1.
The midpoint of the segment with endpoints (4, 8) and (-11, -2) is ______.
Explanation
The midpoint of a segment is found by taking the average of the x-coordinates and the average of the y-coordinates of the endpoints. For the given endpoints (4, 8) and (-11, -2), the average of the x-coordinates is (4 + (-11))/2 = -7/2 and the average of the y-coordinates is (8 + (-2))/2 = 3. Therefore, the midpoint is (-7/2, 3). Additionally, this can also be written as (-3.5, 3) since -7/2 is equal to -3.5.
2.
The distance between the points (2,1) and (6,4) is:
Explanation
[distance formula: square root of ((x1-x2)^2 + (y1-y2)^2))
3.
Write the equation of the circle with center (3, 7) and containing the point (11, 13).
Explanation
1. Use the distance formula to find the radius of the circle.
2. Substitute the values into the equation of a circle:
The equation of a circle with center (h, k) and radius r is (x-h)^2 + (y-k)^2 = r^2
As r=10, x=3, and y=7, your formula will be [hopefully posted in the answer box, and] (x-3)^2+(y-7)^2=100.
4.
1.A/n _______ is the set of points in a plane such that the sum of the distances from any point on the _____ to two fixd points (the foci) is the constant sum.
2.A/n _______ is the set of points in a plane such that the difference of the distances from P to fixed points F1 and F2, the foci, is constant.
3.A/n _______ is the set of all points in a plane that are an equal distance from both a fixed point, the focus, and a fixed line, the directrix.
Explanation
So the formulas for these superbly amazing conic sections are (besides the circle; we've already done that one):
Horizontal Axis Vertical Axis
Ellipse: ((x-h)^2)/a^2 + ((y-k)^2)/b^2=1 ((x-h)^2)/b^2 + ((y-k)^2)/a^2=1
Hyperbola: ((x-h)^2)/a^2 - ((y-k)^2)/b^2=1 ((y-k)^2)/a^2 - ((x-h)^2)/b^2=1
Parabola: x-h = (1/(4p))(y-k)^2 y-k = (1/(4p))(x-h)^2
In all of those formulas, co-vertices are represented by b, vertices by a, and foci by c.
5.
MATCHING:
So, we've already got the equations of all the basic conics. Now for the location of the vertices, foci, and co-vertices. Y'all will now match:
For a horizontal A, vertices are (h+a, k) and (h-a, k); foci are (h+c, k) and (h-c, k); co-vertices are (h, k+b) and (h, k-b); asymptotes are y-k= (+/-)b/a(x-h)
For a horizontal B, the focus is (h+p, k), and the directrix is x=h-p
For a horizontal C, the vertices are (h+a, k) and (h-a, k); foci are (h+c, k) and (h-c, k); co-vertices are (h, k+b) and (h, k-b).
The below answers are listed as the projected guesses for A, B, and C in that order.
Correct Answer
C. Parabola, Hyperbola, Ellipse
Explanation
The given answer is Parabola, Hyperbola, Ellipse. This is because the first set of information describes a parabola, with the focus and directrix given. The second set of information describes a hyperbola, with the vertices, foci, and co-vertices given. The third set of information describes an ellipse, with the vertices, foci, and co-vertices given. Therefore, the correct order is Parabola, Hyperbola, Ellipse.
6.
Write an equation in standard form for the below ellipse with center (0,0):
an ellipse with vertex (0,8) and co-vertex (3,0).
No spaces, and recognize all 'squared' symbols as I have been, for instances x^{2 }is equal to x^2.
Correct Answer
y^2/64+x^2/9=1
Explanation
Step 1. The vertex is on the y-axis.
Step 2. Identify the values of a and b ; a=8, b=3.
Step 3. Write the equation.
7.
Classifying Conic Sections:
B^{2}-4AC=0 in the form of Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 is a parabola.
Correct Answer
A. True
Explanation
The equation B^2 - 4AC = 0 is the discriminant of a quadratic equation, and when it is in the form of Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, it represents a conic section. When the discriminant is zero, it indicates that the conic section is a parabola. Therefore, the given statement "B2-4AC=0 in the form of Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 is a parabola" is true.
8.
Classifying Conic Sections:
B^{2}-4AC=0 in the form of Ax^2 + Bxy + Cy^2 + Dx + Ey + F > 0 is a hyperbola.
Correct Answer
A. Yes
Explanation
The equation B2-4AC=0 in the given form represents a conic section. A hyperbola is a type of conic section, so if the equation satisfies this condition, it can be classified as a hyperbola. Therefore, the correct answer is Yes.
9.
Classifying Conic Sections:
B^{2}-4AC < 0 and either B is not equal to 0 or A is not equal to C (in the form of Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0) is a circle.
Correct Answer
B. False--- no way man, that's an ellipse!
Explanation
You bet it is an ellipse!
10.
