1.
In a group of 16 students, 12 take art and 8 take music. One student takes neither art nor music.
The Venn diagram below shows the events art and music. The values p, q, r and s represent
numbers of students.
(a)
(i)
Write down the value of s.
(ii)
Find the value of q.
(iii)
Write down the value of p and of r.
(5)
(b)
(i)
A student is selected at random. Given that the student takes music, write down the
probability the student takes art.
(ii)
Hence, show that taking music and taking art are not independent events.
(4)
(c)
Two students are selected at random, one after the other. Find the probability that the first
student takes only music and the second student takes only art.
(4)
(Total 13 marks)
2.
(a)
(i)
s=1
(ii)
evidence of appropriate approach
e.g. 21–16, 12 + 8 – q =15
q=5
IB Questionbank Maths SL
A1
N1
(M1)
A1
N2
1
(b)
(iii)
p = 7, r = 3
(i)
P(art|music) =
(ii)
METHOD 1
Part 
12 

16 
A1A1
N2
5
8
A2
N2
3

4
A1
evidence of correct reasoning
5
R1
3 5

4 8
e.g.
the events are not independent
AG
N0
METHOD 2
P(art) × P(music) =
96  3 
 
256  8 
evidence of correct reasoning
e.g.
R1
12 8
5
 
16 16 16
the events are not independent
(c)
A1
3
= (seen anywhere)
16
7
P(second takes only art)= (seen anywhere)
15
evidence of valid approach
3 7

e.g.
16 15
21  7 
P(music and art)=
 
240  80 
P(first takes only music) =
AG
N0
4
N2
4
A1
A1
(M1)
A1
[13]
IB Questionbank Maths SL
2
3.
The Venn diagram below shows events A and B where P(A) = 0.3, P( A  B) = 0.6 and
P(A ∩ B) = 0.1. The values m, n, p and q are probabilities.
(a)
(i)
Write down the value of n.
(ii)
Find the value of m, of p, and of q.
(4)
(b)
Find P(B′).
(2)
(Total 6 marks)
4.
(a)
(b)
(i)
n = 0.1
A1
(ii)
m = 0.2, p = 0.3, q = 0.4
A1A1A1
N1
N3
4
N2
2
appropriate approach
e.g. P(B′) =1– P(B), m + q, 1–(n + p)
(M1)
P(B′) = 0.6
A1
[6]
5.
A random variable X is distributed normally with a mean of 20 and variance 9.
(a)
Find P(X ≤ 24.5).
(3)
(b)
Let P(X ≤ k) = 0.85.
(i)
Represent this information on the following diagram.
IB Questionbank Maths SL
3
(ii)
Find the value of k.
(5)
(Total 8 marks)
6.
(a)
σ=3
(A1)
evidence of attempt to find P(X ≤ 24.5)
(M1)
e.g. z =1.5,
24.5  20
3
P(X ≤ 24.5) = 0.933
(b)
A1
N3
A1A1
Note: Award A1 with shading that clearly extends to right of the
mean, A1 for any correct label, either k, area or their value of k
N2
3
(i)
(ii)
z = 1.03(64338)
attempt to set up an equation
k  20
k  20
 1.0364 ,
 0.85
e.g.
3
3
k = 23.1
(A1)
(M1)
A1
N3
5
[8]
7.
A box holds 240 eggs. The probability that an egg is brown is 0.05.
(a)
Find the expected number of brown eggs in the box.
(2)
IB Questionbank Maths SL
4
(b)
Find the probability that there are 15 brown eggs in the box.
(2)
(c)
Find the probability that there are at least 10 brown eggs in the box.
(3)
(Total 7 marks)
8.
(a)
correct substitution into formula for E(X)
(A1)
e.g. 0.05× 240
E(X) =12
(b)
A1
N2
2
N2
2
N3
3
evidence of recognizing binomial probability (may be seen in part (a)) (M1)
 240 
 (0.05)15 (0.95)225, X ~ B(240,0.05)
e.g. 
 15 
P(X =15) = 0.0733
(c)
A1
P(X ≤ 9) = 0.236
(A1)
evidence of valid approach
(M1)
e.g. using complement, summing probabilities
P(X ≥10) = 0.764
A1
[7]
IB Questionbank Maths SL
5
9.
Let f(x) = 8x – 2x2. Part of the graph of f is shown below.
(a)
Find the x-intercepts of the graph.
(4)
(b)
(i)
Write down the equation of the axis of symmetry.
(ii)
Find the y-coordinate of the vertex.
(3)
(Total 7 marks)
10.
(a)
evidence of setting function to zero
e.g. f(x) = 0, 8x = 2x2
evidence of correct working
e.g. 0 = 2x(4 – x),
(i)
x = 2 (must be equation)
IB Questionbank Maths SL
A1
 8  64
4
x-intercepts are at 4 and 0 (accept (4, 0) and (0, 0), or x = 4, x = 0)
(b)
(M1)
A1A1 N1N1
A1
N1
6
(ii)
substituting x = 2 into f(x)
y=8
(M1)
A1
N2
[7]
11.
The straight line with equation y =
(a)
3
x makes an acute angle θ with the x-axis.
4
Write down the value of tan θ.
(1)
(b)
Find the value of
(i)
sin 2θ;
(ii)
cos 2θ.
(6)
(Total 7 marks)
12.
(a)
tan θ =
(b)
(i)
(ii)
3
3 
 do not accept x 
4
4 
A1
3
4
, cos θ =
5
5
correct substitution
 3  4 
e.g. sin 2θ = 2   
 5  5 
24
sin 2θ =
25
sin θ =
N1
(A1)(A1)
A1
A1
correct substitution
N3
A1
2
2
 3  4 3
e.g. cos 2θ = 1 – 2   ,     
5  5 5
7
cos 2θ =
25
2
A1
N1
[7]
IB Questionbank Maths SL
7
13.
The graph of y = p cos qx + r, for –5 ≤ x ≤ 14, is shown below.
There is a minimum point at (0, –3) and a maximum point at (4, 7).
(a)
Find the value of
(i)
p;
(ii)
q;
(iii)
r.
(6)
(b)
The equation y = k has exactly two solutions. Write down the value of k.
(1)
(Total 7 marks)
14.
(a)
(i)
(ii)
evidence of finding the amplitude
73
e.g.
, amplitude = 5
2
p = –5
(M1)
period = 8
(A1)
 2π π 
 
