SAT Math 1 & 2 Subject Tests
Chapter 20
Level 2 Practice Test Form B Answers and Explanations
1 E The statement xy ≠ 0 means that neither x nor y is zero. To find the value of rearrange the equation so that is isolated on one side of the equals sign; whatever”s on the other side will be the answer. In this case, the easiest way to isolate is to divide each side of the equation by x, getting 3 = , and then divide both sides by 0.3, getting = . Your calculator will tell you that is equal to 10.
2 B To find the increase in f(x) as x goes from 2 to 3, calculate f(2) and f(3) by plugging those numbers into the definition of the function. You”ll find that f(2) = 0.0589 and f(3) = 1.4308. The increase in f(x) is the difference between these two numbers, 1.3719.
3 A
Since you are given the two intercepts, draw a rough sketch like the one above and approximate. The slope is positive, but less than 1, since the line has a slope smaller than 45°. Only answer choice (A) matches your description. You could also notice that since the y-intercept is 3, the equation of the line (in slope-intercept form) must look like y = mx + 3, which narrows it down to (A) or (E). But (E) has a negative slope, which doesn”t fit your sketch, so it”s (A).
4 A PITA. Starting with (C), Plug In the values from the answer choices for a in the original expression. If the value makes the equation true, you”ve got the right answer. If not, then pick another answer choice and try again.
5 D An arithmetic series is one that increases by a constant added amount. From 4 to 15 is a total increase of 11, which happens from the second term in the arithmetic series to the tenth, taking 8 steps. This means that each step is worth one-eighth of 11, , or 1.375. That”s the constant amount added to each term in the series to get the next term. To find the first term in the series, just take one step backward from the second term, that is, subtract 1.375 from 4. You get 2.625.
You can also solve this equation by using the formula for the nth term of an arithmetic sequence: a_{n} = a_{1} + (n − 1)d. You”d start by figuring out d (the difference between any two consecutive terms) just as you did above, finding that d = 1.375. Then, take one of the terms you”re given (for example, a_{2} = 4, in which case n = 2) and use these values to fill in the formula, 4 = a_{1} + (2 − 1)(1.375). Then just solve for a_{1}. Once again you”ll find that a_{1} = 2.625.
6 C This simple function question just requires you to plug 6 into g(x). You can start by eliminating (A) and (B), because the entire function is contained within an absolute value sign, so it can”t produce negative values. To find the exact value of g(6), Plug In the number. You get |5(6)^{2} − (6)^{3}|, or |−36|, which equals 36.
7 E
The line y = x is shown above. A graph symmetrical across this line will look like it is reflected in this line as though it were a mirror. Another way to think about it is that the two halves of a curve symmetrical across the line y = x would meet perfectly if you folded the paper along that line. Of the five choices, only (E) has this kind of symmetry.
8 B Just use your calculator.
A = sin^{−1} =
sin^{−1} =
sin^{−1}(0.3220)=
18.8° =
9 B This is the equation of a parabola which opens upward. The minimum value will be the y-value of the vertex, which you can find using the vertex formula. The x-coordinate of the vertex is given by x = which gives you x = 6 in this case. Plug this value back into the equation to get the y-coordinate of the vertex, (6)^{2} − 6(6) + 11 = −7. The function”s minimum value is −7.
10 D This is a great question for Plugging In. Plug In a couple of simple values, like x = 3 and y = 5, and solve. |3 − 5| + |5 − 3| = |− 2| + |2| = 2 + 2 = 4 The correct answer must also equal 4. Using x = 3 and y = 5, you”ll find that only (D) gives you the correct value.
11 D Plugging In works very well here. Find simple values that make the original equation true, like A = 90° and B = 0°, so sin 90° = 1 and cos 0° = 1. Then go through the answer choices to find the one that is also true when A = 90° and B = 0°. In this case, only (D) works. In fact, (D) states a basic trigonometric identity, sin A = cos (90° − A).
12 C To figure out a probability, divide the number of things you”re looking for (in this case, flawed automobiles) by the total (all automobiles produced).
Since you”re calculating the odds for the entire three-month period, you have to add up the two columns of the chart. You find that 49 flawed automobiles were produced out of a total of 1,624. Divide 49 by 1,624 to find the probability, 0.03.
13 C
The easiest way to find the possible ramp lengths is to find the shortest and longest legal lengths. The length of the ramp is the hypotenuse of a right triangle. Using the SOHCAHTOA definition of sine (sin = ), you can set up equations to find the lengths of a 5°-ramp and a 7°-ramp. sin 5° = and sin 7° = . Solve for H in each case, and you”ll find that the shortest possible hypotenuse has a length of 24.62, while the longest has a length of 34.42. Only (C) is between these limits.
