## SAT Math 1 & 2 Subject Tests

**Chapter 17**

**Level 1 Practice Test Form A Answers and Explanations**

**1** **B** Make the bases on both sides of the equation the same. 81 is the same as 3^{4}. So

3^{4x} = 3^{4}, which means that *x* = 1.

**2** **A** The fraction is the reciprocal of ; it’s just flipped over. To find the numerical value of , just flip the numerical value of , which is 0.625. Your calculator will tell you that = 1.6.

**3** **C** PITA, starting with (C). Use your calculator to see which value makes the equation true. The first one you try, (C), is correct.

To solve the problem algebraically, isolate *x* one step at a time. First, multiply through by 3 on the right: *x* − 3 = 3 − 3*x*. Next, add 3*x* to each side, and then add 3 to each side, to get 4*x* = 6. Divide each side by 4 to get *x* = , or 1.5.

**4** **D** This one’s much clearer if you draw it. You can think of segments *AC* and *BD* as overlapping segments, where *BC* is the amount of the overlap. The lengths of *AC* and *BD* add up to 27, but it’s only a distance of 21 from *A* to *D*. The difference, a distance of 6, is the overlap. That’s the length of *BC*.

**5** **D** Notice that the *x*-coordinate in both points is the same. So you just have to find the difference in the *y*-coordinates. The difference is 5 − (−12) = 17.

**6** **A** The value at which a line intersects the *y*-axis is called the *y*-intercept. That’s the *y*-coordinate of the point of intersection; the *x*-coordinate is zero at every point on the *y*-axis, so eliminate answers (D) and (E). If you put the line 3*y* + 5 = *x* − 1 into the form *y* = *mx* + *b*, then *b* will be the *y*-intercept. The rearranged equation looks like *y* = *x* − 2. The *y*-intercept is −2, and you know the *x*-coordinate must be 0, so the point of intersection has the coordinates (0, −2).

**7** **C**

Figure 1

The fact that *m* and *n* are parallel tells you that Fred’s theorem is at work here. That means that all the big angles are equal, all the small angles are equal, and any big angle plus any small angle equals 180°. The big angles measure 125°, so the small ones must measure 55°. Don’t choose (B); it’s a partial answer! Because *d* and *f* are both small angles, *d* + *f* = 110°.

**8** **B** Translate the words into math: = 2. Now peel away. First cube both sides: = 8. Now square both sides: *x* = 64. PITA works well, too.

**9** **E** When you’re given two equations extremely similar in form, you’re probably looking at classic ETS-style simultaneous equations. The best way to solve these? Rack ’em, stack ’em, add or subtract ’em! (Isn’t that satisfying?) In this case, adding the two equations cancels out the *b*term, leaving you with the equation 8*a* = 16, so *a* = 2.

**10** **E** Plug −3 into the function: (−3^{2}) − 3(−3) = 18.

**11** **E** Remember that the absolute value of something can be thought of as the distance on a number line between that value and zero. If the absolute value of some value is greater than 3, then that value must be more than 3 away from zero. In this case, you’re dealing with *y*-coordinates and you want all points 3 units or more from the *x*-axis, where *y* = 0. Only (E) fits the bill. If this kind of reasoning is unclear to you, just try picking points in the shaded region of each graph, and seeing whether they make the equation |*y*| ≥ 3 true. Any answer choice whose graph contains “illegal” points can be eliminated.

**12** **E** Direct variation between two quantities means that they always have the same quotient. In this case, it means that must always equal 5. To find the value of *m* when *n* = 2.2, set up the equation = 5, and solve for *m*. You’ll find that *m* = 11.

**13** **B** To find the slope of the line easily, get its equation into the form *y* = *mx* + *b*, where *m* will be the value of the slope. To express 3*y* − 5 = 7 − 2*x* in this form, just isolate *y*. You’ll find that *y* = − *x* + 4. Here, the slope of the line (*m*) is −.

