Mathematics Test - 5 - SAT MCQ

Given function: f(x) = 3x + n
Substitute f(2) = 0: f(2) = 3(2) + n = 0
Simplif y: 6 + n = 0
Subtract 6: n = -6
Therefore, the function is f(x) = 3x - 6.
Evaluate f(n): f(n) = f(-6) = 3(-6) - 6 = -18 - 6 = -24

First, notice that the question is only asking us to find values of x, so it’s a good idea to substitute in order to eliminate y from the system. x - 3y = -2 
Substitute y = 5/x: 
Multiply by x and simplify: x² - 15 = -2x
Add 2x: x² + 2x - 15 = 0
Factor using Sum-Product Method: (x - 3)(x + 5) = 0
Therefore, the values of x that satisfy the original system also satisfy the equation (x - 3)(x + 5) = 0.

f (3) = (3²)^-2b = 3
Exponential Law #8: 3^-4b = 3¹
Exponential Law #10: -4b = 1
Divide by -4: b = 

6x + 9 = 30
To solve in one step, just divide
both sides by 3: 2x + 3 = 10
Most students waste time solving for x, which will work, but takes longer: 6x + 9 = 30
Subtract 9: 6x = 21
Divide by 6: x = 3.5
Evaluate 2x + 3 by
substituting x = 3.5: 2x + 3 = 2(3.5) + 3 = 7 + 3 = 10

C(n) = an + b
Since this expression is linear in n (the input variable, which represents the number of necklaces produced), the constant a represents the slope of this line, which in turn represents the “unit rate of increase,” in other words, the increase in total cost for each individual necklace produced.
The constant b represents the “y-intercept” of this line, which in this case means the costs when n = 0 (that is, the fixed costs before any necklaces are produced).

Any point that intersects the y-axis has an x-value of 0. So, to find point A, plug in 0 for x and solve for y:
y = 2x² - 16x + 14
Plug in 0 for x: y = 2(0)² - 16(0) + 14 = 14
Any point that intersects the x-axis has a y-value of 0. So, to find points B and C, plug in 0 for y and solve for x:
y = 2x² - 16x + 14
Substitute 0 for y: 0 = 2x² - 16x + 14
Divide by 2: 0 = x² - 8x + 7
Factor: 0 = (x - 7)(x - 1)
Use the Zero Product Property: x = 7 and x = 1
If we connect these three points, we get a triangle with a height of 14 (from y = 0 to y = 14) and a base of 6 (from x = 1 to x = 7).
Use the triangle area formula 

When faced with a system of equations, notice whether the two equations can be combined in a simple way-either by subtracting or adding the corresponding sides-to get the expression the question is asking for.
a - b = 10
a - 2b = 8
Subtract corresponding sides: b = 2

This question tests your ability to translate words into algebraic expressions. Systematically translate the sentence phrase by phrase.
The product of two numbers, a and b is 6 greater than their sum.
Translation: ab = 6 + a + b
Use commutative law of equality on right side: ab = a + b + 6

Let R = the number of red marbles, W = the number of white marbles, and B = the number of blue marbles. If the jar contains twice as many red marbles as white marbles, then R = 2W. If the jar contains three times as many white marbles as blue marbles, then W = 3B. We can substitute numbers to these equations to solve the problem. Let’s say B = 10. This means there are 3(10) = 30 white marbles and 2(30) = 60 red marbles. The total number of marbles is therefore 10 + 30 + 60 = 100, and the number of non-red marbles is therefore 10 + 30 = 40 marbles, so the probability that the marble is not red is 40/100 = 2/5.

Substitute 1/m for x: 
Simplify the denominator:
 
Divide by multiplying by the reciprocal:

Dividing each side of the equation 3r = 18 by 3 gives r  = 6.
Substituting 6 for r in the expression 6r + 3 gives 6(6) + 3 = 39.
Alternatively, the expression 6r + 3 can be rewritten as 2(3r) + 3.
Substituting 18 for 3r in the expression 2(3r) + 3 yields 2(18) + 3 = 36 + 3 = 39.

