ARCO SAT Subject Math Level 1 Practice Test.pdf

7/23/2019 ARCO SAT Subject Math Level 1 Practice Test.pdf

1/29

Lesson 8198

www.petersons .com/arco ARCO SAT II Subject Tests

1. Ifx= (b+ 3)2and b= 4, thenx=

(A) 49(B) 1(C) 0(D) 1(E) 49

2. Iff(x) = 4x 1 and 1


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Mathematics Level IC/IIC Subject Tests 199

ARCO SAT II Subject Tests www.petersons .com/arco

7. What is the length of the line segment betweenthe points (2, 3) and (4, 4)?

(A) 1.41(B) 3.16(C) 4.47

(D) 7.07(E) 7.62

8. In Figure 1, if thenRS=

(A)

(B)

(C)

(D)

(E)

Figure 1

9. In Figure 2, which of the following points fallswithin the shaded area?

(A) (5,0)(B) (5,1)(C) (4,2)(D) (4,3)(E) (3,4)

Figure 2

10. If 4


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Lesson 8200


13. In Figure 4, if the circle has a radius of 3, whatis the length of minor arc PR?

(A)

(B)

(C)

(D)

(E) 3

Figure 4

14. What is the slope of the line perpendicular to

the line whose equation is ?

(A) 1.41(B) 1.18(C) .85(D) .53(E) .21

15. The number (73)36has how many digits when

multiplied out?

(A) 12(B) 36(C) 37(D) 67(E) 68

16. What is the least positive integerxfor which12 xand 15 xwill be nonzero and haveopposite signs?

(A) 3(B) 4(C) 7(D) 11(E) 13

17. The solution set to the pair of equations:

mx+ ny= 15nx+ my= 13isx= 3 andy= 1. What are the values of mand n?

(A) m= 5n= 3

(B) m= 4n= 3

(C) m= 3

n= 4(D) m= 3

n= 5(E) m= 2

n= 6

18. The lengths of the sides of quadrilateral Qareall integers. If three of the sides have lengthsof 3, 4, and 5, then the maximum length of thefourth side is

(A) 13(B) 12(C) 11(D) 7(E) 2

19. In Figure 5, ifABDis a right isosceles triangle,thenx=

(A) 25

(B) 30(C)

(D) 45(E) 60

Figure 5
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20. Ifxyz0, then

(A) 4xyz

(B)

(C)

(D)

(E)

21. Iff(x) = x2 3 and g(x) = 3 x2, what is thevalue off(f(g(7)))?

(A) 46(B) 2119(C) 73207

(D) 4490164(E) 7295398

22. A polygon Qwith a certain perimeter Pwillhave its greatest area when all of its sides havethe same length. What is the maximum areaof a rectangle with a perimeter of Punits?

(A)

(B)

(C) P2

(D) 2P2

(E) 4P2

23. Which of the following graphs is NOT thegraph of a function?

(A)

(B)

(C)

(D)

(E)

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Lesson 8202


24. In the figure above, if sin , then JL =

(A) 26.83(B) 13.42(C) 6.71(D) 1.12(E) 0.37

25. An equation for the circle with its center at theorigin and passing through the point (1,2) is

(A)

(B) x2+y2= 3(C) x2+y2= 5(D) x2+y2= 9(E) x2+y2= 25

26. How many integers are in the solution set of|1 3x| < 5?

(A) None(B) One(C) Two(D) Three(E) Infinitely many

27. If x, y, and z are positive integers such that4x+ 6y=z, thenzmust be divisible by

(A) 2(B) 4(C) 6(D) 10(E) 24

28. If the points (2,4), (3,4), and (3, 2) areconnected to form a triangle, the area of thetriangle is

(A)

(B) 6(C) 12(D) 15(E) 24

29. If i2= 1 and if k= 2 + i, then k2=

(A) 1(B) 3 + 4i(C) 4 + 3i(D) 6 + 7i

(E) 9 + 12i30. If a line contains the points (2, 1) and (4,4),

then thexintercept is

(A) 4

(B)

(C) 0

(D)

(E)

31. In Figure 7, if the radius of the circle is r, then

the ratio =

(A)

(B)

(C)

(D)

(E)

Figure 7

32. f() = sin24 + cos24 , findf(72)