**14** **D** A quick review of exponent rules—when raising powers to powers, multiply exponents; when multiplying powers of the same base, add exponents; and when dividing powers of the same base, subtract exponents. For this problem, you have to do all three. Take the steps one at a time, following the rule of PEMDAS.

**15** **A** Work from the inside out. *g*(2) = 2^{2} + 7 = 11. Now put in 11 for *x*: *f*(11) = 5 − 2(11) = −17.

**16** **D** With few exceptions, logic questions on the Math Level 1 test you on the contrapositive. The idea is that, given a statement like “If *A*, then *B*,” the only thing you automatically know is the contrapositive: “If not *B*, then not *A*.” Here, you’re told that no doctoral candidates graduate. In Cookie Monster language, that would read “If doctoral student, then no graduate.” The contrapositive would be “If graduate, then not doctoral student.” That’s almost exactly what (D) says (since there are only engineers and doctoral candidates at this school, not being a doctoral candidate is the same as being an engineer). The other answer choices all talk about subjects you don’t know anything about, like the number of students or the quality of their scholarly skills. You can eliminate those answer choices.

**17** **D**

Figure 2

There are a few simple rules for lines tangent to circles. Most important, a tangent line is always perpendicular to the radius it meets. That makes both Δ*APO* and Δ*BPO* right triangles. In each triangle, segment *PO* is the hypotenuse, so it’s impossible for *PB* to be longer than *PO*. Statement I is therefore not true, and (A), (C), and (E) can be eliminated. Both of the remaining answers, (B) and (D), contain Statement II, so it must be true. Concentrate on Statement III. Both ∠*PAO* and ∠*PBO* are right angles, so *a* + *d* = 180. Since the other two angles, ∠*APB* and ∠*AOB*, complete a quadrilateral, (*x* + *y*) and (*b* + *c*) must also add up to 180° (making a total of 360° in the quadrilateral). Statement III must also be true, and (D) is correct.

**18** **E** Plug In *x* = 1,000. Then Rodney has earned 4 × 1,000 = $4,000 and spent $200, so he has made a total of $3,800. That’s (E).

**19** **E** All the sides of a square are the same. So each side must be 15. Since the area of a square is (side)^{2}, the area must be 225.

**20** **C** Plug In 0.5 for *n*. Now *n*^{2} = 0.25, = 0.707, |*n*| = 0.5, −*n* = −0.5, and = 2. This makes (A), (B), (D), and (E) true. (C) is false because = *n* whenever *n* is positive.

**21** **D** Move the equation around so that it’s in *y* = *mx* + *b* formula: *y* = *x* − 8. So an equation perpendicular would have a slope of −. The only one is (D).

**22** **C** You can just factor this one, and then cancel.

Alternatively, you could use Plugging In.

**23** **E** Use the formula for surface area of a sphere with radius *r*. In case you forget the equation, it is given in the reference information at the beginning of the test: *S* = 4π*r*^{2}. You are given *S*, so write 75 = 4π*r*^{2}, and therefore *r* = 2.443. The formula for volume of a sphere with radius *r* is also given in the reference info; it’s *V* = π*r*^{3} . So plug *r* = 2.443 into the formula. *V* = 61.075.

**24** **C** Plug In 45° for each angle; this makes them complementary and congruent. Next, draw four lines intersecting at the same point, forming eight 45° angles; angles directly across from each other are vertical, and angles next to each other are adjacent. This shows that (A), (B), (D), and (E) can be true of acute angles. Acute angles measure *less* than 90°, so there’s no way for two of them to add up to 180° and be supplementary.

**25** **B** To simplify the expression , follow PEMDAS and do the exponent first—you’ll need to use FOIL to square the binomial on top of the fraction. You should get . Since *i* is the square root of −1, the value of *i*^{2} is −1, and = . Then just divide by 2 and get the expression in its simplest form, 4 − 3*i*.

