Gibilisco S. Everyday Math Demystified: A Self-Teaching Guide

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Given this information, what is the possible number of combinations

of 8 objects taken 2 at a time?

(a) 40,320

(b) 720

(d) 28

(e) 24

17. Fig. Test 2-2 shows the results of a hypothetical experiment with

eight possible outcomes, called A through H. This illustration is an

example of

(a) a vertical bar graph

(b) a point-to-point graph

(d) a histogram

(e) linear interpolation

18. In Fig. Test 2-2, the vertical, dashed line passes through a point

corresponding to

(a) the mean

(b) the median

(d) the distribution factor

(e) the deviation

19. What, if anything, is wrong with the graph of Fig. Test 2-2?

(a) If the frequency values of the points are added up, the sum should

be equal to 100, but it obviously isn’t.

Fig. Test 2-2. Illustration for Part Two Test Questions 17 through 19.

Test: Part 2 191

(b) A set of events with outcomes like this cannot possibly occur.

values to either side should decrease steadily.

(d) The frequency values should increase steadily as you move

toward the right in the graph.

(e) There is nothing wrong with this graph.

20. What is the solution to the equation 3x þ 4y ¼ 12?

(a) 3

(b) 4

(d) 4

(e) There is no unique solution to this equation.

21. The equation 3x

þ 5x þ 3 ¼ 0 is an example of

(a) a linear equation

(b) a quadratic equation

(d) a quartic equation

(e) a multivariable equation

22. In statistics, a discrete variable

(a) can attain speciﬁc, deﬁned values

(b) can attain inﬁnitely many values within a deﬁned span

(d) can attain only non-negative integer values

(e) can never have a value greater than 1

23. Fill in the blank to make the following sentence true: ‘‘In a statistical

distribution having discrete values for the independent variable,

the highest cumulative absolute frequency number should be equal

to the

————————

of all the individual absolute frequency

numbers.’’

(a) average

(b) sum

(d) product

(e) largest

24. In the equation 4x

þ 8x

 17x  6 ¼ 0, the numbers 4, 8, 17, and

6 are called the

(a) factors

(b) solutions

PART 2 Finding Unknowns

192

(d) complex numbers

(e) imaginary numbers

25. If two outcomes H

and H

are complementary, then the probability,

expressed as a percentage, of one outcome is equal to

(a) the probability of the other outcome

(b) 100% plus the probability of the other outcome

(d) 100% times the probability of the other outcome

(e) 100% divided by the probability of the other outcome

26. What can be said about the equation (x  3)(x þ 2)(x þ 6) ¼ 0?

(a) It is a linear equation.

(b) It is a quadratic equation.

(d) It is a quartic equation.

(e) None of the above.

27. Consider the equation (x  3)(x þ 2)(x þ 6) ¼ 0. By examining this, it

is apparent that

(a) there are no real-number solutions

(b) there exists exactly one real-number solution

(d) there exist exactly three diﬀerent real-number solutions

(e) there exist inﬁnitely many diﬀerent real-number solutions

28. Suppose the two lines graphed in Fig. Test 2-3 represent linear

equations. It appears that for the equation of line A, when y ¼ 0,

the value of x

(a) is not deﬁned

Fig. Test 2-3. Illustration for Part Two Test Questions 28 through 31.

Test: Part 2 193

(b) is positive

(d) can be any real number

(e) requires more information to be deﬁned

29. Suppose the two lines graphed in Fig. Test 2-3 represent linear equa-

tions. It appears that for the equation of line A, when y is positive,

the value of x

(a) is not deﬁned

(b) is positive

(d) can be any real number

(e) requires more information to be deﬁned

30. Suppose the two lines graphed in Fig. Test 2-3 represent linear

equations. It appears that for the equation of line B, when x >  1,

the value of y

(a) is not deﬁned

(b) is positive

(d) can be any real number

(e) requires more information to be deﬁned

31. Suppose the two lines graphed in Fig. Test 2-3 represent linear

equations. It appears that the two equations, considered simulta-

neously:

(a) have no solutions

(b) have a single solution

(d) have inﬁnitely many solutions

(e) requires more information to be deﬁned

32. The term fuzzy truth is used to describe

(a) a theory in which there are degrees of truth that span a range

(b) standard deviation

(d) the probability fallacy

(e) any continuous distribution

33. What is the solution set of the equation x

 12x ¼35?