(A) .71(B) .22(C) 1.0(D) 1.26(E) 4.0


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33. Iff(x) = 3x 2 and g(f(x)) =x, then g(x) =

(A) 3x+ 2(B) 2 3x

(C)

(D)

(E)

34. In Figure 8, ifAC// GEand GF=xand FE=y,

then the ratio =

(A)

(B)

(C)

(D)

(E)

Figure 8

35. If , then c=

(A)

(B) ab

(C)

(D)

(E)

36. Ifx3y2z< 0, then it must be true that

(A) x3< 0(B) z< 0(C) xy< 0(D) xz< 0

(E) yz< 037. If the slope of a line is 3 and theyintercept is

2, then thexintercept of the line is

(A)

(B)

(C) 1

(D)

(E)

38. For the right triangle in Figure 9, all of thefollowing statements are true EXCEPT:

(A) sin

(B) tan

(C) cos

(D) sin = cos (E) cot = tan

Figure 9

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8/29

Lesson 8204


39. Three candidates for president of the StudentCouncilAshley, Jos, and Kimmust eachbe scheduled for a single 10minute addressto the entire student body. If the order of thepresentations is determined randomly, how

many different orders are possible?(A) 3(B) 6(C) 9(D) 12(E) 27

40. Ifx0 then =

(A) 22x

(B) 4x

(C) 42x

(D) 41x

(E) 8x

All SareM.No PareM.

41. Which of the following conclusions can belogically deduced from the two statementsabove?

(A) All Sare P.(B) AllMare S.(C) Some Sare notM.(D) SomeMare P.

(E) No Pare S.

42. Cube Qhas volume V. In terms of V, a cubewith edges only onefourth the length of thoseof Qwill have a volume of

(A)

(B)

(C)

(D)

(E)

43. If is an acute angle and cos = , b> 0 and

c> 0 and bc, then sin =

(A)

(B)

(C)

(D)

(E)

44. If a cube has an edge of length 2, what isthe distance from any vertex to the center ofthe cube?

(A)

(B)

(C)

(D)

(E)

45. Ifx2+ ax+ bx+ ab= 0, andx+ b= 2, then

x+ a=

(A) 8(B) 4(C) 2(D) 0(E) 2


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46. Figure 10 shows two right circular cylinders,Cand C. If r= krand h= kh, then what is

the ratio of: ?

(A)

(B) (C) k

(D)

(E) k3

Figure 10

47. If the circumference of a circle is 1, what is itsarea?

(A) .08(B) .79(C) 1.27(D) 3.14(E) 6.28

48. What are the coordinates of the point of in-tersection of the lines having the followingequations:

(A)

(B)

(C)

(D)

(E)

49. In Figure 12, the radius of the circles is 1.What is the perimeter of the shaded part ofthe figure?

(A)

(B)

(C)

(D)

(E)

Figure 12

50. If , for what value ofxis

f(x) undefined?

(A) 4(B) 2(C) 0

(D)

(E) 2

STOP

IF YOU FINISH BEFORE TIME IS CALLED,

YOU MAY CHECK YOUR WORK ON THIS

TEST ONLY. DO NOT WORK ON ANY

OTHER TEST IN THIS BOOK.


10/29

Lesson 8206


ANSWER KEY

1. D2. B3. D

4. E5. B6. C7. D8. D9. E

10. D

11. B12. E13. C

14. C15. E16. E17. B18. C19. A20. E

21. D22. A23. A

24. B25. C26. D27. A28. D29. B30. A

31. B32. C33. E

34. D35. D36. D37. B38. C39. B40. A

41. E42. C43. E

44. B45. D46. E47. A48. E49. A50. E

EXPLANATORY ANSWERS

1. The correct answer is (D). This first problem is, as we would anticipate, an easy one. Simply sub-stitute 4 for bin the equation:

x= (4+3)2

x= (1)2

x= 1

2. The correct answer is (B). Find the minimum and maximum values off(x) by testing the limits ofx. Forx= 1:

f(x) = 4(1) 1 = 3

Forx= 3:

f(x) = 4(3) 1 = 12 1 = 11

Therefore,f(x) must be greater than 3 but less than 11.

