Mathematics
GRADE
: 11
COMPILED BY
: Mr Motsumi
ACKNOWLEDGED BY
: Mr chobokoane
: Mr mochoane
: Mr motahane
EMAIL ADDREDD
: thapelo4you@gmail.com
CELL NUMBER
: 083 4270 612
Marie Curie
Nothing in life is to be feared, it is only to be understood. Now is the time to understand
more, so that we may fear less.
NOVEMBER EXAMINATION
PAPER 1:
DURATION
: 3 hours
TOTAL MARKS : 150
This examination paper will consist of the
following sections:
Algebra, equations and
45±3 marks
inequalities
Number patterns
25±3 marks
Finance, growth and decay
15±3 marks
Functions and graphs
45±3 marks
Probability
LEVEL 1
LEVEL 2
LEVEL 3
LEVEL 4
PAPER 2:
DURATION
: 3 hours
TOTAL MARKS : 150
This examination paper will consist of
the following sections:
Statistics
20±3 marks
Analytical Geometry
Trigonometry
Euclidean Geometry and
Measurement
30±3 marks
50±3 marks
50±3 marks
20±3 marks
QUESTION PAPER FORMAT
Knowledge
Routine
Complex
Problem solving
20%
35%
30%
15%
2
Compiled By: Mr Motsumi
EXPONENTS AND SURDS
The basics on the laws of exponents that you have studied in Grade 10 is extremely important.
EXPONENTIAL LAWS
1.
2.
Law
a m .a n  a m n
EXPLANATION
When the base are the same and we are multiplying we add
the exponents.
When the base are the same and we are dividing we subtract
the exponents.
am
 a mn
n
a
3.
a   a
4.
abm  a mb m
5.
am
a

 
bm
b
m n
mn
 
 an
m
m
The exponent outside the brackets multiplies the inside
exponent.
The exponent outside the brackets multiplies every exponent
inside the brackets, if and only if what is inside the brackets is
the product (multiplication, ab  ). E.g. not a  b  or
a  b  .
The exponent outside the brackets multiplies every exponent
inside the brackets, if and only if what is inside the brackets is
a
b
ab
.
ab
the quotient (division,   ). E.g. not 
EXPONENTIAL DEFINITIONS (COROLLARIES FROM THE LAWS)
1.
2.
3.
4.
5.
6.
7.
8.
9.
Definition
a0  1
1a  1 , with a  R
1
x n  n
x
a
n
ax  n
x
ax n 
1
ax n
1
 xn
x n
a
 ax n
x n
1
xn

a
ax n
1
n
 ax 
n
ax 
n
10.
a
 
b
b
 
a
11.
a m b n

b n a m
n
3
Compiled By: Mr Motsumi
ROOTS AND SURDS
m
The Root Definition:
n
am  a n
It is extremely important that you are able to work with the Root Definition in both directions.
SURDS
When a root is irrational it is referred to as a surd. In order words a surd is the root of a number that
cannot be determined exactly. Thus
2 , 5 , 3 20 are surds but 4 , 36 and 3 27 are not.
There are two surd laws for a ˃ 0, b ˃ 0, n  2, n  natural numbers.
Law 1 (multiplication law):
Law 2 (division law):
n
a .n b  n ab or n ab  n a .n b
n
a
n
b
n
a
a na
or n

b
b nb
EQUATIONS WITH RATIONAL EXPONENTS

x
1
even number
 negative number
 Whenever you have equations of the form x
m
n
m
 number, always rewrite the expression x n
m
 1n 
in the form  x  .
 
SIMPLE SURDS EQUATIONS
1.
Principle 1: The expression
be non-real.
2.
Principle 2:
3.
Principle 3:
A will only be real if A  0. If A ˂ 0 , then the expression will
A  negative number
 A  A
2
4
Compiled By: Mr Motsumi
PART 1:
Simplify the following expressions without the use of a calculator:
3
1.
81x 
3.
 a3 
 
 2 
4
3
4
(2)
2.
(1)
4.
2
9.
(4)
64 x
p 2  q 2   p  q 2
 p  q2
1
3 
if p  q
(3)
(3)
3 2 x 5
3 x 1.2 x 1
6x
6.
8.
10 n 1.12 n 1
8 n.15 n 1
2  3
2
(4)
3
 2a 3 b 2 
  2 
 3b 
x2 2
7.
2 
(4)
x 1 3
5
5.
5

4
6
16
x


 81 x 


10.
(4)
(4)
x 1
22x  3x
(4)
n 1
11.
 1 
 3   3 x 8
y 
4
xy  2 n 3


2,3 
x 1 3
(4)
13.
12 x  9 x 1
4 x 1  27 x
(3)
15.
3 x 3.12 x 3
2 2 x 6.9 x
17.
3m 4  6.3m1
7.3m 2
19.
2 a 1  2 a 1
2a
21.
3 x  3 x 1
3x
23.
9a 4  6a 3
3a 2
12.
316x .12 3 2 x
(4)
14.
36 n  2  8
2 n  24 2 n1
(4)
(3)
16.
2 x 3  3.2 x 1
2 x2
(4)
(4)
18.
 2(3a 2 ) 0  (31 ) 2
(3)
(3)
(3)
(2)
20.
1
 
 12
3x  x  2 x 2 


(3)
22.
1
5
 1

4 x 4  x 4  2 x 4 


(3)
24.
2 x  4  6.2 x 1
2 x  2.5
(3)
1
2
5
Compiled By: Mr Motsumi
Find the value of the following:
1.
2 x 1  2 x 1
If 5  10, determine the value of
5  10 x
2.
Find the value of:
3.
Calculate the sum of the digits of 2 2015  5 2019 .
4.
Calculate, without using a calculator
x
10 x 3 if
(5)
10 x  1,5
(2)
(4)
2100  2 99
(4)
PART 2:
Simplify completely without the use of a calculator:
1.


4 3 5 3 5

(2)
2.
3.
x  2 x  1. x  2 x  1
(4)
4.
5.
72 x 2  98x 2  2 288x 2
(3)
6.
(3)
8.
16 x 7  25 x 7
7.
9.
3

3 3 243  3 9
(3)
16  3 54  3  128
(4)
27m 6  48m 6
x
12m 6
50  8
(3)
10.
6 2010  10 2011
4 2010  15 2009
(2)
12.
2018
7 2
11.
32018  32016
32017
13.
If x 
3 a
2
and y 
4 a
2
2
Rewrite the following expression as a power of x :
15.
Calculate a and b if
16.
Show, WITHOUT using a calculator, that
17.
If
18.
WITHOUT using a calculator, show that
2  a and
2 2014  2 2013
2 4026
, determine the value of x  y 
14.
x x x x
8
(2)
4 x 1
22x
3. 48 
3

x7
(3)
(4)
(4)
(3)
(4)
7 2014  7 2012
 a 7 b and a is not a multiple of 7.
12
(4)
10  6 640  4 810  40  20
(4)
 
12
3  b, express the following in terms of a and b : 108  18.
2
1 2

8
8
 2
(2)
(4)
6
Compiled By: Mr Motsumi
PART 3:
Solve for x :
1.
16 x  8
3.
5x 3  4  0
5.
3x 5  2  0
7.
5
4
(3)
2.
96  3x
(4)
4.
4x 2  7
(2)
(4)
6.
4.3 x  36
(3)
27 x  x  33 x  9
(3)
8.
27 3 .81 2  9 x
(4)
9.
2.5 x  50
(3)
10.
2 x  0,125
(2)
11.
0,5 x. 1 
(5)
12.
x 4 8
13.
4x  2x  2
(4)
14.
2  16 x 2  0
(3)
15.
4 x  8  9.2 x
(4)
16.
9  243
(3)
17.
x3  4
(2)
18.
2 x 3  3.2 x 1  104
(5)
19.
2 2 x  6.2 x  16
(4)
20.
23 x1  23 x  12
(4)
21.
2 x x  2 27
(4)
22.
3 x  4  (3)1 x
(4)
23.
5x 
1
125
(2)
24.
3.5 2 x 1  5 2 x  2
(4)
25.
54 x 3.100 2 x1  50000
(4)
26.
2x  5 x  3
(6)
27.
2.3 x 1  3.3 x 1  104
(4)
28.
2.2  12  2  0 (3)
29.
2x  3 x  2  0
(4)
30.
3x 5  5 x 5  2  0
31.
x  23  64
(4)
32.
(3)
3
2
4
2
2
2
9
 10
16
2


3
(3)

x
1
3
x
2
x
1
x 3  512
(4)
(4)
1
33.
x  3x 2  4
(6)
7
Compiled By: Mr Motsumi
PART 4:
Solve for x :
3  3x
(4)
2.
x 1  3  x
(6)
3  26 x  3x
(5)
4.
x 1  3  x  4
(6)
5.
2x  1  x  1
(5)
6.
x 3 4  5
(4)
7.
x x2  4
(6)
8.
4x  3  x
(4)
9.
1  2 x  5x  1  0
(5)
10.
2 x  x2
(4)
1.
x5 
3.
11.
x 1 1  x
(5)
12.
6 x 2  15  x  1
(5)
13.
3x  5 x  2
(6)
14.
x2  5  2 x
(4)
15.
3 x  x4
(4)
16.
2 x 3  x 3
(5)
17.
4 x  11  1
(1)
18.
4 x  11  2  x
(4)
19.
7 x  2  2x  0
(5)
20.
2x  1  5
(3)
21.1
Solve for x if k  4. (Leave your answer in the simplest surds form),
(5)
21.2
Write down the minimum value of k .
(1)
21.
22.
23.
Given:
Given:
k
t x  
x  12  4 , where k is real number.
3x  5
x3
22.1
For which values of x will
22.2
Solve for x if t x   1.
3x  5
be real?
x3
(3)
(4)
Given:
f x   2 x  1
23.1
Write down the domain of f .
(1)
23.2
Solve for x if f x   2 x  1
(5)
8
Compiled By: Mr Motsumi
EQUATIONS AND INEQUALITIES
COMPLITING THE SQUARE
Summary of the basic procedure for completing the square:
1.
“Take Out” the coefficient of x 2 for the first two terms if necessary.
2.
1

Add and subtract   the coeffient of x 
2

3.
Factorise the perfect square trinomial and multiply.
2
Determining the maximum and minimum values of quadratic expressions.
Completing the square is an effective technique for determining the maximum and minimum value of
a quadratic expression.