**26** **C**

Figure 3

The diagram of this triangle gives you the length of the hypotenuse and the measure of an angle, and asks for the length of the side opposite that angle. That’s enough to set up a simple equation using the SOHCAHTOA definition of the sine—sin *θ* = . Plugging the values from this triangle into the equation gets you sin 38° = . Use your calculator to find the value of sin 38°, and you get the equation 0.61566 = . Multiply both sides by 4 to get *x* = 2.4626.

**27** **A** The range of a function is the set of values the function can produce; on a graph, the range corresponds to the *y*-coordinates of the curve. Looking at this graph, you’ll see that it seems to continue downward (in the negative *y*-direction) forever. The range doesn’t seem to have a minimum value. (D) and (E) can therefore be eliminated. The graph has an apparent maximum value of 2. That makes (A) a strong contender. The function’s graph is also a continuous curve; that means it occupies a range of values, not just a few specific ones. Only (A) describes such a range of values.

**28** **B** Use three average pies:

**29** **B** This one isn’t really vulnerable to shortcuts or techniques. You pretty much have to visualize the situation described in each answer choice and find the one that produces exactly eight points of intersection. When a cube is inscribed in a sphere, each of the cube’s 8 corners touches the inside of the sphere.

**30** **B** The median of ten numbers will be the average of the fifth and sixth numbers. Because the numbers are distinct, the fifth and sixth numbers cannot be the same, so the median will be between them. This makes II impossible—eliminate (C) and (E). Also, there is no mode because there are no repeated numbers. This makes III impossible—eliminate (D). Some creative Plugging In can make I work: 1, 2, 3, 4, 5, 6, 7, 8, 9, and 15 have a mean of 6, for example.

**31** **E** Since there are variables in the answer choices, you should Plug In. Try *x* = 3. We are trying to find *f*(3 + 4) = *f*(7) = 2(7)^{2} + 2, which is 100, our target number. Now plug 3 in for *x* in the answer choices, to see which answer choice hits the target. Only (E) works.

**32** **A** PITA. Plug In values from the answer choices, and see which one makes the equation true. Only (A) does it. When solving this question algebraically, the trick is to isolate *x* one step at a time. The correct order is below.

Only (A) represents a possible value of *x*.

**33** **D** Plug In 30° for *θ* and use your calculator; this makes the whole expression equal 0.25, which is sin^{2}(30°). (Remember that this means to find sin(30°) first, which is 0.5, and then square that result.) If you want to see how to simplify the expression, read on…

The best place to start is usually with any tangent functions in the expression, using the fact that tan *θ* = :

Once you’ve simplified it this far, you can use a basic trigonometric identity to simplify the expression one step further: 1 − cos^{2} *θ* = sin^{2} *θ*.

**34** **B** Plug In! Suppose *x* = 2. Then *f*(*x*) = 64 and *f*(−*x*) = −64, making Statement I not true. You can eliminate any answer choice containing I—(A), (C), and (E). Both of the remaining answer choices contain Statement II, so it must be true. Go straight to Statement III: *f*(2) = 32 and *f* = 2. Statement III is also false, so the correct answer is (B).

**35** **E** Gaps in the domain occur in two situations—when taking even roots of negative numbers, and when dividing by zero. The denominator of the function will be zero if *x* is zero, so that’s one hole in the domain. Eliminate (A), (B), and (D) because they include 0. The other problems occur when the expression inside the square root sign is negative, or zero, since you can’t divide by zero. So we know that *x* + 10 > 0, which means *x* > −10. (E) points out both situations.

**36** **D** In a 3-digit number containing no zeros, there are nine possibilities for the first digit (1−9); nine possibilities for the second digit (1−9); and nine possibilities for the third digit (1−9). That makes a total of 9 × 9 × 9 possible 3-digit numbers, or 729.

**37** **B** At all points on the *y*-axis of the coordinate plane, *x* = 0, so eliminate (D) and (E) right away. Then PITA. In this case, plug each point (*x*, *y*) from the answer choices into the equation, to see if it makes the equation true. If the first point in the answer choice works, then try the other point. If they both work, you’ve found the right answer. If either point fails, cross off that answer choice. Only the two points in (B) fit the given equation. Notice that (D) gives you the *x*-intercepts.