(a) {12, 35}

(b) {12, 35}

PART 2 Finding Unknowns

194

(d) {1, 12, 35}

(e) None of the above.

34. The mean for a discrete variable in a distribution is

(a) the mathematical average of the values

(b) the middle value

(d) the square root of the product of all the values

(e) the reciprocal of the sum of the values

35. In statistics, the frequency of an outcome is

(a) the number of cycles per second during which the outcome occurs

(b) the likelihood that an outcome will occur more than once

(d) the number of times the outcome occurs within a sample

(e) an undeﬁned term

36. What is the solution set of the equation (x  4)

¼ 0?

(a) {4}

(b) {4}

(d) 1

(e) It is impossible to tell because this equation is not in standard or

factored form.

37. Consider the following pair of equations, in which the intent is to

ﬁnd a solution or solutions, if any, for x and y simultaneously:

3x  4y ¼ 8

6x þ 2y ¼4

This is an example of a set of

(a) two-by-two linear equations

(b) two-by-two quadratic equations

(d) nonlinear equations

(e) redundant equations

38. In Fig. Test 2-4, suppose that curves A and B represent equations

for y versus x. How many solutions are there to both equations

considered simultaneously?

(a) None.

(b) One.

Test: Part 2 195

(d) Inﬁnitely many.

(e) More information is needed to answer this question.

39. In Fig. Test 2-4, suppose curve A represents the equation x þ y ¼ 0,

while curve B represents the equation x

 2y ¼ 0. Which of the

following ordered pairs, representing values (x,y), is a solution to

both of these equations considered simultaneously?

(a) (0,0)

(b) (1,1)

(d) (2,4)

(e) None of the above.

40. In Fig. Test 2-4, suppose that curves A and B represent equations for

y versus x. What can be said in general about these equations?

(a) Neither of them is linear.

(b) One of them is linear.

(d) Both of them are quadratic.

(e) Both of them are quartic.

Fig. Test 2-4. Illustration for Part Two Test Questions 38 through 40.

PART 2 Finding Unknowns

196

PART THREE

Shapes and

Places

197

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CHAPTER

Geometry on

the Flats

Geometry is widely used in science and engineering. The ancient Egyptians

and Greeks used geometry to calculate the diameter of the earth and the

distance to the moon, and scientists have been busy with it ever since. You

do not have to memorize all the formulas in this chapter, but it’s good to

be comfortable with this sort of stuﬀ. So roll up your sleeves and make

sure your brain is working.

Fundamental Rules

Here are some of the most important principles of plane geometry, which

involves the behavior of points, lines, and various ﬁgures conﬁned to a

199

ﬂat, two-dimensional (2D) surface. There are lots of illustrations to help you

envision how these rules and formulas work.

TWO POINT PRINCIPLE

Suppose that P and Q are two distinct geometric points. Then the following

statements are true, as shown in Fig. 9-1:

P and Q lie on a common straight line L.

L is the only straight line on which both points lie.

THREE POINT PRINCIPLE

Let P, Q, and R be three distinct points, not all of which lie on a straight line.

Then the following statements are true:

P, Q, and R all lie in a common Euclidean (ﬂat) plane S.

S is the only Euclidean plane in which all three points lie.

PRINCIPLE OF n POINTS

Let P

, P

, ..., P

be n distinct points, not all of which lie in the same

Euclidean (that is, ‘‘non-warped’’) space of n  1 dimensions. Then the

following statements are true:

, P

, ..., P

all lie in a common Euclidean space U of n

dimensions.

U is the only n-dimensional Euclidean space in which all n points lie.

DISTANCE NOTATION

The distance between any two distinct points P and Q, as measured from

P towards Q along the straight line connecting them, is symbolized by

writing PQ.

Fig. 9-1. Two point principle.

PART 3 Shapes and Places

200