maths

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GRADE 11
BILINGUAL PURE MATHEMATICS
Name:_______________
SEMESTER 1
Textbook: Advanced Maths as core for Edexcel
(C1C2) Rosemary Emanuel & John Wood
HW=homework (5 min)
SQ=1 or 2 short questions
(5 min)
ST=short test(2 per
semester 10marks)
Max 20 minutes
Oral
UNIT 1 Algebra
Quadratic Equations & Functions (Pg. 41 to 63)
Chapter 3 (C1)
Introduction – Pre-knowledge needed for grade 11 basics
Solving linear equations
Solve quadratic equations by :
1.
Factoring and applications
2.
Formula
3.
Completing the square
4.
Applications
Sketching Quadratic Graphs
1.
Max/min
2.
Shape
3.
Turning Point(vertex) and axis of symmetry
4.
Intercepts with x and y axes
Nature of roots (working with the discriminant)
Review
Booklet
3A Page 46
3B Pg. 55
3B Pg. 56
3B Pg. 55
3B Pg. 56
Solving Inequalities (Pg. 65 to 70)
Chapter 4 (C1)
Solving inequalities:
1.
Linear
2.
Quadratic
Revise
4B Pg. 67
4C Pg. 70
4D Pg. 71
Solving Simultaneous equations (Pg. 72 – 87)
Chapter 5 (C1)
1.
2 Linear
2.
1 linear & 1 quadratic
Intersecting linear & Quadratic graphs, finding the point and dealing
with. 3 cases of the discriminate related to this
Review
SS = selfstudy
Lessons
1
2
1
2
SS
Total 3
24/9/17
6
5A,B Pg. 75, 78
5C Pg. 83
5D Pg. 87
1
2
2
1/10/17
7
5E Pg. 87
SQ, Oral
SS
Total 5
8/10/17
8
3C Pg. 59
3D Pg. 62
3E Pg. 63
2
1
2
1
1
2
SS
Total 10
Chapter 18 (C2)
Revise exponents
Definition – log exp
Rules:
;
WS
18A Pg. 313
18B Pg. 316
1
WS
1
)
Special cases:
3
( )
Exponential function: Graphs
Relationship between (log & exp).
1
Week
28/8/17
1
UNIT 2
Exponents & Logs (Pg. 312 – 323)
(
Predicted
Date
Date
Complete
Change of base
Solving equations
Review
Assessment
18C Pg. 319
18D Pg. 322
18E Pg. 323
HW, ST
UNIT 3 – Geometry
Coordinate Geometry (Pg. 92 – 110)
Chapter 6 (C1)
Revision: Coordinates, midpoint, gradient(+/-),
(drawing
and writing an equation), length of line segment joining 2 points.
6A Pg. 96
3
6B Pg. 105
2
6B Pg. 105
6B Pg. 106
1
2
Gradient of a line including: m=tanθ
Special gradients (i.e. 0, 1, -1 and ∞)
Parallel & perpendicular lines(
Determine if points lie on a line or curve
Equation of a straight line using 3 different formulae
(
)
1
2
SS
1
Total 9
15/10/17
9
12/11/17
13
26/11/17
15
)
(
)
Forms of equations of straight lines:
1
6B Pg. 106
Sketching straight lines
Applications on straight lines
Applications in co-ordinate geometry, use all previous knowledge
Review Self-study
Circle graph equation
(
)
(
)
Tangents and normal to circle.
Assessment & tests
6B Pg. 107
6C Pg. 109
6D Pg. 111
13A,B Pg. 204
2
1
3
SS
1
HW, Oral
Total 15
UNIT 4 – Trigonometry
Solving triangles, radians and applications
Page 252 – 253 and 282 – 302
Radians definition
radian degree
Angles and quadrants (
Length of an Arc (
)
Area of a sector (
Area of a triangle (
Area of segment (
Solving triangles:
Sin and cosine rules
Assessment
Chapter 16 (C2)
Chapter 17 (C2)
16A Pg. 255 [leave
1
out angles greater than
360°/2π or less than 0° ]
)
)
)
)
17A Pg. 284, 285
17A Pg. 284, 285
1
2
17B Pg. 289, 290
17B Pg. 289, 290
17C, 17D, 17E, 17F
Pg. 291 – 302
SQ, Oral
2
1
5
Total 12
Trig functions for any angle Page 254 - 268
Chapter 16 (C2)
radian degree
Trig ratio’s in all 4 quadrants
(including reduction formulae:
)
Special angles – special triangles (angles of any size)
16A Pg. 255 angles
< 0°, >360°
16B Pg. 266 - 268;
2
1
16B Pg. 266 - 268
2
Graphs of trig functions
Period, asymptotes, symmetry
Transformation of trig graphs
( ) ( ) ( )
(
i.e.:
(
)
16B Pg. 266 – 268
2
16B Pg. 266 – 268
3
HW, SQ, ST
2
Total 10
)
Self-study, Assessment & tests
SEMESTER 1 EXAM
3
LESSON PLAN
GRADE: 11 bilingual
YEAR: 2017 DATE: Aug/September
MATHEMATICS TEXTBOOK: Advanced Maths C1C2 as core for Edexcel
SECTION: Quadratics Equations
PERIOD
Week
1
CLASS WORK



