Math Readiness Test

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Math Readiness Test Fall 2012 (for Engineering Students)Math Readiness Test Practice

Madhavi Sivan

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Math Readiness Test Fall 2012 (for Engineering Students), Math Readiness Test PracticeMadhavi Sivan, 9/9/12 at 6:01 PM

Question 1:Score 0/1

Your response Correct response

What is the equation of the basic function shown

in the graph below?

What is the equation of the basic function shown

in the graph below?

Incorrect
http://gethelp%28%27student%27%2C%27https//maple-ta.uwaterloo.ca:443/mapleta/');http://aboutpopup%28%27https//maple-ta.uwaterloo.ca:443/mapleta/');http://aboutpopup%28%27https//maple-ta.uwaterloo.ca:443/mapleta/');http://aboutpopup%28%27https//maple-ta.uwaterloo.ca:443/mapleta/');http://aboutpopup%28%27https//maple-ta.uwaterloo.ca:443/mapleta/');http://gethelp%28%27student%27%2C%27https//maple-ta.uwaterloo.ca:443/mapleta/');


2/37

No answer (0%) x^3

Total grade: 0.01/1 = 0%

Comment:

The function is an excellent example of an odd function, which is any function satisfying .

An even function on the other hand is any function satisfying , such as .

Odd functions have the property that we can rotate their curves about the origin by and get the same curve.

Even functions have the property that we can reflect their curves across the -axis and get the same curve.



The expression

simplfies to:

(0%)

The expression

simplfies to:

Incorrect



3/37


4/37

.

No answer (0%)

.

x^2+x*y+y^2

Incorrect


Comment:

We simplify by factoring the denominator:

by the formula for a difference of squares, which is .

Then we're left with:

To deal with the numerator we can factor a difference of cubes by the following formula:

which gives us:

And thus



If is the midpoint of the line

segment which joins and

what is the value of ?

(0%)

If is the midpoint of the line

segment which joins and

what is the value of ?

13Incorrect


Comment:

Given two (distinct) points and ,

the midpoint of the line segment joining them is simply the point: .


5/37

The question is now simple, since by this and so .



Calculate the coordinates of the pointswhere the line through the point

with slope intersects the circle

with centre the origin and radius .

The leftmost point of intersection is:

(0%) , (0%)

The rightmost point of intersection is:

(0%) , (0%)

Calculate the coordinates of the pointswhere the line through the point

with slope intersects the circle

with centre the origin and radius .

The leftmost point of intersection is:

-4 , 3

The rightmost point of intersection is:

5 , 0

Incorrect

Total grade: 0.01/4 + 0.01/4 + 0.01/4 + 0.01/4 = 0% + 0% + 0% + 0%

Comment:

We first find the equation of the line using the point-slope formula:

In general, a line passing through the point with slope satisfies the equation:

.

And so the equation of our line is given by

Now we find the equation of our circle:

Our circle is centered at the origin and thus the equation is simply

.

Since any points of intersection must satisy both equations we can substitute


6/37

into our circle equation:

Expand the left-hand side and collect like terms, then divide by the coefficient of to get:

Thus roots exist at .

Substitute these values of into the equation of the line to solve for :

When , , thus a point of intersection is .

When , , thus another point of intersection is .


Solve: where .

On the image below click on the region that best fits your answer. There are several regions, somouse over the map to see them all before selecting your answer. Incorrect

Your Answer

Correct

Answer

Comment:

To solve the inequality , first bring all terms to one side:

Then factor to get:

This product is positive whenever 1 or 3 of the factors are positive, which is easily seen by sketching on anumber line.


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x - - + +

x - 2 - - - +

x + 2 - + + +

+--------+-----------------+-----------------+--------+

-10 -2 0 +2 10

Thus, the expression is positive on the intervals and .



Solve:

(0%)

Solve:

Incorrect


Comment:

For , both bases are powers of 3 ,

so use the basic property of exponents to re-write the equation as:

We solve the equation and get .



Solve .

Enter an exact answer.

