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In the complex plane, the horizontal axis is called the real axis and the vertical axis is called the imaginary axis. The complex number a + bi graphed in the complex plane is comparable to the point (a, b) graphed in the standard (x, y) coordinate plane. The modulus of the complex number a + bi is given by:
\[\sqrt{a^2 + b^2}\]
Question:
Which of the complex numbers z1, z2, z3, z4 and z5 below has the greatest modulus?
Answer:
Okay, so what is this question really asking?
All it is asking is for you to find the largest value of the square root of the sum of the squares of our coordinate points, or find:
\[\sqrt{x^2 + y^2}\]
The challenge is that we are not given the real x and y values of the coordinate points. So we have to estimate the coordinate points of our z points.
Because we are working with squares, negatives are not a factor, so we can eliminate the smaller numbers as we are just taking a number, positive or negative and taking the square of it. Let us estimate our coordinates: â€‹ z1 = (4, 5) z2 = (2, 1) z3 = (2, 3) z4 = (2, 2) z5 = (4, 2) We are looking for whichever point has the largest combination of its coordinate points. At a glance, the two points with the largest coordinates are z1 and z5. Let's find the modulus of z5, and smaller of the two.
\[\sqrt{4^2 + (2)^2}\]
\[\sqrt{16 + 4}\]
\[\sqrt{20}\]
\[4.5\]
And the modulus of z1:
\[\sqrt{(4)^2 + 5^2}\]
\[\sqrt{16 + 25}\]
\[\sqrt{41}\]
\[6.4\]
We can see that the modulus of z1, 6.4, is higher than that of z5.
Final answer: F, z1.
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Question:Consider the functions f(x) = sqrt(x) and g(x) = 7x + b. In the standard (x,y) coordinate plane, y = f(g(x)) passes through (4,6). What is the value of b?
A. 8 B. 8 C. 25 D. 26 E. 4  7. sqrt(6)
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Question:
N.B: This question is for SAT Subject Test Math Level 2
\[if \ log_a 5 = x \ and \ log_a 7 = y, \ then \ log_a \sqrt{1.4} = \]
(A) (xy)/2
(B) x/2  y (C) (x + y)/2 (D) (y  x)/2 â€‹(E) y/(2x) Answer:
The answer is D.
\[log_a\sqrt{1.4} = log_a\sqrt{\frac{7}{5}} = \frac{1}{2} (log_a 7  log_a 5) = \frac{1}{2} (y  x) \]
â€‹
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Two whole numbers have a greatest common factor of 8 and a least common multiple of 48. Which of the following pairs of whole numbers will satisfy the given conditions?
F. 4 and 9 G. 5 and 10 H. 10 and 16 J. 14 and 20 K. 16 and 24
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A group of students conducted several experiments using a variety of nonstick cookware, a spring scale, and several different weighted objects. Their goal was to determine which brand of cookware products had the best nonstick surface by measuring the coefficient of static friction, which is a measure of how resistant a stationary object is to movement. Experiment 1 A student connected the spring scale to a weighted object that was placed inside a piece of nonstick cookware as shown in Figure 1. The students planned to calculate the coefficient of static friction by determining the force required to disturb an object from rest. During the experiment, one student anchored the nonstick cookware be holding tightly to the handle while the other student attached a weighted, smooth steel object to the spring scale. The student pulled on the spring until the object began to move. A third student recorded the force in newtons, N, indicated on the spring scale at the moment the object began to move across the nonstick surface. This procedure was repeated for 3 different brands of cookware; each brand of cookware was tested with various weighted objects. The coefficient of static friction was calculated by dividing the average force required to move the object by its weight (mass × g, the gravitational constant). The results are shown in Table 1. Experiment 2
The students performed an experiment similar to Experiment 1, except three different brands of cooking spray were applied to the same cookware surface before the weights were put in place. The results are shown in Table 2.