14 A This question is simpler than it looks. The constant term −n represents the y-intercept (where x = 0) of the whole complicated function. Just look to see where the function crosses the y-axis. It does so at y = 50. That means that −n, the y-intercept, equals 50. Just plain n, however, equals −50. Watch out for (C), a trap answer!
15 A Here”s a classic limit question. You can”t just plug the x-value in the question into the function, because it makes the denominator equal zero, which means the function is undefined at that point. To find the limit of the function approaching that point (assuming the limit exists), try to cancel out the troublesome term. First, factor the top and bottom of the function, . As you can see, the term (x + 2) occurs in the numerator and the denominator, so you can cancel it out. You”re left with . Now you can plug x = −2 into the function without producing an undefined quantity. The result, −1.67, is the limit of the function as x approaches −2.
16 B This is a great PITA question. When you receive an inheritance in Titheland, you get the first 1,000 florins free and clear. After that, you get only 35% of the remaining amount; the government keeps the other 65%. To find the right answer, take the numbers from the answer choices and see which one would give you 2,500 florins after taxes.
Answer choice (C) is 4,475 florins. Starting with that amount, you”d get 1,000 florins free and clear, and 3,475 would be taxed. The government would take 65% of 3,475, or 2,258.75. That would leave you with 1,216.25 plus the first 1,000, for a total of 2,216.25 florins after taxes—not enough. Your next step is to select the next larger answer choice and try again. Answer choice (B) is 5,286 florins, which will give you 1,000 untaxed and 4,286 taxed. That means the government takes 65% of 4,286, or 2,785.9 florins, leaving you with 1,500.1. Add the untaxed 1,000, and you”ve got a total of 2,500.1 florins after taxes—right on the money.
17 B PITA. Plug each answer choice into the equation for t, to see which one makes d = 10. (B) works.
18 C Plug In 1 for k in the expression 3k − 2; then repeat for 2, 3, 4, etc., all the way up to 10, and add up all the results. This gives us 1 + 4 + 7 +…+ 28. This is clearly an arithmetic sequence where you keep adding 3. The formula for the sum of the first n terms of an arithmetic sequence is sum = n which, in this case, gives us sum = 10 The answer is (C).
19 B It may not be obvious at first, but this is a natural logarithm question. The equation e^{x} = 5 can be rewritten in logarithmic form: log_{e}5 = x. A logarithm to the base e is called a natural logarithm, which can be written as 1n 5 = x. And that”s something you can just punch into your calculator. The natural logarithm of 5 is 1.60944, which rounds to 1.61. The correct answer is (B). You can also PITA, starting with (C), to see which value, when plugged in for x, makes the equation true.
20 E The greatest distance within a rectangular solid is the length of the long diagonal—the line between diagonally opposite corners, through the center of the solid. The length of this line can be determined using the Super Pythagorean theorem, a^{2} + b^{2} + c^{2} = d^{2}, where a, b, and c are the dimensions of the solid, and d is the length of the diagonal. To find the possible coordinates of the box, use the Super Pythagorean theorem on each of the answer choices, and find the one that gives a diagonal of 12. Only (E) produces the right number, 4^{2} + 8^{2} + 8^{2} = d^{2}, and so d = , or 12.
21 E The variables in the answer choices tell you that this is a perfect Plugging In question. Pick a rate for each runner. Say runner A travels 100 feet every minute (a = 100) and runner B travels 1 foot every second (b = 1). In one hour, that means A travels 60 × 100 feet, or 6,000 feet, while B travels 3,600 × 1 foot, or 3,600 feet. In this case, A travels 2,400 feet farther than B. This makes 2,400 your target number—the number the correct answer will equal. Only (E) equals 2,400 using these values.
22 B This question is easier when you draw the triangle. This right triangle has a hypotenuse of 13 and legs of 5 and 12. The smallest angle will clearly be the angle opposite the side of length 5. Knowing this, you can use SOHCAHTOA to figure out the angle (let”s call it θ). sin θ = = 0.3846. Take the inverse sine of both sides, and you get θ = 22.62°.
23 B Plugging In makes this question easy. Suppose, for example, that x = 2. To evaluate the expression g(f(2)), start on the inside, f(2) = , or . Then, work with the outside function, g() = + 1 or 3 + 1, which equals 4. That”s the value of g(f(2)). The correct answer choice will be the one that gives the same value (4) when you Plug In x = 2. Answer choice (B) is the only one that does.