Classifying Conic Sections:
B^{2}-4AC < 0 and either B=0, and A=C (in the form of Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0) is a circle.
Correct Answer
A. True
Explanation
If B2-4AC < 0 and either B=0 and A=C, then the equation is in the form of a circle. This is because the condition B2-4AC < 0 ensures that the equation represents a non-degenerate conic section (not a line or a point), and when B=0 and A=C, the equation simplifies to the standard form of a circle. Therefore, the statement is true.
11.
Let's talk about nonlinear system of equations; rhar is a system in which at least one of the equations is not linear; a class of which is the conic sectiosn. You can solve these bad boys by graphing, substitution, and elimination, and the points you'll finally settle down with are the intersection points between the equations in your system. It appears that most of them are conic section and some other sort of linear or nonlinear equation.
Correct Answer
C. Oh, dear. Next is probability.
12.
Combination: order does not matter.
Permutation: order does indeed matter.
_{n}C_{r} = n! / r!(n-r)! <-- that is indeed a combination formula. Permutation's easier, do you know that one?
What is the probability of an event happening?
Answer only the second question please.
Correct Answer
B. Number of favorable outcomes / number of outcomes in the sample space
Explanation
The correct answer is "number of favorable outcomes / number of outcomes in the sample space". This is the formula for calculating the probability of an event happening. It represents the ratio of the number of favorable outcomes (outcomes that satisfy the event) to the total number of outcomes in the sample space (all possible outcomes).
13.
Each student received a 4-digit code to use the library computers, with no digit repeated. Manu received the code 7654. What was the probability that he would receive a code of consecutive numbers?
Correct Answer
1/360
14/540
Explanation
1. Order is important, so this is a permutation.
2. Find the number of outcomes in the sample space. The sample space is the number of permutations of 4 of 10 digits; 10P4.
3. Then you just count out the seven numbers that are consecutive and the reverse of these to get 14/10P4, or `14/540, or 1/360.
14.
You roll a dice (basic; numbered 1-6) twice. What is the probability that you will roll one number which is a prime number (not 1), and then roll a number greater than four but less than seven?
Correct Answer
1/6
Explanation
If A and B are independent events, then P(A and B) = P(A) x P(B).
2, 3, 5, are prime numbers (1/2)
5, 6 form a 2 in 6 chance (1/3)
1/3 x 1/2 is 1/6.
15.
Find the probability that the red cube shows a 5 and that the sum of the red cube and the orange cube is less than or equal to 8.
Correct Answer
B. 1/12
Explanation
Dependent events: P(A and B) = P(A) x P(B given A)
1/6 that red cube shows a 5
6, 7, 8,
16.
Inclusive events are events that have one or more outcome in common.
A simple event is an event that describes a single outcome.
A compound event is an event made up of one or more simple events.
Mutually exclusive events are events that cannot both occur in the same trial of an experiment (like rolling a 1 and a 2 on the same roll).
Correct Answer
B. False
Explanation
Just kidding, guys. Everything's true except that the third line, where it should be, "A compound event is an event made up of TWO or more simple events."
17.
Find the probability with a general dice of rolling at least one 4 when rolling 2 dice:
Correct Answer
C. 11/36
Explanation
P(4 or 4) = P(4) + P(4) - P(4 and 4), given the formula P (A or B) = P (A) + P (B) - P(A and B).
18.
Round to the second decimal place:
There are 4 students in a music-listening-to club. Each student randomly chooses an album from a list of 8 titles. What is the probability that at least 2 of the students in the group choose the same book?
Correct Answer
0.59
59%
Explanation
Use the complement!
P(all choose different) = number of ways 4 students can choose different albums / total number of ways 4 students can choose albums
=8P4/8^4
=0.41015625
1 - 0.41015625 = about 0.59
19.
Find the mode and median in your lovely and very intelligent head.
Then, find the weighted average of stars for a movie.
But, ignore the category saying below 0 stars; this will involve taking 0.07 out. Keep it in mind, kay?
Round to two decimal places.
Correct Answer
3.32
Explanation
0.14(5)+0.32(4)+0.24(3)+0.16(2)+0.07(1)+0.01(0)=3.09
3.09 divided by 0.93 is then equal to 3.32258065
20.
Final question! *There are no binomial distributions on our final, she said so herself.
Take the data values
1, 1, 6, 4, 8, 3, 6, and 3.
Then find the standard deviation of the data.
Round to four decimal places.
Correct Answer
2.3452
Explanation
1, 1, 6, 4, 8, 3, 6, and 3.
Mean: 4
Find the difference between the mean and each value:
3, 3, 2, 0, 4, 1, 2, and 1.
Square each of these values and then add them together:
9+9+4+0+16+1+4+1=44
Then divide that number by the number of data values:
44/8=5.5
That's the variance. For the standard deviation, take the square root of the variance;
2.34520788
Rounded:
2.3452