Some important principles can be deduced from the expression a x  p

2 will have minimum value of 0

2 will have maximum value of 0
If a ˃ 0 , the expression a x  p
If a ˂ 0 , the expression a x  p
2 :
NATURE OF ROOTS
Determine the nature of the roots of a quadratic equation by considering   b 2  4ac is summarized
in the following table:
b 2  4ac
 ˂0
0
 ˃ 0 and   perfect square
Roots
Non-real
Real
Real, rational and unequal
 ˃ 0 and   perfect square
Real, irrational and unequal
0
Real, rational and equal
9
Compiled By: Mr Motsumi
WORD PROBLEMS
Solving word problems requires you to translate words into mathematical statements. The table below
contains typical examples of words translated into mathematical statements.
IN ENGLISH
The sum of two numbers
The difference between numbers
The product of two numbers
The quotient of two numbers
A number is increased by 5
A number is decreased by 5
6 times a number
A number 8 more than b
A number 8 less than b
A number’s square
The sum of three consecutive numbers
An even number
An odd number
IN MATHEMATICAL TERMS
ab
a b
ab  a  b
a b
b5
b5
6a
b8
b 8
a2
a 1  a  a 1
2a
2a  1 or 2a  1
A few things to consider when attempting word problems
 Read through the problem a few time and highlight key numbers and information, especially
comparisons.
 Let the quality you are required to determine be equal to x (or any other letter).
 Mate sure that you work with the same units. For example choose one of the following:
minutes or hours, cm or m, km/h or m/s; etc.
 Set up an equation with the information gained so as to solve for x (or in some instances y
as well).
 Basic general knowledge:
Profit  Selling price - Cost price
Total cost  Cost per item  number of items
distance
distance
; distance  speed  time or time 
speed
time
Area and perimeter formula for circle, triangles and rectangles.
 Conclusion by stating your answer clearly. Don’t just leave your answer as x  ...
speed 
10
Compiled By: Mr Motsumi
PART 1:
Solve for x (correct to TWO decimal places where necessary):
1.
x  23x  6  0
(2)
2.
x 2  5x  2
(4)
3.
x  3x  1   x  1
(4)
4.
x 2  3x  1
(5)
5.
x  22  1
(3)
6.
2 x 2  11x  4  0
(4)
7.
2 x  3x  7  0
(2)
8.
7 x 2  3x  2  0
(3)
9.
x2 x  1  0
(2)
10.
5x 2  2 x  6  0
(3)
11.
x2 x  5  0
(2)
12.
3  x 2 x  1  1
(5)
13.
x  22 x  1  2
(3)
14.
4  xx  3  5
(5)
15.
xx  1  6
(3)
16.
3x 2  4 x  8
(4)
17.
2 x  1x  4  0
(2)
18.
3x 2  x  5
(4)
19.
3x 2  5 x  2
(3)
20.
x
2
5
x
(4)
21.
x  92x  1  0
(3)
22.
x 2  x  13  0
(4)
23.
2 x  33  x   4
(3)
24.
2 x 2  3x  3
(4)
25.
x 2  x  12  0
(3)
26. (a) 2 x 2  5x  11  0
(3)
26. (b) 2 x 3  5x 2  11x  0
(2)
27.
x 2  2 x  35  0
(3)
28.
3x 2  4 x  0
(2)
29.
x6
30.
3x 2  7 x  0
(2)
31.
5 x 2  3x  6
(4)
32.
x 2  5x  6  0
(3)
33.
 3x 2  4 x  2  0
(3)
34.
x  24  x   0
(2)
35.
3x 2  2 x  14
(4)
36.
x 2  9 x  20  0
(3)
37.
3x 2  5 x  4
(4)
38.
Given: x  33x  1  m
2
2
0 ; x0
x
(4)
38.1
Solve for x if m  0
(2)
38.2
Solve for x, round to two decimal places, if m  6.
(5)
11
Compiled By: Mr Motsumi
PART 2:
1.
For which value(s) of m will the equation 2 xx 1  m  x have non-real roots?
(5)
2.
For which values of k will the roots of the equation kx2  2kx  3 be equal if k  0 ?
(5)
3.
Determine the values of k for which the equation x
4.
For which value of k will the equation x 2  x  k have no real roots?
(4)
5.
If f x   0 has roots x for which values of k will the roots be
equal?
(3)
6.
The roots of the equation f x   0 are x 
2
 3x  k  1  0 has real roots.
4  16  4m m  5
2m
Determine the value of m for which the roots will be non-real.
7.
8.
10.
11.
(4)
Determine:
7.1
The nature of the roots of 2 x 2  3x  5  0 without solving the equation
(3)
7.2
For which value(s) of k will x 2  2 x  k  0 have real roots?
(4)
7.3
The value of m if x  3 is a root of 2 x 2  mx  5m  0.
(2)
Discuss, without solving the equation, the nature of the roots for the equation:
3x 2  5 x  3  0
9.
(4)
(2)
If x  2 is a root of kx2  4 x  24 , determine the
9.1
value of k
(2)
9.2
other root
(3)
Given:
f x   x 2  5 x  2
10.1
Solve for x if f x   0
(3)
10.2
For which values of c will f x   c have no real roots?
(4)
The solution of quadratic equation are given by x 
 2  2p  5
for which value(s)
7
of p will this equation have:
12.
11.1
Two equal solutions
(2)
11.2
No real solutions
(1)
Given:
f x   3x  1  5 and g x   3
12.1
Is it possible for f x   g x  ? Give a reason for your answer.
(2)
12.2
Determine the value(s) of k for which f x   g x   k has TWO unequal
real roots.
(2)
2
12
Compiled By: Mr Motsumi
13.
The roots of a quadratic are given by x 
where k   3;2;1;0;1;2; 3
14.
 5  20  8k
,
6
13.1
Write down TWO values of k for which the roots will be rational.
(2)
13.2
Write down ONE value of k for which the roots will be non-real.
(1)
Given the quadratic equation:
3x 2  kx  2  0
1
is one root of the equation.
3
14.1
Find the value of k if x 
14.2
Determine the value of the other root.
(2)
(3)
PART 3:
Solve for x and y in the following simultaneous equations:
1.
y  2 x  1  0 and xy  2 y  x
2.
x  y  3 and 2 x
3.
2 x  y  3 and 27 3  3 y 1
4.
x  y  2  0 and x
5.
y  3x  2 and y
6.
2
2
 3x  10
 2 y 2  5xy
(7)
(8)
x
(6)
 y2  4
(6)
2
 9 x 2  16
(5)
2 y  x  1 and x
2
 y 2  3xy  y  0
(6)
7.
y  x  2 and x
2
 3xy  8  0
(6)
8.
2 y  x  1 and y  3x
 x 3
(7)
9.
xy  6  0 and x  3 y  3  0
(6)
10.
x  y  1 and x y  1  3xy  x
11.
2 x  y  3 and x
12.
y  2  2 x and 2 x
13.
2 x  y  4 and y  x
14.
x  2 y  3 and x
15.
x  2 y  1 and x
2

2

2
2
(6)
 6 x  y 2  2 y  30  0
(7)
 2  y2
(6)
2
 3
(5)
2
 3xy  10
(6)
2
 2 y  3xy  6
(6)
2
13
Compiled By: Mr Motsumi
16.
x  2 y  3 and xy  20
17.
x  y  3 and x
18.
3x  y 2  x  52  0
19.
3 x 10  33 x and y
20.
Consider the equation:
21.
2
(6)
 xy  2 y 2  7  0
2
(5)
(4)
 x  20
(5)
x 2  5xy  6 y 2  0
x
.
y
20.1
Calculate the values of the ratio
(3)
20.2
Hence, calculate the values of x and y if x  y  8
Given:
2 x  2 x2  5 y  20
21.1
Express 2 x in terms of y.
(2)
21.2
How many solutions for x will the equation have if y  4 ?
(2)
21.3
Solve for x if y is the largest possible integer value for which
2 x  2 x2  5 y  20 will have solutions.
(5)
(3)
PART 4:
Solve for x :
1.
x 2  3x  3
(5)
2.
 xx  9  14
(4)
3.
3x 2  x
0
3
(4)
4.
2x 2  7 x  4  0
(3)
5.
x2  9
(3)
6.
x 2  16
(4)
7.
 3x  p x  q   0 where p; q   (3)
8.
x 2  4 x  21
(5)
9.
2x 2  7 x  4  0
(4)
10.
x 2  2x  8  0
(3)
11.
 x 2  5x  4
(5)
12.
3x  22  3x
(6)
13.
x 2  2 x
(3)
14.
4 x 2  1  5x
(4)
15.
x 2  7x  0
(3)
16.
 3x  7x  5  0
(4)
17.
x4  x   0
(3)
18.
7 x 2  18x  9  0
(4)
14
Compiled By: Mr Motsumi
TRIGONOMETRY
Part 1
1.
In the diagram below, Tx ; 15 is a point in the Cartesian plane such that
OT  17 units. P- 2 ; a  lies on OT. K is a point on the positive x  axis
and TÔK   .
Determine, with the aid of the diagram, the following:
2.
1.1
The value of x
(2)
1.2
tan 
(1)
1.3
cos180   
(2)
1.4
sin 2 
(2)
1.5
The value of a
(3)
In the diagram below, R is a point in the first quadrant such that TÔR   . RO is
3
extended to P such that OP  2 RO and TÔP   . It is given that sin   .
5
WITHOUT using a calculator, determine:
2.1
The value of tan 
(3)
2.2
The value of sin 
(3)
2.3
The coordinates of P
(4)
15
Compiled By: Mr Motsumi
3.
In the diagram below, P1 ; 3 is a point on the Cartesian plane, OP  r and
XÔP   .
4.
3.1
Make use of the diagram to calculate the value of  .
(2)
3.2
Calculate the length of OP. Leave the answer in surd form.
(2)
3.3
Determine the values of the following, without using a calculator:
(a)
sin 
(1)
(b)
cos180   
(2)
In the diagram below, P- 8 ; t  is a point in the Cartesian plane such that
OP  17 units and reflex XÔP   .
4.1
Calculate the value of t.
4.2
Determine the value of each of the following WITHOUT using calculate:
(2)
(a)
cos  
(2)
(b)
1 sin 
(2)
(c)
tan 
(1)
(d)
1  2 sin 2 
(3)
16
Compiled By: Mr Motsumi
5.
In the diagram below, P6 ; k  is a point in the first quadrant. P6 ; k  is a point in
first quadrant. PÔT   and OT  2 . It is further given that
5.1
6.
5 cos   2  0
Determine, without the use of a calculator:
(a)
tan  in terms of k .
(1)
(b)
The value of k .
(4)

In the diagram below, P is the point a ; 

21 such that OP  5 units.
Reflex BÔP   as indicated.
6.1
Calculate the numerical value of c.
6.2
Determine without the use of calculator, the numerical value of the
following:
(2)
a)
cos 
(1)
b)
tan  sin 2 
(2)
c)
2 sin  . cos 
(2)
17
Compiled By: Mr Motsumi
7.
In the diagram below similar triangles OPR and OQT are presented. O is the
origin. R and T are points on the x  axis.
Determine the following (leave answers in surd form if necessary)
8.
7.1
sin 90   
(3)
7.2
the value of a.
(2)
In the diagram, P3 ; t  is a point in the Cartesian plane. OP 
a reflex angle.
34 and HÔP   is
Without using a calculator, determine the value of:
8.1
t
(2)
8.2
tan 
(1)
8.3
2 cos 2   1
(3)
8.4
sin 
(1)
18
Compiled By: Mr Motsumi
9.