(0%)

Solve .


(1+ln(8))/5Incorrect


Comment:

Solve the equation as follows:

(Taking on both sides)



8/37


Solve for in the logarithmic equation

.


(0%)


.


8

Incorrect


Comment:

We want to solve the logartihmic equation .

Since the two logarithmic terms on the left-hand side do not have the same base, use the change of base formula

to get:

Now that we have a sum of logarithmic terms with the same base, use the formula

to get:

Then the original equation becomes:

We can solve for the roots of this quadratic using the quadratic formula, which says that the roots of any quadratic


9/37

are given by .

In our quadratic we have , and .

Thus our quadratic has roots at:

And thus is the solution.



Determine for which values of the

quadratic has two

equal real roots

This quadratic has two equal real rootswhenever

No answer (0%)

Enter your answer as a list seperatedby commas.


quadratic has two

equal real roots


0, 4


Incorrect


Comment:

By the quadratic formula, the roots of the quadratic are given by:

and we call the discriminant.

When the quadratic has two real roots.

When the two roots are equal. This is called a double root.

When there are no real roots (since we would be looking for the root of a negative number).

In the given quadratic we have and .

So for this quadratic to have two equal real roots we need to have:

and this equality holds when or .



10/37


Express in terms of the other variables in the

diagram. Simplify your answer as far as possible.

No answer (0%)



r*t/sqrt(r^2 - h^2)

Incorrect


Comment:

Let us denote the height of the bigger triangle as .

We make use of the fact that both the triangles are similar and hence the ratio of their sides are the same. Thismeans that:

We can now solve for using the Pythagorean Theorem:


11/37



What is the center of the circle

?

The center (0%) , (0%) .


?

The center 2 , -1 . Incorrect

Total grade: 0.01/2 + 0.01/2 = 0% + 0%

Comment:

We need to put the given equation into the standard form of the equation of a circle.

For a circle of radius centered at ,

the equation of the circle is .

Begin by bringing the constant to the right-hand side:

Separate the and terms and complete the square on the resulting quadratics:

Thus, this is a circle centered at with radius .



Simplify

as far as possible.

No answer (0%)

Simplify

as far as possible.

1 Incorrect



12/37

Comment:

Simply the expression as follows:

(by factoring common terms)

(recall that )



Find , in radians, such that

, where

.

(0%)


, where

.

Incorrect


Comment:

Begin by isolating :

Using your knowledge of (and possibly a special triangle) we have:

since



When Sean was twelve years old, hewas twice as old as Tanya.

Tanya is now 12 years old.

How old is Sean now ?

(0%)

When Sean was twelve years old, hewas twice as old as Tanya.

Tanya is now 12 years old.

How old is Sean now ?

18Incorrect


13/37


Comment:

When Sean was twelve years old, Tanya was years old.

Tanya is now 12 years old, so 6 years have passed.Thus, Sean is now years old.


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Your response

Which of the following is the graph of the function ?

Correct


14/37

(100%)Comment:

Comment:

The absolute value function measures the distance of a number from .

This is why we also use the notation for the Euclidean norm (distance from the origin) of the point

in the -plane (i.e. the cartesian plane or the -plane).


Your response

The expression simplifies to:

(100%)

Correct

Comment:


15/37

Take a common denominator on top

Multiply out, gather like terms.

Invert and multiply

This expression occurs when you are finding the derivative of "from first principles". Notice that if you let h go

to 0 the result is .


Your response

If , then

and(100%)

Correct

Comment:

Anytime you have an equation of the form where you actually have two equations:

and .

So is equivalent to

and .

Solving:

and

and .


Your response


16/37

The expression factors to:

(x-4)*( x^2+4x+16) (100%) Correct

Comment:

is a difference of cubes.

The Difference of Cubes formula is: .

So we take and (so that and ) and we get:

.


Your response

If two lines are parallel, they have

the same slope.(100%) Correct

Comment:

Parallel lines have the same slope.

They do not have the samey-intercept, unless they are both the same line.



