1

I Strategy For Studying FE Exam Morning Subjects

• Math 15%• Engineering Probability and Statistics 7%• Chemistry 9%• Computers 7%• Ethics 7%• Engineering Economics 8%• Statics and Dynamics 10%• Strength of materials 7% • Material Properties 7%

FE Exam Morning Subjects

• Fluid 7%• Electricity and Magnetism 9%• Thermodynamics 7%

FE Exam Afternoon Subjects• Advanced Engineering Mathematics 10%• Engineering Probability and Statistics 9%• Biology 5%• Engineering Economics 10%• Application of Engineering Mechanics 13%• Engineering Materials 11%• Fluid 15%• Electricity 12%• Thermodynamics and heat transfer 15%

Combined• Math 12.5%• Fluid 11%• Thermodynamics 11%• Electricity 10.5%• Engineering Economics 9%• Material Properties 9% • Eng. Probability and Statistics 8%• Statics and Dynamics 8%• Strength of materials 8% • Chemistry 4.5%• Computers 3.5%• Ethics 3.5%• Biology 2.5%

Subject % in EIT HoursMath 12.5 16Thermodynamics 11 16Fluid Mechanics 11 16Electricity 10.5 16Engineering Economy 9 8Material 9 8Probability 8 8Strength of Material 8 8Dynamics +Calculator 4+? 8Biology+Chemistry 7 8Computer+measurement+control 3.5 4Strategy for taking the exam 4

2

Strategy for StudyingA) Study Smart Not Hard

B) Focus On the Primary Topics• Math• Fluids• Thermodynamics• Electricity

Is passing the EIT Exam possible?

Passing the exam is possible, just focus.In order to pass the EIT exam a student should spend a good portion of their time studying the primary topics, that are mentioned in the previous slide, and score at least 80% on the midterm given in this class. Don’t let fear or anxiety get in the way of studying; as it did to our friend in the Diagram below.

• After teaching the course on a particular topic several additional problems will be placed in the web side. Study those

• Once the midterm is completed, students are strongly encouraged to spend at least 5 hours per week on the primary courses in addition to the new material that is taught.

PRACTICE MAKES PERFECT!

It is important to keep track of how long it takes to complete the problem sets. This will prepare you for the actual exam so that you can manage your time wisely.

Remember, the test has a time limit!Combined

• Math 12.5%• Fluid 11%• Thermodynamics 11%• Electricity 10.5%

• Engineering Economics 9%• Material Properties 9% • Eng. Probability and Statistics 8%• Statics and Dynamics 8%• Strength of materials 8%

• Chemistry 4.5%• Computers 3.5%• Ethics 3.5%• Biology 2.5%

3

Try to focus on mastering at least two of the following topics:

• Engineering Economics 9%• Material Properties 9% • Eng. Probability and Statistics 8%• Statics and Dynamics 8%• Strength of materials 8%

(Note: It is strongly suggested that you select Engineering Economy as one of your topics)

Once you have mastered your topics, spend at least two hours a week reviewing the lectures and studying the extra problems available on the webpage that relate to your topics, mentioned in the previous slide. Also remember towork on the 4 primary topics as well.

Study, Study, and then Study again!

When the Midterm is Complete

A) Every week spend a minimum of 5 hours reviewing the topics or attempting problems in 4 of the primary subjects. This is in addition to the new materials which are taught every week.

B) For 2 subjects which you picked in the second group , spend an additional 2 hours a week reviewing.

C) Work on the extra problems from our website that can be used to accomplish both A) and B).

(pages 19-20)

• SI UNITS (The Metric System)

Converting Units

• English System

English System• Length = feet (ft)

• Weight = pounds (lbf)

• Time = seconds (s)

• For a system, the unit mass is given in lbm or slug

Basic Metric System

• Length = meter (m)

• Mass = kilograms (kg)

• Time = second (s)

Weight is measured in Newtons

4

• A table for metric prefixes and fundamental constants is given on page 19. When having to convert units, use page 20 to look up the conversion factors.

• To Convert Temperatures

• F = 1.8(C) + 32• C= (F - 32)/1.8• R= F + 459.69• K= C + 273.15

• Commonly used Equivalents

• 1 gallon of water weighs 8.34 lbf• 1 cubic foot of water weighs 62.4 lbf• 1 cubic inch of mercury weighs 0.491 lbf• The mass of 1 cubic meter of water 1.000kg

Equation of a Straight Line(page 21 first column)

Equation of a LineThe equation of a line in y-intercept form is

y = mx + b

Slope Y intercept

Slope

12

12XXYY

m−−

===X in Change

Yin ChangeSlope

Using the diagram above what is the equation of the line?