3. The correct answer is (D). There are at least two ways to approach this problem. One is to solve fork. Begin by crossmultiplying:

5(k+ 6) = 7(k+ 4)5k+ 30 = 7k+ 282 = 2kk= 1

You could also solve the problem by substituting the answer values into the equation for k. Obvi-

ously, the values provided by (A), (B), and (C) will require more complex calculations than those pro-

vided by (D) and (E). So start with (D):

Since 1 = 1, this proves that (D) is correct.


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4. The correct answer is (E). This problem requires you to think about the concept of an average. Anaverage is the sum of the values to be averaged divided by the number of values in the average.Therefore:

Which means the total value of the 13 items is:

Sum = $12 13 = $156

The sum of the 12 items that remain after the one is removed is:

The difference between the two sums is the value of the item removed:

$156 $132 = $24

5. The correct answer is (B).

In right triangle I:

In right triangle II:

6. The correct answer is (C). First rearrange the equation to isolate the 5th root term

Then raise each side of the equation to the 5th power and solve forx.


12/29

Lesson 8208


7. The correct answer is (D). Using the distance formula

8. The correct answer is (D). This is a good exercise in problem solving. Since Cannot be deter-mined is not an answer choice, we can be certain that we have sufficient information to findRS. Wealready know:

And the figure shows:

QS+RT= QR+RS+RS + STAll we have to do is find a way to manipulate these symbols until we isolateRS:

(QR+RS) + (RS+ ST) = QS+RTRS+ (QR+RS+ ST) = (QS+RT)

RS= (QS+RT) (QR+RS+ ST)Since QR+RS+ ST= QT:

RS= (QS+RT) QT

And, using the values provided:

If you tend to reason spatially, you might just have looked at the figure and seen:

Then you would see that the length of QSandRTtogether exceeds that of QTby the length ofRS. So,just add QSandRTand subtract QT.


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9. The correct answer is (E). There are several ways of reaching the correct answer here. One, sincethe line that bounds the lower portion of the shaded area includes the points (0,0) and (5,5), it is thegraph of the equationx=y. For a point to be in the shaded region, theycoordinate must be greaterthan thexcoordinate. Only (E) meets this requirement. Or, you might reason less formally that theline contains points in which the rise is equal to the run. So you are looking for a point for whichthe rise is greater than the run. Or you might just choose to plot the points on the diagram.

10. The correct answer is (D). To handle this item, just ask yourself how to obtain the minimum andmaximum values forxy. Sincex> 4 andy> 6, the lower limit forxyis 24. And sincex< 12 andy< 8,the upper limit forxyis 96.

11. The correct answer is (B). Since the triangles are equilateral, the unmarked angles with vertices atthe common point have degree measures of 60. Then, the sum of the degree measures of all six of theangles with vertices at the common point (those marked x,y, andz, and those not marked) is 360.Therefore:

x+ 60 +y+ 60 +z+ 60 = 360x+y+z+ 180 = 360x+y+z= 180

12. The correct answer is (E). This is a defined operation problem. Dont let the Greek letter throwyou. Just substitute 2 forxand 1 foryinto the expressionxyyx:

(2)( 1) (1) (2) = 2 + 1 + 2 = 5

13. The correct answer is (C). For this problem, you have to know that the degree measure of an inscribedangle is onehalf that of the intercepted arc. (Or, an inscribed angle intercepts twice the arc.) Since theinscribed angle has a measure of 30 degrees, the minor arc PRhas a measure of 60 degrees. And since the

total measure of the circle is 360 degrees, minor arc PRis60

360

1

6= of the circumference.

PR C=6

And since C= 2r

PR r

=2

6

And r= 3:

PR=

=

= 2 3

6

6

6


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Lesson 8210


14. The correct answer is (C). To find the slope of a line express it in y= mx+ bform, where mrepre-sents the slope.

A perpendicular line would have a slope that is the negative reciprocal of . The negative reciprocal is

15. The correct answer is (E). (73)36= 1.20122113364 E 67 or a similar expression depending on yourparticular calculator. The calculator is using scientific notation to express a huge number.

1.2011 . . . 1067means move the decimal point 67 places to the right, resulting in a 68 digit number.

16. The correct answer is (E). Here is a good problem on which to use one of our alternative strategies.Test answer choices:

(A) 12 3 = 9 Wrong! Both expressions have the same sign.15 3 = 12

(B) 12 4 = 8 Ditto!15 4 = 11

(C) 12 11 = 1 Ditto!