17/37

Your response

How many points of intersection are there between the curves

and ?

(100%)

Correct

Comment:

The curves intersect where the and values for both are the same.

Setting the 's equal we get:

Gathering like terms gives:

This factors to:

Giving solutions:

.

Since these are equal, there is only one point of intersection for these two curves.


Your response

The solution set for the inequality is:

(100%)Correct

Comment:

To solve the inequality , first factor:

This product is negative whenever 1 or 3 of the factors are positive, which is easily seen by sketching on a numberline:

x - - + +x - 4 - - - +

x + 4 - + + +

+--------+-----------------+-----------------+--------+

-4 0 +4

Thus, the expression is negative in the intervals (-,-4) and (0,4).



18/37

Your response

Solve:

(100%)

Correct

Comment:

The equation is solved with the following steps:


Your response

(100%)Correct

Comment:

Use the logarithmic property .

Question 10:Score 1/1Your response


.

Enter an exact answer.Correct


19/37

-12 (100%)

Comment:

We know that .

And so we have

Thus,


Your response

The domain of the function is a closed interval

with endpoints and . Find the length of the interval, .

12 (100%)Correct

Comment:

We know that the square root function is only defined for non-negative real numbers,

so we must determine when the quadratic is greater than or equal to .

The quadratic is negative and thus positive between its roots, which are clearly and .

Hence, the domain of is the closed interval .

Thus, .







Incorrect


20/37

((t^2)*(r^2))/(r^2-h^2) (0%) r*t/sqrt(r^2 - h^2)


Comment:

Let us denote the height of the bigger triangle as .

We make use of the fact that both the triangles are similar and hence the ratio of their sides are the same. Thismeans that:

We can now solve for using the Pythagorean Theorem:


21/37



Find the perimeter of the region

bounded by the lines

and the circle

shown by

the solid line below.

Enter an exact answer. To enter ,

write Pi.

(52*Pi)/3 (0%)units

Find the perimeter of the region

bounded by the lines

and the circle

shown by

the solid line below.

Enter an exact answer. To enter ,

write Pi.

26+52*Pi/3 units

Incorrect


Comment:

Since the diagram is symmetric about the -axis we can find the perimter in the top half of the diagram and then

multiply by two to get the full perimeter.

Start by finding the points of intersection (in the top half):

Substitute into the circle equation, which gives:

Then expand, simplify, collect like terms, and factor:

or

It is clear from the diagram and can easily be verified that when , , thus a point of intersection is

.

When , we have . Thus another point of intersection is


22/37

.

We want to know the distance from the origin to this last point, which is given by the Euclidean norm:

Also, the distance from the center of the circle to this point is since the point lies on the perimeter of this circle

which has radius .

Thus we have an equilateral triangle with side lengths and with vertices , and

.

Because it is an equilateral triangle we have that the interior angles are all .

Let's have a look at our updated diagram.

On any circle of radius the circumference is given by , and so the circumference of the entire circle is

, and the circumference of just the top half of the circle is .

Let be the length of the circular part of the perimeter in the top half (i.e. the solid line portion of the top half).

The ratio of to the full perimeter of the top half of our circle should be equivalent to the ratio of the angle

(subtended by ) to the full angle of a half-circle, which is .

Hence,

So the circular part of the perimeter for the top half of the diagram has length .

We also need to find the length of the line segment from to from the perimeter of the top

half of our diagram. Since this line segment is a side of the equilateral triangle we constructed, it has length .

Thus the top half of the diagram has perimeter

Thus the whole diagram has perimeter


23/37

Thus, the total perimter is .


Your response

Determine the length of in the following diagram:

Enter an exact answer. To enter a square root, use the sqrt(..) function, e.g.,

is entered as sqrt(2) .

24*sqrt(3) - 24 (100%)

Correct

Comment:Let's have another look at the diagram, and give it a few more labels to work with:

To calculate the angle we can use the triangle sum theorem which says that the sum of the interior angles of any

triangle is .