A) Y = 0.5x + 1 B) y = 2x - 1 C) y = 2x + 2 D) y = 0.5x - 1

A line, AB, makes a 30 degree angle with the x axis. If the angle is measured with respect to the counter clockwise direction, CCW, and crosses the y axis at y = 3 what is the equation of the line AB?

A) y = 0.577x + 3B) y = 1.732x + 3 C) y = 0.476x - 3 D) y = -0.5x + 4

5

Equations for a Straight Line: Other Forms

• If a line has an x-intercept of 4 and a y-intercept of - 6, what is the general form of the equation for this straight line?

A) 4x-3y-18 = 0B) 2x+6y+18 = 0C) 3x-2y-12 = 0D) 3x+5y-15 = 0

))(( 112

121 xx

xxyyyy −

−−

=−

212

21221 )()( yyxxpp −+−=

Examples• If a line passes though the points A (2,3) and B (6,-5),

what is the equation of this line in y-intercept form?

A) y = -2x + 7 B) y = 4x - 5 C) y= - 4x - 9 D) y = x + 6

Parallel and Perpendicular Lines

If we have two lines given by the equations, y = mx + b Y= Mx + c

• The lines are parallel if the slopes are equal:M = m

• The lines are perpendicular if the product of the slopes is negative one:

Mm= - 1

6

The angle of intersection that two lines, AB and CD, make with one another is given by the following expression:

Where α is the angle of intersection, m1 is the slope of line AB and m2 is the slope of line CD.

[ ]⎥⎦

⎤⎢⎣

⎡+−

=)1()(arctan

12

12

mmmmα

Example• Two lines in a coordinate system are parallel to one

another. If one of the lines is given by the equation y= 3x – 25, which of the following is the equation of the other line?

A) y = (1/3)x + 4 B) y = (-1/3)x + 5C) y = 3x - 12 D) y = -3x - 25

• What is the distance between the points A(3,-3,-1) and B(8,-3,0.73)?

A) 5.29 B) 6.87 C) 8 D) 3

Horizontal line• Mathematically: y =b

Vertical Line• Using the equation y = mx – b, we see that a

vertical line, that does not lie on top of the y axis, can be found if we set y = 0. This gives us an equation that looks like x = b/m

Quadratic Equations(page 21 first column)

02 =++ cbxax

roots no 04b If

2 <− ac

abac

204 −===− 21

2 x x b

If

aacbbac

24,04

2 −±−=>− 21

2 x x b

If

Equation:

Conditions:

1) What are the roots of the following equation

A) x = -0.36, -0.79 B) x = 0.36, -0.79C) x = -0.8, -0.2 D) x = -0.36,0.79

2) What are the roots of the following equation

A) x = 1, 5 B) x = 1, 6C) x = -1, -5 D) x = -1, 6

0237 2 =−+ xx

0562 =+− xx

7

Conic Sections(page 21 column 1 and 2)

Circle

Ellipse

Hyperbola

8

Parabola

Eccentricity• Eccentricity measures the deviation that a conic

section has compared to a circle.

Eccentricity

Conic Sections022 =+++++ FEyDxCyBxyAx

Conic Sections022 =+++++ FEyDxCyBxyAx

9

022 =+++++ FEyDxCyBxyAx

curves no is or there point, ellipse, circle, aeither isobject The04

If2 <− ACB

022 =+++++ FEyDxCyBxyAx

022 =++++ FEyDxAyAxcircle. a bemight object the

0B and 0, C A If:Condition =≠=

022 =+++++ FEyDxCyBxyAx

04 If:Condition

2 <− ACB

022 =+++++ FEyDxCyBxyAx

04If :Condition

2 =− ACB

022 =+++++ FEyDxCyBxyAx

04 If :Condition

2 >− ACB

• Condition: If A = B = C = 0 the object is a straight line.

022 =+++++ FEyDxCyBxyAx

10

Examples

0197753 )2 22 =++−++− yxyxyxWhat type of conic sections are represented by the equations given by 1) and 2) , respectively?