P  7 ; 3 and Sa ; b are points on the Cartesian plane, as shown in the diagram
below. PÔR  PÔS   and OS  6.
Determine, WITHOUT using a calculator, the value of:
10.
9.1
tan 
(1)
9.2
sin   
(3)
9.3
a
(4)
In the diagram below, Tx ; p  is a point in the third quadrant and it is given that
sin  
p
1 p2
.
10.1
Show that x  1 .
(3)
10.2
Write cos180    in terms of p in its simplest form.
(2)
10.3
Show that cos 2   sin 2  can be written as
1 p2
.
1 p2
(3)
10.4
tan 
(1)
19
Compiled By: Mr Motsumi
11.
Given:
4 tan   5  0,   0 ; 180. Evaluate without using a calculator:
41 cos   4 sin 150 . cos 180
12.
13.
14.
(5)
3
, where A 90 ; 270 determine, using a diagram, without the use of
7
a calculator, the value of sin  A  30 .
If sin A  
15
and sin   0, use a suitable diagram to determine the following
25
without using a calculator:
If cos   
1.13.1 cos 2   sin 2 
(4)
1.13.2 tan  360
(2)
Given 5 tan   4  0 and  180 ; 360. Use a suitable diagram to determine the
following, without using a calculator:
14.1.1 2 cos180   
14.1.2
15.
(4)
(4)
sin 2   90  sin 2 
(3)
Given: p. sin   4  0 and p. cos   3  0 where p  0
15.1
Explain why  90 ; 180.
15.2
Show that:
15.3
Determine the numerical value of p.
tan   
(3)
4
3
16.
If x  3 sin  and y  3 cos  , determine the value of x
17.
Given that sin   
(2)
(2)
2
 y2 .
(3)
4
and 90    270.
5
WITHOUT using a calculator, determine the value of each of the following in its
simplest form:
18.
17.1
sin   
(2)
17.2
cos 
(2)
17.3
sin   45
(3)
Calculate without the use of a calculator and with the aid of a diagram the value of
3
 12
cos   tan  , if sin   with  90 ; 270 and cos  
5
13
with  0 ; 180
(6)
20
Compiled By: Mr Motsumi
Part 2
Simplify WITHJOUT using a calculator:
1.
sin 120. cos 210. tan 315. cos 27
sin 63. cos 540
(7)
2.
sin 315. tan 210. sin 190
cos 100. sin 120
(6)
3.
sin 143. cos127  sin 53. cos 37
(5)
4.
cos 350 sin 40  cos 440 cos 40
(5)
5.
sin 75  cos 752
(5)
6.
cos 225. sin 135  sin 330
tan 225
(6)
7.
tan150. sin 300. sin 10
cos 225. sin 135. cos 80
(7)
8.
cos 100
 tan 2 120
sin  10
(6)
9.
cos 180. sin 225. cos 80
sin 170. tan135
(6)
10.
sin 190 cos 225 tan 390
cos 100 sin 135
(7)
11.
sin90  x . cos180  x   tan x. cos x. sinx  180
(6)
12.
cos90  x . sin  x 
cos 2 180  x 
(5)
13.
sin 117
 cos x . tan180  x . sin 360  x 
cos 27
(6)
14.
sin  x . tanx  360. sin 450  x 
 cos 2 x  180
cos 180
(8)
15.
1  cos 2 A
4 cos90  A
(3)
16.
sin 360  x . tan x 
cos180  x . sin 2 A  cos 2 A


(6)
21
Compiled By: Mr Motsumi
17.
sin   360sin 90    tan  
cos90   
(5)
18.
sin 180  x   2 cos90  x  cos x
2 cos 2 360  x   cos x 
(6)
19.
sin 180   . sin 540   . cos  90
tan  . sin 2 360   
(7)
20.
sin 90  x . tan180  x 
cos x . sin 180  x 
(6)
21.
cos180  x . tan360  x . sin 2 90  x 
 sin 2 x
sin 180  x 
(6)
22.
tan180   .sin 2 90     cos  180.sin 
(6)
Part 3
1.
2.
3.
4.
If sin 29  p determine the following in terms of p :
1.1
cos 29
(3)
1.2
tan 569
(2)
1.3
1  cos 2 61
(2)
If sin 17  a , WITHOUT using a calculator, express the following in terms of a :
2.1
tan17
(3)
2.2
sin 107
(2)
2.3
(4)
cos 2 253  sin 2 557
If cos 23  p , express, without the use of a calculator, the following in terms of p :
3.1
cos 203
(2)
3.2
sin 293
(3)
If sin 40  p, write the following in terms of p :
4.1
cos 50
(3)
4.2
tan 220
(3)
22
Compiled By: Mr Motsumi
5.
6.
7.
8.
9.
10.
If cos 20 
1
, use a sketch to find the following in terms of p :
p
5.1
sin 70
(2)
5.2
cos 200
(2)
5.3
tan160
(2)
If cos 37  a, determine in terms of a, the value of the following:
6.1
sin 37
(3)
6.2
sin 53
(3)
If sin 12  m and cos 13  n, determine the following in terms of m and / or n
7.1
2 cos 2 12  1
(3)
7.2
sin 25
(4)
If cos 26  r , determine the following in terms of r , in its simplest form:
8.1
1  2 sin 2 26
(3)
8.2
tan 71
(6)
If sin 31  p, determine the following, without using a calculator, in terms of p :
9.1
sin 149
(2)
9.2
cos 59
(2)
9.3
2 cos 2 31  1
(2)
Given: sin 16  p
Determine the following in terms of p, without using a calculator.
11.
12.
10.1
sin 196
(2)
10.2
cos 16
(2)
Given that sin 23  k , determine, in its simples form, the value of each of the
following in terms of k , WITHOUT using a calculator:
11.1
sin 203
(2)
11.2
11.3
cos 23
(3)
(2)
tan 23
If sin 32  t , determine in terms of t , the value of the following:
12.1
12.2
12.3
tan 58
sin 212
2 cos 2 32 1
(2)
(2)
(2)
23
Compiled By: Mr Motsumi
13.
If sin 28  a and cos 23  b, determine the following in terms of a and/or b :
13.1
cos 28
(2)
13.2
sin 4
(4)
Part 4
Prove the following identities:
1.
1
cos 

 tan 
cos  1  sin 
(5)
2.
sin   tan  . cos 2 
 tan 
cos   1  sin 2 
(4)
3.

1 
 tan y 
 1  cos 2 y  tan y
tan
y



(6)
4.
sin x
1  cos x
2


1  cos x
sin x
sin x
(5)
5.
 1
1 
1  cos 

 

1  cos 
 sin  tan  
(5)
6.
1
1

2
cos x  1cos x  1 tan x. cos 2 x
(4)
7.
1
1
2


1  cos x 1  cos x sin x tan x
(6)

2
8.
cos A  cos180  A.sin 2 A  cos 3 A
(4)
9.
2 tan x  2 sin x. cos x
 tan x
2 sin 2 x
(5)
10.
cos 2 x sin 2 x  cos 4 x
 1  sin x
1  sin x
(4)
11.
cos 2 90   
1

1
cos    sin 90    cos  cos 
(6)
12.
sin 90  x . cos x. tan x 
 sin x
cos180  x 
(4)
13.
2 sin 2 x
cos x

2 tan x  2 sin x cos x sin x
(4)
24
Compiled By: Mr Motsumi
14.
15.
16.
Consider the identity:
14.1
Prove the identity.
(4)
14.2
Determine the value of x, x 0 ; 180 for which the identity is undefined.
(2)
18.
19.
20.
cos x
cos x

 2 tan x
1  sin x 1  sin x
Given the identity:
15.1
Prove the identity.
(5)
15.2
If x  180 ; 180, give 2 values of x for which the identity is undefined.
(2)
Consider the identity:
1
 tan 2 A  1
2
cos A
16.1
Prove the identity.
16.2
Hence, express tan A in terms of p if
cos A 
17.
sin 2 x
1  cos x

2
cos x
cos x  cos x
(3)
1
, p  0 and 90  A  360 .
p
cos x
1  sin x
2


1  sin x
cos x
cos x
17.1
Prove the identity.
17.2
For which values of x in the interval 0  x  360 will the identity in
QUESTION 17.1 be undefined?
8
4
4


2
sin A 1  cos A 1  cos A
18.1
Prove that
18.2
For which value(s) of x in the interval 0  x  360 is the identity in
QUESTION 18.1 undefined?
1  tan x cos x  sin x

1  tan x cos x  sin x
19.1
Prove that
19.2
For which value(s) of x in the interval 0  x  180 is the identity in
QUESTION 19.1 undefined?
Given:
sin x  cos x 2 
1  sin x
2
(3)
(5)
(2)
(5)
(3)
(3)
(2)
1
 2 tan x
cos 2 x
20.1
Prove the above identity.
(5)
20.2
For which values of x in the interval 0 ; 360 is the above identity
undefined?
(3)
25
Compiled By: Mr Motsumi
Part 5
Determine the general solution of the following:
1.
3
cos x  0,2  0
4
(5)
2.
sinx  30  cos 2 x
(5)
3.
5 sin2 x  10  3  0
(6)
4.
sin x  3 cos x  0
(4)
(6)
6.
sin 2 x  4 cos 2 x
(5)
5.
tan   x 
1
1
if x 2  2  1
x
x
7.
2 sin x. cos x  cos x
(6)
8.
sin x . 2 cos x  1  0
(6)
9.
2 cos 2 x  5 sin x  4
(6)
10.
8 cos 2 x  2 cos x  1  0
(6)
11.
cos  
1
5