WORKBOOK AND TEXTBOOK
Pre-knowledge: expressions,
multiplication, factorisation
Linear equations
Solve
and
Booklet pages 5-9
Worksheet pages 9-11
Page 10 & Ex 3A no 1i,k,q, no
2a,b,f,h,k,l,n
1-2

Look at basic linear equations and then at
fractions
Quadratic equations
Do a worksheet on basic factorization
Explain common factors, diff of two
squares, trinomials
Worksheet pages 12-16
3



Factorization in equations
Do in class 3B no 1a,d,g,t, no 2b, no 8b,
Also show two ways to solve equations
like
Ex. 3B no 1k, l, o, r, 2 e, g, 7 c, d, 8 c, e
4-5




Consolidate quadratic factorization
Ex. 3B no 8 a, b, d, f

Completing the square for trinomials that
do not factorize
Ex 3B no 3 a, b, c, d, e, i, j


Revise the quadratic formula
Any quadratic equation can be solved
by using the quadratic formula!!
Ex 3B no 6 a, d, j,

Applications of quadratic equations
Ex 3B no 9 and 10
5
6-7
8
9
4
PRE-KNOWLEDGE FROM IGCSE NEEDED FOR THIS SECTION:
Expressions
5
6
Quadratic expressions:
Perfect squares:
(
)
(
)
(
)
(
Difference of perfect squares factorised:
)
(
)(
)
Trinomials:
Three different types:
Examples: (Factorise)
(
)(
) add the factors
(
)(
) add the factors
(
)(
) look for difference
Take factors of the last term
and add
them until the sum gives the coefficient of the center
term.
(
)(
). Because the last
term is +, both signs in the brackets will be the same as
the sign of the center term
Take factors of the last term
and add
them until the sum gives the coefficient of the center
term.
(
)(
). Because the last
term is +, both signs in the brackets will be the same as
the sign of the center term
Take factors of the last term - and look for
the difference between the two factors to be the value
of the coefficient of the center term.
(
). Because the last term is -,
)(
one signs in the brackets will be + and one sign will be (
)(
)
7
In quadratics
where the
is not 1, the factors of the first term and the last term
must give the coefficient of the center term.
An easy method to factorise quadratics of this type looks as follows:
: multiply the coefficient of
(
and the constant with each other,
.
)
: then find two factors of
that will add up to the coefficient of , that is
or
: then breakdown the center term as
and rewrite the quadratic as:
(hint: it is easier to solve if you write the –coefficient first
if there is a negative last term)
: put brackets around the first two terms and brackets around the last two
terms (
)
)
(
: take out common factors from each bracket if there are any
(
)
)
(
: then take out the common brackets and put the other factors in the second
bracket as follows:
(
: multiply the coefficient of
(
)(
)
and the constant with each other,
.
)
: then find two factors of 18 that will add up to the coefficient of , that is
: then breakdown the center term as
and rewrite the quadratic as:
: put brackets around the first two terms and brackets around the last two
terms (
)
) NOTE: if you put your own brackets after a – then
(
the sign in the bracket must change.
: take out common factors from each bracket if there are any
(
)
(
)
: then take out the common brackets and put the other factors in the second
bracket as follows:
(
8
)(
)
When factoring quadratics, first always look if there is not a common factor to take out before you
look at difference of two squares or trinomials
Linear Equations:
Linear equations can be written in the form:
9
3
4
10
11
Quadratic equations:
Quadratic equations can be written in the form:
12
The Quadratic formula
This formula can be used to solve x for any quadratic equation
These are called EXACT solutions, as exact
solutions have no decimal places
13
Completing the square:
The quadratic equation
can be rewritten as
Complete the square in an expression: Rewrite
Method: multiply the coefficient of
(
(
)
(
)
by and square the answer
( )
)
add
In bracket
(
)
______________________________________________________
First take out a common factor of 2, as completing the square can only happen when you have
(
)
Method: multiply the coefficient of
(
)
(
by and square the answer
)
add
In bracket
14
(
)
(
)
Now multiply the common factor with the two terms in the bracket
(
)
_______________________________________________________
Complete the square in an equation: Rewrite
(
)
and then solve for
Method: multiply the coefficient of
(
by and square the answer
( )
)
add
In bracket
(
)
(
)
√
√
or
√
√
√
________________________________________________________
As the equation = 0, first divide by 2, as completing the square can only happen when you have
Method: multiply the coefficient of
15
by and square the answer
(
)
(
)
add
In bracket
(
)
(
)
√
√
or
√
√
√
√
Complete the square of the following expressions:
a
b
c
Solve for x, by completing the square method:
a
√
b
c
16
√
LESSON PLAN
GRADE: 11 bilingual
YEAR: 2017 DATE: September
MATHEMATICS TEXTBOOK: Advanced Maths C1C2 as core for Edexcel
SECTION: Parabola graphs
PERIOD
1
2
CLASS WORK
HOME WORK


Sketch parabola graphs
Show that the stationary point or vertex
can be determined by differentiation as
well as the completing the square
Worksheet pages 19