0943453 )1 22 =+++++ yxyxyx

A) circle B) ellipseC) hyperbola D) parabola

A Closer Look at Circles

022 =+++++ FEyDxCyBxyAx

022 =++++ FEyDxAyAxcircle. a bemight object the

0B and 0, C A If:Condition =≠=

222 )()( rkyhx =−+−022 =++++ FEyDxAyAx

Tangential Length from an Arbitrary Point P(x’,y’)

(page 22 column 1)

2222 )'()'( rkyhxt −−+−=

A Closer Look at an Ellipse

11

022 =+++++ FEyDxCyBxyAx

04 IF2 <− ACB

1)()(

2

20

2

20 =

−+

−

byy

axx

222 bac −=

0kh when 0) ,c ( at is focus The

==±

acabe /)/(1 2 =−==tyeccentrici

1)()(2

2

2

2

=−

+−

bky

ahx 1)()(

2

2

2

2

=−

+−

aky

bhx

1) Find the radius of the circle given by the following equation:

A) r=1 B) r=2C) r=3 D) r=1.5

2) What conic section is represented by the following equation:

A) circle B) ellipseC) point D) No curves

044222 =++++ yxyx

054222 =++++ yxyx

0144916 22 =−+ yx

3) What is the major and minor axis of an ellipse given by the equation below?

A) 4,3 B) 8, 6C) 2, 3 D) none of above

4) Where is the location of the foci for the ellipse described by the equation in the previous problem?

A) -2.64, 0 and 2.64, 0 B) 0, -5 and 0, 5C) 4, 3 and -4, -3 D) 0,2.64 and 0, -2.64

12

A Closer Look at Parabolas

• Concept of Solar Cooking system2ab-Symmetry x of Axis

2

=

++= cbxaxy

khxay +−= 2)(Where A(h,k) is thevertex of parabola

cbxaxy ++= 2

} px ±=⎯⎯←

} py ±=⎯⎯←

Directrix of a Parabola

left opens parbola

right opens parbola

down opens parbola

up opens parbola

2

2

2

2

41

41

41

41

yp

x

yp

x

xp

y

xp

y

−=

=

−=

=

2)(4

1 hxp

ky −±

=−

2)(4

1 kyp

hx −±

=− • Every parabola is defined by a point, called the focus, and a line not through the focus, called the directrix.

• A parabola is the locus of points that is equal to the distance between the focus and directrix.

13

cbxaxy ++= 2

} px ±=⎯⎯←

} py ±=⎯⎯←

Directrix of a Parabola

left opens parbola

right opens parbola

down opens parbola

up opens parbola

2

2

2

2

41

41

41

41

yp

x

yp

x

xp

y

xp

y

−=

=

−=

=

Find the focus point and directrix of Parabola

4

2xy =1

144±=

=⇒=y

pp

•Since a > 0 •Directrix y=-1

If a parabola has a vertex that is located at the point (6, 2) and has a directrix at x = 4.0 , which one of the following is the equation of the parabola?

A) (y - 2)2 = 6(x - 6) B) (x - 6)2 = 12(y - 2) C) (y - 2)2 = 8(x - 6) D) (y - 6)2 = 12(x - 2)

A Closer Look at HyperbolasThe hyperboloid is the design standard for all nuclear cooling towers.

By designing these cooling tower engineers can overcome the following problems:

A) The structure can withstand high windB) Structure is build with as little materialas possible

acabe

bky

ahx

/1:

1)()(

2

2

2

2

2

2

=+=

=−

−−

tyEccentrici

C,0)(ator e,0), a (at FocusFoci

±±=

14

A hyperbola is the set of all points P(x, y) in the plane such that

| PF1 - PF2 | = 2aAgain F1 and F2 are focus points.

Components of a Hyperbola

1)()(2

2

2

2

222

=−

−−

+=

bky

ahx

baC

1)()(2

2

2

2

222

=−

−−

+=

bhx

aky

baC

• A plane at an angle of 60 degrees, measured clock wise with respect to a horizontal plane, intersects the left side of a right circular cone. If the plane exits the cone by cutting through the base, what kind of conic section is created?

A) Ellipse B) CircleC) Parabola D) Hyperbola

15

parabola

A plane at an angle of 60 degrees, measured clock wise with respect to a horizontal plane, intersects the left side of a right circular cone. If the plane exits through the ride side of the cone what kind of conic section is created?