cos  6
(6)
12.
8 sin 2 x  6 cos x  9
(7)
13.
3 sin x  2 tan x
(6)
14.
tan x sin x  cos x tan x  0
(7)
15.
2 sin x. cos x  4 cos x  2  cos x
(6)
16.
2 cos x  3 tan x
(7)
Part 6
1.
Consider the functions f x   cos
1.1
1.2
1.3
1.4
x
and g x   x  60 for x  180 ; 180
4
Write down the amplitude of the following:
1.1.1
f
(2)
1.1.2
g
(2)
Write down the period of the following:
1.2.1
f
(2)
1.2.2
g
(2)
State the range of f if the graph of f undergoes a vertical shift of
1 unit upwards.
(3)
Write down the new equation of g if it is shifted 60 horizontally to
the left.
(2)
26
Compiled By: Mr Motsumi
2.
Given: f x   2 sin x and g x   tan x for x  90 ; 180
2.1
3.
Show by calculations that f and g intersect at x  60 ; 0 ; 60 and 180
for x  90 ; 180.
(7)
2.3
With reference to the graphs, write down the values of x , for which:
2.3.1
f x   g x 
(3)
2.3.2
g is undefined
(2)
2.3.3
f x  45  2
(2)
Consider the functions f x   sinx  30 and g x   cos 2 x
3.1
Write down the period of g
(1)
3.2
State the range of f
(2)
3.3
Draw the graphs of f and g for x  90 ; 180 . Clearly show ALL
Intercepts with the axes, turning points and end points.
(5)
Write down the x  coordinates of the points of intersection of f and g in the
interval x  90 ; 180 .
(3)
Given: f x    cos45  x  and g x   tan x  for x  90  x  180
Sketch the graphs of f and g , showing clearly all intercepts with axes and
asymptotes.
(6)
4.2
For which values of x is f x   g x   0 for x  90 ; 90
(2)
4.3
Write the equation of hx  if hx    f x  45.
(2)
4.1
5.
(6)
2.2
3.4
4.
Sketch the graph of f and g , showing clearly all intercepts with the axes and
asymptotes.
Given: f x  
1
tan x and g x   sin 2 x
2
5.1
Draw the graph of f and g for x  90 ; 180. Show all the turning points
and intercepts with the axes. Clearly show the asymptotes using dotted lines.
(6)
5.2
Determine the values of x , for x  90 ; 180, for which f x   g x 
(6)
5.3
Write down the period of g 2 x 
(1)
27
Compiled By: Mr Motsumi
6.
Given: f x    sin x and g x   cosx  30
6.1
Write down the maximum value of 3 . g x 
6.2
Sketch the graphs of f and g on the same system of axes for
x  180 ; 180
6.3
6.4.2
7.
Write down the equation of h if h is the translation of g by 60 to
the right and 1 unit up.
(2)
Determine the maximum value of hx   f x .
(2)
Explain why the reflection of f in the x  axis and the reflection of f in the
y  axis will both results in the same graph.
(2)
Given: f x   tan x and g x   sinx  45
7.1
Draw the sketch graphs of f and g on the same set of axes for
x  90 ; 180.
7.2
7.3
8.
(4)
Answer the following questions:
6.4.1
6.5
(6)
Use your graph to determine the values of x, for x  180 ; 180, for which:
g x   f x   0
6.4
(1)
(6)
Use your graph to determine the values of x for which:
7.2.1
g x   f x   1
(2)
7.2.2
g x   f x  for x  90 ; 90
(2)
7.2.3
State the period of y  f 2x .
(1)
Write down the equation of h if h is obtained from g by translation of 30
to the left and 2 units down.
Consider:
(2)
g x   4 cosx  30
8.1
Determine the amplitude of g x .
(2)
8.2
Write the maximum value of g x .
(1)
8.3
Determine the range of g x   1.
(2)
8.4
The graph of g is shifted 60 to the left and then reflected about the
x  axis to form a new graph h. Determine the equation of h in its
simplest form.
(3)
28
Compiled By: Mr Motsumi
9.
The graphs of the functions f x   a cos b and g x   c tan for
  180 ; 180 are sketched below. The graphs intersect at P56 ; 1,6 and Q.
9.1
Write down the range of f .
(2)
9.2
If M 45 ; - 1 lies on g , determine the value of c.
(1)
9.3
Write down the values of a and b.
(2)
9.4
Determine the coordinates of Q.
(2)
9.5
K lies on f such that KM is parallel to the y  axis. Calculate the
length of KM.
(2)
If the system of axes is shifted 45 to the left and the graphs remain fixed,
write down the equation that is now represented by graph f .
(2)
9.6
10.
The diagram below shows the sketch graphs of f x   a cos bx and
g x   p sin x  r  for x   90 ; 180
10.1
Write down the values of a, b, p, and r.
(4)
10.2
Use the graph to determine the values of x for which f x   g x   0.
(2)
10.3
Write down the period of f .
(1)
10.4
Write down the equation of h if h is obtained by first moving the graph of g
45 to the right and then doubling its period.
(2)
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Compiled By: Mr Motsumi
11.
In the diagram below, the graph f x   cosx  p  and g x   q sin x are shown
for the interval  180  x  180 .
11.1
Determine the values of p and q.
11.2
The graphs intersect at A 22.5 ; 0,38 and B. Determine the coordinates of B. (2)
11.3
11.4
12.
Determine the value(s) of x in the interval  180  x  180 for which
f x   g x   0 .
(2)
(2)
The graph of f is shifted 30 to the left to obtain a new graph h .
11.4.1 Write down the equation of h in its simplest form.
(2)
11.4.2 Write down the value of x for which h has a minimum in the
interval  180  x  180 .
(1)
In the diagram below the graphs of f x   a cos bx and g x   sinx  p  are
drawn for x  180 ; 180.
12.1
Write down the values of a, b and p.
(3)
12.2
For which values of x in the given interval does the graph of f increase as
the graph of g increases?
(2)
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Compiled By: Mr Motsumi
13.
12.3
Write down the period of f 2x .
(2)
12.4
Determine the minimum value of h if hx   3 f x   1.
(2)
12.5
Describe how the graph of g must be transformed to form the graph k , where
(2)
In the diagram, the graph of f x   tan bx is drawn for the interval  90  x  135
13.1
Determine the value of b.
13.2
Determine the values of x in the interval 0  x  135 for which
13.3
14.
hx    cos x.
(1)
f x   1.
(2)
Graph h is defined as hx   tan bx  55 . Write down the equations
of the asymptotes of h in the interval  90  x  135 .
(2)
Given below is the graph of f x   sinx  45 , for x  90 ; 180.
14.1
Write down the range of f .
14.2
On the same set of axes, sketch the graph of g x   tan x for x  90 ; 180 .
Show ALL intercepts with the axes as well as asymptotes and end points.
(3)
(1)
31
Compiled By: Mr Motsumi
15.
15.3
Write down the period of g .
(1)
15.4
Write down the value(s) of x for which f x   g x  for x  90 ; 180.
(2)
15.5
For which value(s) of x is f x .g x   0 for x  90 ; 180?
(2)
15.6
Write down the equation of hx  if hx  is a result of shifting f x  such
that its minimum value is zero.
(1)
In the diagram, the graph of g x   cosx  60 is drawn for the interval
 150  x  120 .
On the same system of axes, draw the graph of k x    sin x for the
interval  150  x  120 . Show ALL the intercepts with the axes as well
as the coordinates of the turning points and end points of the graph.
(4)
15.2
Determine the minimum value of hx   cosx  60  3.
(2)
15.3
Solve the equation cosx  60  sin x  0 for the interval
15.1
 150  x  120
15.4
15.5
(6)
Determine the values of x for the interval  150  x  120 , for which
cosx  60  sin x  0.
(2)
The function g can also be defined as y   sinx    , where  is an
acute angle. Determine the value of  .
(2)
32
Compiled By: Mr Motsumi
16.
In the diagram the graph of f x   cos x and g x   sinx  b are drawn for the
interval  180  x  90.
16.1
Write down the value of b.
(1)
16.2
Write down the period of g .
(1)
16.3
Write down the value(s) of x in the interval  180  x  90 for which
f x   g x   0.
(2)
16.4
For which values of x in the interval  180  x  90 is sin90  x   g x ?
(3)
16.5
The graph of h is obtained by shifting f 3 units upwards. Determine the
range of h.
(2)
Part 7
1.
In the figure below, acute-angled ABC is drawn having C at the origin.
1.1
Prove that c 2  a 2  b 2  2ab cos C.
(6)
1.2
Prove that
sin A sin C

