Mathematics Test - 2 - SAT MCQ

For this question, we need to know two Laws of Exponentials from Chapter 9: Law #8 and Law #9. First, we use Law #9 to translate the radicals into exponents.
Given equation:
Apply Law of Exponentials #9:
Apply Law of Exponentials #9 again: 
Apply Law of Exponentials #8: m³ = n^1/4
Raise to the 1/3 power: 
Apply Law of Exponentials #8 again: m = n^1/12

In order to solve this without a calculator, we need to know how to analyze this problem in terms of the unit circle. First, let’s solve for cos x: 4 cos x = 1
Divide by 4: cos x = 1/4
the cosine of any angle corresponds to the x-coordinate of the corresponding point for that angle on the unit circle:

Notice that there are exactly two points on the unit circle that have an x-coordinate of 1/4. Now let’s think about the angle. We are told that x goes from 0 to 3π. Remember that a full trip around the circle is 2π radians; therefore, a journey from x = 0 to x = 3π is 1.5 trips around the circle counterclockwise starting from the positive x-axis. If you trace with your finger 1.5 times around the circle starting from the point (1, 0), you’ll hit our “points of interest” exactly three times.

Given inequality: 
Multiply by 2x: 12 + 1 < 2x
Simplify: 13 < 2x
Divide by 2: 6.5 < x
The smallest integer that is greater than 6.5 is 7.

Since ABCD is a rectangle, we can find the length of its diagonal using the Pythagorean Theorem: 10² + 24² = d². Even better, we can notice that the two legs are in a 5:12 ratio, and therefore triangle BCD is a 5-12-13 triangle. In either case, we find that DB = 26. Since DB is also a diameter of the circle, the radius of the circle is 26/2 = 13, and therefore, the area of the circle is πr² = π(13)² = 169π.

If AB = BC, then triangle ABC is isosceles and therefore the two base angles are congruent and the triangle has a vertical axis of symmetry at the line x = 3. This implies that the slopes of linesandare opposites. We can calculate the slope of BC from its endpoints:

Therefore, the slope of  is 3, and so mn = (3)(-3) = -9.

Let a = # of adult tickets sold, and c = # of child tickets sold. If 300 tickets were sold altogether: c + a = 300
The revenue for a adult tickets sold at $5 each is $5a, and the revenue for c child tickets sold at $3 each is $3c. Since the total revenue is $1,400: 5a + 3c = 1,400

The general equation of a parabola in the xy-plane is y = a(x - h)² + k, in which (h, k) is the vertex. Now let's express each choice in precisely this form.
(A) y = (x - 3)2 + 2 y = 1(x - 3)2 + 2 a = 1, h = 3, k = 2
(B) y = 2(x - 3)2 y = 2(x - 3)2 + 0 a = 2, h = 3, k = 0
(C) y = 2x2 - 3 y = 2(x - 0)2 - 3 a = 2, h = 0, k = -3
(D) y = 3x2 + 2 y = 3(x - 0)2 + 2 a = 3, h = 0, k = 2
If this vertex is on the x-axis, then k = 0. The only equation in which k = 0 is (B).

A(2 - i) = 2 + i
Divide by (2 - i): 
Multiply numerator and denominator by the conjugate (2 + i): 
FOIL: 
Combine terms: 
Substitute i² = -1:  
Simplify: 
Combine Terms: 
Distribute to express in standard a + bi form: 

Set up a proportion: 9/25 = x/225
Cross multiply: 2,025 = 25x
Divide by 25: 81 = x

The sum of the measures if the interior angles of any polygon is (n - 2)180°, where n is the number of sides in the polygon. Since this is a 5-sided polygon, the sum of its interior angles is (5 - 2)(180°) = 3(180°) = 540°. Therefore the average of these measures is 540°/5 = 108°.

If the Giants' win-loss is 2 : 3, then they won 2n games and lost 3n games, where n is some unknown integer. (For instance, perhaps they won 2 games and lost 3, in which case n = 1, or perhaps they won 20 games and lost 30, in which case n = 10, etc.) This means that the total number of games they played is 2n + 3n = 5n. Since they won 120 games,
5n = 120
Divide by 5: n = 24
Therefore they won 2n = (2)(24) = 48 games and lost 3n = (3)(24) = 72 games, and so they lost 72 - 48 = 24 more games than they won.

Given equation: 
Add 
Combine the fractions into one: 
Multiply by n² + 3: n² - 9 + k = n² + 3
Subtract n²: -9 + k = 3
Add 9: k = 12

Choice B is correct. To fit the scenario described, 30 must be twice as large as x. This can be written as 2x = 30.
Choices A, C, and D are incorrect. These equations do not correctly relate the numbers and variables described in the stem. For example, the expression in choice C states that 30 is half as large as x, not twice as large as x.