Sketch more graphs, and look at graphs
that do not cut the x-axis
Ex 3C page 59 no 2 b, c, e
Quadratic graphs:
Parabola graphs from IGCSE were sketched using points
In grade 11 you will sketch parabola graphs using x-intercepts, y-intercept, turning
point
17
x-intercepts:
Axis of symmetry and vertex / turning point:
Turning point = (
)
(
)
18
Determine whether a point is on the graph of a parabola (quadratic equation)
19
LESSON PLAN
GRADE: 11 bilingual
YEAR: 2017 DATE: September
MATHEMATICS TEXTBOOK: Advanced Maths C1C2 as core for Edexcel
SECTION: Nature of roots
PERIOD
1
2

CLASS WORK
HOME WORK

Link quadratic formula and the nature of
the roots of an equation
On page 22 do the 4 questions.

Look at sketching parabola and nature of
roots given
Work out the unknown value if roots are
given
On pages 24 answer the questions.

When you solve a quadratic equation in the form
, the solutions of
are
called the roots or zeroes.

The graphs of a quadratic equation look as follows:
OR

Any quadratic equation in the form
formula:

can be solved by using the quadratic
√
The NATURE OF THE ROOTS are all about the values underneath the √
equation. The answer of
in the
will be called the discriminant or delta and we show it
as .
o
If the answer under the square root is 4, e.g. √ then the answers from the square
root will be ±2. The 4 is a perfect square, so the answers will be real and rational.
o
If the answer under the square root is 5, e.g. √ then the answers from the square
root will be ±2.236067977. The 5 is not a perfect square, so the answers will be real
and irrational.
o
If the answer under the square root is -6, e.g. √
your calculator. The answer will thus be non-real.
20
then the answers will be an error on
The three questions you must answer when talking about the nature of the roots are:
o
Real OR non-real
o
Rational OR irrational
o
Equal OR unequal
SUMMARY:

: Real roots
<0: non-real roots
> 0 : real, unequal roots
=0 : real, equal roots
 = perfect square : real, rational
roots

perfect square : real, irrational
roots
21
√
Roots :
22
Nature of roots
Sketches of parabola graphs (quadratic equations) and
the nature of the roots (x-intercepts):
 - delta is:
Roots: nature of and
how many xintercepts
Two real roots
>0

The answer under the
√ is a positive
value.
x-intercepts
Parabola cuts the xaxis at two separate
places
y
0
x
= 0

One real root, as the
Parabola only touches
two answers (roots) of the x-axis yonce.
the equation will be
The answer under the the same.
√ is equal to zero.
0
<0

Non-real roots
The answer under the
√ is a negative
value.
x
Parabola will not touch
or go through the xaxis
y
0
23
x
Examples of the types of questions that can be asked:
o A.) Determine the nature of the roots of the following equations.
o B.) Determine the value of an unknown (e.g. k, m, t) if the nature of the roots is
known.
o C.) Show OR prove that the nature of the roots are:………….. (e.g. real, rational)
To answer any of the above types of questions the quadratic equation must always be
written in standard form:
A. Determine the nature of the roots of the following equations.
Without solving the equations, determine the nature of the roots of:
(equal, unequal, rational, irrational or non-real)
Example:
Exercise: Determine the nature of the roots without solving the equations
(1)
(2)
B. Determine the value of an unknown (e.g. k, m, t) if the nature of the roots as
known.
Example: For which value(s) of
will the equation of
Exercise: For which value(s) of
have equal roots?
will the equation of
have equal roots?
C. Show OR prove that the nature of the roots are…………..:
Remember:
( )
(
Example: Show that the roots of
)
will be real for all real values of
Hint: use the “completing the square method” on the expression of the discriminant
Exercise: Prove that for all real values of
the roots of (
will be non-real.
24
)
(
)
LESSON PLAN
GRADE: 11 bilingual
YEAR: 2017 DATE: September
MATHEMATICS TEXTBOOK: Advanced Maths C1C2 as core for Edexcel
SECTION: Quadratic inequalities
PERIOD
1
2
CLASS WORK
HOME WORK

Linear inequalities

Quadratic inequalities in the book-explain Do all the questions in the book on
also the special cases and things we do
pages 28-30
not do in inequalities
Stress the concept of positive shaped
parabola graphs

Ex 4B page 67 no no 1 k,l,m and n
3c and d
Linear inequalities:

Solving algebraic inequalities is just like solving equations, add, subtract, multiply or
divide to, or from for example both sides.