Using the inner triangle, we have

.

Now using the outer triangle to find , we have

.

Then the length of is given by .

Calculate the length of the base as follows:

And so .

Question 15:Score 0.75/1


Find all values of , in radians, such

that ,

where .

Enterexact answers as a list

separated by commas. To enter ,

write Pi. You must be explicit about

Find all values of , in radians, such

that ,

where .

Enterexact answers as a list

separated by commas. To enter ,

write Pi. You must be explicit about

Incorrect


24/37

multiplication, so use * to multiply.

0, 2*Pi/3 , 2*Pi (75%)

multiplication, so use * to multiply.

0, 2/3*Pi, 4/3*Pi,2*Pi

Total grade: 0.751/1 = 75%Comment:

To solve where ,

we must remember a fundamental identity to simplify this problem:

If we substitute into our equation we get:

.

Now let .

Then we have a quadratic equation we can factor which gives:

So either ,

or .

Thus the equation holds for .



A wire in length is cut

into two parts, one of which has

length . The piece of length

is formed into a circle, and theother piece is formed into asquare. Find a function that

expresses the total area

enclosed by the circle and the

square.

Enter an exact expression for the

area. To enter write Pi.

x^2/4*Pi +((32-x)/4)^2 (0%)

A wire in length is cut into two parts,

one of which has length . The piece of

length is formed into a circle, and the other

piece is formed into a square. Find a function

that expresses the total area enclosed

by the circle and the square.

Enter an exact expression for the area. To

enter write Pi.

(8-x/4)^2+x^2/(4*Pi)

Incorrect


Comment:


25/37

To solve this, find the area of the square and circle seperately, then add them together.

The perimeter of the square has length ,

and hence each side of the square has length .

The area of the square is then given by

Now we relate the circumference of the circle, which we know to be , to the radius of the circle:

Recall the formula for the circumference of a circle is , and ,

thus .

The area of a circle is given by .

And so we have .

Thus, the total area is given by

.


26/37


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Yourresponse

Correct response

Which ofthefollowing isthe graph of

the function

?

(0%)Comment:

Which of the following is the graph of the function ?

Incorrect


27/37


Comment:

The function is sometimes called the reciprocal function.

Notice the vertical asymptote at , and the horizontal asymptote at as approaches

positive/negative infinity.

This plot helps us visualize what happens to fractions as the denominator becomes large or small.



The expression

simplifies to:

(0%)

The expression

simplifies to:

Incorrect


Comment:

The "trick" here is to use theDifference of Squares formula: a2

- b2

= (a - b)(a + b) and recognize that you have the"(a - b)" term. Thus:


28/37

Applying DOS formula to the top



Solve .

Enter your answer as a list of numbersseperated by commas.

No answer (0%)

Solve .

Enter your answer as a list of numbersseperated by commas.

14, 0Incorrect


Comment:

implies

or

Solving these gives or respectively.



Simplify as far as possible:

.

No answer (0%)

Simplify as far as possible:

.

x^2+x*y+y^2

Incorrect


Comment:

We simplify by factoring the denominator:

by the formula for a difference of squares, which is .

Then we're left with:


29/37

To deal with the numerator we can factor a difference of cubes by the following formula:

which gives us:

And thus




(0%)


the same slope.Incorrect


Comment:

Parallel lines have the same slope.

They do not have the samey-intercept, unless they are both the same line.


30/37



Find the points of intersection betweenthe curves

and.

Enterexact answers.

Leftmost point: (0%) (0%)

Rightmost point: (0%)

(0%)

Find the points of intersection betweenthe curves

and.

Enterexact answers.

Leftmost point: -3 -32

Rightmost point: -1 -10

Incorrect

Total grade: 0.01/4 + 0.01/4 + 0.01/4 + 0.01/4 = 0% + 0% + 0% + 0%

Comment:

The curves intersect where the and values for both are the same.

First rearrange the equations to be in terms of :

Setting the 's equal we get:

Giving solutions:

and

Substituting thex-values back into either equation to find the correspondingy-values gives

and respectively.