Choice C is correct. If x − b is a factor of f(x), then f(b) must equal 0. Based on the table, f(4) = 0. Therefore, x − 4 must be a factor of f(x).
Choice A is incorrect because f(2) ≠ 0; choice B is incorrect because no information is given about the value of f(3), so x − 3 may or may not be a factor of f(x); and choice D is incorrect because f(5) ≠ 0.

Choice A is correct. The parabola with equation y = α(x − 2)(x + 4) crosses the x-axis at the points (−4, 0) and (2, 0). The x-coordinate of the vertex of the parabola is halfway between the x-coordinates of (−4, 0) and (2, 0). Thus, the x-coordinate of the vertex is = −1. This is the value of c. To find the y-coordinate of the vertex, substitute −1 for x in y = α(x − 2)(x + 4): y = α(x − 2)(x + 4) = α(−1 − 2)(−1 + 4) = α(−3)(3) = −9α.
Therefore, the value of d is −9α.
Choice B is incorrect because the value of the constant term in the equation is not the y-coordinate of the vertex, unless there were no linear terms in the quadratic. Choice C is incorrect and may be the result of a sign error in finding the x-coordinate of the vertex. Choice D is incorrect because the negative of the coefficient of the linear term in the quadratic is not the y-coordinate of the vertex.

Choice B is correct. Of the 25 people who entered the contest, there are 8 females under age 40 and 2 males age 40 or older. Therefore, the probability that the contest winner will be either a female under age 40 or a male age 40 or older is 
Choice A is incorrect and may be the result of dividing 8 by 2, instead of adding 8 to 2, to find the probability. Choice C is incorrect; it is the probability that the contest winner will be either a female under age 40 or a female age 40 or older. Choice D is incorrect and may be the result of multiplying 8 and 2, instead of adding 8 and 2, to find the probability.

Choice D is correct. To solve the equation for w, multiply both sides of the equation by the reciprocal of 3/5 , which is 5/3 =. This gives (5/3) · 3/5 w = 4/5 · (5/3), which simplifies to w = 20/9.
Choices A, B, and C are incorrect and may be the result of errors in arithmetic when simplifying the given equation.

Choice D is correct. A zero of a function corresponds to an x-intercept of the graph of the function in the xy-plane. Therefore, the complete graph of the function f, which has five distinct zeros, must have five x-intercepts. Only the graph in choice D has five x-intercepts, and therefore, this is the only one of the given graphs that could be the complete graph of f in the xy-plane.
Choices A, B, and C are incorrect. The number of x-intercepts of each of these graphs is not equal to five; therefore, none of these graphs could be the complete graph of f, which has five distinct zeros.

Choice B is correct. The quantity of the product supplied to the market is given by the function S(P) = 1/2 P + 40. If the price P of the product increases by $10, the effect on the quantity of the product supplied can be determined by substituting P + 10 for P as the argument in the function. This gives S(P + 10) = 1/2(P + 10) + 40 = 1/2P + 45, which shows that S(P + 10) = S(P) + 5.
Therefore, the quantity supplied to the market will increase by 5 units when the price of the product is increased by $10.
Alternatively, look at the coefficient of P in the linear function S. This is the slope of the graph of the function, where P is on the horizontal axis and S(P) is on the vertical axis. Since the slope is 1/2, for every increase of 1 in P, there will be an increase of 1/2 in S(P), and therefore, an increase of 10 in P will yield an increase of 5 in S(P).
Choice A is incorrect. If the quantity supplied decreases as the price of the product increases, the function S(P) would be decreasing, but S(P) = 1/2P + 40 is an increasing function. Choice C is incorrect and may be the result of assuming the slope of the graph of S(P) is equal to 1. Choice D is incorrect and may be the result of confusing the y-intercept of the graph of S(P) with the slope, and then adding 10 to the y-intercept.

Choice B is correct. One of the three numbers is x; let the other two numbers be y and z. Since the sum of three numbers is 855, the equation x + y + z = 855 is true. The statement that x is 50% more than the sum of the other two numbers can be represented as x = 1.5(y + z), or x/1.5 = y + z. Substituting  x/1.5 for y + z in x + y + z = 855 gives x + x/1.5 = 855. This last equation can be rewritten as x + 2x/3 = 855, or 5x/3 = 855. Therefore, x equals 3/5 × 855 = 513.
Choices A, C, and D are incorrect and may be the result of calculation errors.

