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1 If  , then  (A)  (B)  (C)  (D)  (E) 

Choice (A) is correct. If  ,

then  . Multiplying each side of this equation by   yields  .Choice (B) is not correct. If   

were equal to  , then   would be equal

to  . However,   is equal to  , not .Choice (C) is not correct. If   

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were equal to  , then   would

be equal to  . However,   is equal to  , not  .Choice (D) is not correct. If   

were equal to  , then   would be equal

to  .

However,   is equal to  , not  .Choice (E) is not correct. If   

were equal to  , then   would be equal

to  . However,   is equal to  , not  .

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2 In the figure above,  ,  , and   lie on the same line.   is the center of the larger circle, and   is the center of the smaller circle. If the radius of the larger circle is  , what is the radius of the smaller circle?

(A)  (B)  (C)  (D)  (E) 

Choice (B) is correct. Since   is the center of the larger

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circle, segment   is a radius of the larger circle, and so  .Since   is the center of the smaller circle, segment   is a radius of the smaller circle, as is segment  . Since  ,  , and   lie on the same line,  . Since they are radii of the same circle,   and   have the same length. Therefore,  , and so the radius of the smaller circle is  .Choice (A) is not correct. If the radius of the smaller circle were  , then   would equal . But since  ,  , and   lie on the same line,  , and the radius of the larger circle would be  . But the radius of

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the larger circle is  , not  , and so the radius of the smaller circle cannot be  .Choice (C) is not correct. The radius of the smaller circle must be less than the radius of the larger circle, which is  . Thus the radius of the smaller circle is not  .Choice (D) is not correct. The radius of the smaller circle is half, not double, the radius of the larger circle, which is  . Thus the radius of the smaller circle is  , not  .Choice (E) is not correct. If the radius of the smaller circle were  , then the diameter of the larger circle would be  . But this is not what the

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question asks. The radius of the smaller circle must be less than the radius of the larger circle, which is  . Thus the radius of the smaller circle is not  .

3 Roy planted corn on   of his land. If he planted   acres of corn, how many acres of land does he have?

(A)  (B)  (C)  (D)  (E) 

Choice (D) is correct. Roy

planted corn on   of his land, and he planted   acres of corn. Therefore,   acres

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represents   of his land. If Roy has a total of   acres of land, then  . Solving for   gives  , and so Roy has   acres of land.Choice (A) is not correct. If Roy

had   acres of land, then   of his land planted with corn would be   acres, not   acres.Choice (B) is not correct. If Roy

had   acres of land, then   of his land planted with corn would be   acres, not   acres.Choice (C) is not correct. If Roy

had   acres of land, then   of his land planted with corn would be   acres, not   acres.

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Choice (E) is not correct. If Roy

had   acres of land, then   of   of his land planted with corn would be   acres, not   acres.

4 The first number in the list above is  . Which of the following gives a rule for finding each successive number in the list?

(A) Add   to the preceding number. (B) Take   of the preceding number

and then add   to that result. (C) Double the preceding number and

then subtract   from that result. (D) Subtract   from the preceding

number and then double that result. (E) Triple the preceding number and

then subtract   from that result.

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Choice (C) is correct. In each case, doubling the preceding number and then subtracting   from that result gives the successive term in the list:

  

So this rule works for all the numbers in the list.Choice (A) is not correct. Although adding   to   gives  , adding   to   gives  , not  . So this rule does not work for all the numbers in the list.Choice (B) is not correct. Although taking   of    and

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then adding  gives  , taking   of   and then adding   gives  , not  . So this rule does not work for all the numbers in the list.Choice (D) is not correct. Subtracting   from   and then doubling the result gives  , not  . So this cannot be the correct rule.Choice (E) is not correct. Although tripling   and then subtracting  from the result gives  , tripling   and then subtracting   gives  , not  . So this rule does not work for all the numbers in the list.

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5 The two semicircles in the figure above have centers   and  , respectively. If  , what is the total length of the darkened curve?

(A)  (B)  (C)  (D)  (E) 

Choice (C) is correct. The darkened curve intersects line   in three points. Let   be the point of intersection that lies on  , as shown in the

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figure below.

