CMA Math Camp Question Booklet
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CMA
Math Camp
Question Booklet
8:30 am - 4:30 pm
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CMA Math Camp Question Booklet
FORMAT OF THE EXAMINATIONS
The examination will consist of two papers: Paper 01, an objective type paper based on the Core
Objectives and Paper 02, an essay or problem solving type paper based on both the Core and
Optional Objectives.
Paper 01
(1 hour 30 minutes)
The Paper will consist of 60 multiple-choice items, sampling the
Core as follows:
Sections
No. of items
Computation
6
Number Theory
4
Consumer Arithmetic
8
Sets
4
Measurement
8
Statistics
6
Algebra
9
Relations, Functions and Graphs
6
Geometry and Trigonometry
9
Total
60
Each item will be allocated one mark.
Paper 02
The Paper will consist of two sections.
(2 hours and 40 minutes)
Section I: 90 marks
The section will consist of 8 compulsory structured and problem-solving
type questions based on the Core.
The marks allocated to the topics are:
Sections
No. of marks
Sets
5
Consumer Arithmetic and Computation
10
Measurement
10
Statistics
10
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CMA Math Camp Question Booklet
Algebra
15
Relations, Functions and Graphs
10
Geometry and Trigonometry
20
*Combination question/ investigation
10
Total
90
* Combination question/investigation may be set on any
combination of objectives in the Core including Number Theory.
Section II:
30 marks
This section will consist of 3 structured or problem-solving questions
based mainly on the Optional Objectives of the syllabus. There will be
1 question from each of the Sections Algebra and Relations, Functions
and Graphs; Measurement and Geometry and Trigonometry; and Vectors
and Matrices.
Candidates will be required to answer any two questions.
question will be allocated 15 marks.
Each
The optional questions will be set as follows:
Section II Chose only 2
Question 9:
ALGEBRA AND RELATIONS, FUNCTIONS AND GRAPHS
Question 10:
MEASUREMENT AND GEOMETRY AND TRIGONOMETRY
Question11:
VECTORS AND MATRICES
Math Studying & Test Taking Tips
1. Always read math problems completely before beginning any
calculations. If you "glance" too quickly at a problem, you may
misunderstand what really needs to be done to complete the
problem.
2. Whenever possible, draw a diagram. Even though you may be able
to visualize the situation mentally, a hand drawn diagram will allow
you to label the picture, to add auxiliary lines, and to view the
situation from different perspectives.
3. Know your calculator! If you must borrow a calculator from your teacher, be sure that
you have used that "brand" of calculator on previous occasions. If you are not familiar
with how a particular calculator works, your calculations may be incorrect.
4. If you know that your answer to a question is incorrect, and you cannot find your
mistake, start over on a clean piece of paper. Oftentimes when you try to correct a
problem, you continually overlook the mistake. Starting over on a clean piece of paper
will let you focus on the question, not on trying to find the error.
5. Do not feel that you must use every number in a problem when doing your calculations.
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CMA Math Camp Question Booklet
Some mathematics problems have "extra" information. These questions are testing your
ability to recognize the needed information, as well as your mathematical skills.
6. Be sure that you are working in the same units of measure when performing
calculations. If a problem involves inches, feet AND yards, be sure to make the
appropriate conversions so that all of your values are in the same unit of measure (for
example, change all values to feet).
7. Be sure that your answer "makes sense" (or is logical). For example, if a question asks
you to find the number of feet in a drawing and your answer comes out to be a negative
number, know that this answer is incorrect. (Distance is a positive concept - we cannot
measure negative feet.)
8. Remember, that it may be necessary to "solve" for additional information in a problem
before being able to arrive at the final answer. These questions are called "two-step"
problems and are testing your ability to recognize what information is needed to arrive
at an answer.
9. If time permits, go back and resolve the more difficult problems on the test on a separate
piece of paper. If these "new" answers are the same as your previous answers, chances
are good that your solution is correct.
10. Remain confident! Do not get flustered! Focus on what you DO know, not on what you
do not know. You know a LOT of math!!
