Mathematics Test - 3 - SAT MCQ

Distribute: 3x/12 + 4/12
Simplify: x/4 + 1/3

To solve this without a calculator, you must be able to evaluate a few low powers of i.
that i⁰ = 1, i¹ = i, i² = -1, i³ = -i, and i⁴ = 1.
Therefore i3 + i = -i + i = 0.
Now, it’s just a matter of finding the choice that does NOT equal 0.
(A) (2i)² + 4 = -4 + 4 = 0
(B) 2 -  2i⁴ = 2 - 2 = 0
(C) 2i² - 2 = -2 - 2 = -4
(D) i⁴ - 1 = 1 - 1 = 0
Therefore, the correct answer is (C).

First, drawing a line as shown in the diagram shows that the figure is composed of two rectangles, but the height of the smaller one is unknown. Let’s call it x. The area of the larger rectangle is (3)(1) = 3, and the area of the smaller rectangle is (1)(x) = x. Clearly, t he area of t he figure must be the sum of these two areas Area = 16/5 = 3 + x
Subtract 3: 
Therefore, the perimeter of the figure is just the sum of the lengths of its sides. If we travel around the figure clockwise from the leftmost side, we get a perimeter of 

• The sum of the measures if the interior angles of a triangle is 180°, therefore m ∠BED + 90° + 50° = 180°, and so m ∠BED = 40°.
• Since ∠AEC is vertical to ∠BED, it must also have a measure of 40°, and so 40 + x + x = 180
• Simplify: 40 + 2x = 180
• Subtract 40: 2x = 140
• Divide by 2: x = 70

If Niko is the 8th oldest person in the class, then there are 7 students older than he is. If he is the 12th youngest person, then there are 11 students younger than he is. Therefore, there are 18 students in addition to him, for a total of 19 students.

To answer this question, we must evaluate each of the three functions for an input of ½:

Express right side in terms of a common denominator: 
Combine terms on right into one fraction: 
Combine terms: 
Multiple by x + 3: 

Given equation: y + x = k(x - 1)
Subtract x: y = k(x - 1) - x
Distribute: y = kx - k - x
Collect like terms: y = (k - 1)x - k
The slope of this line is k - 1 and its y-intercept is -k. If k > 2, then k - 1 > 1, and -k < -2. In other words, the slope of the line is greater than 1 and the y-intercept is less than -2. The only graph with these features is the one in choice (B).

Let's fill in the table with the information we're given and work our way to the value the question asks us to find. First, use the information in the FAVORABLE column to determine how many women viewed the politician favorably:
26 + w = 59
Subtract 26: w = 33
Next, go to the WOMEN row: 33 + x + 13 = 89
Combine terms: 46 + x = 89
Subtract 46: x = 43

We want to find the slope of the line of best fit because it represents the average annual increase in revenue per store. Although the question asks about the years 2004 and 2012, we can choose ANY two points on this line to find its slope. We should choose points on the line of best fit that are easy to calculate with, such as (2005, $300,000) and (2011, $600,000).

We might begin by plugging in a number for p. Let’s say p = 120 cells to start. We are told that after one hour the population decreased by 1/3. Since 1/3 f 120 is 
40, the population decreased by 40 and the population was then 120 - 40 = 80 cells. In the second hour, the population increased by 40%. Increasing a number by 40% is equivalent to it by 1.40 (because it becomes 140% of what it was), so the population was then 80(1.40) = 112 cells. In the third hour, the population increased by 50%, so it became 112(1.50) = 168 cells.
Substituting p = 120 into each of the answer choices yields (A) 1.3p = 1.3(120) = 156, (B) 1.4p = 1.4(120) = 168, (C) 1.5p = 1.5(120) = 180, and (D) 1.6p = 1.6(120) = 192. Therefore the answer is (B).
Alternately, you can solve this problem algebraically: p(2/3)(1.40)(1.50) = 1.40p.