Dividing both sides of the quadratic equation 4x² − 8x − 12 = 0 by 4 yields  x² − 2x − 3 = 0. The equation x² − 2x − 3 = 0 can be factored as (x + 1)(x − 3) = 0. This equation is true when x + 1 = 0 or x − 3 = 0. Solving for x gives the solutions to the original quadratic equation: x = −1 and x = 3.
Choices A and C are incorrect because −3 is not a solution of 4x² − 8x − 12 = 0: 4(−3)² − 8(−3) − 12 = 36 + 24 − 12 ≠ 0. Choice D is incorrect because 1 is not a solution of 4x² − 8x − 12 = 0: 4(1)² − 8(1) − 12 = 4 − 8 − 12 ≠ 0.

The constant term 331.4 in S(T) = 0.6T + 331.4 is the value of S when T = 0. The value T = 0 corresponds to a temperature of 0°C. Since S(T) represents the speed of sound, 331.4 is the speed of sound, in meters per second, when the temperature is 0°C.
Choice B is incorrect. When T = 0.6°C, S(T) = 0.6(0.6) + 331.4 = 331.76, not 331.4, meters per second. Choice C is incorrect. Based on the given formula, the speed of sound increases by 0.6 meters per second for every increase of temperature by 1°C, as shown by the equation 0.6(T + 1) + 331.4 = (0.6T + 331.4) + 0.6. Choice D is incorrect. An increase in the speed of sound, in meters per second, that corresponds to an increase of 0.6°C is 0.6(0.6) = 0.36.

When n is increased by 1, t increases by the coefficient of n, which is 1.
Choices A, C, and D are incorrect and likely result from a conceptual error when interpreting the equation. 

Use substitution to create a one-variable equation that can be solved for x. The second equation gives that y = 2x. Substituting 2x for y in the first equation gives Dividing both sides of this equation by 1/2 yields (2x + 2x )= 21 . Combining like terms results in 4x = 21. Finally, dividing both sides by 4 gives x = 21/4 = 5.25. Either 21/4 or 5.25 can be gridded as the correct answer.

The first expression (1.5x − 2.4)² can be rewritten as (1.5x − 2.4)(1.5x − 2.4). Applying the distributive property to this product yields (2.25x² − 3.6x − 3.6x + 5.76) − (5.2x² − 6.4). This difference can be rewritten as (2.25x² − 3.6x − 3.6x + 5.76) + (−1)(5.2x² − 6.4). Distributing the factor of −1 through the second expression yields 2.25x² − 3.6x − 3.6x + 5.76 − 5.2x² + 6.4. Regrouping like terms, the expression becomes (2.25x² − 5.2x²) + (−3.6x − 3.6x) + (5.76 + 6.4). Combining like terms yields −2.95x² − 7.2x + 12.16.
Choices A, B, and D are incorrect and likely result from errors made when applying the distributive property or combining the resulting like terms.

Any point (x, y) that is a solution to the given system of inequalities must satisfy both inequalities in the system. Since the second inequality in the system can be rewritten as y < x - 1, the system is equivalent to the following system.
y < 3x +1
y < x-1
Since 3x + 1 > x - 1 for x > -1 and 3x + 1 < x - 1 for x < -1, it follows that y < x - 1 for x > -1 and y < 3x + 1 for x < -1. Of the given choices, only (2, -1) satisfies these conditions because -1 < 2 - 1 = 1.
Alternate approach: Substituting (2, -1) into the first inequality gives -1 < 3(2) + 1, or -1 < 7, which is a true statement. Substituting (2, -1) into the second inequality gives 2 - (-1) > 1, or 3 > 1, which is a true statement. Therefore, since (2, -1) satisfies both inequalities, it is a solution to the system.
Choice A is incorrect because substituting -2 for x and -1 for y in the first inequality gives -1 < 3(-2) + 1, or -1 < -5, which is false. Choice B is incorrect because substituting -1 for x and 3 for y in the first inequality gives 3 < 3(-1) + 1, or 3 < -2, which is false. Choice C is incorrect because substituting 1 for x and 5 for y in the first inequality gives 5 < 3(1) + 1, or 5 < 4, which is false.

