math readiness test
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7/31/2019 Math Readiness Test
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Math Readiness Test Fall 2012 (for Engineering Students)Math Readiness Test Practice
Madhavi Sivan
20455645
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StudentAboutQuit & Save
Math Readiness T 7778 viewdetails Math Readiness T
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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/'); -
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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.
Question 2:Score 0/1
Your response Correct response
The expression
simplfies to:
(0%)
The expression
simplfies to:
Incorrect
Total grade: 0.01/1 = 0%
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.
No answer (0%)
.
x^2+x*y+y^2
Incorrect
Total grade: 0.01/1 = 0%
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
Question 5:Score 0/1
Your response Correct response
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
Total grade: 0.01/1 = 0%
Comment:
Given two (distinct) points and ,
the midpoint of the line segment joining them is simply the point: .
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The question is now simple, since by this and so .
Question 6:Score 0/1
Your response Correct response
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
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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 .
Question 7:Score 0/1
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 .
Question 8:Score 0/1
Your response Correct response
Solve:
(0%)
Solve:
Incorrect
Total grade: 0.01/1 = 0%
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 .
Question 9:Score 0/1
Your response Correct response
Solve .
Enter an exact answer.
(0%)
Solve .
Enter an exact answer.
(1+ln(8))/5Incorrect
Total grade: 0.01/1 = 0%
Comment:
Solve the equation as follows:
(Taking on both sides)
Question 10:Score 0/1
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Your response Correct response
Solve for in the logarithmic equation
.
Enter an exact answer.
(0%)
Solve for in the logarithmic equation
.
Enter an exact answer.
8
Incorrect
Total grade: 0.01/1 = 0%
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
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are given by .
In our quadratic we have , and .
Thus our quadratic has roots at:
And thus is the solution.
Question 11:Score 0/1
Your response Correct response
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.
Determine for which values of the
quadratic has two
equal real roots
This quadratic has two equal real rootswhenever
0, 4
Enter your answer as a list seperatedby commas.
Incorrect
Total grade: 0.01/1 = 0%
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 .
Question 12:Score 0/1
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Your response Correct response
Express in terms of the other variables in the
diagram. Simplify your answer as far as possible.
No answer (0%)
Express in terms of the other variables in the
diagram. Simplify your answer as far as possible.
r*t/sqrt(r^2 - h^2)
Incorrect
Total grade: 0.01/1 = 0%
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:
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Question 13:Score 0/1
Your response Correct response
What is the center of the circle
?
The center (0%) , (0%) .
What is the center of the circle
?
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 .
Question 14:Score 0/1
Your response Correct response
Simplify
as far as possible.
No answer (0%)
Simplify
as far as possible.
1 Incorrect
Total grade: 0.01/1 = 0%
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Comment:
Simply the expression as follows:
(by factoring common terms)
(recall that )
Question 15:Score 0/1
Your response Correct response
Find , in radians, such that
, where
.
(0%)
Find , in radians, such that
, where
.
Incorrect
Total grade: 0.01/1 = 0%
Comment:
Begin by isolating :
Using your knowledge of (and possibly a special triangle) we have:
since
Question 16:Score 0/1
Your response Correct response
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
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Total grade: 0.01/1 = 0%
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.
Math Readiness Test Fall 2012 (for Engineering Students)Math Readiness Test Practice
Madhavi Sivan
20455645
View Details View GradeHelp
StudentAboutQuit & Save
Math Readiness T 7778 viewdetails Math Readiness T
Feedback: Details Report[PRINT]
Math Readiness Test Fall 2012 (for Engineering Students), Math Readiness Test PracticeMadhavi Sivan, 9/9/12 at 6:03 PM
Question 1:Score 1/1
Your response
Which of the following is the graph of the function ?
Correct
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/'); -
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(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).
Question 2:Score 1/1
Your response
The expression simplifies to:
(100%)
Correct
Comment:
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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 .
Question 3:Score 1/1
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 .
Question 4:Score 1/1
Your response
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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:
.
Question 5:Score 1/1
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.
Question 6:Score 1/1
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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.
Question 7:Score 1/1
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).
Question 8:Score 1/1
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Your response
Solve:
(100%)
Correct
Comment:
The equation is solved with the following steps:
Question 9:Score 1/1
Your response
(100%)Correct
Comment:
Use the logarithmic property .
Question 10:Score 1/1Your response
Solve for in the logarithmic equation
.
Enter an exact answer.Correct
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-12 (100%)
Comment:
We know that .
And so we have
Thus,
Question 11:Score 1/1
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, .
Question 12:Score 0/1
Your response Correct response
Express in terms of the other variables in the
diagram. Simplify your answer as far as possible.
Express in terms of the other variables in the
diagram. Simplify your answer as far as possible.