11. When asked to "show work" or "justify your answer", don't be lazy. Write down
EVERYTHING about the problem, including the work you did on your calculator.
Include diagrams, calculations, equations, and explanations written in complete
sentences. Now is the time to "show off" what you really can do with this problem.
12. If you are "stuck" on a particular problem, go on with the rest of the test. Oftentimes,
while solving a new problem, you will get an idea as to how to attack that difficult
problem.
13. If you simply cannot determine the answer to a question, make a guess. Think about the
problem and the information you know to be true. Make a guess that will be logical
based upon the conditions of the problem.
14. In certain problems, you may be able to "guess" at an approximate (or reasonable)
answer. After you perform your calculations, see if your final answer is close to your
guess.
CXC NEW MATH SYLLABUS
1. CAPE INTEGRATED MATHEMATICS SYLLABUS Effective for examinations from
May-June 2016. Required for all 6th form student for September 2015.
2. CSEC MATHEMATICS SYLLABUS Effective for examinations from May-June 2018
with an S.B.A. / Internal Assessment component. Required for all 4th for student for
September 2016.
3. New C.X.C. E-marking format
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CMA Math Camp Question Booklet
Consumer Arithmetic
A ratio is a relationship between two numbers of the
same kind (e.g., objects, persons, students, spoonfuls,
units of whatever identical dimension), expressed as "a to
b" or a:b, sometimes expressed arithmetically as a
dimensionless quotient of the two that explicitly indicates
how many times ...
1. Concrete tiles are made using buckets of cement, sand and gravel mix in the ration
1: 4: 6.
( 2 marks)
(a) How many buckets of gravel are needed for 4 buckets of cement?
(b) If 20 buckets of sand are used, how many buckets of EACH of the following will
be needed?
i. Cement
ii. Gravel
( 3 marks)
2. (a)
A sum of money is shared between Andre and Raymond in the ratio 2: 5. Andre
received $600. How much money was shared altogether?
( 2 marks)
(b)
Express the following in a ratio in its simplest form $1035.00, $345.00 and
$2760.00
( 2 marks)
3.
( 3 marks)
In Hire purchase a purchaser agrees to pay for goods in
parts () installments or a percentage over a number of
months. Normally a deposit is made.
4. The cash price for a laptop is $1299. It can be bought on hire purchase by making a
deposit of $350 and 10 monthly payments of $120 each.
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(a) What is the Total hire purchase price for the laptop?
( 2 marks)
(b) How much is saved by buying the laptop cash?
( 1 mark)
CMA Math Camp Question Booklet
Overtime is the amount of time someone works
beyond normal working hours. Example time and a half on
weekends and double time on Sundays
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CMA Math Camp Question Booklet
10. A credit union pays 8 % per annum compound interest on
all fixed deposits. A customer deposited $24 000 in an account.
Calculate the TOTAL amount of money in the account at the
end of two years.
( 4 marks)
11.
The simple interest on $36 000 for 5 years is $13 500. What is the rate for this deposit?
( 3 marks)
Use the concept of stronger currency
12.
The table shows some rates of exchange.
US $1.00 = EC $2.70
TT $1.00 = EC $0.40
Calculate the value of
(a)
(b)
(c)
EC $1 in TT $
US $80 in EC $
TT $ 648 in US $
( 1 mark)
( 1 mark)
( 3 marks)
Measurement
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2.
(b)
Usain ran 100 meters in 9.72 seconds. Calculate his average speed in
(i)
m/s
(ii)
km / h
( 3 marks)
To convert from the map to the ground we multiply by the scale and then change to the required
units.
To convert from the ground to the map we divide by the scale and then change to the required
units.
Note: 100 000 cm = 1 km
The map 1 : 1250 mean 1 cm on the map is equal to 1250 cm on the ground.
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CMA Math Camp Question Booklet
(b)
(i) L and M are two tracking stations. State in cm the distance LM on the map.
( 1 mark)
(ii) Calculate the ACTUAL distance in km from L to M on the island.
( 2 marks)
(c)
(i) The area shaded on the map is a forest reserve. By counting the squares
estimate in cm2 , the area of the forest reserve as shown on the map. ( 2 marks)
(ii) Calculate in km2, the ACTUAL area of the forest reserve.
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( 2 marks)
CMA Math Camp Question Booklet
Algebra
1. Solve the simultaneous equations
2x  3y  9
( 4 marks)
3x  y  8
2. Write as a single fraction in its lowest terms
x  2 x 1