The exception is if you multiply or divide both sides by a negative value, then you must
turn the inequality sign around.
Example: Solve for
25
Quadratic Inequalities:
Terminology:

When a value is bigger than 0, then we have a positive interval

When a value is smaller than 0, then we have a negative interval
Steps:

If there is a
in an inequality, place all the values on the one side so that the other
side is

.
Arrange the variables in descending order, and then factorise to find the zeros ( roots
or x-intercepts)

Sketch a x-axis and indicate the roots on the x-axis:

Now choose a value in between the two x-intercepts, and substitute the value into the
inequality, and see whether the answer is positive or negative. Then take values on the
left and right of the x-intercepts to find the signs of the intervals.
Examples:
1.
Solve for
Solution:
(
)(
)
So the -intercepts are:
1
5
Choose a value between 1 and 5, and substitute it into the equation
the value of the sign, for example:
26
to find
, now indicate it on the number line
( )
( )
1
5
Choose a value smaller than 1, and substitute it into the equation
to find
the value of the sign, for example:
, now indicate it on the number line
( )
( )
+
1
5
Choose a value bigger than 5, and substitute it into the equation
to find
the value of the sign, for example:
, now indicate it on the number line
( )
( )
-
+
From the question (
1
)(
)
+
5
we can see that we have to look for the positive
intervals, so the final answer:
2.
Solve for
Solution:
(
)
(
)(
)
So the -intercepts are:
-7
1
Choose a value between -7 and 1, and substitute it into the equation
to find
the value of the sign, for example:
, now indicate it on the number line
( )
( )
-7
1
Choose a value smaller than -7, and substitute it into the equation
the value of the sign, for example:
(
)
(
)
, now indicate it on the number line
+
-7
1
27
to find
Choose a value bigger than 1, and substitute it into the equation
to find the
value of the sign, for example:
( )
, now indicate it on the number line
( )
-
+
-7
From the question (
)(
+
1
we can see that we have to look for the negative
)
intervals, so the final answer:
Exercise:
Solve for
(1)
(2)
(3)
(4)
(
)(
)
SPECIAL CASES:
(1)
Solve for
(
)
As (any base)² is always positive, we can enter any value in the place of , and the
answer after we squared will be greater than 0 (positive).
In this case there is only one x-intercept of the quadratic equation
2
If you substitute a value less than 2 or a value bigger than 2 into the equation, bith
answers will be positive
+
+
2
So, the final answer:
(2)
Solve for
(
)
As (any base)² is always positive, we can enter any value in the place of , and the
answer after we squared will be greater than 0 (positive).
In this case there is only one x-intercept of the quadratic equation
2
28
If you substitute a value less than 2 or a value bigger than 2 into the equation, bith
answers will be positive
+
+
2
There are no places where the interval is negative, and there is one solution where
it is = to zero, so the final answer is:
(3)
Solve for
If there is only
in an inequality, then we use the signs of the values to determine
what the sign of the unknown must be:
+
-
?
Which sign must be divided into a ( + ) to give us a ( – ) answer? ______________
So the denominator must be
Final solution:
______________________
Note: here we can’t have = 0 as the unknown is in the denominator!
(4)
Solve for
If there is only
in an inequality, then we use the signs of the values to determine
what the sign of the unknown must be:
?
+
-
Which sign must be divided by a ( -) to give us a ( + ) answer? ______________
So the numerator must be
Final solution:
(4)
______________________
can be solved in the same way as (
29
)(
)
All that you have to remember is that the denominator can’t = 0 in a fraction!!!
DO NOT DO THE FOLLOWING IN INEQUALITIES!!!!!
(1)
If you are given:
but if
do NOT square root both sides, as:
√
will be
was a negative value, then the sign should change to the opposite sign!!!!
So rather take all to one side so that the other side has a 0, and then factorise.
(2)
If you have
do NOT multiply with
whether the
on both sides, as you do not know
is positive or negative, and if it was negative the sign should have
changed. Rather take all to one side, make other side 0, put all on the same
denominator on the side where the fraction is, and then solve the inequality.
Exercise: Solve for
(1)
(2)
(3)
30
(4)
,
LESSON PLAN
GRADE: 11 bilingual
YEAR: 2017 DATE: October
MATHEMATICS TEXTBOOK: Advanced Maths C1C2 as core for Edexcel
SECTION: Simultaneous equations
PERIOD
CLASS WORK

Revise linear simultaneous equations,
using the elimination method as well as
the substitution method Ex 5B no 1a
Ex 5B no 1 b and c – use only the
substitution method

Do a problem on substitution with
fractions – first with linear equations,
and then with linear and quadratic
equations
Ex 5C no 1 a,f , 2d, 3c
This is one of the most important
skills in maths and will be in both
exams, so practice more questions!!

Three cases of the nature of roots of
the intersections of linear and quadratic
equations
Page 33
1
2-3
4-5
HOME WORK
Two linear equations
Different sign we add
Same sign we subtract
_________________________________________________________
31
One linear and one quadratic equation
32
Nature of roots questions can also be asked when working with the intersections of graphs:
Exercise:
(Exam question) Determine how many intersections the curves will have with each other
Simultaneous equations and then discriminant value
33
Quadratic Applications
34
LESSON PLAN
GRADE: 11 bilingual
YEAR: 2017 DATE: October
MATHEMATICS TEXTBOOK: Advanced Maths C1C2 as core for Edexcel
SECTION: Exponents and Logs
PERIOD
CLASS WORK

1-2

HOME WORK
Revise exponent rules, do applications of
exponents and surds and the relationship
between the two.
Concentrate on negative exponents and
fraction exponents and roots
Worksheets pages 36-39
3