**38** **A** Plug In 1 for *BC* and for *AC*. Rotating around *BC* makes the radius and the height 1, so the volume is π ()^{2} (1). Rotating around *AC* makes the radius 1 and the height , so the volume is π(1)^{2} (). To find the ratio of the two volumes, either cancel common factors or enter each expression into your calculator. Your calculator will give you volumes of 3.1416 and 1.8138; their ratio is 1.7321, which is just about .

**39** **C**

There are two ways to solve this one. The elegant way is to notice that *c* is a central angle intersecting the arc *YZ*. Since *a* is an inscribed angle that intersects the same arc, it must measure half of angle *c*. Therefore 2*a* = *c* (see __Chapter 5__ for central versus inscribed angles in circles).

If that doesn’t jump out at you, then just Plug In using triangle rules. Suppose, for example, that *c* = 50. The other two angles in ∆*OYZ* would then need to add up to 130°; since Δ*OYZ* is isosceles (two sides are radii), each of the other angles must equal 65°. That tells you that *d* = 65. Meanwhile, elsewhere in the triangle, the Rule of 180° tells you that *b* = 130, because *c* = 50. Then, since Δ*OZX* is also isosceles (two sides are radii), the other two angles must each measure 25°. That means that *a* = 25. This is just one possible set of numbers. Any number would work as long as you obeyed the rules of geometry. Go to your answer choices using *a* = 25, *b* = 130, *c* = 50, and *d* = 65. Using these values, you’ll find that the only answer choice that equals *c* is (C).

**40** **A** This is a repeated percent-change question, basically the same as a compound interest question about a bank account. Remember the formula, Final = Original × (1 + Rate)^{# of changes}. This colony has an original size of 1,250 and increases by 8% (.08) every month. In two years, it will make 24 of these increases. That’s all you need to fill in the formula, which would look like the following: Final = 1,250 × (1 + 0.08)^{24}. Paying careful attention to the order of operations, run that through your calculator. You should find that Final = 7,926.4759. That’s very close to (A).

**41** **A**

Figure 7

Both segments *OA* and *OB* have lengths of 4, because they’re both radii. The ratio you’re given, *OD* = 3*DB*, tells you that the lengths of *OD* and *DB* are 3 and 1, respectively. Consequently, you know you’re looking at a right triangle with legs of lengths 3 and 4, so the length of the hypotenuse must be 5. Using the SOHCAHTOA definition of the sine function, you can determine that sin ∠*A* = , or 0.6.

**42** **D** PITA. In this situation, pick an easy number that is in some, but not all of the ranges in the answer choices. This will allow you to cross off the greatest number of answers. Let’s try *x* = 1. This gives you |(1)^{3} − 8| ≤ 5, which means 7 ≤ 5. But that is false, so eliminate any answer choices that include 1 in their ranges: (A), (B), and (C). Now, the trick is to try a value that is in one of the two remaining answer choices, but not the other. Use *x* = 2. This produces |(2)^{3} − 8| ≤ 5, which simplifies to 0 ≤ 5. Since that’s true, the correct answer must contain 2, and you should cross off any answer that doesn’t contain 2. Cross off (E), and pick (D).

**43** **C** The best way to attack this question is just to try it. You’ll find that no matter how you arrange the circles and connect them, the resulting polygon (which will be a hexagon) always has a perimeter of 24. That’s because the polygon is always made up of two radii from each circle, for a total of 12 radii—each with a length of 2.

**44** **C** To find the *y*-intercept, just make *x* = 0 and solve for *y*. *y* = 3, so it’s the point (0, 3). To find the *x*-intercept, make *y* = 0 and solve for *x*. *x* = 6, so it’s the point (6, 0). To find the distance between the two points, sketch their positions on the coordinate plane. You’ll see they form a right triangle with legs of lengths 3 and 6, in which the hypotenuse represents the distance between the two points. Use the Pythagorean theorem to find the length of the hypotenuse, 6.7082. That’s the distance between the intercepts.