Let p = the price per share of the stock. The cost of 200 of these shares (before commission) is therefore 200p. With a 3% commission, the cost becomes (1.03)(200p)
(1.03)(200p) = $3,399
Divide by 1.03: 200p = $3,300
Divide by 200: p = $16.50 per share

Choice A is correct. Substituting 25 for y in the equation y = (x − 11)² gives 25 = (x − 11)².
It follows that x − 11 = 5 or x − 11 = −5, so the x-coordinates of the two points of intersection are x = 16 and x = 6, respectively.
Since both points of intersection have a y-coordinate of 25, it follows that the two points are (16, 25) and (6, 25).
Since these points lie on the horizontal line y = 25, the distance between these points is the positive difference of the x-coordinates: 16 − 6 = 10.
Choices B, C, and D are incorrect and may be the result of an error in solving the quadratic equation that results when substituting 25 for y in the given quadratic equation.

Choice D is correct. If C is graphed against F, the slope of the graph is equal to 5/9 degrees Celsius/degrees Fahrenheit, which means that for an increase of 1 degree Fahrenheit, the increase is 5/9 of 1 degree Celsius. Thus, statement I is true. This is the equivalent to saying that an increase of 1 degree Celsius is equal to an increase of 9/5 degrees Fahrenheit. Since 9/5 = 1.8, statement II is true. On the other hand, statement III is not true, since a temperature increase of 9/5 degrees Fahrenheit, not 5/9 degree Fahrenheit, is equal to a temperature increase of 1 degree Celsius.
Choices A, B, and C are incorrect because each of these choices omits a true statement or includes a false statement.

The correct answer is 3/5 or .6.
Triangle ABC is a right triangle with its right angle at B. Thus,is the hypotenuse of right triangle ABC, andand are the legs of right triangle ABC. By the Pythagorean theorem, Since triangle DEF is similar to triangle ABC, with vertex F corresponding to vertex C, the measure of angle F equals the measure of angle C. Thus, sinF = sinC. From the side lengths of triangle ABC, sinC = opposite side/hypotenuse = AB/AC = 12/20 = 3/5 . Therefore, sinF = 3/5. Either 3/5 or its decimal equivalent, .6, may be gridded as the correct answer.

Choice D is correct. On Mercury, the acceleration due to gravity is 3.6 m/sec². Substituting 3.6 for g and 90 for m in the formula W = mg gives W = 90(3.6) = 324 newtons.

Choice A is incorrect and may be the result of dividing 90 by 3.6. Choice B is incorrect and may be the result of subtracting 3.6 from 90 and rounding to the nearest whole number. Choice C is incorrect because an object with a weight of 101 newtons on Mercury would have a mass of about 28 kilograms, not 90 kilograms.

Choice A is the correct answer. Experimental research is a method used to study a small group of people and generalize the results to a larger population. However, in order to make a generalization involving cause and effect:
- The population must be well defined.
- The participants must be selected at random.
- The participants must be randomly assigned to treatment groups.
When these conditions are met, the results of the study can be generalized to the population with a conclusion about cause and effect. In this study, all conditions are met and the population from which the participants were selected are people with poor eyesight. Therefore, a general conclusion can be drawn about the effect of Treatment X on the population of people with poor eyesight.
Choice B is incorrect. The study did not include all available treatments, so no conclusion can be made about the relative effectiveness of all available treatments. Choice C is incorrect. The participants were selected at random from a large population of people with poor eyesight. Therefore, the results can be generalized only to that population and not to anyone in general. Also, the conclusion is too strong: an experimental study might show that people are likely to be helped by a treatment, but it cannot show that anyone who takes the treatment will be helped. Choice D is incorrect. 

This conclusion is too strong. The study shows that Treatment X is likely to improve the eyesight of people with poor eyesight, but it cannot show that the treatment definitely will cause improvement in eyesight for every person. Furthermore, since the people undergoing the treatment in the study were selected from people with poor eyesight, the results can be generalized only to this population, not to all people.

The sum of (a² − 1) and (a + 1) can be rewritten as (a² − 1) + (a + 1), or a² − 1 + a + 1, which is equal to a² + a + 0. Therefore, the sum of the two expressions is equal to a² + a.
Choices B and D are incorrect. Since neither of the two expressions has a term with a³, the sum of the two expressions cannot have the term a³ when simplified. Choice C is incorrect. This choice may result from mistakenly adding the terms a² and a to get 2a².