3
4
3. (a)
( 3 marks)
Factorise completely
2 x3  8 x
2ax  2ay  bx  by
( 6 marks)
3x  5 x  2
2
(i)
9
Make C the subject of the formula F  C  32
5
(ii)
Given that F = 113, calculate the value of C
(b)
(c)
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( 3 marks)
The binary operation * is defined by a * b   a  b   2ab . Calculate the value of
2
(i)
3*4
( 2 marks)
(ii)
2*  3* 4 
( 2 marks)
CMA Math Camp Question Booklet
4.
5.
6.
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CMA Math Camp Question Booklet
7.
8.
5  2x  9
(i)
Solve the inequality:
( 2 marks)
(ii)
If x is a whole number , determine the smallest value of x which satisfies
the inequality above.
( 1 mark)
Solve the pair of simultaneous equations y  x 2  x  3 and y  6  3x ( 6 marks)
For f ( x)  ax 2  bx  c
h
b
2a
k
4ac  b 2
4a
f ( x )  a ( x  h) 2  k
b  b2  4ac
, every quadratic has 2 roots and
2a
their nature is based on the sign or value of b 2  4ac
The roots of f ( x)  ax 2  bx  c  0 are x 
9.
Express the function f ( x)  4 x 2  8 x  2 in the from a( x  h) 2  k where a, h and k are
( 3 marks)
Express the quadratic function 1  6x  x in the from k  a( x  h) 2 where a, h
constants.
10.
(a)
2
( 3 marks)
and k are constants.
(b)
Determine the roots of 1  6 x  x  0 correct to 2 decimal places
( 4 marks)
2
Statistics and Probability
Mean for grouped data
Find the Midpoint of each class interval
and multiply them by each corresponding
frequency
x
 f  x
f
Cumulative Frequency Curve
Plot the upper class limit or boundaries
on the horizontal axis against each
corresponding cumulative frequency on
the vertical axis. Draw a smooth curve
though the points.
Frequency Polygon
Plot the class Midpoint of each class
interval against each corresponding
frequency. Use a ruler to connect
successive points and close the Polygon at
both ends.
Histogram
Plot the class boundaries of each class
interval on the horizontal axis against each
corresponding frequency on the vertical
axis
Pie Chart
Determine the proportionate sector angle
for each data set and use a protractor to
measure these angles
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CMA Math Camp Question Booklet
1.
2.
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3.
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4.
5.
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CMA Math Camp Question Booklet
Matrices Part 1
(see end of booklet for notes on matrix
transformation first)
1.
( 4 marks)
(b)
(c)
2.
7 2 
The matrix M is defined as M  
 Determine the value of p for which the
 p 1
matrix M does NOT have an inverse .
( 2 marks)
 x 4
The matrix L is defined as L  
 Determine the value of x for which the
1 x
matrix L is singular.
( 3 marks)
3 1
The matrix N is defined as N  