A) Ellipse B) CircleC) Parabola D) Hyperbola

Ellipes A Sphere and the Distance Between Two Points in Space

(page 22 column 2)

)()()(212121

zzyyxxd −+−+−=

• The equation for a sphere is given by:

• The distance between two points is given by:

16

Logarithms(page 22 column 2)

Mathematically: b = loga(x).

Which can also be written as x = ab

Definition: “a” is a positive, real number and is the base of the logarithm.

The Natural Logarithm

x = ln(b)

Which can also be written as b = ex.

Using our knowledge from the previous slide, we see

ln(b) = loge(b).

Which is the same as,

log(x) = log10(x)

Properties of Logarithms

yxyx

byxxy

xcx

cx

nb

b

c

xc

nb

loglog)/log(01log

1loglogloglog

loglog

loglog

log

−===

+==

=

=

)(anti

yxxy 10 as writtenbe can == 10log

Complex Numbers(page 23 column 1)

12 −=i

All complex are comprised of a real part and an imaginary part, that is denoted by i.

Complex numbers are often written in the form a + bi, where a and b are real numbers, and i gives the imaginary component.

Imaginary numbers are just an extension of real numbers.

Complex NumbersComplex numbers are often used in electronics since they allow the topic of alternating signals to be more easily understood and analyzed.

17

• Real part = V0 cos θ• Imaginary part = V0 sin θ

The complete complex value can then be described as, V0 cos (2πft) + j٠sin (2πft)

or V0Exp(j2πft)

Euler Relation

• Multiplying and Dividing Complex Numbers

Rationalizing Complex Number

Step 1:

Step 2:

18

Complex Number: Other Forms• Cartesian: Z= a+ bj

• Polar:

Which is related by:

)/(tan 1

22

ba

bar

rerZ j

−=

+=

=∠=

θ

θ θ

where

Polar Coordinates

)sin()cos(

θθ

ryrx

==

[ ])sin(cos θθ iriyx +=+

)sin()cos( θθθ iei +=

)sin()cos( θθθ ie i −=−

2)cos(

θθ

θii ee −+

=

iee ii

2)sin(

θθ

θ−−

=

Examples1) What is the value of log 50?

A) .34 B) 2.88D) 0.71 D) 1.69

2) Which of the following is the polar form of 3+4i?

A) 3(cos(23.17)+sin23,17)i)B) 3(cos(43.17)+sin43,17)i)C) 4(cos(23.17)+sin23,17)i)D) 5(cos(53.13)+sin53.13)i)

3) What is the value of x in the expression below?

A) 31234 B) 50118C)51970 D) 40234

0005.055.3

=−

x

5197010715.4log3.3log7.0

10log45loglog7.0105x0005.0

71.4

40.7-55.3

==

=⇒=−=−

×=⇒= −−

xxx

xx

19

Trigonometry(page 22 and 23)

eHypotenousOpposite

==ABACSin )(α

eHypotenousAdjacent

==ABBCCos )(α

AdjacentOpposite

==BCAC)tan(α

)cos()sin() tan(

)tan(1)cot(

)cos(1)sec(

)Sin(1)csc(

:relationsportant Im

ααα

αα

αα

αα

==

==

Trigonometric Identities

)tan()tan(1)tan()tan()tan(

)sin()()cos()cos()cos()()cos()cos()()(

1)()( 22

βαβαβα

βαβαβαβαβαβα

αα

−+

=+

−=++=+

=+

SinSinSinSin

CosSin

sin(A + B) = sin A cos B + cos A sin Bsin(A - B) = sin A cos B - cos A sin Bcos(A + B) = cos A cos B - sin A sin Bcos(A - B) = cos A cos B + sin A sin Btan(A + B) = (tan A + tan B)/(1 - tan A tan B)tan(A - B) = (tan A - tan B)/(1 + tan A tan B)

sin A + sin B = 2 sin ½(A + B) cos ½(A - B)sin A - sin B = 2 cos ½(A + B) sin ½(A - B)cos A + cos B = 2 cos ½(A + B) cos ½(A - B)cos A - cos B = - 2 sin ½(A + B) sin ½(A - B)tan A + tan B = sin (A + B)/(cos A cos B)tan A - tan B = sin (A - B)/(cos A cos B)

More Trigonometric Identities

)(tan1)tan(2)2tan(

)(21)()(

)()(2)2(

2

222

QQQ

QSinQSinQCos

QCosQSinQSin

−=

−=−

=

1) Which of the following expressions is equal to sin(2θ)?

