)
“Does not accept H0”
(A1)(ft)

(R1)(ft)
2
crit
< 8.12 or p-value < 0.05
Note: Allow “Reject H0” or equivalent. Follow through from
their χ2 statistic. Award (R1)(ft) for comparing the appropriate
values. The (A1)(ft) can be awarded only if the conclusion is
valid according to the comparison given. If no reason given or
if reason is wrong the two marks are lost.
(b)
(i)
98
(= 0.49, 49%)
(A1)(A1)(G2)
200
Note: (A1) for numerator, (A1) for denominator.
IB Questionbank Mathematical Studies 3rd edition
30
NOV 2008
(ii)
(c)
(i)
(ii)
15
(= 0.333, 33.3%)
(A1)(A1)(G2)
45
Note: (A1) for numerator, (A1) for denominator.
98 97

= 0.239 (23.9 %)
(A1)(M1)(A1) (G3)
200 199
Note: (A1) for correct probabilities seen, (M1) for multiplying
two probabilities, (A1) for correct answer.
1–
102 101

= 0.741 (74.1 %)
200 199
(M1)(M1)
(A1)(ft)(G2)
Note: (M1) for showing the product, (M1) for using the
probability of the complement, (A1) for correct answer. Follow
through for consistent use of with replacement.
OR
98 97 98 102 102 98





= 0.741 (74.1 %)
200 199 200 199 200 199
(M1)(M1)
(A1)(ft)(G2)
Note: (M1) for adding three products of fractions (or
equivalent), (M1) for using the correct fractions, (A1) for
correct answer Follow through for consistent use of with
replacement.
[18]
39.
(a)
T(0) = 20 + 70 × 2.72–0.4×0 = 90
(M1)(A1)(AG)
Note: (M1) for taking m = 0, (A1) for substituting 0 into the
formula. For the A mark to be awarded 90 must be justified by
correct method.
(b)
(i)
21.3
IB Questionbank Mathematical Studies 3rd edition
(A1)
31
NOV 2008
(ii)
(A4)(ft)
Note: Scales and labels (A1).
Smooth curve (A1). All points correct including the y-intercept
(A2), 1 point incorrect (A1), otherwise (A0).
Follow through from their value of s.
(iii)
m = 1.7 minutes (Accept ±0.2)
(A2)(ft)
Note: Follow through from candidate’s graph. Accept answers
in minutes and seconds if consistent with graph.
If answer incorrect and correct line(s) seen on graph award
(M1)(A0)
(iv)
20 °C
(A1)(ft)
The curve behaves asymptotically to the line y = 20 or similar. (A1)
OR
The room temperature is 20 or similar
OR
When m is a very large number the term 70 × 2.72–0.4m tends
to zero or similar.
Note: Follow through from their graph if appropriate
IB Questionbank Mathematical Studies 3rd edition
32
NOV 2008
(c)
(See graph above)
(A1)(A1)
Note: (A1) for correct line, (A1) for domain.
If line not drawn on same set of axes award at most (A1)(A0)
(d)
It indicates by how much the temperature increases per minute.
(e)
(i)
m = 3.8 (Accept ±0.1)
(A2)(ft)
Note: Follow through from candidate’s graph. Accept answers
in minutes and seconds if consistent with graph.
If answer incorrect and correct line(s) seen on graph award
(M1)(A0)
(ii)
3.8 < m < 6
(A1)(A1)(ft)
Note: (A1) for m > 3.8 and (A1) for m ≤ 6
Follow through from candidate’s answer to part (e)(i). If
candidate was already penalized in (c) for domain and does not
state m ≤ 6 then award (A2)(ft).
(A1)
[18]
40.
(a)
(i)
y=0
(ii)
(0, –2)
Note: Award (A1)(A0) if brackets missing.
(A1)
(A1)(A1)
OR
x = 0, y = –2
Note: If coordinates reversed award (A0)(A1)(ft).
Two coordinates must be given.
IB Questionbank Mathematical Studies 3rd edition
(A1)(A1)
33
NOV 2008
(b)
(A4)
Note: (A1) for appropriate window. Some indication of scale on
the x-axis must be present (for example ticks). Labels not
required. (A1) for smooth curve and shape, (A1) for maximum
and minimum in approximately correct position, (A1) for x and
y intercepts found in (a) in approximately correct position.
(c)
(d)
(e)
dy
= 3x2 + 3x – 6
(A1)(A1)(A1)
dx
Note: (A1) for each correct term. Award (A1)(A1)(A0) at most
if any other term is present.
(i)
3 × 4 + 3 × 2 – 6 = 12
(M1)(A1)(AG)
Note: (M1) for using the derivative and substituting x = 2. (A1)
for correct (and clear) substitution. The 12 must be seen.
(ii)
Gradient of L2 is 12 (can be implied)
(A1)
2
3x + 3x – 6 = 12
(M1)
x = –3
(A1)(G2)
Note: (M1) for equating the derivative to 12 or showing a
sketch of the derivative together with a line at y = 12 or a table
of values showing the 12 in the derivative column.
(iii)
(A1) for L1 correctly drawn at approx the correct point
(A1)
(A1) for L2 correctly drawn at approx the correct point
(A1)
(A1) for 2 parallel lines
(A1)
Note: If lines are not labelled award at most (A1)(A1)(A0).
Do not accept 2 horizontal or 2 vertical parallel lines
(i)
b=1
IB Questionbank Mathematical Studies 3rd edition
(G2)
34
NOV 2008
(ii)
The curve is decreasing.
(A1)
Note: Accept any valid description.
(iii)
y=8
(A1)(A1)(G2)
Note: (A1) for “y = a constant”, (A1) for 8.
[23]
IB Questionbank Mathematical Studies 3rd edition
35