.
a
c
(5)
1.3
Hence, deduce that 1  cos C 
a  b  c a  b  c 
2ab
(4)
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Compiled By: Mr Motsumi
2.
Quadrilateral ABCD is drawn with BC  235 m and AB  90,52 m. It is also given
that AD̂B  31,23; DÂB  109,16 and CB̂D  48,88.
Determine the length of:
3.
2.1
BD
(3)
2.2
CD
(3)
In the diagram, PR is the diameter of the circle. Triangle PQR is drawn with vertex Q
outside the circle. R̂   , PR  QR  2 y and PQ  y.
3.1
Determine the value of cos  .
3.2
If QR cuts the circumference of the circle at T, determine PT in terms of y
and  .
(4)
(3)
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Compiled By: Mr Motsumi
4.
5.
In the diagram below, LK̂N   , KL̂N   and KL  x meters. KL and MN are
perpendicular to LM.
4.1
Determine the distance NL in terms of  ,  and x.
(3)
4.2
Show that MN 
x sin  . cos 
sin    
(6)
4.3
Given   76 ,   72 and x  48 metres :
4.3.1
Calculate the length of MN.
(2)
4.3.2
Calculate the area of KLM if LN  88 m.
(3)
In PQR, P̂  132, PQ  27,2 cm and QR  73,2 cm.
5.1
Calculate the size of R̂ .
(3)
5.2
Calculate the area of PQR.
(3)
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Compiled By: Mr Motsumi
6.
AB and DE are two towers in the same horizontal plane. The angle of elevation from a
man at point C to E, the top of tower DE is 43. From point C the angle of elevation
of A, the top of tower AB, is 37. AB is 13 m high and DE  20 m.
6.1
6.2
7.
8.
Calculate the following:
6.1.1
AC
(2)
6.1.2
CE
(2)
Hence, calculate the distance between the top of the building that is AE.
(3)
In the diagram below: AB  3 units ; AD  9,5 units ; Â  112 CB̂D  BD̂C  67
7.1
Show, by calculation, that BD  10,98 units
(3)
7.2
Hence calculate the perimeter of BCD.
(5)
7.3
Calculate the area BCD.
(2)
In the diagram, P̂  67, PQ  3 cm and PR  9,2 cm. Determine the length of QR.
(3)
36
Compiled By: Mr Motsumi
9.
10.
In the diagram below, DĈB   , AC  h units and AĈB   .
9.1.1
Determine size of AĈD in terms of  and  .
(1)
9.1.2
Prove that AD 
h sin    
cos 
(4)
9.1.3
Determine the length of AD if h  17 units,   58 and   23.
(2)
9.1.4
Calculate the area of ADC.
(3)
In PQR, QR  3 units, PR  x units, PQ  2 x units and PQ̂R   .
10.1
Show that cos  
10.2
If x  2,4 units :
10.3
x2  3
4x
(3)
10.2.1 Calculate 
(3)
10.2.2 Calculate the area of PQR
(2)
Calculate the values of x for which the triangle exists.
(4)
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Compiled By: Mr Motsumi
11.
TRS is a secant of the circle, and SU is a tangent at U. TU  16 cm, TR  12 cm
and T̂  41.
Calculate the following:
12.
11.1
Length of UR, correct to two decimals.
(3)
11.2
Size of Û 2
(3)
11.3
Length of secant TRS
(5)
In the diagram below, PQ is a straight line 1 500 m long. RS is a vertical tower 158 m
high with P, Q and S points in the same horizontal plane. The angles of elevation of R
from P and Q are 25 and  . SP̂Q  30.
12.1
Determine the length of PS.
(3)
12.2
Determine the length of SQ.
(3)
12.3
Hence, find the value of  .
(3)
12.4
Determine the area of SPQ.
(4)
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Compiled By: Mr Motsumi
13.
14.
CP is a radio mast. CA and CB are cables used to support the mast. B and A are on
the same level as P and CAB is formed. CA  11 m, CB  8 m and BA  10 m.
13.1
Determine CÂB.
(4)
13.2
Hence, determine the length of the mast CP.
(2)
13.3
Determine the area of CAB.
(2)
In the diagram KN represents a vertical tower, of height h metres, standing on a
horizontal plane LMN. The angle of elevation of K, as seen from L, is w. NL̂M  y
NM̂L  z .
h
tan w
14.1
Show that LN 
14.2
Hence, prove that LM 
h sin  y  z 
tan w sin z
(4)
14.3
Calculate LM if h  38 m, w  21, w  52 and z  59.
(2)
(1)
39
Compiled By: Mr Motsumi
EUCLIDEAN GEOMETRY
O
A
O
A
B
M
B
M
AM  MB (line from centre  to chord)
OM  AB (line from centre to midpt of chord)
E
E
x
G
x
x
O
2x
AÔB  2AÊB ( at centre  2   at circumfere nce)
B
A
B
A
AÊB  AĜB (s in the same seg)
F
M
90o
E
x
x
B
O
A
A
M̂  90(s in semi circle )
E
Ê  F̂ (equal chords; equal s)
E
G
G
x
x
x
y
A
D
B
Ê  B̂  180 o (Opp s of cyclic quad)
A
B
Ê  B̂ (ext  of cyclic quad)
40
Compiled By: Mr Motsumi
P
x
A
P
Q
x
A
B
C
B
CB̂Q  P̂ (tan chord theorem)
PB  PA (tans from same pt)
A
x
B
x
D
C
ABCD is a cyclic quad (converse s in the same seg)
P
A
x
Q
D
x
B
x
B
C
x
C
CB is Tangent to circle BPQ (Converse tan chord theorem)
ABCD is a cyclic quad (converse ext  of cyclic quad)
A
x
D
O
x+y=180o
A
B
AB̂O  90 (tan  radius )
C
B
y
C
ABCD is a cyclic quad (Converse opp s of cyclic quad )
41
Compiled By: Mr Motsumi
Part 1
1.
2.
Complete the statements below by filling in the missing word(s) to make the statements
Correct.
1.1
The angle between a tangent and a chord is …
(1)
1.2
The exterior angle of a cyclic quadrilateral is equal to …
(1)
1.3
The line drawn from the centre of the circle perpendicular to the chord …
(1)
1.4
The angle subtended by an arc at the centre of a circle is …
(2)
1.5
The angle subtended at the circle by a diameter is …
(1)
1.6
The angle between a tangent to a circle and a chord drawn from the point of
contact is equal to …
(1)
1.7
The angle in a semi-circle is equal to …
(1)
1.8
The opposite angles of a cyclic quadrilateral is …
(1)
1.9
The perpendicular bisector of a chord passes through …
(1)
1.10
The exterior angles of a triangle is equal to ….
(1)
In the diagram below, AST is a tangent to a circle O at S.
RŜT  Ŝ1  23 and QR  RS
Calculate, with reasons, the sizes of:
2.1
QŜR
(4)
2.2
R̂
(2)
2.3
P̂
(2)
2.4
Ô1
(2)
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Compiled By: Mr Motsumi
3.
In the diagram below, QOB is the diameter of the circle with centre O. PR||QB,
QB  RS and PB̂Q  25 . P, R and R are point on the circle.
4.
3.1
Determine, with reasons, three other angles each equal to 25.
3.2
Determine, with reasons:
(6)
(a)
RÔB
(2)
(b)
OR̂T
(2)
(c)
RÔS
(2)
(d)
RP̂Q
(2)
M is the of the circle SVQR having equal chords SV and QR. RP and QP are tangents
to the circle at R and Q respectively such that RP̂Q  70.
4.1
Calculate the size of R̂ 2 .
(4)
4.2
Calculate the size of Q̂1 .
(2)
4.3
Calculate the size of M̂ 2 .
(3)
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Compiled By: Mr Motsumi
5.
6.
In the figure, O is the centre of the circle. A, B, C, D and E lie on the circle
such that chord AB and chord DC are equal in length and AÊB  39 .
5.1
Determine the size of Ô1 .
(2)
5.2
Determine the size of Ô 2 .
(2)
In the figure, QS and PR are diameters of the circle with centre O such that PQ||SR.
PS is produced to T. N is a point on the circle such that Q̂1  Q̂ 2 . SN is drawn.
RS intersects QN at M. Ŝ1  48
6.1
6.2
Determine, with reasons, the size of:
(a)
Q̂1
(3)
(b)
R̂
(2)
(c)
M̂ 1
(2)
Prove that ST is a tangent to the circle passing through M, N and S.
(2)
44
Compiled By: Mr Motsumi
7.
O is the centre of circle TNSPR. PÔS  60 and PS  NT.
Calculate the size of:
8.
7.1
PR̂S
(2)
7.2
NŜT
(2)
In the figure below, DĈO  25 and O is the centre of the circle. A, B, E, C and D
are points on the circumference. Calculate, giving reasons, the sizes of:
8.1
D̂1
(2)
8.2
Ô1
(2)
8.3
 1
(2)
8.4
Ê
(2)
45
Compiled By: Mr Motsumi
9.
In the diagram below, PT is a diameter of the circle with centre O. M and S are
points on the circle on either side of PT. MP, MT, MS and S are drawn.
M̂ 2  37
Calculate, with reasons, the size of:
10.1
9.1
M̂ 1
(2)
9.2
Ô1
(2)
In the diagram below, the circle with centre O passes through M, R, Q and N.
MOR is a straight line and P is the midpoint of chord NQ.
Q̂1  42 and R̂  65
Calculate the size of the following angles, giving reasons for your answers:
10.2
10.1.1
P̂1
(2)
10.1.2
Ô 2
(2)
10.1.3 Ô 3
(4)
10.1.4 M̂
(2)
Will it be possible for O, M, N and P to lie on the circumference of a circle? Justify
your answer.
(5)
46
Compiled By: Mr Motsumi
11.
In the diagram, M is the centre of the circle. A, B, C, K and T lie on the circle.
AT produced and CK produced meet in N. Also NA  NC and B̂  38 .
11.1
12.
Calculate, with reasons, the sizes of the following angles:
(a)
KM̂A
(2)
(b)
T̂2
(2)
(c)
Ĉ
(2)
(d)
K̂ 4
(2)
11.2
Show that NK  NT.
(2)
11.3
Prove that AMKN is a cyclic quadrilateral.
(3)
In the diagram, PQRS is a cyclic quadrilateral. O is the centre of the circle. OS bisects
PŜR such that Ŝ1  Ŝ2  42 . Chord PQ is equal to chord SR.
12.1
Name, giving a reason, another angle equal to 42 .
12.2
Determine, giving reasons, the sizes of:
12.3
12.2.1
PQ̂R
(2)
12.2.2
P̂1
(3)
12.2.3 R̂ 3
(2)
Prove that SP||QR
(2)
47
Compiled By: Mr Motsumi
13.
ABC is a tangent to circle BPQRS at B. PQ||BS. QR  RS. Ŝ1  30 and B̂3  70 .
Calculate, with reasons, the size of the following angles:
14.
13.1
B̂1
(2)
13.2
P̂2
(2)
13.3
R̂
(2)
13.4
Q̂ 2
(3)
In the diagram below, O is the centre of circle TRNM. MP is a tangent to the circle at
M such that RT produced meets MP at P. OM intersects TN at K. K is the midpoint of TN.
PM̂T  40 and M̂ 3  26 .
Calculate, with reasons, the size of:
14.1
TÔM
(4)
14.2
N̂
(4)
14.3
T̂3
(3)
48
Compiled By: Mr Motsumi
15.
In the diagram, TAN is a tangent to the circle at A. ABCD is a cyclic
quadrilateral. BD is drawn and produced to meet the tangent at T. B̂1  30
and D̂ 2  51 . TAN||CB
Calculate, giving reasons, the size of:
16.
15.1
 1
(2)
15.2
T̂
(2)
15.3
B̂ 2
(2)
15.4
Ĉ
(3)
O is the centre of the circle in the diagram with CD parallel to diameter AB. AC is
produced to F and EG is a tangent to the circle. AB̂C  35 and Ĉ 2  54
Calculate, WITH REASONS, the size of the following angles:
16.1
Ê 1
(2)
16.2
Ĉ1
(2)
16.3
Ĉ 3
(2)
16.4
AÊD
(2)
16.5
Ê 3
(3)
49
Compiled By: Mr Motsumi
17.
In the diagram, O is the centre of the circle. A, B, C and D are points on the
circumference of the circle and CB is the diameter of the circle. Chord CA intersect
radius OD at E. AB is drawn. CD||OA and  2  x .
17.1
17.2
17.3
18.
Give reasons why
17.1.1
Ĉ1  x
(1)
17.1.2
Ĉ 2  x
(1)
Determine, giving reasons, the size of the following angles in terms of x.
17.2.1
 1
(3)
17.2.2
Ô1
(2)
17.2.3
Ô 2
(2)
For which value of x will ABOE be a cyclic quadrilateral?
(3)
In the diagram, S is the centre of the circle PQRT. PT is a diameter. RŜT  x  8
and PQ̂R  2 x  40 .
Determine the value of x .
(4)
50
Compiled By: Mr Motsumi
19.
In the circle RMNPQ, SQT is a tangent to the circle and PQ||MR and MN  NP.
R̂ 1  38 and Q̂ 3  62
Determine, with reasons, the sizes of:
20.
19.1
Q̂1
(2)
19.2
R̂ 2
(2)
19.3
N̂
(2)
19.4
M̂ 1
(3)
19.5
P̂2
(3)
19.6
M̂ 2
(2)
In the diagram, BD is a diameter of circle ABCD with centre O. AÔC  128
and D̂1  32 .
Calculate, stating reasons, the numerical values of:
20.1
D̂ 2
(2)
20.2
Ĉ 2
(2)
20.3
BĈD
(2)
20.4
Ĉ 3
(2)
20.5
B̂
(2)
51
Compiled By: Mr Motsumi
21.
In the diagram below, M is the centre of circle PQRS. PM||RS, QR  PR and R̂ 2  28
Determine, giving reasons, the size of the following angles:
22.
21.1
Ŝ 2
(3)
21.2
PŜR
(3)
21.3
Q̂
(2)
21.4
P̂3
(3)
In the diagram below O is the centre of the circle. Quadrilateral ABCO is drawn
with A and C on the circumference. AB is a tangent to the circle at point A and BC
is a tangent to circle at point C. D is the midpoint of chord AC and AD  DC. E is
the point on the circumference of the circle with centre O.
The length of:
AC  8 cm,
OD  3 cm,
BE  x cm and
AB  x  4.
22.1
Write down the size of D̂1 with a reason.
(2)
22.2
Calculate the length of OA.
(2)
22.3
Determine the size of  and give a reason for your answer.
(2)
22.4
Calculate the value of x if x  1 .
(4)
52
Compiled By: Mr Motsumi
23.
24.
O is the centre of the bigger circle. O lies on the circumference of the smaller circle.
OWVU are points on the circumference of the smaller circle. TSVU are points on the
circumference of the bigger circle.
23.1
Name four other angles each equal to x with reasons.
(8)
23.2
Determine the size of Ŵ2 in terms of x with reasons.
(4)
23.3
Prove that WS  WV .
(5)
In the figure, ABCD is a cyclic quadrilateral. AB||DC in circle with centre O.
BC and AD produced meet at M. D 3  x
24.1
Show that MC  MD.
(5)
24.2
If D̂ 3  x, determine the value of M̂, in terms of x.
(2)
24.3
Hence, show that BODM is a cyclic quadrilateral.
(3)
53
Compiled By: Mr Motsumi
25.
A, B, C and D are points on the circumference of the circle in the diagram below.
ECF is a tangent at C, B̂1  B̂ 2 .
26.
25.1
If B̂1  x, find, with reasons, TWO other angles equal to x.
(4)
25.2
Hence, show that DC bisects AĈF.
(2)
In the diagram below, PQ and RS are chords of the circle such that PQ||RS. The
tangent to the circle at Q meets RS produced at T and the tangent as S meets QT
at V. PS and QR intersect at W. QS and PR are drawn.
Let Q̂1  x and R̂ 2  y .
26.1
Write down a reason why QV  VS.
26.2
Write down the following angles in terms of x :
(1)
(a)
Ŝ 2
(2)
(b)
R̂ 1
(2)
(c)
V̂1
(3)
26.3
Show that R̂ 1  Ŝ 4
(4)
26.4
Prove that QVSW is a cyclic quadrilateral.
(4)
26.5
Write down the following angles in terms of y :
26.6
(a)
Q̂ 4
(2)
(b)
T̂
(2)
If M is the centre of the circle, hence prove that PMWR is a cyclic quadrilateral
(2)
54