2) Which of the following expressions is equal to cos(2θ)?

)2Sin( d) )2Cos( c)2Cos b) ))Cos(2Sin( a) 2

θθθθθ 1)( −

)2Sin( d) )2Cos( c)2Cos b) ))Cos(2Sin( a) 2

θθθθθ 1)( −

Examples

20

Law of Sines

Cc

Bb

Aa

sinsinsin==

Law of Cosines

Cabbac cos2222 −+=

If we use the triangle below and using the following values, sin A = a/10 and b = 5 , what is the value of B?

A) B=45 B) B=60C) B=30 D) B=32.7

Cc

Bb

Aa

sinsinsin==

Example 1) If A=90+a and “a” is an acute angle and

For any value of “a” alwaysA) 0<S<1 B) -1<S<0C) S=0 D) S>1

)(cos)()cos()(

22 aASinAASinS

+=

Matrix(page 23 column 2)

In mathematical terms, a matrix is a table of rows and columns that contains numerical values or variables.

The matrices above are 4 x 3 matrices. Rows are stated first then columns.

The value for a particular component is given as the following:

7 is or ) ,A(a 3223

A matrix can either be written using square brackets or parenthesis

21

Addition of Matrices• Two or more matrices can be summed if they contain

the same number of identical dimensions (m and n).

Index Notation:

Example:

Scalar Multiplication • A matrix can be multiplied by a scalar, by

multiplying every element of the matrix by scalar value.

• Example:

Matrix Multiplication• In order to multiply matrices, the number of columns

of left matrix has to be equal to the number of rows of the right matrix.

Identity Matrix• The identity matrix, is a special matrix square matrix,

n by n, in which all main diagonal components are equal to one and the rest of the elements are zero.

Transpose of a Matrix• Mathematically speaking, the transpose of an n-by-m

matrix A, is given by the m-by-n matrix defined as AT . In other words, the transpose of a matrix is when you switch the rows with the columns.

Determinant of Matrix Steps for addressing the determinant of a matrix:

A) Use the calculator.

B) To find the Inverse of a Matrix use the calculator

C) For manual calculations of Co-factors, adjoins and determinants of matrices use the appendix A

22

From the following system of equations, find the values of x, y, and z using matrices.

⎪⎩

⎪⎨

⎧

=++=++=−+

7592432

zyxzyxzyx

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −

794

151121132

zyx

A) For Manual Solution see Appendix AB) For FAST solution CALCULATOR

• 1. For this set of equations

• Find x, and y• A) x=2 y=3 B) x=2, y=4• C) x=1, y=3 D) x=1, y=5• .2. For this set of equations

• A) x=3, y=1, z=4• B) x=2, y=2, z=-1• C) x=3, y=1, z=2• D) x=2, y=3, z=1

• 3. The following Matrix has a special property in that it equals to its own inverse

• Find

• A) 1785 B) 8• C) 2 D) -8

]][2det[ 10A

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

=33806420494082017

A

Scalar

• A scalar is an object that has a magnitude but does not have a direction.

• Examples: Time and Temperature

Vectors(p24 column 1)

• A vector is an object that has both a magnitude and a direction.

23

jAiAA yx ˆˆ+=r Components of a Vector

X Component of Force=Fxi

NFFx 229)40cos(300)40cos( ===

Y component of Vector=FyjNFFy 193)40sin(300)40sin( ===

F=Fxi+Fyj

24

Vector Addition

jbaibajbibjaiaba

jbibb

jaiaa

yyxxyxyx

yx

yx

)()( +++=+++=+

+=

+=

rr

r

r

Vector Subtraction

Resultant of Vectors• The resultant is the sum of two or more vectors.

jiCBAR

jiC

jiB

jiA

25

43

52

4

−=++=

+=

−−=

−=

rrr

r

r

r

Dot Product• The dot product of two vectors is a scalar quantity

that is defined by:

tbaba cos=⋅

Example

kjiakjia7104

645

2

1

++=++=

r

rFind the dot product of the following vectors

102424020. 21 =++=aa rr

Angle Between Two Vectors

21

21)(VVVVangleCos ⋅

=

25

Example• What is the angle between the following two

vectors?

kjiakjia7104

645

2

1

++=++=

r

r

905.04910016361625

424020)cos(21

21 =++++

++=

⋅=

aaaaθ

2.25=θ

Examples1) Two vectors, A and B, are perpendicular to

each other. Vector A is given as A = i + 2j, and the magnitude of the x component of vector B is 3. Which of the following expression correctly gives the components of vector B?

