2024年3月24日发(作者：文昌初二数学试卷及答案)

SAT Subject Test Practice - Results Summary

Mathematics Level 2

1Your answer Omitted!

What is the distance in space between the points with coordinates

(A)

(B)

(C)

(D)

(E)

Explanation

Difficulty: Easy

The correct answer is D.

The distance between the points with coordinates

.

and

,

which simplifies to

2Your answer Omitted!

If

(A)

(B)

(C)

(D)

(E)

, what value does approach as gets infinitely larger?

.

is:

and is given by the

and ?

distance formula:

Therefore, the distance between the points with coordinates

Explanation

Difficulty: Easy

The correct answer is E.

One way to determine the value that approaches as gets infinitely larger is to rewrite the

definition of the function to use only negative powers of and then reason about the behavior of

negative powers of as gets infinitely larger. Since the question is only concerned with what

happens to

expression

as gets infinitely larger, one can assume that is positive. For

. As

, the

is equivalent to the expression

approaches the value

. Thus, as

, so as

gets infinitely

larger, the expression gets infinitely larger, the expression

approaches . approaches the value gets infinitely larger,

Alternatively, one can use a graphing calculator to estimate the height of the horizontal asymptote

for the function

, say, from

. Graph the function

to .

on an interval with “large”

By examining the graph, the

from, say, to

all seem very close to

.

. Graph the function again,

The vary even less from . In fact, to the scale of the coordinate plane shown, the

is nearly indistinguishable from the asymptotic line .graph of the function

This suggests that as gets infinitely larger, approaches , that is, .

Note: The algebraic method is preferable, as it provides a proof that guarantees that the value

approaches is . Although the graphical method worked in this case, it does not provide a

complete justification; for example, the graphical method does not ensure that the graph resembles

a horizontal line for “very large” such as .

3Your answer Omitted!

If

(A)

(B)

(C)

(D)

(E)

Explanation

Difficulty: Easy

The correct answer is A.

By the Factor Theorem,

Therefore, .

by

.

and then find a value

is a factor of only when is a root of

, which simplifies to .

is a factor of , then

that is,

Alternatively, one can perform the division of

for so that the remainder of the division is

Since the remainder is

4Your answer Omitted!

, the value of must satisfy . Therefore, .

Alison deposits into a new savings account that earns percent interest compounded

annually. If Alison makes no additional deposits or withdrawals, how many years will it take for the

amount in the account to double?

(A)

(B)

(C)

(D)

(E)

Explanation

Difficulty: Medium

After year, the amount in the account is equal to

equal to

to find the value of

. After years, the amount is

. You need, and so on. After years, the amount is equal to

for which . There are several ways to solve this

equation. You can use logarithms to solve the equation as follows.

Since , it will take more than

you need to round up to .

Another way to find

years for the amount in the account to double. Thus,

is to use your graphing calculator to graph and

. From the answer choices, you know you need to set the viewing window with values

from to about and values extending just beyond . The of the

point of intersection is approximately . Thus you need to round up to .

5Your answer Omitted!

In the figure above, when

(A)

(B)

(C)

(D)

(E)

Explanation

Difficulty: Medium

The resultant of

is subtracted from , what is the length of the resultant vector?

can be determined by.

The length of the resultant is:

6Your answer Omitted!

In the

(A)

(B)

(C)

(D)

(E)

Explanation

Difficulty: Medium

It is helpful to draw a sketch of the triangle:

-plane, what is the area of a triangle whose vertices are , , and ?

The length of the base of the triangle is

the area of the triangle is

7Your answer Omitted!

and the height of the triangle is

. The correct answer is B.

. Therefore,

A right circular cylinder has radius and height . If and are two points on its surface, what

is the maximum possible straight-line distance between and ?

(A)

(B)

(C)

(D)

(E)

Explanation

Difficulty: Medium

The maximum possible distance occurs when and are on the circumference of opposite bases:

You can use the Pythagorean Theorem:

The correct answer is (B).