15 11 = 4(D) 12 12 = 0 Wrong! Both expressions are said to be nonzero.15 12 = 3

(E) 12 13 = 1 Bingo!15 13 = 2

But how do we prove mathematically that 13 is correct? We dont have to. We eliminated four of the

choices, so the one that remains has to be the correct answer to the test question.


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17. The correct answer is (B). Here you could use either of two approaches. First, you can attack theproblem by treating the equations as a system of simultaneous equations:

3m+ n= 153n+ m= 13

Use the first equation to solve forn:

n= 15 3m

Substitute this expression for n in the second equation:

3(15 3m) + m= 1345 9m+ m= 138m+ 45 = 138m= 32m= 4

And find the value of n:

3(4) + n= 1512 + n= 15n= 3

As an alternative, you could test answer choices. Just substitute the values given for mand nuntil you

find a pair that satisfies both equations.

18. The correct answer is (C). The problems on this test are arranged in ascending order of difficulty.Often, what makes a problem more difficult than another problem is that the insight required to solvethe more difficult problem is more subtle. We know from the fact that this problem is situated morethan one third of the way through the test that it is not a giveaway. Therefore, you should not makethe mistake of reasoning 3 plus 4 plus 5 is equal to 12, therefore the answer is (B). If the problemwere that obvious, it would be one of the first few, not number 18. The correct answer is (C). Since thesum of the lengths of the other three sides is 12, the longest side of integral length that could be used

to close the figure is 11.19. The correct answer is (A). Letyrepresent the degree measure of angleBAD. SinceABCis isosce-

les,y= 45. And since the sum of the degree measures of the interior angles of a triangle is 180:

90 + 45 + 20 +x= 180155 +x= 180x= 25

20. The correct answer is (E). You can solve this problem by applying the rules for manipulating expo-nents. When dividing terms of the same base, you subtract exponents. Therefore, thexterm isxto the

3 2 = 1 power. Theyterm isyto the 2 3 = 1 power, which is1

y. And similarly, thezterm iszto

the 1 2 = 1 power, which is1

z. Since

2

8is

1

4, the final result is .

Alternatively, you could assume some values for the variables. Use those values to evaluate the

complex expression and derive a value. Then, using the same values, evaluate each answer choice

until you find one that generates the same value. You can make it easy on yourself by assuming small

values that are easy to work with, such as 1. But watch out! Since 1 1 = 1 1, you may find that

more than one answer choice seems to be correct. (If you use 1 forx,y, andz, in this problem, you will


16/29

Lesson 8212


find that the complex expression has the value1

4but that (C), (D), and (E) all equal 1/4 as well.) If

this happens, or if you anticipate this may happen, use other numbers. Assume, for example, x = 1,

y= 2, andz= 3. The complex expression has the value:

(Dont do the multiplication. Simplify by cancellation.) Now use the same values in the answer choices:

(A) 4(1)(2)(3) = 24 Wrong!

(B) Wrong!

(C) Wrong!

(D) Wrong!

(E) Correct!

Yes, substituting numbers is likely to take longer here than using the rules of exponents, but at least

you have an alternative available if you cant remember what to do.

21. The correct answer is (D). g(7) means substitute 7forxin the expression g(x) = 3 x2

g(7) = 3 (7)2= 46

f(g(7)) =f(46) which means substitute 46 forxin the expressionf(x) = x2 3

f(46) = (46)2

3 = 2119f(f(g(7))) =f(f(46)) =f(2119) which means substitute 2119 forxin the expressionf(x) = x2 3

f(2119) = (2119)2 3 = 4490164

22. The correct answer is (A). The rectangle with four equal sides is the square. And a square withperimeter Pwill have sides of P/4. The area of this figure is:

You can also solve this problem by assuming a value for P. For example, let the perimeter of square

Pbe 4. On that assumption, each side has a length of 1, and the area of the square is 1. Now, substitute

4 for Pin the answer choices until you find one that generates the value 1:

(A) Correct!

(B) Wrong!

(C) 42= 16 Wrong!

(D) 2(4)2= 32 Wrong!

(E) 4(4)2= 64 Wrong!


