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# 1

Q1: [ 3 Marks]
CLO 1: Perform algebraic manipulations including working with exponents,
radicals, polynomial operations, factoring and algebraic fractions.
Simplify the following expression:

Solution:
( )
( )( )
( )
( ) ( )
1
1
1
1
1 ) 1 (
1 1
1
2 2
2 3
=
+

+ +
+ +

+
+
x x x
x x x
x x
x

( )
( ) ( ) ( ) 1 2
1
1
) 1 (
1
1
2 2
3
2
3
+ +

+ +

+
x x x x
x
x
x
2

Q2: [ 4 Marks]
Present methods for solving first and second degree equations and inequalities
and related topics.
Find the values of (x) that satisfy the following equation:
0 1
2
= + + x x
Are these values (roots) real, imaginary or complex? What is the relation between
these roots?
1 = a , 1 = b , 1 = c
a
ac b b
x
2
4
2

=
1 2
1 1 4 1 1
2

= x
2
3 1
= x
i x .
2
3
2
1
1
|
|
.
|

\
|
+ =
i x .
2
3
2
1
2
|
|
.
|

\
|
=
these values (roots) are complex.
These two roots are complex conjugates.

3

Q3: [ 6 Marks]
Recognize and graph linear functions, rational functions, absolute value
functions and graph inequalities in two variables.
1. Graph the inequality : [ 3 Marks]
x y x 2 3 >
x x y + s 2 3
x y 3 3 s
x y s
x y =
Plot y against x and the region below the line y=x (including line points) is
the solution.

4

2. [ 3 Marks]
The manager of a weekend flea market knows from past experience that if
he charges (x) dollars for a rental space at the market, the number (y) of
spaces he can rent is given by the equation:
x y 2 100 =
a) Sketch a graph of this linear function.
b) What do the slope, the y-intercept and the x-intercept of the graph
represent?
Slope = -2 and it represents the number of spaces per dollar.
y-intercept = 100 and it represents the number of spaces if he charges
(zero dollars.)
x-intercept = 50 dollars for a rental space at the market when the number
of spaces is zero.

5

Q4: [ 5 Marks]
CLO 4: form and solve matrices and operations involving matrices.

Find the value of (x) that satisfies the solution for the following system of linear
equations using Cramers Rule:

3 2 4 = + z y x

z y x 7 1 3 = +
z y x = 9 8 2

|
|
|
.
|

\
|
=
1 9 2
7 3 1
2 4 1
A
, det (A) =1(3-63)-1(4+18)+2(28+6) = -14

|
|
|
.
|

\
|
=
8
1
3
b

|
|
|
.
|

\
|
=
1 9 8
7 3 1
2 4 3
x
A
, det (Ax) = 3(3-63)-1(4+18)+8(28+6) = -180 22 + 272 = 70

5
14
70
=

= =
A
A
x
x

6

Q5: [5 Marks]
CLO 5: Identify limits of algebraic, trigonometric, and composite functions.
1. On what intervals is the following function continuous? [3 Marks]

x
x
x g
2 cos 2
2 sin
) (
+
=

Then find:

x
x
x
2 cos 2
2 sin
lim
4
+

t

Solution:
The function is continuous on the open interval ( + , )
2
1
0 2
1
2
cos 2
2
sin
4
2 cos 2
4
2 sin
2 cos 2
2 sin
lim
4
=
+
=
+
=
|
.
|

\
|
+
|
.
|

\
|

=
+

t
t
t
t
t x
x
x

2. Find the value of the following limit:
[2 Marks]

2
4
2
2
lim

x
x
x

Solution:
( )
) 2 (
2 ) 2 (
lim
2

+
=

x
x x
x

( ) 4 2 2 2
lim
2
= + = +
=

x
x

7

Q6: [6 Marks]
CLO 6: solve for the derivatives of algebraic, trigonometric and composite
functions.
Find the equation of the tangent line to the curve:
2
3 6 x x y =
at the point where x=2.
Also find the point at which the rate of change of (y) with respect to (x) is equal to
zero. Is this point a maximum, a minimum or a turning point?
Solution:
x
dx
dy
6 6 =
= = = |
.
|

\
|
=
6 2 6 6
2 x
dx
dy
slope of tangent to curve at x = 2.
0 12 12 2 3 2 6
2
= = = y
Equation of straight line is:
c mx y + =
( ) 12 2 6 0 = + = c c
12 6 + = x y
1 0 6 6 = = = x x
dx
dy

3 1 3 1 6
2
= = y y so the point is (1,3)
6
2
2
=
dx
y d

Since second derivative is zero, therefore, the point (1,3) is a maximum.
8

Q7: [6 Marks]
CLO 7: find indefinite and definite integrals of algebraic, trigonometric, and
composite functions.
1. Find the area between the curve [3 Marks]
1
2
+ = x y
And the x-axis from x=0 to x=3.
( ) 12 0 ) 3
3
3
(
3
1
3
3
0
3 3
0
2
=
(

+ =
(

+ = + =
}
x
x
dx x A

2. Find the value of the following integral: [3 Marks]

( )dx x x
}

t
0
sin 5
( ) ( ) ( )
(

+ =
(

+ + =
(

+ =
}
) 1 ( 5 1 5
2
0 cos 5 0 ) cos 5
2
( cos 5
2
sin 5
2 2
0
2
0
t
t
t
t
t
x
x
dx x x

10
2
2
=
t

9

Q8: [5 Marks]
CLO 8: Graph and solve the equations of lines, circles, parabolas, ellipses and
Hyperbolas.
Decide which of the following equations represents a straight line, a parabola, a
circle, an ellipse or a hyperbola. Then sketch the graph of the equation that
represents an ellipse.
Circle x y = + 1
2 2

Line Straight x y . 1 + =
Hyperbola x y = 1
2 2

Parabola x y =
2

Ellipse
y x
= + 1
9 16
2 2
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