Compiled By: Mr Motsumi
27.
In the diagram, O is the centre of the circle. Chord AC is perpendicular to radius OD at B.
OB  2 x units and AC  8x units.
Show that the length of BD is 2 x
28.
 5  1 units.
In the diagram, O is the centre of circle ABD. F is a point on chord AB such that
DOF  AB. AB  FD  8 cm and x cm .
Determine the length of the radius of the circle.
29.
(5)
(5)
In the figure below, AB and CD are chords of the circle with centre O. OE  AB.
CF  FD. OE  4 cm, OF  3 cm and CD  8 cm.
29.1
Calculate the length of OD.
(3)
29.2
Hence calculate the length of AB.
(4)
55
Compiled By: Mr Motsumi
30.
In the diagram, AB is a chord of a circle with centre O. D is a perpendicular to AB.
OA  25 cm and CD  18 cm.
Calculate, with reasons, the length of AB.
31.
(5)
In the diagram, O is the centre of the circle. The diameter DE is perpendicular to the
Chord PQ at C. DE  20 cm and CE  2 cm.
Calculate the length of the following with reasons:
32.
31.1
OC
(2)
31.2
PQ
(4)
In the diagram below, AOCD is a diameter of the circle with centre O and chord
BE  30 cm. AOCD  BE and OC  2CD.
Calculate with reasons:
32.1
BC
(2)
32.2
If CD  a units, determine OC in terms of a.
(1)
32.3
Calculate OB.
(1)
32.4
AB (correct to one decimal place).
(3)
32.5
the radius of the circle CAB.
(2)
56
Compiled By: Mr Motsumi
33.
In the diagram below, a circle passing through A, B and C is drawn. O is the centre
of the circle. AB  BC and AĈB  x .
33.1
Give, with reasons, the following in terms of x :
33.1.1
34.
 1
(1)
33.1.2 AÔB
(2)
33.2
Prove that AM̂B  90
(3)
33.3
Determine the length of AM, if it is given that AC  12 units.
(2)
In the diagram, M is the centre of circle and diameter CMPD is perpendicular to
chord AB, AB  4t , PD  t and CP  15 cm.
34.1
Give a reason why AP  2t .
34.2
If it is further given that CAP ||| BDP , calculate:
(1)
(a)
The value of t.
(4)
(b)
The length of the radius of the circle.
(2)
57
Compiled By: Mr Motsumi
35.
In the diagram, TP and TQ are tangents at P and Q respectively to the circle. P, Q and R
lie on the circumference of the circle. TY is drawn parallel to QR with Y on PR. TP̂Q  x .
36.
35.1
Name, with reasons, THREE other angles each equal to x.
(6)
35.2
Prove that TPYQ is a cyclic quadrilateral.
(2)
35.3
Prove that TY bisects PŶQ .
(2)
35.4
Prove that RQY is isosceles.
(4)
35.5
Prove that TY is a tangent to the circle passing through points Y, R and Q.
(2)
35.6
If it is given that PT  50 units, determine the length of PQ in terms of x.
(6)
In the diagram, PA is a tangent to the circle ACBT at A. CT and AD are produced to
meet at P. BT is produced to cut PA at D. AC, CB, AB and AT are joined. AC || BD.
36.1
Prove that ABC ||| ADT
(6)
36.2
Prove that PT is a tangent to the circle ADT at T.
(4)
36.3
Prove that APT ||| TPD
(3)
58
Compiled By: Mr Motsumi
37.
TV and VU are tangents to the circle with centre O at T and U respectively.
TSRUY are points on the circle such that TR is the diameter. X is the midpoint of
chord TU. T̂3  y.
Prove:
38.
37.1
RU||SY
37.2
T̂1 
37.3
TOUV is a cyclic quadrilateral
1
y
2
(5)
(5)
(5)
In the diagram, O is the centre of the circle and P, Q, S and R are points on the circle.
PQ  QS and QR̂S-  y. The tangent at P meets SQ produced at T.
OQ intersects PS at A.
38.1
Give a reason why P̂2  y.
(1)
38.2
Prove that PQ bisects TP̂S.
(4)
38.3
Determine PÔQ in terms of y.
(2)
38.4
Prove that PT is a tangent to the circle that passes through points P, O and A.
(2)
38.5
Prove that OÂP  90.
(5)
59
Compiled By: Mr Motsumi
39.
Tangents PQ and PR touch the circle at Q and R respectively. T is a point on the
circle such that QT  QR. QT and PR are produced and they meet at S. Q̂1  x.
40.
39.1
Name THREE other angles equal to x.
(3)
39.2
Determine, in terms of x, the size of Q̂ 2 .
(2)
39.3
Hence show that TR||QP.
(3)
39.4
Prove that STR ||| SRQ.
(3)
ABCD is a cyclic quadrilateral. AS is a tangent. CBS is a straight line. AD || SC
and AD  BD.
40.1
Name, with reasons, FIVE other angles each equal to x.
(5)
40.2
Prove that ASCD is a parallelogram.
(4)
40.3
Name a triangle in the figure similar to ADB.
(1)
60
Compiled By: Mr Motsumi
41.
42.
FAN is a common tangent to the smaller circle ABCD and the larger circle ARZP.
FP is a tangent to the smaller circle at C. The straight line ABR meets the larger
circle at R.
41.1
Prove that BC||RZ
(4)
41.2
Hence, prove that BC is a tangent to circle ACP.
(3)
41.3
Prove that RZA ||| DPC.
(5)
ABC is a tangent to the circle BFE at B. From C a straight line is drawn parallel to BF
to meet FE produced at D. EC and BD are drawn. Ê1  Ê 2  x and Ĉ 2  y.
42.1
Give a reason why EACH of the following is TRUE:
42.1.1
B̂1  x
(1)
42.1.2
BĈD  B̂1
(1)
42.2
Prove that BCDE is a cyclic quadrilateral.
(2)
42.3
Which TWO other angles are each equal to x ?
(2)
42.4
Prove that B̂ 2  Ĉ1 .
(3)
61
Compiled By: Mr Motsumi
43.
In the diagram below, O is the centre with A, B and T on the circumference,
BP  OB  AO, PTR is a tangent and EP  AP.
Prove:
44.
43.1
TEPB is a cyclic quad
(3)
43.2
ATB ||| APE
(3)
43.3
TP  PE
(6)
43.4
ATB ||| EPB
(5)
In the figure below, SR and RNP are tangents to the circle with centre O at the points
S and N. Radius NO is produced and cuts the circle at M and meets RS produced at T.
44.1
Why is RŜO  90 ?
(1)
44.2
Prove that RNOS is a cyclic quadrilateral.
(4)
44.3
If Ŝ1  x , determine, with reasons, FOUR other angles in the figure which
Equal to x.
44.4
Prove that Ŝ3 
(8)
1
Ô 3
2
(4)
62
Compiled By: Mr Motsumi
NUMBER PATTERN
WORD
Linear Pattern
DEFINITION / DESCRIBTION
Is a number pattern with a constant
difference d  T2  T1  between
consecutive terms
Is a number pattern with a constant
second difference, its 1st difference
between consecutive terms, forms
a linear pattern
Is a linear number pattern with a
constant difference d  T2  T1 
between consecutive terms
Quadratic Pattern
Arithmetic Sequence
FORMULA / SYMBOL
Tn  an  b
Tn  an 2  bn  c
Tn  a  n  1d
LINEAR NUMBER PATTERNS
First term
bc
2b  c
b
4b  c
3bc
b
b
Constant difference
QUADRATIC NUMBER PATTERNS
abc
4a  2b  c
3a  b
9a  3b  c
5a  b
16a  4b  c
7a  b
First
Difference
2a
2a
Second Difference
63
Compiled By: Mr Motsumi
Part 1
1.
2.
3.
Given the sequence:
7 ; 12 ; 17 ; ....
1.1
Write down the next two terms of the sequence.
(2)
1.2
Determine the general term of the sequence in the form of Tn  an  b.
(2)
1.3
Determine if 12 5 will be a term in above sequence.
(3)
1.4
Explain why any positive number ending with a 2 will form part of the
sequence
(2)
Given the finite linear pattern:
12 ; 17 ; 22 ; ... ; 172
2.1
Determine a formula for the n th term of the pattern.
(2)
2.2
Calculate the value of T12 .
(2)
2.3
Determine the number of terms in the pattern.
(2)
Given the pattern:
4 ; 9 ; 16 ; 25 ; ....
Show how you will determine the n th term in this pattern.
4.
5.
6.
7.
Given the linear pattern:
(2)
7 ; 2 ; - 3 ; ....
4.1
Determine the general term, Tn , of the linear pattern.
(2)
4.2
Calculate the value of T20 .
(2)
4.3
Which term in the pattern has a value of  138 ?
(2)
Given the linear pattern:
5 ; - 2 ; - 9 ; ... ; - 289
5.1
Write down the constant first difference.
(1)
5.2
Write down the value of T4 .
(1)
5.3
Calculate the number of terms in the pattern.
(3)
Consider the arithmetic sequence:
12 ; 9 ; 6 ; ....
6.1
Determine the general term, Tn .
(2)
6.2
Determine the 40 th term.
(2)
Given the finite arithmetic sequence:
5 ; 1 ; - 3 ; .... ; - 83 ; - 87
7.1
Write down the forth term T4  of the sequence.
(1)
7.2
Calculate the number of terms in the sequence.
(3)
64
Compiled By: Mr Motsumi
8.
Given the linear pattern:
18 ; 14 ; 10 ; ....
8.1
Write down the fourth term.
(1)
8.2
Determine a formula for the general term of the pattern.
(2)
8.3
Which term of the pattern will have a value of  70 ?
(2)
8.4
If this linear pattern forms the first difference of a quadratic pattern, Q n ,
determine the first difference between Q 509 and Q 510 .
9.
Given the linear pattern:
156 ; 148 ; 140 ; 132 ; ....
9.1
Write down the 5 th term of this number pattern.
(1)
9.2
Determine a general formula for the n th term of this pattern.
(2)
9.3
Which term of this linear number pattern is the first term to be negative?
(3)
9.4
The given linear pattern forms the sequence of first differences of a
quadratic number pattern Tn  an
2
 bn  c with T5  24.
Determine a general formula for Tn .
10.
Given the first four terms of a linear pattern:
(5)
7 ; x ; y ; 30
Calculate the values of x and y .
11.
13.
(4)
6 ; 2 x  1 and 3x  3 are the first three terms of a linear pattern.
Calculate the value of x.
12.
(2)
(3)
The first three terms of a linear number pattern are 2 x  1 ; 4 x  5 ; 3x  3 and hence
determine the:
12.1
value of x
(2)
12.2
next three terms of a pattern
(3)
12.3
n th term of the given pattern
(4)
A linear pattern has a difference of 3 between consecutive terms and its 20 th term is
equal to 64 (that is T20  64 ).
14.
13.1
Determine the value of T22 .
(1)
13.2
Which term in the pattern will be equal to 3T5  2 ?
(4)
The first term of a linear number pattern is 92 and the constant difference is  4.
14.1
14.2
14.3
Write down the values of the second and third terms of the number pattern.
Determine an expression for the n th term of the number pattern.
Determine the value of the eighteenth term.
(1)
(2)
(2)
14.4
If T p  Tq  0, determine the value of  p  q .
(2)
65
Compiled By: Mr Motsumi
Part 2
1.
2.
Given the sequence:
1.1
Write down the value of the next term of the sequence.
(1)
1.2
Determine an expression for the n th term of the sequence.
(5)
1.3
Calculate the value of the first term that is greater than 269.
(4)
Given the quadratic pattern:
4.
5.
6.
244 ; 193 ; 148 ; 109 ; ....
2.1
Write down the next term of the pattern.
(2)
2.2
Determine a formula for the n th term of the pattern.
(4)
2.3
Which term of the pattern will have a value of 508 ?
(4)
2.4
Between which TWO consecutive terms of the quadratic pattern will the first
difference be 453 ?
(3)
Show that all the terms of quadratic pattern are positive.
(4)
2.5
3.
3 ; 9 ; 17 ; 27 ; ....
The sequence
3 ; 9 ; 17 ; 27
is a quadratic pattern.
3.1
Determine the next TWO terms of the pattern.
(2)
3.2
Determine an expression for the n th term of the pattern. Tn  ...
(4)
3.3
Which is the first term of the pattern that is greater than 269 ?
(5)
Consider the pattern:
1 ; 4 ; 11 ; 22 ; 37 ; ...
4.1
Calculate a formula for the n th term of the pattern.
(4)
4.2
Use your formula to calculate the 15 th term of the pattern.
(2)
Consider the quadratic pattern: 5 ; 12 ; 29 ; 56 ; ...
5.1
Write down the NEXT TWO terms of the pattern.
(2)
5.2
Prove that the first differences of this pattern will always be odd.
(3)
Consider the quadratic pattern: 3 ; 5 ; 8 ; 12 ; ...
Determine the value of T26 .
7.
(6)
The following number pattern has a constant second difference.
41 ; 43 ; 47 ; 53 ; 61 ; 71 ; 83 ; 97 ; 113 ; 131 ; 151 ; 173 ; 197 ; 223 ; 251 ; ....
7.1
Write down the value of the constant second difference.
(1)
7.2
7.3
Determine an expression for the n th term for the number pattern.
The first forty terms of the number pattern are all prime numbers. Determine
the forty-first term and show that it is not a prime number.
(4)
Determine the units digit of the 49 999 998 th
(2)
7.4
T49 999 998  term.
(2)
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8.
The following number pattern is quadratic:
8.1
Determine the general term of the first differences.
(3)
8.2
Determine the first difference between the 20 th and 21st terms of the
quadratic pattern.
Show that the general term of the quadratic pattern is given by
(2)
Tn  n 2  4n  12 .
(5)
8.3
Which term of the quadratic pattern is equal to 128 ?
Between which TWO terms of the quadratic pattern will the first difference
be 599 ?
8.4
8.5
9.
10.
Give a reason why this is a quadratic pattern.
(2)
9.2
Determine an expression for the n th term of the pattern.
(7)
9.3
Determine which term of the pattern will be equal to 445.
(5)
The number pattern
8 ; 20 ; 38 ; 62 ; ...
‘second difference’ is a constant.
is such that the sequence of
Write down the fifth term of the number pattern.
Determine an algebraic expression that represents the n th term of the
number pattern.
Calculate the 20 th term of the number pattern.
The number pattern
11.1
11.2
11.3
Determine the 5 th number in the pattern.
Derive a formula for the n th number in the pattern.
What is the 100 th number in the pattern?
5 ; 12 ; 29 ; 48 ; 77 ; .... is Tn  3n
The general term
14.
The sequence
(1)
(7)
(1)
1 ; 5 ; 11 ; 19 ; .... is such that the second difference is constant.
2
4 ; 9 ; x ; 37 ; ....
(1)
(7)
(3)
 2 . Is this statement true?
Show working to motive your answer.
13.
(3)
9.1
10.3
12.
(5)
The following pattern is given: - 5 ; - 3 ; 3 ; 3 ; 13 ; ....
10.1
10.2
11.
- 7 ; 0 ; 9 ; 20 ; ....
(4)
is a quadratic sequence.
13.1
Determine the value of x.
(3)
13.2
Now continue to determine the general term of this sequence.
(4)
A certain quadratic pattern has the following characteristics:

T1  p

T2  18

T4  4T1

T3 T2  10
Determine the value of p.
(6)
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15.
16.
The quadratic number pattern:
4 ; p ; 11 ; q ; 22 ; ... has a constant difference of 1.
15.1
Show that p  7 and q  16.
(3)
15.2
Determine the general term, Tn , of the quadratic pattern.
(4)
15.3
Determine the value of n if Tn  32.
(4)
Consider the quadratic pattern:
- 9 ;  6 ; 1 ; 12 ; x ; ...
16.1
Determine the value of x.
(1)
16.2
Determine a formula for the n th term of the pattern.
(4)
16.3
A new pattern, Pn , is formed by adding 3 to each term in the given
quadratic pattern. Write down the general term of Pn in the form
16.4
17.
18.
Pn  an 2  bn  c.
(1)
Which term of the sequence found in QUESTION 16.3 has value of 400 ?
(4)
The following quadratic pattern is given:
15 ; 10 ; 17 ; x ; 7 ; ....
17.1
Calculate the value of x.
(3)
17.2
Determine the n th term of the above pattern.
(4)
17.3
Calculate the value of the 50 th term of the pattern.
(2)
A quadratic pattern has a second term equal to 6, a third term equal to 2 and a fifth
term equal to  18.
19.
18.1
Calculate the second difference of the pattern.
(4)
18.2
Calculate the first term.
(3)
18.3
If the sum of two consecutive terms in the pattern is 1 227, calculate the
Difference between these two terms.
(5)
A given quadratic pattern Tn  an
2
 bn  c has T2  T4  0 and a second
difference of 12. Determine the value of the 3 rd term of the pattern.
20.
21.
(6)
A quadratic pattern has a constant second difference of 2 and T5  T17  29.
20.1
Does this pattern have minimum or maximum value? Justify the answer.
20.2
Determine an expression for the n th term in the form Tn  an
2
(3)
 bn  c. (5)
A quadratic sequence second term is equal to 1, the third term is equal to  6 and
the fifth term is equal to  14.
21.1
21.2
Determine the second difference of this quadratic sequence.
Now continue to determine the first term of this sequence.
(5)
(2)
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Compiled By: Mr Motsumi
Part 3
1.
Write down the next TWO terms of the pattern:
(2)
1 ; 1 ; 2 ; 4 ; 3 ; 7 ; 4 ; ....
2.
The n th term of a number pattern is as following:
Tn  5n  4 if n is an even number
and Tn  n
3.
4.
5.
6.
2
 6 if n is an uneven number.
2.1
Determine the value of T6  T7 .
(3)
2.2
Determine the value of k if Tk  219.
(5)
If n terms of a linear pattern is added the answer is given by S n  2n
2
 60n .
3.1
Show that the first term is 58.
(1)
3.2
Determine the sum of the first 2 terms of the pattern.
(2)
3.3
Calculate the common difference.
(1)
3.4
Determine the value(s) of n for which the sum to n terms is 450.
(3)
Tn  3.2 n1
4.1
Write down the first four terms of the sequence.
(2)
4.2
Which pattern do you notice?
(1)
4.3
What will the sum of the first four terms be?
(1)
4.4
Which term of the above sequence will be equal to 6 144 ?
(5)
Given:
1 1 1
1
; ; ; ..... ;
2 4 8
1 024
5.1
Explain how you will determine the 4 th term of the sequence.
(2)
5.2
Write a formula for the n th term of the sequence.
(2)
5.3
Determine the number of terms in the sequence.
(2)
Consider the following pattern:
6.1
1 ; 3 ; 9 ; 27 ; ....
Write down conjecture in words for the above pattern if the pattern behaves
consistently.
(2)
6.2
Determine an algebraic expression for the n th term of the above pattern.
(3)
6.3
Which term of the pattern has a value of 19 683?
(4)
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Compiled By: Mr Motsumi
Part 4
1.
Consider the following shapes created with black and white tiles:
1.1
Complete the table:
Figure number
Number of shaded tiles
No of white tiles
Total number of tiles
1
4
1
5
2
3
36
9
61
4
11
144
49
113
(5)
1.2
2.
Hence determine a formula for the total number of tiles in the n th figure.
(3)
Grey and white squares are arranged into patterns as indicated below.
Number of grey squares
Pattern 1
5
Pattern 2
13
The number of grey squares in the n th pattern is given by Tn  2n
Pattern 3
25
2
 2n  1.
2.1
How many white squares will be in the FOURTH pattern?
(2)
2.2
Determine the number of white squares in the 157 th pattern.
(3)
2.3
Calculate the largest value of n for which the pattern will have less than
613 grey squares.
(4)
Show that the TOTAL number of squares in the n th pattern is always
an odd number.
(3)
2.4
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3.
4.
Dots are arranged to form a sequence a sequence of patterns as shown below:
3.1
Write down the number of dots in pattern 5.
(1)
3.2
Determine a formula for the number of dots in the n th pattern.
(7)
3.3
Which pattern number has 2 113 dots in it?
(5)
The sketch of the sequence shows tile patterns. Each one is formed by the previous
pattern (black tile) with the grey surrounding tiles.
4.1
5.
Two sequences is formed. Firstly there is the amount of squares that is
being added to each new pattern (the grey squares). Secondly there is the
total amount of squares that forms the pattern. Write down the first six
terms of each pattern.
(6)
In the sketch below, a new line is drawn each time, from the same vertex to
the side opposite. This leads to more and more triangle being formed each time.
5.1
Redraw the table on your diagram sheet below and complete the blank cells
Sketch number
No of internal lines
Total number of triangles
1
0
1
2
1
3
3
2
6
4
(4)
5.2
How many internal lines would be added in the n th sketch?
(1)
5.3
Determine how many triangles would there be in the n th sketch?
(5)
5.4
Which sketch would have 153 triangles?
(4)
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Compiled By: Mr Motsumi
FUNCTIONS
FUNCTIONS
Straight Line
Parabola
Exponential
Hyperbola
y  mx  c
y  ax 2
y  ax
xy  k
y  ax 2  bx  c
y  ab x p  q
y
a
q
x p
y
xa
xb
y  a x  p   q
2