Then   is a radius of the semicircle on the left,   is a radius of the semicircle on the right, and  . Since the darkened curve consists of the two semicircles, its length is the sum of the lengths of the two semicircles: 

. It follows that length of the darkened curve is  , or  .Choice (A) is not correct. The length of a semicircle is given

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by  , where   is the radius of the semicircle. The length of the darkened curve is the sum of the lengths of the two semicircles. Thus if the length of the darkened curve were  , then the sum of the radii of the two semicircles would be  . But the sum of the radii of the two semicircles is given by  , which is equal to  .Choice (B) is not correct. The length of a semicircle is given by  , where   is the radius of the semicircle. The length of the darkened curve is the sum of the lengths of the two semicircles. Thus if the length of the darkened curve were  , then the sum of the radii of the

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two semicircles would be  . But the sum of the radii of the two semicircles is given by  , which is equal to  .Choice (D) is not correct. The length of a semicircle is given by  , where   is the radius of the semicircle. The length of the darkened curve is the sum of the lengths of the two semicircles. Thus if the length of the darkened curve were  , then the sum of the radii of the two semicircles would be  . But the sum of the radii of the two semicircles is given by   which is equal to  .Choice (E) is not correct. The length of a semicircle is given by  , where   is the radius of

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the semicircle. The length of the darkened curve is the sum of the lengths of the two semicircles. Thus if the length of the darkened curve were  , then the sum of the radii of the two semicircles would be  . But the sum of the radii of the two semicircles is given by  , which is equal to  .

6 If   and   are positive numbers and  , then 

(A)  (B)  (C)  (D)  (E) 

Choice (A) is correct. If  , then  . Therefore,  .

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Choice (B) is not correct.

If   were equal to  , then   would equal  . But  , so  . Thus if  , then  . However,   is a positive number. Therefore,   cannot equal  .Choice (C) is not correct. If   were equal to  , then   would equal  . But  , so  . Thus  . This can only be true if  . However,   is a positive number. Therefore,   cannot equal  .Choice (D) is not correct. If   were equal to  , then   would equal  . But  , so  . Thus  . It follows that

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either   or  . It is given that   is a positive number, so   must be  ; then  . However, there are positive values for   and   that satisfy   where   and  . (For example,   can be   and  can be  .)

Therefore,   is not equivalent to  .Choice (E) is not correct. If  , then  . But  , so  . Thus  . This equation simplifies  ; since    is a positive number (and thus  ), it follows that  . Then  . However, there are positive values for   and   that

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satisfy   where   and  . (For example,   can be   and  can be  .) Therefore,   is not equivalent to  .

7

 The table above shows the populations of two countries and their population densities. The number of square miles in the area of Country   is approximately how much greater than the number of

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square miles in the area of Country 

(A)  (B)  (C)  (D)  (E) 

Choice (D) is correct. There are   people in Country with   people per square mile. Thus there are

in Country There are   people in Country  , with   people per square mile. Thus there are 

 in Country   Therefore, the area of Country   is

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approximately   square miles greater than the area of Country   Of the given answer choices,   is closest to Choice (A) is not correct. There are   more people per square mile in Country   than in Country   However, the question asked how many more square miles, not people per square mile, there are in Country  than in Country Choice (B) is not correct. The area of Country   is approximately   square miles greater than the area of Country   Of the given answer choices,  , not  , is closest to 

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Choice (C) is not correct. The area of Country   is approximately 

 square miles greater than the area of Country   Of the given answer choices,  , not  , is closest to Choice (E) is not correct. To find the number of square miles in Countries  and  the population should be divided by, not multiplied by, the number of people per square mile. If the number of people per square mile is computed incorrectly in this way, an incorrect answer of   could result.

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8 If  , which of the following must be true?

(A)  (B)  (C)  (D)  (E) 

Choice (E) is correct. If  , then  . Since the difference   is a positive number, it follows that  .Choice (A) is not correct. If  , it cannot be true that  :   and  .Choice (B) is not correct. The equation   can be rewritten as   and then factored to give  .

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Thus if  , it need not be true that  , as   satisfies the equation  :   and  .Choice (C) is not correct. The equation   can be rewritten as   and then factored to give  . Thus if  , it need not be true that  , as   satisfies the equation  :   and  .Choice (D) is not correct. If   were true, then   would also be true. But it is given that  , so  .

9 Let the function of   be defined by  , where   is a constant. If  ,

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what is the value of   ? (A)  (B)  (C)  (D)  (E) 

Choice (C) is correct. The function   is defined by  , where   is a constant. We have   and   and  . Therefore,  , which simplifies to  . Thus,   and  .Choice (A) is not correct. If   were equal to  , then   would be defined by  . Then   would equal  , and   would equal  . However,  , not  .

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Thus   cannot be  .Choice (B) is not correct. If   were equal to  , then   would be defined by  . Then   would equal  , and   would equal  . However,  ,  not  . Thus   cannot be  .Choice (D) is not correct. If   were equal to  , then   would be defined by  . Then   would equal  , and   would equal  . However,  , not  . Thus   cannot be  .Choice (E) is not correct. If   were equal to  , then   would be defined by  . Then   would equal  , and   would equal  . However,  , not  . Thus   cannot be  .

