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Mathematics SL
Practice Short Problems For New Syllabus – May 2006
Version 3
This paper is a collection of sample questions originally designed by participants at the IBNA Group 5
Workshop in Cinncinati, April 2004. They are meant as practice questions only, and have not been
authorized nor proof-read by IBCA. Users of this material are advised to carefully check that the questions
fit into the new Mathematics SL syllabus. Answers only are provided, whereas students will be expected to
show working on an actual examination.
Graphic display calculators may be used, but set-up working is required where GDCs have been used to get
an answer. Answers should be given exactly, or rounded to 3 s.f., unless otherwise specified in the
question.
A = easy
1.
B = basic
E = hard
(a)
Find the common ratio.
[4]
(b)
Find the fifth term of the sequence.
[2]
1.1
D
(a)
-2
(b)
48
Find the coefficient of x 3 y 8 in the expansion of (3x  2 y 2 ) 7 .
Section:
Difficulty:
Answer:
3.
D = quite difficult
Given the first term of a geometric sequence is 3 and the sum of the
first 10 terms is -1023:
Section:
Difficulty:
Answers:
2.
C = a bit difficult
[6]
1.3
C/D
15 120
Consider the function f : x  e x .
The graph of the function g is a translation of the graph of f three
units to the right.
(a)
Write down an expression for g .
[2]
The graph of the function h is a transformation of the graph of f by
a reflection in the y-axis and a vertical stretch with scale factor 4.
(b)
Write down an expression for h.
[4]
4.
5.
6.
Section:
Difficulty:
2.3 , 2.8
B/C
Answers:
(a)
Let
e x 3
(b)
f : x  x  5 and g : x 
4e  x
1 3
x .
2
(a)
Find g 1 (108) .
(b)
(i)
Write down an expression for ( f  g )( x) .
(ii)
Hence, solve the equation ( f  g )( x) = 27
Section:
Difficulty:
2.1 , (2.2?)
C
Answers:
(a)
f :x
Given
[2]
6
x5
1 3
x 5
2
(b)
,
1
Find an expression for f
(b)
Sketch the graph of
(c)
Write down the domain of f
2.1 , 2.2 , 2.7
C
Answers:
(a)
(c)
x2  5
x ≥ 0.
4
x≥5
(a)
Section:
Difficulty:
(c)
[4]
f
1
.
[2]
.
(b)
[2]
1
.
[2]
right half of parabola, vertex (0,0)
The graph of y  log a ( x  b)  c has the following properties: (include sketch)
- it has a vertical asymptote with equation x = 2;
- it passes through the points with coordinates (3,5) and (5,6).
(a)
Write down the value of b.
[2]
(b)
Find the value of a.
[2]
(c)
Find the value of c.
[2]
Section:
Difficulty:
Answers:
7.
2.3 , 2.7
D
(a)
-2
3
(c)
5
(include sketch of 2 triangles ABC, both with B = 40º, one with C an acute
angle, one with C an obtuse angle)
Given ABˆ C  40 ,
AB = 10
,
AC = 7
(a)
Determine the two possible values for ACˆB .
[3]
(b)
Determine the smaller of the two possible values for CB.
[3]
Section:
Difficulty:
Answers:
8.
(b)
3.6
C/D
(a)
66.7º , 113º
(b)
4.89
The graph of f ( x)  a cos b( x  c) is given below:
(include sketch of curve with axes marked on interval –π/2 < x < 3π/4 and
-3 < y < 3. Marked on the graph should be minima at (-π/4, -2.5) and
(5π/12, -2.5) )
Use the graph to find the values of:
(a)
a
Section:
Difficulty
Answers:
9.
(a)
(b)
3.4
D
(a)
-2.5
b
(c)
(b)
c
3
(c)
-π/4
Find the values of a and b in the matrix equation:
 2  1  a 4   5 1 
  
  

3
 0 4   b  2   3 10 
(b)
[2] [2] [2]
[2]
Find the values of c and d in the matrix equation:
 c  2  4 2   26 8 


  

7
3

3
d
19
17


 

[4]
Section:
Difficulty:
Answers:
10.
4.2
B/C
(a)
a = -1 , b = 3
(b)
Given the system of three linear equations in three unknowns, x, y and z :
(a)
3x
+
4y
+
z
=
5
2x
-
4y
+
2z
=
14
x
+
5y
-
z
=
-6
Write this system as a matrix equation in the form:
 x
 
where X   y 
z
 
AX  B
[1]
A 1 .
(b)
Write down
(c)
Hence, solve the matrix equation to find the values of x, y and z.
Section:
Difficulty:
[2]
(c)
Given
[3]
4.3 , 4.4
B/C
1  x   5 
3 4

   
Answers: (a)  2  4 2  y    14 
 1 5  1 z    6 

   
11.
c=5,d=1
9
12 
 6

1
4 4 
(b)
 4
12 

  14  11  20 
x = 2 , y = -1 , z = 3




a  3i  2 j  4k ,
(a)

Find the magnitude of a .
[3]
(b)

Find a unit vector in the same direction as a .
[3]
Section:
Difficulty:
5.1
B
Answers:
(a)
29
(b)
1
29
3i  2 j  4k 

12.
 1 1
The points O, A and B in 3-dimensions are defined by O (0,0,0) , A  8, , 
 3 2
1

and B  ,9,2  .
2

(a)
(b)
(c)
Write down OA and OB in component form.