q = 0.785  
4
 8
(iii)
(b)
73
2
r=2
r=
k = –3 (accept y = –3)
A1
A1
N2
N2
(A1)
A1
N2
A1
N1
[7]
IB Questionbank Maths SL
8
15.
The following table gives the examination grades for 120 students.
(a)
Grade
Number of students
Cumulative frequency
1
9
9
2
25
34
3
35
p
4
q
109
5
11
120
Find the value of
(i)
p;
(ii)
q.
(4)
(b)
Find the mean grade.
(2)
(c)
Write down the standard deviation.
(1)
(Total 7 marks)
16.
(a)
(i)
(ii)
(b)
(c)
evidence of appropriate approach
e.g. 9 + 25 + 35, 34 + 35
p = 69
(M1)
evidence of valid approach
e.g. 109 – their value of p, 120 – (9 + 25 + 35 + 11)
q = 40
(M1)
evidence of appropriate approach
fx
e.g. substituting into
, division by 120
n
mean = 3.16

1.09
A1
A1
N2
N2
(M1)
A1
N2
A1
N1
[7]
IB Questionbank Maths SL
9
17.
Jan plays a game where she tosses two fair six-sided dice. She wins a prize if the sum of her
scores is 5.
(a)
Jan tosses the two dice once. Find the probability that she wins a prize.
(3)
(b)
Jan tosses the two dice 8 times. Find the probability that she wins 3 prizes.
(2)
(Total 5 marks)
18.
(a)
(b)
36 outcomes (seen anywhere, even in denominator)
(A1)
valid approach of listing ways to get sum of 5, showing at least two pairs
e.g. (1, 4)(2, 3), (1, 4)(4, 1), (1, 4)(4, 1), (2, 3)(3, 2) , lattice diagram
4  1
P(prize) =
 