24 D The roots of an expression are the values that make that expression equal to zero. In this case, there are three roots—π, 3, and e. To figure out which are the greatest and least roots, use your calculator to find the values of π (3.14159…) and e (2.71828…). These are the greatest and least roots, so subtract them to find their difference, 0.42331. (Hint: If your calculator doesn”t have an e key, get one that does.)
25 D Plug In! Suppose that x = 4, for example. Then = = The correct answer will be the one that equals 12 when x = 4.
26 D
A rectangle rotated around one edge generates a cylinder. This rectangle is being rotated around the vertical axis, so the cylinder will have a radius of 7 and a height of 5. Just plug these values into the formula for the volume of a cylinder, V = r^{2}h.
V = π(7)^{2}(5)
= π × 49 × 5
= 245π
= 769.69
27 E To make this problem easier, simplify the original function. You can do this by factoring the top and bottom of the fraction: f(x, y) = You can cancel out an (x − y) term on the top and bottom, producing the simplified function f(x, y) = . Then, to answer the question, just Plug In (−x) for x and (−y) for y:f(−x, −y) = . Then, just multiply by to flip the signs on the top and bottom of this fraction, .
28 D In order to disprove a rule, it”s only necessary to find one exception. That”s what (D) is saying. Even if you didn”t know that, however, there are still ways to eliminate wrong answers here. Use POE and some common sense. Answer choices (A), (C), and (E) are all pretty much impossible—they each require you to accomplish an enormous or even endless task. Answer choice (B) is more reasonable, but it doesn”t relate to the question, which asks about numbers less than 5. Answer choice (D) is the only one that is both possible and relevant to the question.
29 E Careful with this one. Given three corners of a rectangle, you know for sure where the fourth one is. But this is just a parallelogram—capable of having many different shapes. You could place the fourth vertex at (−3, 3), (3, −3), or (7, 3), and still have a parallelogram. You don”t know for sure where the fourth vertex is.
30 A An expression is undefined when its denominator equals zero. To find out what values might do that wicked deed, factor the expression. You”ll find it equals . Two values will make this expression undefined—x = −1 and x = −4, both of which make the denominator equal to zero. (A) is correct. [Note: It”s true that the term (x + 4) cancels out of the factored expression. That doesn”t mean that the original expression is defined at x = −4, however. It just means that you can calculate the limit of the expression as x approaches −4. Don”t confuse the existence of a limit with the defined/undefined status of an expression.] You can also PITA to see which values from the answer choices cause problems for the expression. In this case, pick an easy value that shows up in 2 or 3 answer choices, so you can eliminate the greatest number of them. Try x = 0. Plugging this in gives you −, that is fine. So eliminate answer choices that contain 0: Cross off (C) and (E). Let”s try x = −1. This causes the denominator to become 0, so you know that −1 is in the correct answer. Therefore, eliminate answer choices that don”t contain −1: Cross off (D). The only difference between (A) and (B) is −4. When you test x = −4, the denominator again becomes 0, so −4 must be in the correct answer, which is (A).
31 C In a I, II, III question, tackle the statements one at a time and remember the process of elimination. The expression in Statement I is always positive, because x^{2} must be positive and adding 1 can only increase it. That means that (B) and (D) can be eliminated, because they don”t include Statement I. Statement II is trickier. The sine of an expression can be anywhere from −1 to 1, inclusive. That means that most values of (1 − sin x) will be positive. If sin x = 1, however, then the expression equals zero—not a positive value. Statement II is out, and you can eliminate (E). Finally, Statement III can be simplified. π(π^{x − 1}) = π^{x − 1 + 1} = π^{x}. Since π is positive, and no exponent can change the sign of a base, π^{x} is always positive. (A) is out, and (C) is the correct answer. Graphing each function on your calculator may be easier, as long as you look carefully when you check Statement II.
32 A
Figure 5
Changing the sign of x in the expression f(x) will flip the function”s graph around the x-axis. Changing the sign of the whole function to produce y = −f(−x) will flip the graph over the y-axis as well. The graph in answer choice (A) represents the original graph flipped both horizontally and vertically.
33 D It”s helpful to draw this one. The wire has a slope of , meaning that it rises 2 feet for every 5 feet it runs. Since its total rise is 48, or (2 × 24), its total run must be 120, or (5 × 24). Don”t pick answer choice (B), though. The question asks for the length of the wire, not the distance between the anchor and antenna. The wire”s length is the hypotenuse of a right triangle with legs 48 and 120. The Pythagorean theorem will tell you that its length is 129.24.