 2 6
(a)
Find the inverse of N.
(b)
1 0
Show that NN 1  I  

0 1
(c)
 x   12 
Hence, calculate the value of x and of y for which M      ( 6 marks)
 y   8 
3.
 a 4  2 4   2 0 
Calculate the values of a and b such that 



 1 b  1 3   0 2 
4.
3 2
 1 3 
Given that P  
 and Q  
,
1 4
 0 2
(a)
5.
P  2Q
(b)
( 3 marks)
Evaluate
PQ
( 4 marks)
Using a matrix method, find the values of x and y in the equation s
4 x  2 y  8 and x  y  3
( 4 marks)
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CMA Math Camp Question Booklet
Matrices Part 2 (Transformation Geometry)
Matrix x Object = Image
Matrix Inverse x Image = Object
1.
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CMA Math Camp Question Booklet
3.
(b)
(c)
 0 1 
The Matrix J  
 represents a single transformation. The image of the point P
1 0 
under transformation J is (5, 4).
(i)
Describe J completely.
( 2 marks)
(ii)
Determine the coordinates of P.
( 4 marks)
(i)
Write down a matrix H, or size 2x2 which represents an enlargement of scale
factor -3 about the origin.
( 1 mark)
(ii)
Determine the coordinates of the point (5, -7) under the combined transformation,
H followed by J.
( 3 marks)
Vectors
1.The diagram below shows position vectors OP and OQ .
(a)
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 x
Write as a column vector in the form  
 y
OP
(i)
( 1 mark)
(ii)
OQ
( 1 mark)
(iii)
PQ
( 1 mark)
CMA Math Camp Question Booklet
the magnitude of PQ
(b)
PQ
(c)
Given that k OP  10 , determine the value of k.
2.
3.
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( 2 marks)
( 2 marks)
CMA Math Camp Question Booklet
4.
5.
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6.
Function, Relations and Graph
(see end of booklet for notes on quadratic graphs first)
1. (a)
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3.
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4.
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CMA Math Camp Question Booklet
5.
Completing the square example
(a) Express 2 x 2  3 x  1 in the form a  x  h   k , where a, h and k are real numbers.
2
(3 marks)
(b) Using your answer above or otherwise calculate
i. the minimum value of 2 x 2  3 x  1
ii. the value of x for which the minimum occurs
(2 marks)
(c) Sketch the graph of y  2 x 2  3x  1 , clearly showing
i. the coordinates of the minimum point
ii. the value of the y-intercept
iii. the values of x where the graph cuts the axis
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(4 marks)
CMA Math Camp Question Booklet
6.
Trigonometry
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1.
2.
3.
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4.
5.
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6.
7.
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8.
Speed (velocity), Distance and Time Graph
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CMA Math Camp Question Booklet
4.
(ii) What is the gradient of the graph during the Second stage? What is the car doing
during this stage?
(iii)
Calculate the distance traveled by the car on the Third stage of the journey.
Summary of Coordinate Geometry Formulas
If A ( x1, y1) and B( x2, y2,), then
distance d, from A to B =
Midpoint, M, of AB =
slope, m, of
Following is a list of the equations of lines:
Ax + By = C
A, B, and C are real numbers
A and B are not both zero
y − y1 = m ( x − x1)
Point-slope form:
( x1, y1) is a point on the line and m is the slope of the line
Slope-intercept form: y = mx + c
m is the slope of the line and b is the y-intercept value
Standard form:
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Coordinate Geometry
1.
2.
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3.
4.
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5.
Transformation Geometry see end of booklet for notes
1.
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2.
3.
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4.
5.
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6.
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7.
8.
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Construction
1.
2.
3.
4.
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5.
6.
7.
Sequence, Series and Patterns
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CMA Math Camp Question Booklet
Additional Things you need to know
Relations, Functions and Graphs
1. For ax 2  bx  c  0
a.
C is the y-intercept or the value at which the graph cuts the y-axis
b.
If a is positive the graph of the function takes the shape of a “U”
and has a minimum value
c.
If a is negative the graph of the function takes the shape of a “
a maximum value
2. For a( x  h) 2  k
a. k is the min or max value
b. x+h = 0
therefore x=-h
is the axis of symmetry or the value of x for which the max or min occurs.
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h
b
b
therefore x 
2a
2a
k
4ac  b 2
4a
” and has
CMA Math Camp Question Booklet
3. a( x  h) 2  k the minimum point or the maximum POINT is (x, y) which is (–h, k)
f ( x)  y
4.
5. the x-intercept occurs when y=0
6. the y-intercept occurs when x=0
7. The x solutions of the quadratic equation is called the ROOTS of the equation.
8. The set of x values which defines the function is called the DOMAIN a  x  b
while the set of y values is called the RANGE of the function. a  y  b
TRANSLATION Transformation
A translation is a geometry transformation is which an object slides along a straight line and
changes its position without turning. All points move in the same direction and the same
distance. The image is the same shape and size as its object. All points on the image undergoes a
 x
change by a translation vector in the form T   
 y
 x 
T  
 y
Movement horizontally: left if negative and right if positive
Movement horizontally: down if negative and up if positive
Object + Translation = Image
P T  P'
 x1   x   x1 x  also
     