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23. The correct answer is (A). The defining characteristic of a function is for each element in thedomain, there is only one element in the range. Since the x coordinates are the domain, and theycoordinates are the range, for each x coordinate there can be no more than one ycoordinate. Thegraphs of (B), (C), (D), and (E) all fit this description. (A) does not. For the relationship described by(A), every value ofx (except for one) has twoy values.

24. The correct answer is (B).

25. The correct answer is (C). Here is a problem for which I would go directly to the alternative attackstrategy. Instead of trying to figure out which equation correctly describes the circle in question, Iwould simply put the pair (1,2) into the equations. The one that works is the right answer:

(A)

(B) 12

+ 22

= 3 Wrong!(C) 12 + 22= 5 Correct!

So we know that (C)is the right answer.

26. The correct answer is (D). They are 1, 0, and 1:

|1 3(1) | = |1 + 3| = 4|1 3(0) | = |1 0| = 1|1 3(1) | = |1 3| = 2

If, however, you try an integer larger than 1:

|1 3(2) | = |1 6| = 5

which is not less than 5, or one that is less than 1:

|1 3(2) | = |1 + 6| = 7

27. The correct answer is (A). One way of attacking this problem is to factor the expression 4x+ 6y:

4x+ 6y= 2(2x+ 3y)

This shows that 4x + 6y is divisible by 2 and thatz must also be divisible by 2. Alternatively, you can

just assume some numbers for x and y. For example, letxandyboth be 1:

4(1) + 6(1) = 10

Since 10 is divisible by 2 and 10 but not by 4, 6, or 24, we can eliminate all choices except (A) and

(D). Now try another set of numbers, say,x= 1 andy= 2:

4(1) + 6(2) = 1616 is divisible by 2 but not by 10, so we eliminate (D). By the process of elimination, we have

established that (A)must be the correct choice.


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Lesson 8214


28. The correct answer is (D). For this problem, you may want to sketch a figure:

Since the figure is a right triangle, we can use the adjacent sides as altitude and base:

(It makes no difference which you consider the altitude and which you consider the base.)

29. The correct answer is (B). First find the value of k2in terms of i:

k2= (2 + i)2= (2 + i)(2 + i) = 4 + 2i+ 2i+ i2= 4 + 4i+ i2

Since i2= 1:

4 + 4i+ i2= 4 + 4i+ (1) = 3 + 4i

30. The correct answer is (A). One way to attack this problem is to try to find the values for the generalequationy= mx+ b, where mis the slope of the line and btheyintercept. Since the line contains thepoints (2,1) and (4,4), the value of mis:

Now, to find b, use one of the sets of points already known:

The equation of the line is:

The line crosses thexaxis where the value ofy is 0:


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Alternatively, you can quickly sketch the line:

Now look at the choices. There is only one choice available that could be the value forxwheny= 0.

And that is (A).

31. The correct answer is (B). Before we do any math on this problem, use common sense to eliminatesome choices. The question asks for the ratio of the area of the larger figure to the area of the smallerfigure. That means that the numerator of the fraction must be larger than the denominator of thefraction. What about (D) and (E)? They must be wrong. Now for the math. Since ris the radius of thecircle, the larger square has a side of length 2r.

So the larger square has an area of 4r2. The diameter of the circle, which is also the diagonal of the

smaller square, is 2r. So the smaller square has sides ofr :

And the area of the smaller square is 2r2. So the ratio asked for is:


20/29

Lesson 8216


32. The correct answer is (C). sin2x+ cos2x= 1 for all values ofx, including 4 .

33. The correct answer is (E). One way to solve this problem is to figure out what mathematical opera-tions would be needed in g(x) to produce x from 3x 2. First, you need xrather than 3x, so thefunction would have to include dividing 3xby 3. And you dont want 2 in the final result, so it should

also include adding 2 to the result off(x). The expression that describes this procedure is

And we confirm that this is correct by performing this onf(x):

Alternatively, you could substitute the result of performingf(x) onxinto each answer choice until you

find one that produces the valuex:

(A) 3(3x 2) + 2 = 9x 6 + 2 = 9x 4 Wrong!

(B) 2 3(3x 2) = 2 9x+ 6 = 8 9xWrong!

(C) Wrong!

(D) Wrong!

(E) Correct!