Discuss log laws
Ex 18B no 1e,f,g,h,k,l, no 2c,d,i,j,k,l
4



Exponential graphs
Asymptotes
transformations
Ex 18 D no 10

Ex 18C no 1a,c,d,f,

The use of the calculator and logs. What
does
,
Log x, lg x, and ln x mean on the
calculator.
The laws / rules of logs


Change of base rule
Rules:
and
Ex 18C no 2b,c,d,e

Equations of logs and inequalities
Ex 18D no 1a,d,e,f,i, 2a,d,e,3,a,b,g,h,
4a,b
5&6
7
8&9
35
36
37
Example: exponents are used in algebraic solving of fractions
38
Exponential equations
x as a base
x as a power
Quadratic equations in exponents

  
NOTE:


( )
Let 

(


Then 

Factorise: (
)(
)









0


39

)
Inside
Roots or surds
Exponents can be written as roots:
power
√
Outside
power
Logarithms (logs)
40
Expanding – reversing the laws
41
Exponential equations where the bases can’t be written as the same base, can be solved using logs
Solve for x:
Finding the value of a log-expression or log-equation can also be solved by changing the base
Example:
42
Example:
Solving log equations
43
Exponential graph:
All exponential graphs have a horizontal asymptote. In the equation
the equation of the asymptote is
. An asymptote is a straight line towards which the graph goes but it will never touch or intersect the line
44
LESSON PLAN
GRADE: 11 bilingual
YEAR: 2017 DATE: October/Nov
MATHEMATICS TEXTBOOK: Advanced Maths C1C2 as core for Edexcel
SECTION: Coordinate Geometry
PERIOD
CLASS WORK
HOME WORK
1

Distance formula
Ex 6A no 1a,d,e,h
2

Midpoint of line segment
Ex 6A no 2a,d,e,h,4,7

Gradients of line, parallel and
perpendicular lines
m=0,m=-1,m=1,m=, sketching straight
lines
m=tanθ angle of inclination
when does a point lie on a curve or
straight line?
Ex 6B no 1a,b,c,i,2 all, 3a,b,
Ex 6B no 9b,c,8b,c,11 all, 4 (any 3)


Equations of lines – 3 different equations
Do the examples as well as when lines are
parallel and perpendicular to other lines
Ex 6B no 14a,d,15b,d,16a,c

Applications of straight lines and coordinate geometry
Circle, tangents and normals
13A no 1a,b,c,2a,b,3a,b,4a,b
13B no 2a,3a

3-4
5-6
7-8



45
Distance formula
46
Midpoint of a line segment
47
1
2
48
Gradient of a line
49
50
51
Equations of straight lines
52
53
2
3
4
Sketching straight lines
54
Circles:
We use the distance formula to work out the length of the radius from the centre of the circle
to the circumference of the circle.
Circle centre at (0,0)
Circle centre at (a,b) - somewhere in the four
quadrants
(
55
)
(
)
Finding equation of a circle
Finding centre coordinates and radius

A tangent is a straight line that is always perpendicular to the radius of the circle.

A normal is perpendicular to a tangent and the normal
is the equation of the radius of the circle.
Tangent
56
LESSON PLAN
GRADE: 11 bilingual
YEAR: 2017 DATE: November
MATHEMATICS TEXTBOOK: Advanced Maths C1C2 as core for Edexcel
SECTION: Trigonometry
PERIOD
1
2
3,4,5
CLASS WORK
HOME WORK

Convert radians to degrees and vice versa
Ex 16A all

Length of an arc in a circle
Ex 17A no 3, 4, 5

Area of segment, sector and triangle
Ex 17A no 2 all,
Ex 17B no 1 a,b,c 4a,b

Ex 16B no 1 all the negative angles, 2
all, 3, 4.
Worksheet page 63-64
Ex 16B no 5-13
Worksheet page 75
6-9

10-11


CAST Diagram – draw and explain all 4
quadrants
Reduction formulae, angles bigger than
360°, negative angles
Special angles
Solving triangles: sine, cosine, area rules
12-13