 y1   y   y1 y 
P ' T  P
P ' P  T
REFLECTION Transformation
A reflection is the transformation which maps an object onto its image by means of folding in a
mirror line. The image of the object is laterally inverted
Properties of Reflection


The object and its image are congruent
The sense is changed (laterally inverted) or the orientation is changed (clockwise to anticlockwise or vice versa)
To carry out a reflection:
1.
2.
3.
4.
5.
The object
the mirror line
each point on the object is reflected in the mirror line
a point on the mirror line when reflected remains the same point (invariant)
each point on the object when matched with its corresponding point on the image
is the same distance from the mirror line.
6. the line joining a corresponding point on the object to the image is perpendicular
to the mirror line.
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CMA Math Camp Question Booklet
Standard reflections
Image Point
1. In the x-axis
the point ( x, y )  ( x,  y )
2. in the y-axis
the point ( x, y )  (  x, y )
3. in the line y=x
( x, y )  ( y , x )
4. in the line y=-x
( x, y )  (  y ,  x )
5. in the origin
( x , y )  (  x,  y )
Reflection matrices
1 0 


 0 1 
 1 0 


 0 1
0 1


1 0
 0 1 


 1 0 
 1 0 


 0 1 
ENLARGEMENT Transformation
An enlargement is a geometric transformation in which a plane figure is mapped onto a similar
shape by means of a scale factor or multiplier and a centre of enlargement. The scale factor tells
you amount of times the image is the object.
k 0
E 0,k   

0 k
Enlargement with centre O, and scale factor k.
Scale Factor > 1
If the scale factor of Enlargement is positive the object and its image are on the same side as the
centre of enlargement as shown below. The image is magnified.
C’
C
O
B
A
B’
A’
From the above diagram
The Scale Factor of Enlargement
A ' B ' A ' C ' B ' C ' image
k



,
AB
AC
BC
object
which implies that A ' B '  k  AB  or A ' C '  k  AC  and also B ' C '  k  BC 
Image length = scale factor x Object Length
Also k 
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OA ' OB ' OC '


Image Distance = scale factor x Object Distance
OA OB OC
CMA Math Camp Question Booklet
OA '  k  OA OB '  k  OB  OC '  k  OC 
NB: Area of Image = k 2 x AREA of Object
Negative Scale Factor
If the scale factor of Enlargement is negative the object and its image are on the opposite sides of
the centre of enlargement as shown below. The image is magnified.
B’
C
A
Object
Image
B
C’
A’
o
The image undergoes a 180 Rotation about the centre of Enlargement
ROTATION Transformation
A rotation is the transformation which maps an object onto its image by means of TURNING or
rotation about a fixed point (the centre of rotation) through a given angle (angle of rotation) in
anti-clockwise direction (+rotation)
Properties of Rotation
 The object and its image are congruent
 The sense does not change
To carry out a rotation:
1. The object
2. the centre of rotation
3. the angle of rotation
Standard Rotations
1. 180 degrees
Image Point
Reflection matrices
the point ( x, y )  ( x,  y )
 1 0 


 0 1 
the point ( x, y )  ( y,  x)
 0 1


 1 0 
the point ( x, y )  (  y, x)
 0 1 


1 0 
Same as reflection in the origin
2. 90 degrees clockwise
Same as 270 ANTICLOCWISE
3. 90 degrees anticlockwise
Same as 270 CLOCWISE
4. GENERAL ROTATION
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 cos 
R 0,   
 sin 
 sin  

cos  
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