34 B To start, you can do some useful elimination. The statement > θ > 2π tells you that you”re working in the fourth quadrant of the unit circle, where the tangent is negative. You can immediately eliminate (C), (D), and (E). If sec θ = 4, then cos θ = 0.25, because, by definition, sec θ= . Here you have to be careful. If you take the inverse cosine of 0.25, your calculator will display a value whose cosine equals 0.25—in radians, you should get 1.3181; but remember that different angles can produce the same cosine. In this case, you know that > θ > 2π which means that the angle in question is in the fourth quadrant of the unit circle. The angle in that quadrant with an equivalent cosine can be expressed in radians as −1.3181, or as 2π − 1.3181 = 4.9651. Take the tangent of either of these values, and you”ll get −3.8730.
35 C Drawing this one is helpful. You”ll find that the radii of the two circles have to add up to the distance between the circles” centers. You can find that distance using the distance formula, d = You”ll find that the two centers are separated by a distance of 8.6023. Since one circle has a radius of 4, the other must have a radius of 4.6023.
36 C The equations in the beginning of this question can be rearranged into the Law of Sines. A little algebraic manipulation gets you = and = . These equations can be combined into = = which is the Law of Sines. This tells you that the lengths of the triangle”s sides are in a ratio of 7:10:4. So, you can call the sides 7x, 10x, and 4x. They add up to 16, so 21x = 16, and x = 0.7619; side a has length 7x, which equals 5.3333. The answer is (C).
37 E
x |
h(x) |
−1 |
0 |
0 |
3 |
1 |
0 |
2 |
3 |
Test each expression with the values from the table. The easiest one to use is 0; when you make x = 0, the function should equal 3. Only (D) and (E) equal 3 when x = 0. Then, notice that the function is equal to zero when x = 1 or −1. These values must be roots of the function. Only (E) contains both 1 and −1 as roots. It”s the right answer.
38 A This is a tricky simultaneous-equations problem. After some experimentation, you might notice that the second and third equations can be added together, a + b + 2c = n + 8, which is very similar to the first equation, a + b + 2c = 7. From this, you can determine that n + 8 = 7. If n is any value other than −1, this is impossible—no values of a, b, and c can make this system of equations true. There is no solution for this system if n ≠ −1.
39 D
Figure 6
Note: Figure not drawn to scale.
When you know all three sides of a triangle, and you need to determine the measures of the angles, it”s time to use the Law of Cosines, c^{2} = a^{2} + b^{2} − 2ab cos C. Just Plug In the lengths of the sides, making sure that you make c the side opposite θ. = (12)^{2} + (5)^{2} −2(12)(5) cosθ. Simplifying this gives you the equation cos θ = 0.3667. Taking the inverse cosine of both sides shows that θ = 68.4898°.
40 D The formula for the determinant of a 3 × 3 matrix tells you that A = lqu + mrs + npt − mpu − lrt − nqs. This is easy to see if you take the first two columns of the matrix and recopy them to the right of the original matrix, and make diagonal lines connecting the elements. Now do the same thing with the second matrix, and you find that the determinant is 8lqu + 8mrs +8npt − 8mpu − 8lrt − 8nqs. Factor out 8, and it”s clear that this determinant is 8A. The answer is (D).
Want an easier way to solve this? Go to the MATRIX menu on your graphing calculator and enter a random 3 × 3 matrix, like . Your calculator will tell you that the determinant of this matrix is 30. Now enter the matrix with each entry doubled: . The determinant of this matrix is 240. This is 8 times the determinant of the original matrix, making (D) the answer once again.
41 A This is a little tricky; you”ve got to pay attention to that underlined word, “increased.” The values in the equation g(x) = A[sin (Bx + C)] + D that determine amplitude and period are A and B, respectively. Quantities C and D do not have to change to alter the amplitude or period, so (C) and (E) are out. The trick is that while you increase the amplitude by increasing A, you increase the period by decreasing B. If the amplitude and period of the curve both increase, that means that A increases and B decreases.
42 A All the fancy language in this question basically boils down to this: List M and list N each contains 20 elements; each element in list M is larger than the corresponding element in list N.
Once you have a clear idea of what that means, tackle the answer choices one at a time. Since the middle two values of M are bigger than the middle two values of N, the median of M must be greater, and (A) is correct. If you don”t see this right away, you can always eliminate the other answer choices. You can eliminate (B) because it”s quite possible that the largest value in list N is larger than the smallest value in list M. You don”t know anything about the modes of the two lists, so cross off (C). For (D), suppose that each element in M is exactly 1 greater than the corresponding element in N. The ranges would be identical, so you can cross off (D). In that same situation, the standard deviations of the two lists would be identical, so you can cross off (E).
