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 first ordered pair, x = 0 and y = 2, does not satisfy the equations in (A), (B), or (D), so those choices can be eliminated. You should also confirm that the equation in (C), y = 2x + 2, is satisfied by all four ordered pairs.

• 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:

One way to tackle this question is simply to simplify the expression for f(7), and then see which choice gives the same expression.
f(7) = k(7 + 6)(7 - 1) = k(13)(6) = 78k
Evaluate (A): f(-78) = k(-78 + 6)(-78 - 1) = k(-72)(-79) = 5,688k
Evaluate (B): f(-12) = k(-12 + 6)(-12 - 1) = k(-6)(-13) = 78k
Evaluate (C): f(-2) = k(-2 + 6)(-2 - 1) = k(4)(-3) = -12k
Evaluate (D): f(78) = k(78 + 6)(78 - 1) = k(84)(77) = 6,468k
This shows that f(-12) is equal to f(7). Alternately, you might just make a quick sketch of the parabola and take advantage of the symmetry:

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).

First, we should simplify the first equation: 
Subtract 1/3y: 
Multiply by 12: 6x - 4y = 1.2
This equation represents a line with slope of 6/4 = 3/2. The second equation, 6x - 4y = k, also represents a line with slope 6/4 = 3/2. In order for t his system of equat ions to have at least one solution, these two lines must have an intersection. How can two lines with the same slope intersect? They must be identical lines, and therefore intersect in all of their points. If this is the case, then k must equal 1.2.

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.

The graph of the quadratic y = ax² + bx + c is a parabola. If a < 0, the parabola is “open-down” like a frowny-face. The only graph with this feature is (C).

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 C is correct. Multiplying each side of by x(x + 20) gives 15x = 5(x + 20). Distributing the 5 over the values within the parentheses yields 15x = 5x + 100, and then subtracting 5x from each side gives 10x = 100. Finally, dividing both sides by 10 gives x = 10. Therefore, the value of x/5 is 10/5 = 2.
Choice A is incorrect because it is the value of x, not x/5. Choices B and D are incorrect and may be the result of errors in arithmetic operations on the given equation.

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 B is correct. Since 7 percent of the 562 juniors is 0.07(562) and 5 percent of the 602 seniors is 0.05(602), the expression 0.07(562) + 0.05(602) can be evaluated to determine the total number of juniors and seniors inducted into the honor society. Of the given choices, 69 is closest to the value of the expression.
Choice A is incorrect and may be the result of adding the number of juniors and seniors and the percentages given and then using the expression (0.07 + 0.05)(562 + 602). Choices C and D are incorrect and may be the result of finding either only the number of juniors inducted or only the number of seniors inducted.

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 B is correct. To answer this question, find the point in the graph that represents Michael’s 34-minute swim and then compare the actual heart rate for that swim with the expected heart rate as defined by the line of best fit. To find the point that represents Michael’s swim that took 34 minutes, look along the vertical line of the graph that is marked “34” on the horizontal axis. That vertical line intersects only one point in the scatterplot, at 148 beats per minute. On the other hand, the line of best fit intersects the vertical line representing 34 minutes at 150 beats per minute. Therefore, for the swim that took 34 minutes, Michael’s actual heart rate was 150 − 148 = 2 beats per minute less than predicted by the line of best fit.
Choices A, C, and D are incorrect and may be the result of misreading the scale of the graph.

The volume of right circular cylinder A is given by the expression πr²h, where r is the radius of its circular base and h is its height. The volume of a cylinder with twice the radius and half the height of cylinder A is given by π(2r)²(1/2)h which is equivalent to 4πr²(1/2) h = 2πr²h. Therefore, the volume is twice the volume of cylinder A, or 2 × 22 = 44. Choice A is incorrect and likely results from not multiplying the radius of cylinder A by 2. Choice B is incorrect and likely results from not squaring the 2 in 2r when applying the volume formula. Choice D is incorrect and likely results from a conceptual error. 

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. 

 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.

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 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.