The solution set for the

inequality

is:(0%)

The solution set for the inequality

is:

Incorrect


Comment:

To solve the inequality , first factor:


31/37

This product is negative whenever 1 or 3 of the factors are positive, which is easily seen by sketching on a numberline:

x - - + +

x - 4 - - - +

x + 4 - + + +

+--------+-----------------+-----------------+--------+

-4 0 +4

Thus, the expression is negative in the intervals (-,-4) and (0,4).



Solve:

(0%)

Solve:

Incorrect


Comment:

The equation is solved with the following steps:



(0%)Comment: Incorrect



32/37

Comment:

This is a basic property of logarithms.

Start with , , and (all by the definition of ) .

Then.




.


(0%)


.


64

Incorrect


Comment:

We want to solve the logartihmic equation .

Since the two logarithmic terms on the left-hand side do not have the same base, use the change of base formula

to get:

Now that we have a sum of logarithmic terms with the same base, use the formula

to get:

Then the original equation becomes:


33/37

We can solve for the roots of this quadratic using the quadratic formula, which says that the roots of any quadratic

are given by .

In our quadratic we have , and .

Thus our quadratic has roots at:

And thus is the solution.




quadratic has two

equal real roots


No answer (0%)



quadratic has two

equal real roots


0, 4


Incorrect


Comment:

By the quadratic formula, the roots of the quadratic are given by:

and we call the discriminant.

When the quadratic has two real roots.

When the two roots are equal. This is called a double root.

When there are no real roots (since we would be looking for the root of a negative number).

In the given quadratic we have and .

So for this quadratic to have two equal real roots we need to have:


34/37

and this equality holds when or .



In triangle shown below, is

the midpoint of .

If and

, find .

(0%)

In triangle shown below, is

the midpoint of .

If and

, find .

60

Incorrect

Total grade: 0.01/1 = 0%Comment:

We are given that and .

Also, since is the midpoint , we know that .

, then using , we find .

Since , the triangle on the right is isosceles and .

Now, , so and the triangle on the left is also isosceles.

Thus, , so




?

The center (0%) , (0%) .


?

The center 7 , 9 . Incorrect

Total grade: 0.01/2 + 0.01/2 = 0% + 0%

Comment:

We need to put the given equation into the standard form of the equation of a circle.

For a circle of radius centered at ,

the equation of the circle is .

Begin by bringing the constant to the right-hand side:


35/37

Separate the and terms and complete the square on the resulting quadratics:

Thus, this is a circle centered at with radius .



If and is between

and , what is the value of ?


(0%)

If and is between

and , what is the value of ?


7/25

Incorrect


Comment:

Since is between and (which is and ), and ,

we can draw a right triangle diagram with as one of the angles:

Calculate the length of the hypotenuse using the Pythagorean Theorem:

Hence, .




, where

.


, where

. Incorrect


36/37

(0%)


Comment:

Begin by isolating :

Using your knowledge of (and possibly a special triangle) we have:

since



A wire in length is cut

into two parts, one of which has

length . The piece of length

is formed into a circle, and the

other piece is formed into asquare. Find a function that

expresses the total area

enclosed by the circle and thesquare.

Enter an exact expression for

the area. To enter write Pi.

No answer (0%)

A wire in length is cut into two parts,

one of which has length . The piece of

length is formed into a circle, and the other

piece is formed into a square. Find a function

that expresses the total area enclosed

by the circle and the square.

Enter an exact expression for the area. To

enter write Pi.

(11-x/4)^2+x^2/(4*Pi)

Incorrect


Comment:

To solve this, find the area of the square and circle seperately, then add them together.

The perimeter of the square has length ,

and hence each side of the square has length .

The area of the square is then given by

Now we relate the circumference of the circle, which we know to be , to the radius of the circle:

Recall the formula for the circumference of a circle is , and ,


37/37

thus .

The area of a circle is given by .

And so we have .

Thus, the total area is given by

.

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