An even easier approach is to assign a value forx. Letx= 2. Thenf(x) = 3(2) 2 = 4. Now use the

result off(x), 4, in the choices. The one that generates the value 2 (which isx) is the winner:

(A) 3(4) + 2 = 14 Wrong!

(B) 2 3(4) = 10 Wrong!

(C) Wrong!

(D) Wrong!

(E) Correct!

34. The correct answer is (D).There are two insights required to handle this problem. First, BEGissimilar to BCA, and BEFis similar to BCD. This means that the ratio area BEF:area BEGisthe same as the ratio area of BCD:area BCA. So if we find the first ratio, we also have thesecond. Second, sideBG, because it is perpendicular to side GE, is an altitude of triangleBEGand

of triangleBEF:


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The formula for finding the area of a triangle is1

2altitude base. So the areas of the triangles are:

Therefore, the ratio is:

Area

Area

BEF

BEG

BG y

BG x y

y

x y=

( )( )( ) +( )

=+

1

2

1

2

And that is the ratio of the areas of the two larger triangles as well.

35. The correct answer is (D). One way to attack this item is to do the indicated operations:

So:

Cross multiply:

c(b+ a) = ab

And divide by (b+ a)

Alternatively, you could assign some values to aand b. Let a= 1 and b= 2:

That means:

And


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Lesson 8218


Now substitute the values a= 1 and b= 2 into the choices until you find the one that produces 2/3:

(A) Wrong!

(B) (1)(2) = 2 Wrong!

(C) Wrong!

(D) Right!

(E) Wrong!

36. The correct answer is (D). This question tests your understanding of the properties of numbers.First, since the entire expression is less than zero, we know that none of the variables is 0. (Otherwise,the expression would be equal to and not less than 0.) In addition, we know thaty2must be positive.

And this means that eitherxorzbut not bothmust be negative! This is what (D) means.Alternatively, you could assign sets of numbers tox,y, andz, until you manage to eliminate all but one choice.

37. The correct answer is (B). You can attack this problem by using the general equationy= mx+ b,where m is the slope and bis theyintercept. The line intercepts thexaxis wherey= 0:

0 = 3x+ 2

3x= 2

38. The correct answer is (C). Test each statement:

(A)

(B)

(C)

(D)

(E)


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39. The correct answer is (B). One solution to this problem is to recognize that since the question asksabout different orders, this is an appropriate problem for the formula used to calculate permutations:3! = 3 2 1 = 6. Even if you didnt recall the formula, dont panic. You should be able to count thenumber of possibilities on your fingers:

1. AJK

2. AKJ

3. KAJ

4. KJA

5. JAK

6. JKA

And thats all there is to it.

40. The correct answer is (A). Remember that the rules of exponents can be applied only to terms oflike bases. Here, the numerator has a base of 8, and the denominator a base of 2. Before we canmanipulate the expression, it will be necessary to change one or the other term. There are several

different routes available to us. For example:

82x= (8)2x= (2 2 2)2x= (22x)(22x)(22x) = 22x+ 2x+ 2x= 26x

Now we can complete our division:

26x24x= 26x 4x= 22x

Or:

(8)2x= (23)2x= (2)(3)(2x)= 26x

And complete the division as shown above. Or you could choose to work with the denominator. I

think, however, that the best approach to this problem is just to assume a value forx. Sayx = 1:

If we substitute 1 forxinto the answer choices, the correct choice will generate the value 4:

(A) 22(1)= 4 Correct!

(B)

(C) 42(1)= 16 Wrong!

(D) 41 1= 40= 1 Wrong!

(E)


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Lesson 8220


41. The correct answer is (E). A good way to attack this problem is to use Venn or circle diagrams. Thefirst statement can be represented as follows:

The S circle contains all items that are Ss, and the M circle all those that are Ms. Notice that the S circle

is entirely contained within the M circle. The second statement can be added to the diagram as follows:

The fact that there is no overlap between the P and M circles shows that No P are M. Now we examine

the answer choices. The correct answer is (E). There is no overlap between the P and S circles.