Choice A is correct. If a system of two linear equations has no solution, then the lines represented by the equations in the coordinate plane are parallel. The equation kx − 3y = 4 can be rewritten as y =where k/3 is the slope of the line, and the equation 4x − 5y = 7 can be rewritten as y = where 4/5 is the slope of the line. If two lines are parallel, then the slopes of the line are equal. Therefore, 4/5 = k/3, or k = 12/5. (Since the y-intercepts of the lines represented by the equations are and the lines are parallel, not identical.)
Choices B, C, and D are incorrect and may be the result of a computational error when rewriting the equations or solving the equation representing the equality of the slopes for k.

Choice A is correct. Dividing each side of the given equation by 3 gives the equivalent equation x² + 4x + 2 = 0. Then using the quadratic formula, with α = 1, b = 4, and c = 2, gives the solutions x = −2 ± √2.
Choices B, C, and D are incorrect and may be the result of errors when applying the quadratic formula.

The correct answer is 370. A system of equations can be used where h represents the number of calories in a hamburger and f represents the number of calories in an order of fries. The equation 2h + 3f = 1700 represents the fact that 2 hamburgers and 3 orders of fries contain a total of 1700 calories, and the equation h = f + 50 represents the fact that one hamburger contains 50 more calories than an order of fries. Substituting f + 50 for h in 2h + 3f = 1700 gives 2(f + 50) + 3f = 1700.
This equation can be solved as follows:
2f + 100 + 3f = 1700
5f + 100 = 1700
5f = 1600
f = 320
The number of calories in an order of fries is 320, so the number of calories in a hamburger is 50 more than 320, or 370.

Choice C is correct. The graph of y = f(n) in the coordinate plane is a line that passes through each of the points given in the table. From the table, one can see that an increase of 1 unit in n results in an increase of 3 units in f(n); for example, f(2) − f(1) = 1 − (−2) = 3. Therefore, the graph of y = f(n) in the coordinate plane is a line with slope 3. Only choice C is a line with slope 3. The y-intercept of the line is the value of f(0). Since an increase of 1 unit in n results in an increase of 3 units in f(n), it follows that f(1) − f(0) = 3. Since f(1) = −2, it follows that f(0) = f(1) − 3 = −5. Therefore, the y-intercept of the graph of f(n) is −5, and the slope-intercept equation for f(n) is f(n) = 3n − 5.
Choices A, B, and D are incorrect because each equation has the incorrect slope of the line (the y-intercept in each equation is also incorrect).

Choice C is correct. It is given that 1 ounce of graphene covers 7 football fields. Therefore, 48 ounces can cover 7 × 48 = 336 football fields. If each football field has an area of acres, than 336 football fields have a total area of 336 × = 448 acres. Therefore, of the choices given, 450 acres is closest to the number of acres 48 ounces of graphene could cover.
Choice A is incorrect and may be the result of dividing, instead of multiplying, the number of football fields by. Choice B is incorrect and may be the result of finding the number of football fields, not the number of acres, that can be covered by 48 ounces of graphene. Choice D is incorrect and may be the result of setting up the expressionand then finding only the numerator of the fraction.

Choice D is correct. Let c be the number of students in Mr. Kohl’s class. The conditions described in the question can be represented by the equations n = 3c + 5 and n + 21 = 4c. Substituting 3c + 5 for n in the second equation gives 3c + 5 + 21 = 4c, which can be solved to find c = 26.
Choices A, B, and C are incorrect because the values given for the number of students in the class cannot fulfill both conditions given in the question. For example, if there were 16 students in the class, then the first condition would imply that there are 3(16) + 5 = 53 milliliters of solution in the beaker, but the second condition would imply that there are 4(16) − 21 = 43 milliliters of solution in the beaker. This contradiction shows that there cannot be 16 students in the class.

If x = 9 in the equation  - x = 0, this equation becomes  - 9 = 0, which can be rewritten as  = 9 . Squaring each side of  = 9 gives  = 81, or k = 79. Substituting k = 79 into the equation  - 9 = 0 confirms this is the correct value for k.
Choices A, B, and C are incorrect because substituting any of these values for k in the equation k + 2 - 9 = 0 gives a false statement. For example, if k = 7, the equation becomes  - 9 = 3 - 9 = 0, which is false.

Substituting a² + b² for z and ab for y into the expression 4z + 8y gives 4(a² + b²) + 8ab. Multiplying a² + b² by 4 gives 4a² + 4b² + 8ab, or equivalently 4(a² + 2ab + b²). Since (a² + 2ab + b²) = (a + b)², it follows that 4z + 8y is equivalent to (2a + 2b)².
Choices A, C, and D are incorrect and likely result from errors made when substituting or factoring. 

Since gasoline costs $4 per gallon, and since Alan’s car travels an average of 25 miles per gallon, the expression 4/25 gives the cost, in dollars per mile, to drive the car. Multiplying 4/25 by m gives the cost for Alan to drive m miles in his car. Alan wants to reduce his weekly spending by $5, so setting 4/25 m equal to 5 gives the number of miles, m, by which he must reduce his driving.
Choices A, B, and C are incorrect. Choices A and B transpose the numerator and the denominator in the fraction. The fraction 4/25 would result in the unit miles per dollar, but the question requires a unit of dollars per mile. Choices A and C set the expression equal to 95 instead of 5, a mistake that may result from a misconception that Alan wants to reduce his driving by 5 miles each week; instead, the question says he wants to reduce his weekly expenditure by $5.