36  9 
(M1)
recognizing binomial probability
3
5
 8  1   8 
 1


8
,
e.g. B 
 , binomial pdf,     
 9
 3  9   9 
P(3 prizes) = 0.0426
(M1)
A1
A1
N3
N2
[5]
19.
A standard die is rolled 36 times. The results are shown in the following table.
(a)
Score
1
2
3
4
5
6
Frequency
3
5
4
6
10
8
Write down the standard deviation.
(2)
(b)
Write down the median score.
(1)
(c)
Find the interquartile range.
(3)
(Total 6 marks)
IB Questionbank Maths SL
10
20.
(a)
σ = 1.61
A2
N2
(b)
median = 4.5
A1
N1
(c)
Q1 = 3, Q3 = 5 (may be seen in a box plot)
IQR = 2 (accept any notation that suggests the interval 3 to 5)
(A1)(A1)
A1
N3
[6]
21.
A farmer owns a triangular field ABC. One side of the triangle, [AC], is 104 m, a second side,
[AB], is 65 m and the angle between these two sides is 60°.
(a)
Use the cosine rule to calculate the length of the third side of the field.
(3)
(b)
Given that sin 60°=
3
, find the area of the field in the form 3 p 3 where p is an
2
integer.
(3)
IB Questionbank Maths SL
11
Let D be a point on [BC] such that [AD] bisects the 60° angle. The farmer divides the field into
two parts A1 and A2 by constructing a straight fence [AD] of length x metres, as shown on the
diagram below.
(c)
65x
.
4
(i)
Show that the area of A1 is given by
(ii)
Find a similar expression for the area of A2.
(iii)
Hence, find the value of x in the form q 3 , where q is an integer.
(7)
(d)
(i)
Explain why sinA D̂C  sinA D̂B .
(ii)
Use the result of part (i) and the sine rule to show that
BD 5
 .
DC 8
(5)
(Total 18 marks)
22.
(a)
(b)
using the cosine rule a2 = b2 + c2 – 2bc cos Â
substituting correctly BC2 = 652 + 1042 – 2(65)(104)cos60°
= 4225 + 10 816 – 6760 = 8281
 BC = 91m
finding the area, using
IB Questionbank Maths SL
1
bc sin Aˆ
2
(M1)
A1
A1
N2
(M1)
12
substituting correctly, area =
1
(65)(104)sin60°
2
= 1690 3 (accept p = 1690)
(c)
(i)
(ii)
(iii)
A1
1
A1 =   (65)(x)sin30°
 2
65x
=
4
AG
A1
65x
+ 26x = 1690 3
4
169 x
 1690 3
4
4  1690 3
x=
169
 x = 40 3 (accept q = 40)
(i)
(ii)
Recognizing that supplementary angles have equal sines
e.g. AD̂C = 180° – AD̂B  sinA D̂C  sinA D̂B
using sin rule in ∆ADB and ∆ACD
BD
65
BD
sin30 



substituting correctly
sin30  sinA D̂B
65 sinA D̂B
DC
104
DC
sin30 



and
sin30  sinA D̂C
104 sinA D̂C
since sinA D̂B  sinA D̂C
BD DC
BD 65



65 104
DC 104
BD 5


DC 8
N0
M1
simplifying
(d)
N2
A1
1
A2 =   (104)(x)sin30°
 2
= 26x
stating A1 + A2 = A or substituting
A1
N1
(M1)
A1
A1
A1
N2
R1
(M1)
A1
M1
A1
AG
N0
[18]
IB Questionbank Maths SL
13
23.
The following diagram shows a triangle ABC, where AĈB is 90, AB = 3, AC = 2 and BÂC
is .
(a)
Show that sin  =
5
.
3
(b)
Show that sin 2 =
4 5
.
9
(c)
Find the exact value of cos 2.
(Total 6 marks)
24.
Note:
(a)
Throughout this question, do not accept methods which involve
finding  .
Evidence of correct approach
eg sin  =
sin  =
(b)
BC
, BC  32  2 2  5
AB
5
3
AG
Evidence of using sin 2 = 2 sin  cos 
 5  2 
 
= 2

 3  3 
=
IB Questionbank Maths SL
A1
4 5
9
N0
(M1)
A1
AG
N0
14
(c)
Evidence of using an appropriate formula for cos 2
eg
M1
4 5
4
5  80 
 , 2  1, 1  2  , 1  
9 9
9
9  81 
cos 2 = 
1
9
A2
N2
[6]
IB Questionbank Maths SL
15