8Your answer Omitted!

Note: Figure not drawn to scale.

In the figure above,

(A)

(B)

(C)

(D)

(E)

Explanation

Difficulty: Medium

and the measure of is . What is the value of ?

There are several ways to solve this problem. One way is to use the law of sines. Since ,

the measure of is and the measure of is . Thus, and

. (Make sure your calculator is in degree mode.)

You can also use the law of cosines:

Since is isosceles, you can draw the altitude to the triangle.

9Your answer Omitted!

The function is defined by for .

What is the difference between the maximum and minimum values of ?

(A)

(B)

(C)

(D)

(E)

Explanation

Difficulty: Medium

It is necessary to use your graphing calculator for this question. First graph the function

. It is helpful to resize the viewing window so the

to . On this interval the maximum value of is

. The difference between these two values is

10Your answer Omitted!

Suppose the graph of

represents

(A)

(B)

(C)

(D)

(E)

Explanation

Difficulty: Medium

It may be helpful to draw a graph of and .

is translated units left and unit up. If the resulting graph

?

-values go from

and the minimum value of is

, which rounds to .

, what is the value of

The equation for is . Therefore,

. The correct answer is B.

11Your answer Omitted!

A sequence is recursively defined by

the value of ?

, for . If and , what is

(A)

(B)

(C)

(D)

(E)

Explanation

Difficulty: Medium

The values for and are given. is equal to

. is equal to

.

If your graphing calculator has a sequence mode, you can define the sequence recursively and find

the value of . Let , since the first term is . Define .

Let , since we have to define the first two terms and . Then examining a

graph or table, you can find .

12Your answer Omitted!

The diameter and height of a right circular cylinder are equal. If the volume of the cylinder is ,

what is the height of the cylinder?

(A)

(B)

(C)

(D)

(E)

Explanation

Difficulty: Medium

The correct answer is A.

To determine the height of the cylinder, first express the diameter of the cylinder in terms of the

height, and then express the height in terms of the volume of the cylinder.

The volume of a right circular cylinder is given by

base of the cylinder and

. Thus . Substitute the expression

. Solving for gives

, where is the radius of the circular

for in the volume formula to eliminate :

. Since the volume of the cylinder is , the

. is equal to

is equal to .

is the height of the cylinder. Since the diameter and height are equal,

height of the cylinder is

13Your answer Omitted!

If

(A)

(B)

(C)

(D)

(E)

Explanation

Difficulty: Medium

The correct answer is E.

One way to determine the value of

identity:

identity gives

Another way to determine the value of

trigonometric identity for the sine:

14Your answer Omitted!

A line has parametric equations

the line is

(A)

(B)

(C)

(D)

(E)

Explanation

Difficulty: Medium

The correct answer is B.

, then

.

is to apply the sine of difference of two angles

. Since and

. Therefore,

is to apply the supplementary angle

. Therefore, .

, the

.

and , where is the parameter. The slope of

One way to determine the slope of the line is to compute two points on the line and then use the

slope formula. For example, letting gives the point on the line, and letting gives

the point on the line. Therefore, the slope of the line is equal to

in terms of . Since and

.

,Alternatively, one can express

it follows that . Therefore, the slope of the line is .

15Your answer Omitted!

What is the range of the function defined by

(A) All real numbers

(B) All real numbers except

(C) All real numbers except

(D) All real numbers except

(E) All real numbers between and

Explanation

Difficulty: Medium

The correct answer is D.

The range of the function defined by

for some .

is to solve the equation

. To

is the set of such that

?

One way to determine the range of the function defined by

for

solve

and then determine which

for

correspond to at least one

and then take the

shows that for any

, and that there is no such

, first subtract from both sides to get

. The equation

such that

reciprocal of both sides to get

other than , there is an for

. Therefore, the range of the function defined by

Alternatively, one can reason about the possible values of the term

on any value except , so the expression

is all real numbers except .