The intersecting lines form a triangle, and the angle with measure of x° is an exterior angle of this triangle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles of the triangle. One of these angles has measure of 23° and the other, which is supplementary to the angle with measure 106°, has measure of 180° - 106° = 74°. Therefore, the value of x is 23 + 74 = 97.

The density d of an object can be found by dividing the mass m of the object by its volume V. Symbolically this is expressed by the equation d = m/V. Solving this equation for m yields m = dV.
Choices B, C, and D are incorrect and are likely the result of errors made when translating the definition of density into an algebraic equation and errors made when solving this equation for m. If the equations given in choices B, C, and D are each solved for density d, none of the resulting equations are equivalent to d = m/V.

The graph on the right shows the change in distance from the ground of the mark on the rim over time. The y-intercept of the graph corresponds to the mark’s position at the start of the motion (t = 0); at this moment, the mark is at its highest point from the ground. As the wheel rolls, the mark approaches the ground, its distance from the ground decreasing until it reaches 0—the point where it touches the ground. After that, the mark moves up and away from the ground, its distance from the ground increasing until it reaches its maximum height from the ground. This is the moment when the wheel has completed a full rotation. The remaining part of the graph shows the distance of the mark from the ground during the second rotation of the wheel. Therefore, of the given choices, only choice D is in agreement with the given information.
Choice A is incorrect because the speed at which the wheel is rolling does not change over time, meaning the graph representing the speed would be a horizontal line. Choice B is incorrect because the distance of the wheel from its starting point to its ending point increases continuously; the graph shows a quantity that changes periodically over time, alternately decreasing and increasing. Choice C is incorrect because the distance of the mark from the center of the wheel is constant and equals the radius of the wheel. The graph representing this distance would be a horizontal line, not the curved line of the graph shown.

Choice D is correct. For x > 1 and y > 1, x^1/3 and y^1/2 are equivalent toand √y, respectively. Also, x⁻² and y^-1 are equivalent to 1/x² and 1/y , respectively. Using these equivalences, the given expression can be rewritten as 
Choices A, B, and C are incorrect because these choices are not equivalent to the given expression for x > 1 and y > 1.
For example, for x = 2 and y = 2, the value of the given expression is  the values of the choices, however, are and 1, respectively.

The correct answer is 8. The expression 2x + 8 contains a factor of x + 4. It follows that the original equation can be rewritten as 2(x + 4) = 16. Dividing both sides of the equation by 2 gives x + 4 = 8.

Choice C is correct. Since x = −3 is a solution to the equation, substituting −3 for x gives (−3α + 3)² = 36. Taking the square root of each side of this equation gives the two equations −3α + 3 = 6 and −3α + 3 = −6. Solving each of these for a yields α = −1 and α = 3. Therefore, −1 is a possible value of α.
Choice A is incorrect and may be the result of ignoring the squared expression and solving −3α + 3 = 36 for α. Choice B is incorrect and may be the result of dividing 36 by 2 instead of taking the square root of 36 when solving for a. Choice D is incorrect and may be the result of taking the sum of the value of x, −3, and the constant, 3.

Choice A is correct. The total number of copies of the game the company will ship is 75, so one equation in the system is s + c = 75, which can be written as 75 − s = c. Because each standard edition of the game has a volume of 20 cubic inches and s represents the number of standard edition games, the expression 20s represents the volume of the shipment that comes from standard edition copies of the game. Similarly, the expression 30c represents the volume of the shipment that comes from collector’s edition copies of the games. Because these volumes combined are 1,870 cubic inches, the equation 20s + 30c = 1,870 represents this situation. Therefore, the correct answer is choice A.
Choice B is incorrect. This equation gives the volume of each standard edition game as 30 cubic inches and the volume of each collector's edition game as 20 cubic inches. Choice C is incorrect. This is the result of finding the average volume of the two types of games, using that average volume (25) for both types of games, and assuming that there are 75 more standard editions of the game than there are collector’s editions of the game. Choice D is incorrect. This is the result of assuming that the volume of each standard edition game is 30 cubic inches, that the volume of each collector's edition game is 20 cubic inches, and that there are 75 more standard editions than there are collector’s editions.

Choice D is correct. If b/2 = 10, then multiplying each side of this equation by 2 gives b = 20. Substituting 20 for b in the equation α − b = 12 gives α − 20 = 12. Adding 20 to each side of this equation gives α = 32. Since α = 32 and b = 20, it follows that the value of α + b is 32 + 20, or 52.
Choice A is incorrect. If the value of α + b were less than the value of α − b, it would follow that b is negative. But if b/2 = 10, then b must be positive. This contradiction shows that the value of α + b cannot be 2. Choice B is incorrect. If the value of α + b were equal to the value of α – b, then it would follow that b = 0. However, b cannot equal zero because it is given that b/2 = 10. Choice C is incorrect. This is the value of α, but the question asks for the value of α + b.