42. The correct answer is (C). One way of attacking this problem is to use letters. The formula forfinding the volume of a cube is simply edge cubed. Therefore:

V= e3

And:

And a cube with an edge onefourth of that has an edge of:

. And a volume of:

As an alternative strategy, you could assume some numbers. Assume that the larger cube has an edge

of 4. (Why 4? Because that means the smaller cube has an edge of 1!) The larger cube has a volume

of 4 44 = 64, and the smaller cube a volume of 1 11 = 1. Now, just substitute 64 for Vinto the

answer choices, and the one that generates the value 1 is the correct choice.

43. The correct answer is (E). A sketch will make it easier to keep track of the relationships:


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Notice that I have placed side b in relation to to reflect the cosine relation specified in the problem.I have also designated the third side as a. The sin , therefore, is equal to a/c. And we can use therelationships between the sides of the right triangle to find ain terms of band c:

a2+ b2= c2

And:

a2= c2 b2

Therefore:

Substituting this for a:

44. The correct answer is (B). The neat thing about a cube is that if you have any one feature, e.g.,volume, edge, diagonal of a face, diagonal of the cube, surface area, you can calculate every other

feature. It is for this reason that cubes are often the focus of test problems. Given that the edge has alength of 2, we can use the Pythagorean Theorem to find the length of the diagonal of a face:

Now we can find the length of the diagonal of the cube:


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Lesson 8222


That is the length of the entire diagonal of the cube. The point that is the center of the cube is the

midpoint of the diagonal of the cube and is from each vertex.

45. The correct answer is (D). The trick to this problem is to recognize that:

x2+ ax+ bx+ ab= (x+ a)(x+ b)

Since (x+ a)(x+ b) = 0, one or the other factor must be 0. Sincex+ b= 2,x+ a= 0.

46. The correct answer is (E). The formula for calculating the volume of a cylinder is: V = r2h

So the volume of the larger cylinder is just that. Now we redefine the dimensions of the smaller

cylinder in terms of rand h:

r= krso r= r/k

h= khso h= h/k

And the volume of the smaller cylinder is:

And the ratio is:

As an alternative, you could assume some numbers. Let the radius and height of the larger cylinder be4 and 4, respectively, and those of the smaller cylinder 2 and 2. Since r= krand h= kh, kmust be 2.Now the larger cylinder has a volume of:

V= (4)24 = 64

And the smaller cylinder a volume of:

V= (2)22 = 8

And the ratio of 64to 8is 8 to 1 or simply 8. Now, using k= 2, find an answer choice that has the

value of 8:

(A) Wrong.

(B) Wrong.

(C) 8 Wrong.

(D) Wrong.

(E) 8 Correct.


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47. The correct answer is (A). First use the value of the circumference to solve for the radius

Then use the radius to find the areaor

ris now the length of the altitude of the triangle, and 2rthe length of the base:

Again, the area of the circle is just .

Finally, a little common sense gets you this problem without any math. The triangle, which has an

area of 1, takes up slightly less than half of the circle:

So the correct answer choice must be something thats a bit larger than 2. Only one of the answer

choices qualifies. (B) says, for example, that the triangle is less than1

6of the circle. So (B), and with

it the other choices that are larger, have to be incorrect.


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Lesson 8224


48. The correct answer is (E). The point of intersection will have the coordinates that satisfy both ofthe equations. So treat the equations as a system of simultaneous equations. Using the first equation,solve forx:

Substitute this into the second equation:

And solve fory:

(This allows you to eliminate (A) and (C). And that would be important if you were about to run out

of time.) Now use this value to findx:

49. The correct answer is (A). This is a good exercise in organized problem solving. Look at the figureand ask yourself what you already know. You know the radius of the circle. In addition, you know thatthe perimeter of the shaded area consists of two arcs. There must be some way to use the information

about the radius to find the length of the arcs. Arcs can be measured in terms of length or in terms ofdegrees. Is it possible to find the degree measures of those arcs? Yes:

Since the sides of the triangles are all radii, the triangles must be equilateral. This means that the

degree measure of each arc is 120. Since the circles have a radius of 1, they have circumferences of

2 (1) = 2. And since each arc is a third that long:2

3

. And since there are two such arcs, the

perimeter of the shaded area is 2 2

3

=

4

3

.

50. The correct answer is (E). The function will be undefined when one of the terms in the denomina-tor is 0. (Division by 0 yields an undefined term.) Using 2 will cause (x 2) to be 0, and the entireexpression will be undefined.


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