. The expression can take

can take on any value except . Therefore, the

is all real numbers except .range of the function defined by

16Your answer Omitted!

The table above shows the number of digital cameras that were sold during a three-day sale. The

prices of models , , and were , , and , respectively. Which of the following

matrix representations gives the total income, in dollars, received from the sale of the cameras for

each of the three days?

(A)

(B)

(C)

(D)

(E)

Explanation

Difficulty: Medium

The correct answer is C.

A correct matrix representation must have exactly three entries, each of which represents the total

income, in dollars, for one of the three days. The total income for Day is given by the arithmetic

expression , which is the single entry of the matrix product

; in the same way, the total income for Day is given by

, the single entry of

income for Day is given by

; and the total

, the single entry of

. Therefore, the matrix representation

gives the total income, in dollars, received from the sale of the cameras for each of the three days.

Although it is not necessary to compute the matrix product in order to answer the question

correctly, equals .

17Your answer Omitted!

The right circular cone above is sliced horizontally forming two pieces, each of which has the same

height. What is the ratio of the volume of the smaller piece to the volume of the larger piece?

(A)

(B)

(C)

(D)

(E)

Explanation

Difficulty: Hard

It is helpful to label the figure.

The top piece is a cone whose height is one-half the height of the original cone . Using the

properties of similar right triangles, you should realize the radii of these two cones must be in the

same ratio. So if the top cone has radius , the original cone has radius .

The volume of the top piece is equal to . The volume of the bottom piece is equal to the

volume of the original cone minus the volume of the top piece.

The ratio of the volume of the smaller piece to the volume of the larger piece is .

18Your answer Omitted!

In the figure above,

of ?

(A)

is a regular pentagon with side of length . What is the -coordinate

(B)

(C)

(D)

(E)

Explanation

Difficulty: Hard

The sum of the measures of the interior angles of a regular pentagon is equal to .

Each interior angle has a measure of . Using supplementary angles, has a measure of

. You can use right triangle trigonometry to find the -coordinate of point .

Since , is about . Since the length of each side of the

pentagon is , the -coordinate of point is . Putting the information together tells us that the

-coordinate of point is . The correct answer is (B).

19Your answer Omitted!

For a class test, the mean score was , the median score was , and the standard deviation of

the scores was . The teacher decided to add points to each score due to a grading error. Which

of the following statements must be true for the new scores?

I.

II.

III.

The new mean score is

The new median score is

.

.

.The new standard deviation of the scores is

(A) None

(B) only

(C) only

(D) and only

(E) , , and

Explanation

Difficulty: Hard

For this type of question you need to evaluate each statement separately. Statement is true. If you

add to each number in a data set, the mean will also increase by . Statement is also true. The

relative position of each score will remain the same. Thus, the new median score will be equal to

more than the old median score. Statement is false. Since each new score is more than the old

score, the spread of the scores and the position of the scores relative to the mean remain the same.

Thus, the standard deviation of the new scores is the same as the standard deviation of the old

scores.

20Your answer Omitted!

A game has two spinners. For the first spinner, the probability of landing on blue is

Independently, for the second spinner, the probability of landing on blue is

.

What is the

probability that the first spinner lands on blue and the second spinner does not land on blue?

(A)

(B)

(C)

(D)

(E)

Explanation

Difficulty: Hard

Since the two events are independent, the probability that the first spinner lands on blue and the

second spinner does not land on blue is the product of the two probabilities. The first probability is

given. Since the probability that the second spinner lands on blue is

second spinner does not land on blue is

answer is (E).

21Your answer Omitted!

In January the world’s population was billion. Assuming a growth rate of percent

per year, the world’s population, in billions, for years after can be modeled by the

Therefore,

the probability that the

. The correct

equation

January

(A)

(B)

(C)

(D)

(E)

Explanation

Difficulty: Hard

The correct answer is C.

was

. According to the model, the population growth from January to

According to the model, the world’s population in January

was

to January , in billions, was

.