 Parabola of the form y  a x  p
2  q
Table below summarises the general shapes and positions of parabola function of the

form y  a x  p
2  q. The axes symmetry of are shown by dotted lines.
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Compiled By: Mr Motsumi
 Exponential of the form y  ab
x p
q
Table below summarises the general shapes and positions of parabola function of the
form y  ab
x p
 q for b  0.
 Hyperbola of the form y 
a
q
x p
Table below summarises the general shapes and positions of parabola function of the
form y 
a
 q. The axes symmetry of are shown by dotted lines.
x p
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Compiled By: Mr Motsumi
NOTATION, INCREASE AND DECREASE OF A FUNCTION
 INCREASING & DECREASING
 NOTATION
1.
f x   0
⊕
(above the line y  0 )
(e.g. where y is positive)
⊖
2.
f x   0
⊖
(below the line y  0 )
(e.g. where y is negative)
⊕
f x  . g x   0 ⊖
⊕ ⊖
(one graph lies above y  0 and
one graph lies below y  0 )
3.
f x   g x 
top bottom
(e.g. f x  lies above g x  )
4.
f x   g x 
(point of intersection)
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Compiled By: Mr Motsumi
Part 1
1.
2.
3.
1
5
x
Given: f x     , x  ℝ;
g x  
x
, x  ℝ;
5
h x  
1
 5 , x  0, x  ℝ
x
1.1
Write down the co-ordinates of the y -intercept of the graph f .
(1)
1.2
Give the equations of the asymptotes of f and h.
(3)
1.3
Which of the functions are decreasing?
(2)
1.4
Sketch the graphs of f , g and h on the same system of axes.
Show all asymptotes.
(4)
1.5
Write the equation of the graph obtained by reflecting f in the y -axis.
(1)
1.6
Give the equation of the graph obtained by shifting g vertically
up by five units.
(1)
Given: f x  
1
x  42  2
2
and
g x  
2
1
x 1
2.1
Calculate the co-ordinates of the x -intercept and y -intercept of g .
(3)
2.2
Calculate the co-ordinates of the x -intercept of f .
(3)
2.3
On the same set of axes, sketch the graphs of f and g . Indicate all
intercepts with the axes and the co-ordinates of the turning point of f .
(7)
2.4
Write down the range of g .
(2)
2.5
What is the minimum value of f x .
(1)
2.6
For which values of x will both f x  and g x  increase as x increases?
(2)
Given:
f x   2.3 x  1 and
3.1
Write down the equation of the asymptote of f .
(1)
3.2
3.3
Determine the y -intercept of f . (Give your answer in co-ordinate form)
Find ONE other point on the graph of f .
(2)
(2)
3.4
Sketch the graph of f .
(3)
3.5
3.6
What is the range of f ?
Write down the equations of the asymptotes of g .
(1)
(2)
3.7
3.8
Determine the y -intercept of g .
Write down the equation of the axes of symmetry of g .
(2)
(2)
3.9
3.10
Sketch the graph of g on its own set of axes.
Determine the average gradient of g between the points
(4)
x  2 and x  1.
(3)
g x  
4
x3
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Compiled By: Mr Motsumi
4.
Given the following two functions:
h x  
1
5
x
4.1
Determine the x -intercept of h.
(3)
4.2
Sketch neat graphs of h and g on the same set of axes. Clearly show all
intercepts with the axes as well as asymptote.
(5)
4.3
Write down the equation of the vertical asymptote of h.
(1)
4.4
Determine the coordinates of the point of intersection of h and g . Show all
calculations.
(5)
4.5
4.6
5.
6.
and
g x   x  5
Write down the equation of f if f is the reflection of g about the line
y  4.
(2)
Write down the equation if h is translated so that  2 ; 3 is the new
point of intersection of the asymptotes.
(2)
Given: f x  
8
4
x 8
5.1
Write down the equation of the asymptotes of f .
(2)
5.2
Write down the domain and range of f .
(2)
5.3
Draw the graph of f showing all intercepts and asymptotes.
(4)
5.4
Use your graphs to solve for x, if:
5.4.1
8
 4
x 8
(3)
5.4.2
f x   3
(3)
5.5
Determine the equation of the positive axis of symmetry of f .
(2)
5.6
Determine the equation of g if g x   f x  2  2.
(2)
Given: f x  
3
1
x2
and
g x   2 x  4
6.1
Determine f  3.
(1)
6.2
Determine x if g x   4.
(2)
6.3
Write down the asymptotes of f .
(2)
6.4
Write down the range of g .
(1)
6.5
Determine the coordinates of the x - and y -intercepts of f .
(5)
6.6
Determine the equation of the axes of symmetry of f which has a
negative gradient. Leave your answer in the form y  mx  c.
(2)
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Compiled By: Mr Motsumi
6.7
6.8
Sketch the graphs of f and g on the same system of axes. Clearly show ALL
intercepts with the axes and any asymptotes.
If it is given that f  1  g  1, determine the values of x for which
g x   f x .
7.
8.
Given: f x  
4
2
x3
(2)
and
g x   x  2
7.1
Write down the equation of the asymptotes of f .
(2)
7.2
Determine the x -intercept of f .
(3)
7.3
Determine the x -intercept of f .
(2)
7.4
Sketch the graphs of f and g on the same system of axes. Show clearly
ALL the intercepts with the axes and any asymptotes.
(5)
7.5
Calculate the x -coordinates of the points of intersection of f and g .
(4)
7.6
If x  3, determine the values x for which
7.7
The line y  x  1 cuts f at P1 ; 0 and Q. Write down the
coordinates of Q.
4
 2  x  2.
x3
(2)
(3)
 
Given:
hx   4 2 x  1
8.1
Determine the coordinates of the y -intercepts of h.
(2)
8.2
Explain why h does not have an x -intercept.
(2)
8.3
Draw a sketch graph of h, clearly showing all asymptotes, intercepts
with the axes and at least one other point on h.
8.4
9.
(6)

(3)

Describe the transformation from h to g if g x  4 2
Given: f x  
x

2.
(2)
8
3
x2
9.1
Write down the equation of the asymptotes of f .
(2)
9.2
Calculate the x and y -intercepts of f .
(3)
9.3
Sketch the graph of f . Show clearly the intercepts with the axes and
the asymptotes.
(3)
9.4
If y  x  k is an equation of the line of symmetry of f , calculate the
value of k .
(2)
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Compiled By: Mr Motsumi
10.
Given:
f x   2 x  4
10.1
Write down the equation of the asymptote of f .
(2)
10.2
Determine the x -intercept and y -intercept of f .
(5)
10.3
Draw a neat sketch of f , showing clearly all intercepts with the
axes and asymptotes.
(4)
10.4
Write down the range of f .
(2)
10.5
1
Describe the transformation of f to g if g x      4.
2
x
11.
12.
13.
Given: f x  
2
1
x 1
and
(2)
g x    x 2  x  6
11.1
Determine the x - and y -intercepts of f and g .
(8)
11.2
Write down the equations of asymptotes of f .
(2)
11.3
Determine the coordinates of the turning point of g .
(2)
11.4
Sketch the graphs of f and g on the same set of axes. Clearly indicate
the intercepts, asymptotes and turning point on the sketch.
(4)
11.5
Give the range of g .
(2)
11.6
Determine the equation of the vertical asymptote of f x  2.
(1)
f x  
Given the function:
6
1
x3
12.1
Write down the asymptotes of f .
(2)
12.2
Sketch the graph f x , showing clearly the asymptotes and intercepts.
(6)
12.3
For which values of x is f x   0 ?
(2)
12.4
Determine the average gradient between the points x  2 and x  0
(5)
Given: f x  
3x  2
x2
3x  2
8

3
x2
x2
13.1
Show that
(3)
13.2
Calculate the intercepts of f .
(3)
13.3
Sketch the graph of f .
(5)
13.4
Determine g x  if g x   f x  1.
(2)
13.5
Write down the domain of g x .
(2)
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Compiled By: Mr Motsumi
14.
Given the functions:
g x   
and
1
x  22  2
2
14.1
Write down the co-ordinates of the turning point of g .
(2)
14.2
Calculate the roots of the equation g x   0.
(4)
14.3
Sketch the graphs of y  f x  and y  g x  on the same system of axes.
Clearly indicate all turning points, intercepts with the axes and the intersection
between the graphs.
(4)
14.4
Determine the values of x for which g x   f x .
(4)
14.5
Describe how you would transform the graph of g to obtain the graph of
h x   
15.
f x   2 x  6
1
x  22  1
2
(2)
Given:
f x   2 x 2  x  6
15.1
Calculate the coordinates of the turning point of f .
(4)
15.2
Determine the y -intercept of f .
(1)
15.3
Determine the x -intercept of f .
(4)
15.4
Sketch the graph of f showing clearly all intercepts with the axes
and turning point.
(3)
15.5
Determine the value of k such that f x   k has equal roots.
(2)
15.6
If the graph of f is shifted two units to the right and one unit upwards

to form h, determine the equation of h in the form y  a x  p
16.
Given:
f x    x 2  2 x  1
16.1
Write f x  in the form f x  a x  p
16.2
Determine:
16.3
and
2  q.
(3)
g : 4 y  4x  5
 
2  q
(4)
(a)
f x   0
(3)
(b)
g 0
(2)
(c)
Turning point of f
(1)
Sketch graph of f and g on the same set of axes. Indicating all the
intercepts as calculated in 16.2.
16.4
(4)
Determine the intersection of f and g .
16.5
What name is given to g in relation to f ?
(4)
(1)
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Compiled By: Mr Motsumi
16.6
17.
If hx   f x   k determine all k values for which:
(a)
f  x  touches the x -axis at one point.
(1)
(b)
f does not touch the x -axis.
(1)
Given:
f x    x 2  2 x  3
17.1
Sketch the graphs of f and g on the same set of axes.
(9)
17.2
Determine the average gradient of f between x  3 and x  0.
(3)
17.3
For which value(s) of x is f x .g x   0 ?
(3)
17.4
Determine the value of c such that the x -axis will be a tangent to
the graph of h, where hx   f x   c.
(2)
17.5
Determine the y -intercept of t if t x    g x   1
17.6
The graph of k is a reflection of g about the y -axis. Write down the
and
g x   1 2 x
equation of k .
18.
(2)
(1)
Given:
f x  2 x 2  5x  3
18.1
Write down the coordinates of the y -intercept of f .
(1)
18.2
Determine the coordinates of the x -intercepts of f .
(3)
18.3
Determine the coordinates of the turning point of f .
(3)
18.4
Sketch the graph of y  f x , clearly showing the coordinates of the
turning points and the three intercepts with axes.
(3)
19.
80
Compiled By: Mr Motsumi
Part 2
The sketch below represents the graphs of f x  
2
 1 and g x   dx  e.
x3
Point B 3 ; 6 lies on the graph of g and the two graphs intersect at point A and C.
1.
1.1
Write down the equations of the asymptotes of f .
(2)
1.2
Write down the domain of f .
(2)
1.3
Determine the values of d and e, correct to the nearest integer, if the
graph of g makes an angle of 76 with the x -axis.
(3)
1.4
Determine the coordinates of A and C.
(6)
1.5
For what values of x is g x   f x ?
(3)
1.6
Determine an equation for axis of symmetry of f which has a positive slope.
(3)
3.
Part 3

Sketch the graph of f x  ax
1.

The range of f is   ; 7



a0
b0
2
 bx  c if it also given that:
One root of f is positive and the other root of f is negative.
(4)
81
Compiled By: Mr Motsumi