The linear function that represents the relationship will be in the form r(p) = ap + b, where a and b are constants and r(p) is the monthly rental price, in dollars, of a property that was purchased with p thousands of dollars. According to the table, (70, 515) and (450, 3,365) are ordered pairs that should satisfy the function, which leads to the system of equations below. 

Subtracting side by side the first equation from the second eliminates b and gives 380a = 2,850; solving for a gives a = 2,850/380 = 7.5. Substituting 7.5 for a in the first equation of the system gives 525 + b = 515; solving for b gives b = −10. Therefore, the linear function that represents the relationship is r(p) = 7.5p − 10.
Choices A, B, and C are incorrect because the coefficient of p, or the rate at which the rental price, in dollars, increases for every thousand-dollar increase of the purchase price is different from what is suggested by these choices. For example, the Glenview Street property was purchased for $140,000, but the rental price that each of the functions in these choices provides is significantly off from the rental price given in the table, $1,040. 

The correct answer is 8. Since the line passes through the point (2, 0), its equation is of the form y = m(x − 2). The coordinates of the point (1, 4) must also satisfy this equation. So 4 = m(1 − 2), or m = −4. Substituting −4 for m in the equation of the line gives y = −4(x – 2), or equivalently y = −4x + 8. Therefore, b = 8.
Alternate approach: Given the coordinates of two points through which the line passes, the slope of the line is= −4. So, the equation of the line is of the form y = −4x + b. Since (2, 0) satisfies this equation, 0 = −4(2) + b must be true. Solving this equation for b gives b = 8.

Choice D is correct. The survey was given to a group of people who liked the book, and therefore, the survey results can be applied only to the population of people who liked the book. Choice D is the most appropriate inference from the survey results because it describes a conclusion about people who liked the book, and the results of the survey indicate that most people who like the book disliked the movie.
Choices A, B, and C are incorrect because none of these inferences can be drawn from the survey results. Choices A and B need not be true. The people surveyed all liked the book on which the movie was based, which is not true of all people who go see movies or all people who read books. Thus, the people surveyed are not representative of all people who go see movies or all people who read books. Therefore, the results of this survey cannot appropriately be extended to at least 95% of people who go see movies or to at least 95% of people who read books. Choice C need not be true because the sample includes only people who liked the book, and so the results do not extend to people who dislike the book.

Choice A is correct. To isolate the terms that contain αx and b, 6 can be added to both sides of the equation, which gives 9αx + 9b = 27. Then, both sides of this equation can be divided by 9, which gives αx + b = 3.
Choices B, C, and D are incorrect and may result from computation errors.

Choice D is correct. The figure is a quadrilateral, so the sum of the measures of its interior angles is 360°. The value of x can be found by using the equation 45 + 3x = 360. Subtracting 45 from both sides of the equation results in 3x = 315, and dividing both sides of the resulting equation by 3 yields x = 105. Therefore, the value of x in the figure is 105.
Choice A is incorrect. If the value of x were 45, the sum of the measures of the angles in the figure would be 45 + 3(45), or 180°, but the sum of the measures of the angles in a quadrilateral is 360°. Choice B is incorrect. If the value of x were 90, the sum of the measures of the angles in the figure would be 45 + 3(90), or 315°, but the sum of the measures of the angles in a quadrilateral is 360°. Choice C is incorrect. If the value of x were 100, the sum of the measures of the angles in the figure would be 45 + 3(100), or 345°, but the sum of the measures of the angles in a quadrilateral is 360°.

Choice A is correct. Current’s formula is A = Multiplying each side of the equation by 30 gives 30A = 4 + w. Subtracting 4 from each side of 30A = 4 + w gives w = 30A – 4.
Choices B, C, and D are incorrect and may result from errors in choosing and applying operations to isolate w as one side of the equation in Current’s formula.