Trig graphs
Transformation of graphs
57
the sine-rule: ex 17D no 1a,b,c,d 2a
The cosine-rule ex 17E no 1a,b, 2a,b
The area-rule ex 17F no 17G no 2a,b
Consolidation with word sums: Ex 17F
selected examples
Radians:
58
59
Sector:
Segment:
60
C
C
C
61
TRIGONOMETRY:
CAST DIAGRAM
S
A
T
C
62
Th
The reciprocal functions are:
cosec θ, secθ and cotθ
REDUCTION FORMULAE:
Indicate the reduced function with correct signs (+ or -):
3
Example: sin(180°+θ)= - sinθ . NOTE: all numbers in a
indicate the quadrant number.
cos(180°-θ)=
tanθ=
cot(180°+θ)=
cosec(180°-θ)=
sec(180°-θ)=
sec(360°-θ)=
tan(180°-θ)=
cos(180°+θ)=
secθ=
tan(180°+θ)=
sinθ=
cot(360°-θ)=
sec(180°+θ)=
cosec(360°-θ)=
cosecθ=
cos(360°-θ)=
cotθ=
sin(180°+θ)=
cosθ=
cos(180°-θ)=
sin(360°+θ)=
tan(360°-θ)=
cos(180°+θ)=
cos(θ+360°)=
sin(180°-θ)=
sin(360°-θ)=
tan(360°+θ)=
cosec(180°+θ)=
cot(180°-θ)=
cot(360°+θ)=
63
Reduce the following angles to be acute angles:
3
3
Example: cos 240° = cos(180°+60°)= -cos 60°
cos 120°=
tan 225°=
cos 300°=
sin 20°=
cos 60°=
sin 160°=
tan 315°=
sin 200°=
tan 135°=
sin 340°=
tan 45°=
cos 40°=
EVERY FUNCTION HAS TWO QUADRANTS WHERE IT IS POSITIVE AND TWO QUADRANTS
WHERE IT IS NEGATIVE.
Complete: In which quadrant does θ lie if:
a)
sinθ > 0 and cosθ < 0 __________________________
b)
cosθ > 0 and sinθ < 0 __________________________
c)
tanθ > 0 and sinθ < 0 __________________________
d)
tanθ > 0 and cosθ > 0 __________________________
e)
sinθ =
f)
13 cosθ – 12 = 0 and 180˚< θ < 360˚ __________________________
√
and θ ε [180˚ ; 270˚ ] __________________________
Rules:
Angles bigger than 360
θ is an acute angle which is in the first quadrant, so all angles
90˚
bigger than 360° are firstly in the first quadrant.
sin(360+) = sin 
cos(360+) = cos 
θ
tan(360+) = tan 
180˚
0˚
270˚
Examples: a) sin 780 = sin(2360 + 60) = sin 60
b) tan 405 = tan(1360 + 45) = tan 45
c) cos 960 = cos 240 = cos (180+60) = -cos60
64
NOTE: Angles bigger than 360° are sometimes reduces twice: always first the first quadrant
(because the angle is bigger than 360°), and then the angle must be checked again, and if it is bigger
than 90°, it must be reduced again.
Examples:
1
1
b)
tan 600°
a) tan 480°
2
3
= tan 120°
= tan 240°
= - tan 60°
= tan 60°
Negative angles
-270˚
Quadrant 1
Quadrant 2
-360˚
-180˚
0˚
Quadrant 3
Quadrant 4
Negative angles are measured
clockwise starting from the
-90˚
x-axis
In the first negative quadrant (which is the fourth quadrant for positive angles), only the cos θ
function has a positive ratio.
Rules:
4
sin ( -θ ) = - sin θ
4
 cos ( -θ ) = cos θ
Examples:

4
cos(-20˚) = cos 20˚
4
3
3

cos(-210˚) = cos 210˚ = cos(180˚+30˚)= - cos30˚

sin(-210˚) = - sin 210˚ = - sin(180˚+30˚)= - (-sin30˚)= sin30˚
4
4

3
3
3
3
tan(-210˚) = - tan 210˚ = - tan(180˚+30˚)= - (+tan30˚)= -tan30˚
65
4
tan ( -θ ) = - tan θ
The Unit Circle:
66
67
68
In the sine-rule there is an ambiguous case: When two sides and one angle is given, and the side
opposite the given angle is the shortest of the two sides, then there are two possibilities for the
angle opposite the longer side.
69
70
71
72
Trigonometric graphs:
73
Transformation of graphs
74
2
75
PORTFOLIO WORK:
Name:
Oral Grade 1
Domain
Communication
Taxonomy
Date
Description
Mark
Using language of mathematics
Presenting his or her mathematical thinking coherently and
clearly to peers
Analysing and evaluating the mathematical thinking strategies
of others
Giving accurate answers to questions of knowledge
Giving accurate answers to questions of application
Giving accurate answers to questions of reasoning
Total
Students
grade
1
2
OVER ALL
ORAL
1
2
3
3
1
2
1
10
TOTAL
score
Oral Grade 2
Domain
Communication
Taxonomy
Date
Description
Mark
Using language of mathematics
Presenting his or her mathematical thinking coherently and
clearly to peers
Analysing and evaluating the mathematical thinking strategies
of others
Giving accurate answers to questions of knowledge
Giving accurate answers to questions of application
Giving accurate answers to questions of reasoning
Total
Oral Grade 3
Domain
Communication
Taxonomy
Students
grade
1
2
3
1
2
1
10
Date
Description
Mark
Using language of mathematics
Presenting his or her mathematical thinking coherently and
clearly to peers
Analysing and evaluating the mathematical thinking strategies
of others
Giving accurate answers to questions of knowledge
Giving accurate answers to questions of application
Giving accurate answers to questions of reasoning
Total
76
1
2
3
1
2
1
10
Students
grade
MARK
NAME:___________________________________________ Grade 11
Homework 1 (5 min)
Ministry Portfolio Requirement Exponents and Logs
Solve for
(a)
(b)
(c)
(d)
77
3 Marks
NAME:___________________________________________ Grade 11
Homework 2 (5 min)
Ministry Portfolio Requirement Coordinate Geometry
(a) The points A and B have coordinates (3,-1) and (11,-7) respectively.
a. Find the coordinates of the midpoint of AB
Given that AB is the diameter of a circle:
b. Find the length of the diameter of the circle.
c. What is the equation of the line AB?
(b) The points A and B have coordinates (3,-1) and (6,-4) respectively.
d. Find the coordinates of the C if B is the midpoint of AC
Given that AB is the radius of a circle:
e. Find the length of the radius of the circle.
f. What is the equation of the line AC?
78
3 Marks
NAME:___________________________________________ Grade 11
Homework 3 (5 min)
Ministry Portfolio Requirement
Trig functions and angles 3 Marks
(a) A- Given that  is an acute angle measured in degrees, express in terms of cos:
cos(180°+)
B- Find, as an exact value, the value of tan 120°
(b) A- Given that  is an acute angle measured in degrees, express in terms of cos:
sin(360°+)
B- Find, as an exact value, the value of cos 150°
(c) A- Given that  is an acute angle measured in degrees, express in terms of cos:
tan(180°-)
B- Find, as an exact value, the value of sin 120°
(d) A- Given that  is an acute angle measured in degrees, express in terms of cos:
cos(360°-)
B- Find, as an exact value, the value of tan 240°
79
NAME:___________________________________________ Grade 11
Short Questions 1 (5 min)
Ministry Portfolio Requirement Equations
Solve the simultaneous equations:
(a)
and
(b)
and
(c)
and
(d)
and
80
2 Marks
NAME:___________________________________________ Grade 11
Short Questions 2 (5 min)
Ministry Portfolio Requirement
Trigonometry
2 Marks
(a) Find the area of triangle ABC, rounding your answer to 1 decimal place.
A
102°
12 cm
C
9 cm
B
(b) Find the length of BC in triangle ABC, rounding your answer to 1 decimal place.
A
102°
12 cm
C
9 cm
B
(c) Find the size of angle C in the triangle ABC, rounding your answer to 1 decimal place.
A
102°
C
9 cm
22 cm
B
81
NAME:___________________________________________ Grade 11
Short Questions 3 (5 min)
Ministry Portfolio Requirement
Trigonometry
2 Marks
(a)
-1440°
-1080°
-720°
-360°
0°
360°
720°
1080°
1440°
Write down the equation of the trigonometric graph shown above
(b)
--2π
-π
π
2π
Write down the equation
of the trigonometric
graph on the left
(c)
--2π
-π
π
82
2π
Write down the equation
of the trigonometric
graph on the left
NAME:
Grade 11 (Bilingual: Pure Maths)
MATHEMATICS TEST
Equations
Exponents and logs
Date: October
Time: 20 minutes
Total: 10
Ministry Portfolio Requirement
INSTRUCTIONS:
1. Work neatly and legibly.
2. Answer all questions.
3. Show all your workings within the designated spaces.
4. Write your name in the given space at the top of this page.
The following skills are being tested:
TOPIC
STUDENT COMMENT
Linear equations
Quadratic equations
Simultaneous equations
Exponents and logs
OVERAL IMPRESSION
CORRECTION MARK
Comment:________________________________________________________________
________________________________________________________________________
_____________________________________________________________________________________
Teacher signature:______________________ Parent signature:_________________________________
83
Question 1
Multiple choice questions
Shade the circle next to the correct answer:
1. Factorise
(
)(
(
)
(
)(
)
)
(
)(
)
)(
2. If (
)(
)
then
3. What are the points of intersection of the curves of
(-3,-4) and (3,4)
(-3,4) and (3,-4)
4. Express
(-4,-3) and (4,3)
(-4,3) and (4,-3)
as a single logarithm
(
)
(4 marks)
84
Question 2
Extended response
a) Solve
by completing the square. (Answer to 3 significant digits).
(3 marks)
b) Solve the inequality:
(1 mark)
85
c)
The rectangle has a perimeter of 30 cm, and an area of 44 cm²
Write down the set of simultaneous equations which you should use to find
the length and width of the rectangle.
NOTE: YOU DON’T HAVE TO SOLVE THE TWO EQUATIONS!
x
y
(1 mark)
d) Solve for :
. (Show steps and leave answer correct to 2 decimal places)
( 1 mark)
86
NAME:
Grade 11 (Bilingual: Pure Maths)
MATHEMATICS TEST
Coordinate Geometry
Trigonometry
Date: December
Time: 20 minutes
Total: 10
Ministry Portfolio Requirement
INSTRUCTIONS:
5. Work neatly and legibly.
6. Answer all questions.
7. Show all your workings within the designated spaces.
8. Write your name in the given space at the top of this page.
The following skills are being tested:
TOPIC
STUDENT COMMENT
Coordinate Geometry
Radians and degrees
Trig functions
OVERAL IMPRESSION
CORRECTION MARK
Comment:________________________________________________________________
________________________________________________________________________
_____________________________________________________________________________________
Teacher signature:______________________ Parent signature:_________________________________
87
Question 1
Multiple choice questions
Shade the circle next to the correct answer:
5. Find the length of the line joining the points (-1,-1) and (-3,2)
√
√
√
√
6. The gradient of a straight line joining K( -1 , 2 ) and L( 3 , y ) is 2. What is the value of y?
7. Convert
30°
150°
to degrees
60°
45°
8. The arc AB of a circle with centre O, which subtends an angle of 1.87C at the centre O is 10cm. Find the
radius of the circle.
0.19
5.35
10.7
18.7
(4 marks)
88
Question 2
a) Write your answers for each of the questions in the space provided. Be sure to show all your work
and correct units where applicable.
Extended response
Find the point of intersection of the equations:
(2 marks)
89
b) Find the missing angle  in the triangle ABC below (3 significant places)
A