Find the vector r  2OA  3OB .
Determine if the vectors OA and OB are perpendicular,
parallel, or neither.
Section:
Difficulty:
5.1 , 5.2
C
Answers:
(a)
(b)
(c)
13.
[2]
[2]
[2]

 1 1

1
OA  8i  j  k , OB  i  9 j  2k
3
2
2



 35
79
r
i
j  5k
2
3
1
1 1
8   (9)  (2)  0 
2
2 3
The 50 graduating students at Middleton High School can take up to three
mathematics classes in their final year chosen from Calculus, Statistics and
Algebra. The following breakdown occurs:
7 take all three
12 take Algebra and Calculus
15 take Statistics and Calculus
9 take Statistics and Algebra
21 take Algebra
30 take Calculus
19 take Statistics
(a)
Represent this information on a Venn diagram
(b)
Determine the number of students taking
(i)
(ii)
(iii)
Section:
Difficulty:
Answers:
Calculus only
Statistics but not Calculus
no Mathematics class.
6.8
C
(a)
Venn diagram with 3 circles with numbers marked
n(A∩C∩S’) = 5 , n(A∩S∩C’) = 2 , n(S∩C∩A’) = 8
[3]
[3]
(b)
14.
A multiple choice test is given containing 10 questions. There are 4 choices for
each question and exactly 1 correct response.
(a)
(b)
Determine the probability of obtaining at least 5 correct answers
through random guessing for each question.
(i)
there were 10 questions with 5 choices per question;
(ii)
there were 12 questions with 4 choices per question?
6.10
C
(a)
0.0781
(b) (i) reduced
[2]
(ii) increased
Charlie has a batting average of .378. (ratio of hits to times at bat)
(a)
(b)
(b)
Write down the expected number of hits he will make in
7 times at bat.
[1]
Find the probability that Charlie will hit the next 2 times and
not hit the 3rd time at bat.
[2]
Determine the probability that Charlie will get exactly 3 hits
in his next 7 times at bat.
[3]
Section:
Difficulty:
Answers:
16.
[4]
How will the probability of getting at least five of the questions
correct be changed if:
Section:
Difficulty:
Answers:
15.
n(A∩S’∩C’) = 7 , n(S∩C’∩A’) = 2 , n(C∩A’∩S’) = 10
(i)
10
(ii)
11
(iii) 9
6.9, 6.10
C
(a)
2.65
(b)
0.0889
(c)
0.283
The following relationships apply to dependent events A and B.
p(A) = 0.6 , p(B) = 0.5 , p(B / A) = 0.25.
(a)
Find
p(A / B) ;
[2]
(b)
Find
p(B / A’ ) .
[4]
Section:
Difficulty:
Answers:
6.7
C
(a)
0.3
(b)
0.875
17.
f ( x) 
Given
Section:
Difficulty:
f " ( x) .
7.1 , 7.2
C
4x  8
ex
Answer:
18.
4x
, determine
ex
Consider the region bounded by the graphs of:
y  x3 , x = 0 , y = 0 , x = 2
(a)
On the accompanying sketch of the graphs, shade the region.
[1]
(b)
Find the area of the shaded region.
[2]
(c)
Find the volume of the solid formed when the given region is
revolved around the x-axis.
[3]
Sections:
Difficulty:
Answers:
19.
Determine the exact value of the area between the curves with equations:
y
Section:
Difficulty:
Answer:
20.
7.5 , 7.6
C
(a)
area under curve between x=0 and x=2.
(b)
4
(c)
128π / 7 = 57.4 (3 sf)
2
x
and
x + 2y = 5
7.5
D
15 / 4 – 2 ln4 = 3.75 – 2 ln4
A graph of the derivative of f (x) is given on the interval (-5, 15) showing
zeroes at -3, 4 and 12, maxima at 1 and 12 and a minimum at 8, the largest
absolute value of the function occurring at 1.
(a)
Write down the x-value of any local maximum or minimum of f .
[2]
(b)
Write down the x-value of any points of inflection on the graph of f.
[2]
(c)
Write down the intervals where the graph of f will be concave up.
[2]
Section:
Answers:
7.3
(a) -3 and 4
Difficulty:
D
(b) 1, 8, 12
(c)
(-5, 1) and (8, 12)