**45** **A** The simple, grinding way to do this one is to use the distance formula on the answer choices. Any answer choice that produces a distance other than 25 or 26 can be discarded immediately. Only (A) produces distances of 25 and 26 from the two points given.

**46** **D**

__Note:__ Figure not drawn to scale.

Because this is an isosceles trapezoid, you know that ∠*S* is equal to ∠*T*, and so measures 135° as well. The four angles must total 360°, so ∠*R* and ∠*U* measure 45° each (you can also use Fred’s theorem to figure that out, since the bases of the trapezoid are parallel lines). Divide the trapezoid into a rectangle and two triangles by drawing altitudes to *S* and *T*. Each of the triangles must be a 45°-45°-90° right triangle. Using the proportions of 45°-45°-90°, you can find the length of each of the triangle’s sides: 4, 4, and 4. So the area of each triangle is *b* × *h*, or (4)(7 + 15). The area of the rectangle is 7 × 4 = 28. 28 + 8 + 8 = 44. Or, using the trapezoid area formula, you get (4)(7 + 15) = 44.

**47** **D** No ordinary calculator can work with exponents this big, and there’s no way to spot the biggest values here by looking at them; you’ve got to get tricky. The important fact about this question is that it’s not necessary to find the exact value of any expression merely to compare them. The best way to compare these expressions is to get them into similar forms. To start with, rearrange as many answer choices as possible so that they have exponents of 100. (C) can be expressed as (3^{5})^{100}, or 243^{100}; (D) can be expressed as (4^{4})^{100}, or 256^{100}; and (E) is already there—250^{100}. Suddenly it’s easy to see that (D) is the biggest of the three, and eliminate (C) and (E).

Next, take a look at (A). The exponent 999 is approximately 1,000. The expression is therefore worth a little less than (1.73^{10})^{100}, or (240.14)^{100}. That’s definitely smaller than (D), so you can eliminate (A) as well.

Finally, take a look at (B). The expression 2^{799} is almost equal to 4^{400}. How can you tell? Well, 4^{400} can be written as (2^{2})^{400}, or 2^{800}. That makes it clear that (D) is bigger than (A). Answer choice (D) reigns supreme.

**48** **E** If you have a graphing calculator, press the Y= key and enter the function. If you check the values of the TABLE, you can find that *f*(−3.5) and *f*(2.5) both equal 39. If you don’t have a graphing calculator, you can PITA. It may take a while, but you’ll get it.

**49** **A** To find the probability, first figure out the total number of possibilities, and then figure out how many meet the condition you want. Since there are 6 possible rolls on a fair cube, the total number of possibilities for two rolls is 6 × 6 = 36. Now you need to figure out all the ways to get a product greater than 18. If you roll a 1, 2, or 3 on the first cube, you’re out of luck, since the most you could roll would be 3 × 6 = 18, but you want more than 18. The rolls that will work are 4 × 5, 4 × 6, 5 × 4, 5 × 5, 5 × 6, 6 × 4, 6 × 5, and 6 × 6. That’s 8 rolls out of 36, which is a probability of =0.222.

**50** **B** If log* _{x}*(

*y*

*) =*

^{x}*z*, then

*z*is the exponent that turns

*x*into

*y*

*. If you think about it that way, then it’s clear that*

^{x}*x*raised to the power of

*z*would be

*y*

*. If that doesn’t make sense to you, then review*

^{x}__Chapter 3__.

If that doesn’t work for you, then it’s possible (but a little tricky) to Plug In. Do it this way. Plug 10 in for

*x*so that you’re working with a common logarithm, the kind your calculator can compute. Plug In 2 for

*y*and you get: log

_{10}(2

^{10}) =

*z*. This can be written simply as log 1,024 =

*z*. Your calculator can then compute the value of

*z*.

*z*= 3.0103. You can then compute the value of

*x*

*. You get 1,024. The only answer choice that equals 1,024 is (B).*

^{z}