43 D This problem looks scary, but it”s really just a matter of Plugging In. Each answer choice says that the first term is 3, and then each one gives a different definition of the sequence. In the given sequence a_{0} = 3, a_{1} = 5, a_{2} = 8.333, and a_{3} = 13.889. This obviously isn”t an arithmetic sequence, because the difference between the terms is not consistent. So Plug In values of n to see if the terms you are given fit the equations in each answer choice. If you plug n = 0 into answer choice (A), you get a_{0 + 1} = a_{0} + 2. Well, 5 = 3 + 2, so this seems to work. Now plug n = 1 into answer choice (A). This gives a_{1 + 1} = a_{1} + 2. But 8.333 ≠ 5 + 2, so (A) doesn”t define the sequence. In (B), Plug In n = 0 again, which gives a_{0 + 1} = 2a_{0} − 1. This works, because 5 = 2(3) − 1. But when you try n = 1, you get a_{1 + 1} = 2a_{1} − 1, which is wrong, because 8.333 ≠ 2(5) − 1. Cross off (B). Repeating this process for answer choice (C), Plug In n = 1, which fails. Choice (D) works for all the terms given. It turns out that this is a geometric sequence with a constant factor of
44 D The statement 0≤ n ≤ tells you that you”re working in the first quadrant of the unit circle where both sine and cosine are never negative. The unit also tells you that you”ll be working with angles in radians, not degrees. Make sure your calculator is in the correct mode. The question tells you that the cosine of the cosine of n is 0.8. To find n, just take the inverse cosine of 0.8, and then take the inverse cosine of the result. You should get 0.8717—that”s n. If you get an error, your calculator is probably in degree mode. Finally, take the tangent of 0.8717. You should get 1.1895.
45 C This cylinder has a radius of n (because n is half the diameter) and a height of Just plug these values into the formula for the surface area of a cylinder, SA = 2πr^{2} + 2πrh. You get 2πn^{2} + πn^{2}, or 3πn^{2}.
46 A PITA. Plug In each answer choice for θ to see which one makes the equation true. Make sure your calculator is in radian mode. Only (A) works.
47 A A good way to tackle this one is by trying to disprove each of the answer choices. If you start with (A), you”re lucky. There”s no way to divide 5 by another quantity and get zero; it”s the right answer. Even if you weren”t sure, the other answer choices are pretty easy to disprove. Just set equal to a quantity prohibited by each answer choice, and solve for x. If there”s a real value of x that solves the equation, then the value is in the range after all, and the answer choice is incorrect. Another method is to graph the function on your calculator and see what y-values seem impossible.
48 C csc θ = so rewrite the given equation as = That means sin θ = 3t. Now, Plug In an easy number for θ, such as 30°. So you have sin 30° = 3t. Therefore, t = 0.167. The question asks for cos 30°, which is 0.866, the target number. Now, Plug In 0.167 for t in the answer choices to see which one becomes 0.866. Choice (C) is the only one that comes close.
49 C There”s no sophisticated math here. It”s just an annoying function question with a lot of steps. As usual with I, II, III questions, tackle the statements one at a time and remember the Process of Elimination. Statement I must be true—if x = 0, then the function comes out to which equals 1 no matter what y is (since y can”t be zero). Answer choice (B) can be eliminated since it doesn”t contain Statement I. Statement II must be true; since the value 1 is being plugged into the function in the x and y positions, the function will always equal or 1. Answer choices (A) and (D) can be eliminated because they don”t include Statement II. Finally, Statement III is not necessarily true; f(x, y) = and f(y, x) = . If x and y have different values, then these expressions will not be equal. Answer choice (E) can thus be eliminated, because Statement III is false. That leaves only (C).
50 B Plug In! Since no numerical values are assigned to m and b, you can Plug In whatever you want. For example, say m = −8 and b = 2. The line”s equation is then y = −8x + 2. The x-intercept is and the y-intercept is 2. The triangle formed by this line and the axes has a base of length and a height of length 2 (it”s helpful to sketch this). If you rotate this triangle around the x-axis to generate a cone, the cone will have a radius of 2 and a height of . Plug those values into the formula for the volume of a cone, V = πr^{2}h. You”ll find that the volume equals This immediately eliminates answers (A), (C), and (D). Things look pretty good for (B), but you can”t be sure it”s not (E) until you”ve tried another set of numbers. If you try m = −1 and b = 1, you get a line with the equation y = −x + 1. This line has an x-intercept of 1 and a y-intercept of 1. The cone generated by rotation would then have a radius of 1 and a height of 1. Plug those numbers into the volume formula and you will once again get That”s proof enough, the answer”s clearly (B). That”s the power of Plugging In.