The equation 60h + 10 ≤ 280, where h is the number of hours the boat has been rented, can be written to represent the situation. Subtracting 10 from both sides and then dividing by 60 yields h ≤ 4.5. Since the boat can be rented only for whole numbers of hours, the maximum number of hours for which Maria can rent the boat is 4.

The change in the number of 3-D movies released between any two consecutive years can be found by first estimating the number of 3-D movies released for each of the two years and then finding the positive difference between these two estimates. Between 2003 and 2004, this change is approximately 2 − 2 = 0 movies; between 2008 and 2009, this change is approximately 20 − 8 = 12 movies; between 2009 and 2010, this change is approximately 26 − 20 = 6 movies; and between 2010 and 2011, this change is approximately 46 − 26 = 20 movies. Therefore, of the pairs of consecutive years in the choices, the greatest increase in the number of 3-D movies released occurred during the time period between 2010 and 2011.
Choices A, B, and C are incorrect. Between 2010 and 2011, approximately 20 more 3-D movies were released. The change in the number of 3-D movies released between any of the other pairs of consecutive years is significantly smaller than 20. 

Let n be the number of novels and m be the number of magazines that Sadie purchased. If Sadie purchased a total of 11 novels and magazines, then n + m = 11. It is given that the combined price of 11 novels and magazines is $20. Since each novel sells for $4 and each magazine sells for $1, it follows that 4n + m = 20. So the system of equations below must hold.
4n + m = 20
n + m = 11
Subtracting side by side the second equation from the first equation yields 3n = 9, so n = 3. Therefore, Sadie purchased 3 novels.
Choice A is incorrect. If 2 novels were purchased, then a total of $8 was spent on novels. That leaves $12 to be spent on magazines, which means that 12 magazines would have been purchased. However, Sadie purchased a total of 11 novels and magazines. Choices C and D are incorrect. If 4 novels were purchased, then a total of $16 was spent on novels. That leaves $4 to be spent on magazines, which means that 4 magazines would have been purchased. By the same logic, if Sadie purchased 5 novels, she would have no money at all ($0) to buy magazines. However, Sadie purchased a total of 11 novels and magazines. 

 The equation −2x + 3y = 6 can be rewritten in the slope-intercept form as follows: So the slope of the graph of the given equation is 2/3. In the xy-plane, when two nonvertical lines are perpendicular, the product of their slopes is −1. So, if m is the slope of a line perpendicular to the line with equation  which yields m = Of the given choices, only the equation in choice A can be rewritten in the form  b , for some constant b. Therefore, the graph of the equation in choice A is perpendicular to the graph of the given equation.
Choices B, C, and D are incorrect because the graphs of the equations in these choices have slopes, respectively, of 

Choice B is correct. The first equation can be rewritten as y – x = 3 and the second as x/4 + y = 3, which implies that −x = _x4, and so x = 0. The ordered pair (0, 3) satisfies the first equation and also the second, since 0 + 2(3) = 6 is a true equality.
Alternatively, the first equation can be rewritten as y = x + 3. Substituting x + 3 for y in the second equation gives x/2 + 2(x + 3) = 6. This can be rewritten using the distributive property as x/2 + 2x + 6 = 6. 
It follows that 2x + _x 2 must be 0. Thus, x = 0. Substituting 0 for x in the equation y = x + 3 gives y = 3. Therefore, the ordered pair (0, 3) is the solution to the system of equations shown.
Choice A is incorrect; it satisfies the first equation but not the second. Choices C and D are incorrect because neither satisfies the first equation, x = y − 3.

Choice B is correct. Each of the options is a quadratic expression in vertex form. To rewrite the given expression in this form, the number 9 needs to be added to the first two terms, because x² + 6x + 9 is equivalent to (x + 3)². Rewriting the number 4 as 9 – 5 in the given expression yields x² + 6x + 9 – 5, which is equivalent to (x + 3)² – 5.
Choice A is incorrect. Squaring the binomial and simplifying the expression in option A gives x2 + 6x + 9 + 5. Combining like terms gives x² + 6x + 14, not x² + 6x + 4. Choice C is incorrect. Squaring the binomial and simplifying the expression in choice C gives x² – 6x + 9 + 5. Combining like terms gives x² – 6x + 14, not x² + 6x + 4. Choice D is incorrect. Squaring the binomial and simplifying, the expression in choice D gives x² – 6x + 9 – 5. Combining like terms gives x² – 6x + 4, not x² + 6x + 4.