Incorrect
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((t^2)*(r^2))/(r^2-h^2) (0%) r*t/sqrt(r^2 - h^2)
Total grade: 0.01/1 = 0%
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:
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Question 13:Score 0/1
Your response Correct response
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
Total grade: 0.01/1 = 0%
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
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.
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
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Thus, the total perimter is .
Question 14:Score 1/1
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
Your response Correct response
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
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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 .
Question 16:Score 0/1
Your response Correct response
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
Total grade: 0.01/1 = 0%
Comment:
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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
.
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Math Readiness Test Fall 2012 (for Engineering Students)Math Readiness Test Practice
Madhavi Sivan20455645
View Details View GradeHelp
StudentAboutQuit & Save
Math Readiness T 7778 viewdetails Math Readiness T
Feedback: Details Report
[PRINT]
Math Readiness Test Fall 2012 (for Engineering Students), Math Readiness Test PracticeMadhavi Sivan, 9/9/12 at 8:02 PM
Question 1:Score 0/1
Yourresponse
Correct response
Which ofthefollowing isthe graph of
the function
?
(0%)Comment:
Which of the following is the graph of the function ?
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/'); -
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Total grade: 0.01/1 = 0%
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.
Question 2:Score 0/1
Your response Correct response
The expression
simplifies to:
(0%)
The expression
simplifies to:
Incorrect
Total grade: 0.01/1 = 0%
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:
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Applying DOS formula to the top
Question 3:Score 0/1
Your response Correct response
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
Total grade: 0.01/1 = 0%
Comment:
implies
or
Solving these gives or respectively.
Question 4:Score 0/1
Your response Correct response
Simplify as far as possible:
.
No answer (0%)
Simplify as far as possible:
.
x^2+x*y+y^2
Incorrect
Total grade: 0.01/1 = 0%
Comment:
We simplify by factoring the denominator:
by the formula for a difference of squares, which is .
Then we're left with:
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To deal with the numerator we can factor a difference of cubes by the following formula:
which gives us:
And thus
Question 5:Score 0/1
Your response Correct response
If two lines are parallel, they have
(0%)
If two lines are parallel, they have
the same slope.Incorrect
Total grade: 0.01/1 = 0%
Comment:
Parallel lines have the same slope.
They do not have the samey-intercept, unless they are both the same line.
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30/37
Question 6:Score 0/1
Your response Correct response
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.
Question 7:Score 0/1
Your response Correct response
The solution set for the
inequality
is:(0%)
The solution set for the inequality
is:
Incorrect
Total grade: 0.01/1 = 0%
Comment:
To solve the inequality , first factor:
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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).
Question 8:Score 0/1
Your response Correct response
Solve:
(0%)
Solve:
Incorrect
Total grade: 0.01/1 = 0%
Comment:
The equation is solved with the following steps:
Question 9:Score 0/1
Your response Correct response
(0%)Comment: Incorrect
Total grade: 0.01/1 = 0%
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7/31/2019 Math Readiness Test
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Comment:
This is a basic property of logarithms.
Start with , , and (all by the definition of ) .
Then.
Question 10:Score 0/1
Your response Correct response
Solve for in the logarithmic equation
.
Enter an exact answer.
(0%)
Solve for in the logarithmic equation
.
Enter an exact answer.
64
Incorrect
Total grade: 0.01/1 = 0%
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:
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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.
Question 11:Score 0/1
Your response Correct response
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.
Determine for which values of the
quadratic has two
equal real roots
This quadratic has two equal real rootswhenever
0, 4
Enter your answer as a list seperatedby commas.
Incorrect
Total grade: 0.01/1 = 0%
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:
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and this equality holds when or .
Question 12:Score 0/1
Your response Correct response
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
Question 13:Score 0/1
Your response Correct response
What is the center of the circle
?
The center (0%) , (0%) .
What is the center of the circle
?
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:
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Separate the and terms and complete the square on the resulting quadratics:
Thus, this is a circle centered at with radius .
Question 14:Score 0/1
Your response Correct response
If and is between
and , what is the value of ?
Enter an exact answer.
(0%)
If and is between
and , what is the value of ?
Enter an exact answer.
7/25
Incorrect
Total grade: 0.01/1 = 0%
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, .
Question 15:Score 0/1
Your response Correct response
Find , in radians, such that
, where
.
Find , in radians, such that
, where
. Incorrect
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7/31/2019 Math Readiness Test
36/37
(0%)
Total grade: 0.01/1 = 0%
Comment:
Begin by isolating :
Using your knowledge of (and possibly a special triangle) we have:
since
Question 16:Score 0/1
Your response Correct response
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
Total grade: 0.01/1 = 0%
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 ,
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thus .
The area of a circle is given by .
And so we have .
Thus, the total area is given by
.
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