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10 A number is called "even-odd" if it is halfway between an even integer and an odd integer. If   is an even-odd number, which of the following must be true?

.   is an integer.

.   is even-odd..   is halfway between two

even integers. (A)   only (B)   only (C)   and   only (D)   and   only (E)  ,  , and 

Choice (A) is correct. If   is even-odd, then   is halfway between an even integer, call it  , and an odd integer, call

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it  . Since   is halfway between   and  , it follows

that  . Therefore,  . Since   must be an integer, statement   must be true.Since   is even and   is odd,   is odd and   is not an integer. That is, no even-odd number   can be an integer. Since   is an integer, it cannot be an even-odd number. Therefore, statement   is not true.If the number   is halfway between two even integers, call them   and  , then  . Since   and   are even,   is even, and

thus  is an integer. That is, any number halfway

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between two even integers is an integer. Since   is even-odd, and no even-odd number can be an integer,   is not an integer. Therefore,   cannot be halfway between two even integers. Thus statement   is not true.Since statement   must be true, but statements   and   are not true, the correct answer is choice (A).Choice (B) is not correct. If   is even-odd, then   is halfway between an even integer   and an odd integer  . Thus   and  . Since   must be an integer, statement   must be true, and choice (B) is not

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the answer.Choice (C) is not correct. Statement   is true. If   is even-odd, then   is halfway between an even integer   and an odd integer  . Thus  , and  , an integer.But statement   is not true. Since   is even and   is odd,   is odd and   is not an integer. That is, no even-odd number   can be an integer. Since   is an integer, it cannot be an even-odd number.Thus choice (C) is not the answer.Choice (D) is not correct. If   is even-odd, then   is halfway between an even integer   and

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an odd integer  . Thus  , and  . Since   must be an integer, statement   must be true, and choice (D) is not the answer.Choice (E) is not correct. Statement   is true. If   is even-odd, then   is halfway between an even integer   and

an odd integer  . Thus  , and  , an integer.But statement   is not true. Since   is even and   is odd,   is odd and   is not an integer. That is, no even-odd number   can be an integer. Since   is an integer, it cannot be an even-odd number.Thus choice (E) is not the

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answer.

11 If   is a positive integer, which of the following is NOT equal

to  ? (A)  (B)  (C)  (D)  (E) 

Choice (D) is correct. The expression in this choice,  , can be rewritten as  , or  . The expression   is equal to  , so the two expressions are not equal, as required.Choice (A) is not correct. The expression in this choice,  , is equal to the given

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expression,  , by the laws of

exponents:  .Choice (B) is not correct. The expression in the choice,  , can be rewritten as  , or  .

The given expression,  , is also equal to  .Choice (C) is not correct. The expression in this choice,  , can be rewritten as  , or  . This latter expression is equal to the given expression,  , by the laws of exponents:  .   Choice (E) is not correct. The expression in this choice,  , can be rewritten as the given

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expression,  , since  .

12

 According to the graph above, in which year was the ratio of the number of students enrolled at School   to the number of students enrolled at

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School   the greatest? (A)  (B)  (C)  (D)  (E) 

Choice (E) is correct. In  , there were approximately students enrolled at School   and approximately   students enrolled at school  , so the approximate ratio of the number of students enrolled at School   to the number of students enrolled at School   was  to  , or   to  . In  , the corresponding ratio was   to  , or   to  . In  , the ratio was   to  , or   to  .

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In  , there were an equal number of students enrolled at schools   and  , so the ratio of enrollment was   to  . In  , there were approximately   students enrolled at School   and approximately   enrolled at School  , so the approximate ratio of enrollment at School   to that at School   was   to  . Of the ratios corresponding to these five years, the greatest is   to  , the ratio for  .Choice (A) is not correct. In  , the approximate ratio of the number of students enrolled at School   to the number of students enrolled at

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School   was   to  . In  , this ratio was   to  . Therefore, the ratio was not the greatest in  .Choice (B) is not correct. In  , the approximate ratio of the number of students enrolled at School   to the number of students enrolled at School   was   to  . In  , this ratio was   to  . Therefore, the ratio was not the greatest in  .Choice (C) is not correct. In  , the approximate ratio of the number of students enrolled at School   to the number of students enrolled at School   was   to  . In  , this ratio

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was   to  . Therefore, the ratio was not the greatest in  .Choice (D) is not correct. In  , the ratio of the number of students enrolled at School   to the number of students enrolled at School   was   to . In  , this ratio was   to  . Therefore, the ratio was not the greatest in  .

13

 

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In the figure above,  . Which of the following must equal ?