22Your answer Omitted!

was and in January

. Therefore, according to the model, the population growth from January

, or equivalently,

What is the measure of one of the larger angles of a parallelogram in the

vertices with coordinates , , and ?

(A)

(B)

(C)

(D)

(E)

Explanation

Difficulty: Hard

The correct answer is C.

that has

First, note that the angle of the parallelogram with vertex is one of the two larger angles of

the parallelogram: Looking at the graph of the parallelogram in the makes this

apparent. Alternatively, the sides of the angle of the parallelogram with vertex are a

horizontal line segment with endpoints and and a line segment of positive slope with

endpoints and that intersects the horizontal line segment at its left endpoint , so

the angle must measure more than Since the sum of the measures of the four angles of a

parallelogram equals , the angle with vertex must be one of the larger angles.

One way to determine the measure of the angle of the parallelogram with vertex is to apply

the Law of Cosines to the triangle with vertices , , and . The length of the two

sides of the angle with vertex are and

. Let represent the angle

, so

. Therefore, the measure of one of the

; the length of the side opposite the angle is

with vertex and apply the Law of Cosines:

larger angles of the parallelogram is .

Another way to determine the measure of the angle of the parallelogram with vertex is to

consider the triangle , , and . The measure of the angle of this triangle with vertex

is less than the measure of the angle of the parallelogram with vertex . The angle

of the triangle has opposite side of length and adjacent side of length , so the

measure of this angle is

vertex is

. Therefore, the measure of the angle of the parallelogram with

.

Yet another way to determine the measure of the angle of the parallelogram with vertex is to

use trigonometric relationships to find the measure of one of the smaller angles, and then use the

fact that each pair of a larger and smaller angle is a pair of supplementary angles. Consider the

angle of the parallelogram with vertex ; this angle coincides with the angle at vertex of

the right triangle with vertices at , , and , with opposite side of length

and adjacent side of length , so the measure of this angle is . This angle, together

with the angle of the parallelogram with vertex , form a pair of interior angles on the same

side of a transversal that intersects parallel lines, so the sum of the measures of the pair of angles

equals . Therefore, the measure of the angle of the parallelogram with vertex is

.

23Your answer Omitted!

For some real number , the first three terms of an arithmetic sequence are

, and . What is the numerical value of the fourth term?

(A)

(B)

(C)

(D)

(E)

Explanation

Difficulty: Hard

The correct answer is E.

To determine the numerical value of the fourth term, first determine the value of and then apply

the common difference.

Since are the first three terms of an arithmetic sequence, it must be true that

, that is, Solving for gives .

Substituting for in the expressions of the three first terms of the sequence, one sees that they

are , , and , respectively. The common difference between consecutive terms for this

arithmetic sequence is , and therefore, the fourth term is .

24Your answer Omitted!

In a group of people, percent have brown eyes. Two people are to be selected at random

from the group. What is the probability that neither person selected will have brown eyes?

(A)

(B)

(C)

(D)

(E)

Explanation

Difficulty: Hard

The correct answer is A.

One way to determine the probability that neither person selected will have brown eyes is to count

both the number of ways to choose two people at random from the people who do not have brown

eyes and the number of ways to choose two people at random from all people, and then

compute the ratio of those two numbers.

Since percent of the people have brown eyes, there are people with brown

eyes, and people who do not have brown eyes. The number of ways of choosing two

people, neither of whom has brown eyes, is : there are ways to choose a first person and

, and

ways to choose a second person, but there are ways in which that same pair of people could be

chosen. Similarly, the number of ways of choosing two people at random from the people is

. Therefore, the probability that neither of the two people selected has brown eyes is

.

Another way to determine the probability that neither person selected will have brown eyes is to

multiply the probability of choosing one of the people who does not have brown eyes at random

from the people times the probability of choosing one of the people who does not have brown

eyes at random from the remaining people after one of the people who does not have brown eyes

has been chosen.