A) 3i + j B) 3i - 0.5jC) 3i + 0.5j D) 3i - 1.5j

Cross-product)(θSinbaba =×

a = a1i + a2j + a3kb = b1i + b2j + b3k

ExampleFind the cross-product of the vectors, U = i + 2j + k and V = -j + 2k.

The area of the parallelogram is given by the magnitude of the cross-product of a and b.

)(θSinbaba =×

Gradient, Divergence and Curl(page 24)

kz

jy

i∂∂

+∂∂

+∂∂

=∇

∇

x

:asdefined is , operator,Delvector The

26

Gradient• The gradient of a scalar field is a vector field. The

vector field has the same magnitude and points in the same direction as the greatest rate of increase of the scalar field.

What is a Gradient (intuitively)

• The scalar field is defined by the colors black and red, with black representing highest value.

• The vector field corresponding to the scalar field is represented by the directional arrows.

• The net transfer of heat is in the opposite direction of the temperature gradient.

TkdtdQ

∇−=

Example Divergence

The DivergenceThe divergence of a vector field, is defined as the dot product or scalar product of the del operator with the function.

For example, is we have a vector in rectangular coordinates,

The divergence is given by

321kvjvivv ++=

What is Divergence (intuitively)

Divergence is the flow of a group of objects from a centralized point.

27

The expansion of a fluid flowing with a velocity field F, is capture by the divergence of F.

Curl Tornado and waterspout

The Curl• The curl of a vector function is found by taking

the cross-product or vector product of the del operator with the function.

Your Intuition About CurlThe curl describe the torque that a force field would exert on an object when measured fromthe central point.

CURL Vector Calculus Identities

• The divergence of the curl is equal to zero:

• The curl of a gradient is equal to zero:

28

Laplacian of a VectorThe Laplacian operator is defined as:

Example: The Laplacian of a scalar field f(x, y,z) = xy2 + z3 is given by:

The Laplacian of a scalar field can also be found using the following equation:

Example: Find the Laplacian of the scalar field, f, if the gradient off is given as,

kyzxj

zyxi

yzf 22

1−−=∇

If the gradient of the scalar field is given, to find the Laplacianall we need to do is find the divergence of the gradient or mathematically speaking,

f∇⋅∇

1) Find the radius of a circle described by the equation

A) 1 B) 2.7C) 3 D) 4.2

2) What type of conic section does the following equation represent:

A) ellipse B) hyperbolaC) circle D) Parabola

0424 22 =−−+− yyxx

104263 22 =−++ yyxyx

3) The equation of a straight line passing through (2,3), and (3,2) is:

A) x + 2y = 7 B) x + y = 5C) x – y = 3 D) 2x + y = 7

4) A line AB makes an angle of 45 degrees, measured clock wise, CW, from the line x = 3. What is the equation of AB if theline passes through the origin?

A) y= 3x B) y= (1/3)xC) y= x D) y= x - 3

5) What is the shortest distance between the two points (-2,3) and (4, -5) on a two dimensional graph is

A) 34 B) 10C) 9.6 D) 7

6) Find the roots of

A) 2,0.4,-2.4 B) 1, -0.4, 2.4C)-2, 0.4, -2.4 D) -2, -0.4, -1.4

7) The equation of a line with a slope of 2 and a Y-intercept of 3 is:

A) 0.5y-2x+3=0 B) 0.5y-x=1.5C) y+2x+3=0 D) 2x+y-3=0

0253 =+− xx

29

8) If the angle “a” is an acute angle and x = a + 90 and y = a + 180, what is the value for the equation below for a = 45.

A) S=2 B) S=-1.5 C) S=1.7 D) S= 1.3246

• What is the angle between two straight lines that are defined by the following equations:

y=3x+2 y=4x+7

A) 90deg B) 28.3deg C) 4.399deg D) 5.194

))(

)()()(()cos()sin(

tan2

224

aSinaSinaCosaCos

xxys +

=

Line AB is perpendicular To line CD. If Line AB is represented by:4y+2x=6

Then which one of these might be equation of line CDa) y+2x =9 b) y-2x=9c) 2y +x=5 d) 2y-x=4