(A)  (B)  (C)  (D)  (E) 

Choice (B) is correct. The sum of the degree measures of the angles in a triangle must be  , so   must equal  . We have  . When parallel lines are cut by a transversal, alternate interior angles are equal. Thus  . Vertical angles are equal, so  . It follows that  and  . Therefore,   must equal  .

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Choice (A) is not correct. We are given that  . In addition, from the figure we can conclude that   must equal  . It also must be true that  ,  , and  . One set of values consistent with this information is  ,  ,  ,  ,  , and  . In this case,  . Therefore, it need not be true that   is equal to  .Choice (C) is not correct. We are given that  . In addition, from the figure we can conclude that   must equal  . It also must be true that  ,  , and  . One set of values consistent with this

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information is  ,  ,  , ,  , and  . In this case,  . Therefore, it need not be true that   is equal to  .Choice (D) is not correct. We are given that  . In addition, from the figure we can conclude that   must equal  . It also must be true that  ,  , and  . One set of values consistent with this information is  ,  ,  ,  ,  , and  . In this case,  . Therefore, it need not be true that   is equal to  .Choice (E) is not correct. We are given that  . In addition,

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from the figure we can conclude that   must equal . It also must be true that ,  , and  . One set of values consistent with this information is  ,  ,  ,  ,  , and  . In this case,  . Therefore, it need not be true that   is equal to  .

14 How many different ordered pairs   are there such that   is an even integer, where  , and   is an integer, where  ?

(A)  (B)  (C)  (D)  (E) 

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Choice (C) is correct. There are   even integers   such that  :  ,  ,  , and  . There are   integers   such that  :  ,  ,  ,  , and  . Therefore, by the multiplication rule for counting, there are   different ordered pairs   such that   is an even integer, where   and   is an integer, where  .Choice (A) is not correct. There are   even integers   such that  . There are only   even integers   such that  . But the question states   can be any integer such that  . Thus, there are   possible values for  ,

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not  , and   possible ordered pairs  , not  .Choice (B) is not correct. There are  , not  , even integers   such that  . There are  , not  , integers  such that  . Thus, there are   possible ordered pairs   not  .Choice (D) is not correct. There are  , not  , even integers   such that  . There are  , not  , integers   such that  . Thus, there are   possible ordered pairs  , not  .Choice (E) is not correct. Although  , there are  , not  , even integers   such that  , and there are  , not  , integers   such

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that  . Thus, there are   possible ordered pairs  , not  .

15 The function above can be used to model the population of a certain endangered species of animal. If   gives the number of the species living   decades after the year  , which of the following is true about the population of the species from   to  ?  

(A) It increased by about  . (B) It increased by about  . (C) It decreased by about  . (D) It decreased by about  . (E) It decreased by about  .

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Choice (C) is correct. The population of the species living in   is given by  , because   is   decades after the year  . The value of   is . The population of the species in   is given by  , because   is   decades after the year  . The value of   is . Therefore, the population decreased by approximately   from   to  . Of the answer options, choice (C), "It decreased by about  ," most closely describes how the population of the species changed from   to  . Choice (A) is not correct. The population of the species

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decreased, not increased, from   to  .Choice (B) is not correct. The population of the species decreased, not increased, from   to  .Choice (D) is not correct. The population of the species decreased from   to  , but not by as much as  . The decrease was closer to  .Choice (E) is not correct. The population of the species decreased from   to  , but not by as much as  . The decrease was closer to  .

16 A sphere of radius   inside a cube touches each one of the six sides of the cube. What is

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the volume of the cube, in terms of  ?

(A)  (B)  (C)  (D)  (E) 

Choice (E) is correct. Because of the symmetry of the sphere and the symmetry of the cube, the sphere touches each of the sides of the cube in the center of that side. Thus, a line segment drawn from the point where the sphere touches one face of the cube to where the sphere touches the opposite face of the cube will be perpendicular to those two faces of the cube and will pass

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through the center of the sphere.It follows that the length of each edge of the cube is equal to twice the radius of the sphere, or  . The volume of the cube is therefore equal to   or  .Choice (A) is not correct. The length of each edge of the cube is equal to the diameter of the sphere, not its radius. Therefore, the volume of the cube is  , not  .Choice (B) is not correct. The length of each edge of the cube is equal to twice the radius of the sphere, or  . Therefore, the volume of the cube is equal

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to  , which is equal to  , not  . Choice (C) is not correct. The length of each edge of the cube is equal to twice the radius of the sphere, or  . Therefore, the volume of the cube is equal to  , which is equal to  , not  .Choice (D) is not correct. The

volume of the sphere is  , but the question asks for the volume of the cube, which is  .