Since percent of the people have brown eyes, the probability of choosing one of the people

who does not have brown eyes at random from the people is . If one of the

people who does not have brown eyes has been chosen, there remain people who do not have

brown eyes out of a total of people; the probability of choosing one of the people who does not

have brown eyes at random from the people is

.

25Your answer Omitted!

If

(A)

(B)

(C)

(D)

(E)

Explanation

Difficulty: Hard

The correct answer is E.

One way to determine the value of

, start with the equation

. Isolate

the equation to get

Another way to determine the value of

Let and solve for

to get

.

is to find a formula for

: cubing both sides gives

and then evaluate at

, so , and

is to solve the equation for . Since

, what is ?

. Therefore, if two people are to be selected

from the group at random, the probability that neither person selected will have brown eyes is

, and cube both sides to get

, and apply the cube root to both sides of

. Therefore,

26Your answer Omitted!

, and .

Which of the following equations best models the data in the table above?

(A)

(B)

(C)

(D)

(E)

Explanation

Difficulty: Hard

The correct answer is D.

One way to determine which of the equations best models the data in the table is to use a calculator

that has a statistics mode to compute an exponential regression for the data.

The specific steps to be followed depend on the model of calculator, but can be summarized as

follows: Enter the statistics mode, edit the list of ordered pairs to include only the four points given

in the table and perform an exponential regression. The coefficients are, approximately, for the

constant and for the base, which indicates that the exponential equation is the

result of performing the exponential regression. If the calculator reports a correlation, it should be a

number that is very close , to which indicates that the data very closely matches the exponential

equation. Therefore, of the given models, best fits the data.

Alternatively, without using a calculator that has a statistics mode, one can reason about the data

given in the table.

The data indicates that as increases, increases; thus, options A and B cannot be candidates for

such a relationship. Evaluating options C, D and E at shows that option D is the one

that gives a value of that is closest to In the same way, evaluating options C, D and E at

each of the other given data points shows that option D is a better model for that one data point

than either option C or option E. Therefore, is the best of the given models for the

data.

27Your answer Omitted!

The linear regression model above is based on an analysis of nutritional data from 14 varieties of

cereal bars to relate the percent of calories from fat to the percent of calories from

carbohydrates . Based on this model, which of the following statements must be true?

I.

II.

There is a positive correlation between and .

When percent of calories are from fat, the predicted percent of calories from

carbohydrates is approximately .

The slope indicates that as increases by , decreases by .III.

(A) II only

(B) I and II only

(C) I and III only

(D) II and III only

(E) I, II, and III

Explanation

Difficulty: Hard

The correct answer is D.

Statement I is false: Since , high values of are associated with low values

of which indicates that there is a negative correlation between and .

Statement II is true: When percent of calories are from fat,

of calories from carbohydrates is

Statement III is true: Since the slope of the regression line is

increases by ; that is, decreases by

28Your answer Omitted!

The number of hours of daylight, , in Hartsville can be modeled by , where

and the predicted percent

.

, as increases by ,

.

is the number of days after March . The day with the greatest number of hours of daylight has

how many more daylight hours than May ? (March and May have days each. April and June

have days each.)

(A)

(B)

(C)

hr

hr

hr

(D)

(E)

hr

hr

Explanation

Difficulty: Hard

The correct answer is A.

To determine how many more daylight hours the day with the greatest number of hours of daylight

has than May , find the maximum number of daylight hours possible for any day and then

subtract from that the number of daylight hours for May .

To find the greatest number of daylight hours possible for any day, notice that the expression

is maximized when , which corresponds to , so

. However, for this problem, must be a whole number, as it represents a count

of days after March . From the shape of the graph of the sine function, either

corresponds to the day with the greatest number of hours of daylight, and since

, the expression

or

is maximized

when days after March . (It is not required to find the day on which the greatest number

of hours of daylight occurs, but it is days after March ,that is, June .)

Since May is

is .

days after March , the number of hours of daylight for May

Therefore, the day with the greatest number of hours of daylight has

more daylight hours than May .


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