Choice B is correct. The graph of a quadratic function in the xy-plane is a parabola. The axis of symmetry of the parabola passes through the vertex of the parabola. Therefore, the vertex of the parabola and the midpoint of the segment between the two x-intercepts of the graph have the same x-coordinate. Since f(–3) = f(–1) = 0, the x-coordinate of the vertex is = –2. Of the shown intervals, only the interval in choice B contains –2.
Choices A, C, and D are incorrect and may result from either calculation errors or misidentification of the graph’s x-intercepts.

The correct answer is 30. It is given that the measure of ∠QPR is 60°. Angle MPR and ∠QPR are collinear and therefore are supplementary angles. This means that the sum of the two angle measures is 180°, and so the measure of ∠MPR is 120°. The sum of the angles in a triangle is 180°. Subtracting the measure of ∠MPR from 180° yields the sum of the other angles in the triangle MPR. Since 180 − 120 = 60, the sum of the measures of ∠QMR and ∠NRM is 60°. It is given that MP = PR, so it follows that triangle MPR is isosceles. Therefore ∠QMR and ∠NRM must be congruent. Since the sum of the measure of these two angles is 60°, it follows that the measure of each angle is 30°.
An alternate approach would be to use the exterior angle theorem, noting that the measure of ∠QPR is equal to the sum of the measures of ∠QMR and ∠NRM. Since both angles are equal, each of them has a measure of 30°.

Choice A is correct. The slope of the line of best fit is negative, meaning as the distance of planetoids from the Sun increases, the density of the planetoids decreases. Therefore, planetoids that are more distant from the Sun tend to have lesser densities.
Choice B is incorrect because as the distance of planetoids from the sun increases, the density of the planetoids decreases. Choice C is incorrect. For example, according to the line of best fit, a planetoid that is 0.8 AU from the Sun has a density of 5 g/cm³, but a planetoid that is twice as far from the Sun with a distance of 1.6 AU has a density of 4.25 g/cm³. However, the density of 4.25 g/cm³ is not half the density of 5 g/cm³. Choice D is incorrect because there is a relationship between the distance from a planetoid to the Sun and density, as shown by the line of best fit.

Choice B is correct. Let x be the price, in dollars, of the jacket before sales tax. The price of the jacket after the 6% sales tax is added was $53. This can be expressed by the equation x + 0.06x = 53, or 1.06x = 53. Dividing each side of this equation by 1.06 gives x = 50. Therefore, the price of the jacket before sales tax was $50.
Choices A, C, and D are incorrect and may be the result of computation errors.

Choice A is correct. The y-intercept of the graph of y = 19.99 + 1.50x in the xy-plane is the point on the graph with an x-coordinate equal to 0. In the model represented by the equation, the x-coordinate represents the number of miles a rental truck is driven during a one-day rental, and so the y-intercept represents the charge, in dollars, for the rental when the truck is driven 0 miles; that is, the y-intercept represents the cost, in dollars, of the flat fee. Since the y-intercept of the graph of y = 19.99 + 1.50x is (0, 19.99), the y-intercept represents a flat fee of $19.99 in terms of the model.
Choice B is incorrect. The slope of the graph of y = 19.99 + 1.50x in the xy-plane, not the y-intercept, represents a driving charge per mile of $1.50 in terms of the model. Choice C is incorrect. Since the coefficient of x in the equation is 1.50, the charge per mile for driving the rental truck is $1.50, not $19.99. Choice D is incorrect. The sum of 19.99 and 1.50, which is 21.49, represents the cost, in dollars, for renting the truck for one day and driving the truck 1 mile; however, the total daily charges for renting the truck does not need to be $21.49.

Choice B is correct. The median of a set of numbers is the middle value of the set values when ordered from least to greatest. If the percents in the table are ordered from least to greatest, the middle value is 27.9%. The difference between 27.9% and 26.95% is 0.95%.
Choice A is incorrect and may be the result of calculation errors or not finding the median of the data in the table correctly. Choice C is incorrect and may be the result of finding the mean instead of the median. Choice D is incorrect and may be the result of using the middle value of the unordered list.