18cm
47°
B
C
11.2cm
(2 marks)
90
c) Find the value of x
120°
(x+2) cm
(x-2) cm
(2x-2) cm
(2 marks)
91
NAME:
Section
Marks
Multiple Choice
______
24
Extended
Response
______
36
Date: January
Total
______
60
Time: 2 hours 30 minutes
Grade
Bilingual Diploma
Mathematics Grade 11
Instructions to Candidates
Answer ALL the questions.
You must write your answer to each question in the space following the question.
When a calculator is used, the answer should be given to an appropriate degree of accuracy.
Information for Candidates
Full marks may be obtained for ALL questions.
The marks for individual questions, and the parts of questions are shown in round brackets: e.g.(2).
There are 16 questions in this question paper. The total for this paper is 60.
Advice to Candidates
You must ensure that your answers to parts of questions are clearly labelled.
You should show sufficient working to make your methods clear to the Examiner.
Answers without working may not gain full credit.
92
Section A
Multiple choice: For each of the questions below shade the box with the correct answer
1. The points A and B have the coordinates ( -1 , -1 ) and ( -2 , 4) respectively.
The line
t passes through the point A and is perpendicular to the line AB.
a) Find the midpoint of the line AB
(
(
)
)
(
)
(
)
(2)
b) Find the equation of line
t in the form
, where
are integers.
(2)
93
2. The equation
Find the value of
, where
is a constant, has equal roots.
2
-2
3. Find the coordinates of the point of intersection of the line with equation
(7 , 4)
(-7 , 4)
(-4 , 15)
(-4 , 7)
(2)
and the line
(2)
94
4. Find, in questions a) and b), giving your answer(s) correct to 3 significant figures where appropriate, the
value of for which
a)
1
-2
-3
-4
(2)
b)
(
)
)
(2)
95
5.
Railway track
A
B
Path=70m
44 m
44 m
1.84
C
C
The shape ABC shown is a design of a railway track with a straight path of 70m that connects two points A and B on
the track. A and B are equidistant from point C. The size of angle ACB is 1.84 radians.
Find:
a) The length of the arc AB, to the nearest m.
81
1781
40
70
(2)
b) The shortest distance from C to the path AB, to the nearest m.
56
27
54
28
(2)
c) The area of sector ABC, to the nearest m²
81
1781
40
70
(2)
96
6. The diagram shows a sketch of the curve
for
Y
1
-270°
-180°
-90°
0°
90°
180°
270° x
-1
The exact value of
a) Write down the exact value of
i.
ii.
(
)
i)
ii)
i)-
ii) -
i)-
ii)
i)
ii) -
(2)
b)
What is the correct equation of the graph below?
-4π
-2π
O
2π
4π
6π
(
(
8π
)
)
(2)
97
7. The circle A has centre ( -2 , 3 ) and passes through the point ( 2 , 0 )
Find the equation for A.
(
(
)
)
(
(
)
)
(
(
)
)
(
(
)
)
(2)
Section B
Extended response – please answer the questions in the space below. Be sure to show all your work and correct
units where applicable.
1.
a)
Work out (
00
)
(2)
b)
If
and
find the value of
(1)
98
c)
Evaluate
without using your calculator.
(2)
2.
a) Convert
to degrees. ( to 3 significant figures)
b) Convert 266° to radian measure. ( to 3 significant figures)
(2)
3.
Determine the exact value of: (show all working)
a)
(
)
(2)
b)
(
)
(2)
99
c) If
and
√
, in which quadrant is
(1)
4.
a) Solve for
by completing the square
(4)
b) Find the range of values of .
(2)
100
5.
Find the number of points of intersection of
and
(2)
6.
a) Find the distance between K( 1 , 4 ) and L( -2, -1 ) , ( to 3 significant figures)
(2)
101
b) Find the equation of the line KL in the form
(3)
7.
The equation of a circle is (
)
a) Find the centre coordinates and the value of the radius.
b) Determine whether the point ( 1 , 5 ) lies on the circle.
(3)
102
8.
The points A( -3 , 1 ), B( 1 , 2 ), C( 0 , -1 ), D( -4 , -2 ) are given. Show that ABCD is a parallelogram.
(2)
S
The diagram shows a plan for a patio.
9.
√ m
P
Q
4m
4m
The patio PRQS is in the shape of a sector of a circle with
centre R and radius 4 m.
It is also given that the length of the straight line PQ is √ m.
R
a)
a) Show through calculations that the size of angle PRQ is 2.17 radians
(2)
103
b) Find the area of patio PRQS ( to 3 significant figures)
(2)
c) Determine the area of the segment PQS ( to 3 significant figures)
(2)
(End of Paper)
104
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