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Directions:In the passages that follow, some words and phrases are underlined and numbered. In the answer column, you will find alternatives for the words and phrases that are underlined. Choose the alternative that you think is best, and fill in the corresponding bubble on your answer sheet. If you think that the original version is best, choose “NO CHANGE,” which will always be either answer choice A or F. You will also find questions about a particular section of the passage, or about the entire passage. These questions will be identified either by an underlined portion or by a number in a box. Look for the answer that clearly expresses the idea, is consistent with the style and tone of the passage, and makes the correct use of standard written English. Read the passage through once before answering the questions. For some questions, you should read beyond the indicated portion before you answer. Questions:Need for Speed As an avid skier and inline skater, I thought I had already achieved some incredible downhill speeds. On a recent trip to Quebec City, nevertheless [1], my concept of how fast humans can move was radically altered. It was Carnaval season, the time when people from across the province and the world flocking [2] to the old walled city for two weeks of food, drink, revelry, and winter sports. Normally, I go to Carnaval looking for the rare thrill, all the better if it requires a helmet and my signature on a release of liability. This time for me, it was fullcontact downhill iceskating. [3], [4] The course looked a lot like a bobsled run. From the top of the mountain a sturdy metal chute descended that wound left and right on its way down. About eight inches of icepack covered the metal surface, which was wetted twice daily to maintain an ideal slickness. If by the time you reach the end of the chute you still haven’t regained your footing, there’s a line of meterthick foam padding [4] to absorb your crash. [1]. A. NO CHANGE B. thus C. therefore D. though [2]. F. NO CHANGE G. flock H. flocked by J. are flocking [3]. A. NO CHANGE B. It was fullcontact for me this time, I was iceskating downhill. C. Iceskating downhill this time for me, it was fullcontact. D. This time, I was fullcontact iceskating downhill. [4]. At this point, the author wants to add a sentence to the paragraph that further illustrates his adventurous nature. Which of the options does this best? F. I certainly didn’t know what I had signed up for! G. Downhill iceskating sounded much more exciting than normal iceskating! H. I could easily have been injured, but the thrill I got was worth the risk! J. I normally wouldn’t sign up for such a thing, but anything goes at Carnaval! [5]. A. NO CHANGE B. footing; there’s a line of meterthick foam padding C. footing there’s a line of meterthick foam padding, D. footing; there’s a line of meterthick foam padding, Answers:1. The best answer is D. This sentence states something that seems at odds with the preceding sentence. Only “though,” answer choice D, reflects this relationship. 2. The best answer is G. “Carnaval season” indicates that the event comes around at least once a year. The verb “flock” should be in simple present tense to reflect how the clause is a general statement. 3. The best answer is A. The sentence is clearest as written. Answer choices B and C are commaspliced runon sentences. The subjectpredicate pair in answer choice D does not make sense. 4. The best answer is H. Reminding the reader of the dangers the author likes to face emphasizes his “adventurous nature,” as the question puts it. The other answer choices do not seem wellmatched to the “extreme” theme the author maintains throughout the passage. 5. The best answer is A. The major clue here is the initial “If” clause in the sentence. It must be followed by a comma, then a second clause; the other answer choices may be eliminated.
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ProblemA total of f men went on a fishing trip. Each of the r boats that were used to carry the fishermen could accommodate a maximum number of m passengers. If one boat had 5 open spots and the remaining boats were filled to capacity, which of the following expresses the relationship among f , r, and m?
F. rm+5=f G. rm−5=f H. r+m+5=f J. rf = m + 5 K. rf = m − 5
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ProblemSamantha is making gluten free brownies for the family picnic. If the recipe calls for 2 ½ cups of cocoa to serve 4 people, how many cups will he need if there will be 60 people at the picnic? SolutionSamantha's recipe calls for 2 ½ cups of cocoa to serve 4 people. Or, in other words, > 4 people require 2 ½ cups of cocoa. > 1 person would then require (2 ½) / 4cups of cocoa, which is 0.625 cup of cocoa. > So 60 people would require 0.625 x 60 cups of cocoa = 37.5 cups of cocoa. Answer: 37.5 cups of cocoa. Ingredients
Instructions
Source: http://www.kingarthurflour.com/recipes/glutenfreebrowniesrecipe
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Alexa was attempting to make muffins with blueberries. Her recipe required 3/4 cup of sugar and 1/8 cup of butter. Alexa accidentally put a whole cup of butter in the mix. Problem:A.
B. This got Alexa wondering how she could remedy similar mistakes if she were to dump in a single cup of some of the other ingredients. Assume she wants to keep the ratios the same.
Solution:A.
1. The ratio of cups of sugar to cups of butter is 3/4 : 1/8. If we multiply both numbers in the ratio by 8, we get an equivalent ratio that involves 1 cup of butter. 8 X 3/4 = 6, and 8 X 1/8 = 1 In other words, In other words, 3/4 : 1/8 is equivalent to 6:1, and so six cups of sugar is needed if there is one cup of butter. 2. In the previous part we saw that we have 8 times as much butter, so all the ingredients need to be increased by a factor of 8 for all the ratios to be the same in the new mixture. 3. The ratio of cups of blueberries to cups of butter is 3/8 : 1/8 in the original recipe, so Alexa will need to add 8 × 3/8 = 3 cups of blueberries to his new mixture. B. 1. The ratio of cups of sugar to cups of blueberries is 3/4 : 3/8. If we multiply both numbers in the ratio by 8/3, we get an equivalent ratio. 8/3 X 3/4 = 2 and 8/3 X 3/8 = 1 Since 3/4 : 3/8 is equivalent to 2 : 1, two cups of sugar is needed if there is one cup of blueberries. 2. The ratio of cups of butter to cups of sugar is 1/8 : 3/4. If we multiply both numbers in the ratio by 4/3, we get an equivalent ratio. 4/3 × 1/8=1/6 and 4/3× 3/4 = 1. In other words, 1/8 : 3/4 is equivalent to 1/6 : 1, and 1/6 cup of butter is needed if there is one cup of sugar. 3. The ratio of cups of blueberries to cups of sugar is 3/8 : 3/4. If we multiply both numbers in the ratio by 4/3, we get an equivalent ratio. 4/3×3/8=1/2 and 4/3×3/4=1. Since 3/8 : 3/4 is equivalent to 1/2:1, Alexa would need 1/2 cup of blueberries if there is one cup of sugar.
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Problem:
An interior designer is creating a custom coffee table for a client. The top of the table is a glass triangle that needs to balance on a single support. If the coordinates of the vertices of the triangle are at (3, 6), (5, 2), and (7, 10), at what point should the support be placed?
