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# Three Dimensions and Up Geometry Practice Test

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By McGraw-Hill Professional
Updated on Oct 3, 2011

## Three Dimensions and Up Geometry Practice Test

Directions: You may draw diagrams or use a calculator if necessary. A good score is at least 38 correct.

It’s best to have a friend check your score the first time, so you won’t memorize the answers if you want to take the test again.

#### Practice

1. Two straight lines that are not parallel, and that do not lie in the same plane, are called

(a) orthogonal

(b) perpendicular

(c) non-Euclidean

(d) normal

(e) skew

2. The equation of the unit circle in polar coordinates is

(a) r = θ

(b) r = 1

(c) r = 0

(d) θ = 1

(e) θ = 0

3. The tangent function y = tan x is defined for all the following values of x except:

(a) x = 0°

(b) x = 45°

(c) x = 90°

(d) x = 135°

(e) x = 180°

4. Suppose a vector a begins at the point (2,3) and ends at the point (−5,6). In standard form, this vector is

(a) a = (−3,9)

(b) a = (7,−3)

(c) a = (−10,18)

(d) a = (0,0)

(e) none of the above

5. Refer to Fig. Test 2-1. Suppose that all the flat faces of the 3D figure are parallelograms, but none of the angles x, y , and z are right angles. If this is the case, we can nevertheless truthfully say that the figure is

(a) a parallelogram

(b) a rhombus

(c) a rectangular prism

(d) a parallelepiped

(e) none of the above

Fig. Test 2-1 . Illustration for Questions 5, 6, and 7 in the test for Part Two.

6. Refer to Fig. Test 2-1. Imagine that the lengths of the edges s 1 , s 2 , and s 3 remain constant. Suppose the angles x, y , and z are all right angles at first, but they uniformly and gradually decrease as the 3D object is “squashed down.” What happens to the volume of the 3D object?

(a) It increases

(b) It remains constant

(c) It decreases

(d) Nothing, because the scenario described is impossible

7. Refer to Fig. Test 2-1. The overall surface area, A , of the 3D object shown is given by the following equation:

A = 2 s 1 s 2 sin x + 2 s 1 s 3 sin y + 2 s 2 s 3 sin z

where sin x represents the sine of angle x , sin y represents the sine of angle y , and sin z represents the sine of angle z . Suppose we start with a situation where x , y , and z are all right angles. Then, without changing the lengths s 1 , s 2 , or s 3 , we “squash” the object so all three angles x, y , and z measure 30°. What happens to the overall surface area of the object?

(a) It does not change

(b) It decreases by a factor of 2

(c) It decreases by a factor of 4

(d) It decreases by a factor of 8

8. The graph of 4 r = 3 θ in the mathematician’s polar coordinate system looks like

(a) a spiral

(b) a cardioid

(c) a circle

(d) a three-leafed rose

(e) a four-leafed rose

9. The distance between (0,0,0,0,0) and (1,1,1,1,1) in Cartesian five-space is

(a) equal to 1

(b) equal to the square root of 2

(c) equal to the fifth root of 2

(d) equal to the square root of 5

(e) equal to the fifth root of 5

10. In order to uniquely define a point in 4D time-space

(a) four coordinates in time must be identified

(b) four coordinates in space must be identified

(c) three coordinates in space and one coordinate in time must be specified

(d) three coordinates in time and one coordinate in space must be specified

(e) two coordinates in time and two coordinates in space must be specified

11. Suppose two non-perpendicular planes intersect. The angle at which the planes intersect can be defined in two ways: as an acute angle u or as an obtuse angle v . If u and v are expressed in radians, then

(a) u + v = 1

(b) u + v = π /2

(c) u + v = π

(d) u + v = 2 π

(e) none of the above

12. Refer to Fig. Test 2-2. The coefficients a, b , and c of vector m = ( a,b,c ), which is normal (perpendicular) to plane W , are sufficient to uniquely determine

(a) the equation of plane W

(b) the orientation of plane W

(c) whether or not plane W passes through the origin

(d) all of the above (a), (b), and (c)

(e) none of the above (a), (b), or (c)

Fig. Test 2-2 . Illustration for Questions 12, 13, and 14 in the test for Part Two.

13. Refer to Fig. Test 2-2. Suppose we know the coefficients a, b , and c of vector m = ( a,b,c ), which is normal to plane W . Suppose we also know the coordinates ( x 0 , y 0 , z 0 ) of point P . This information is sufficient to uniquely determine

(a) the equation of plane W

(b) the orientation of plane W

(c) whether or not plane W passes through the origin

(d) all of the above (a), (b), and (c)

(e) none of the above (a), (b), or (c)

14. Refer to Fig. Test 2-2. Suppose we know the coefficients a, b , and c of vector m = ( a,b,c ), which is normal to plane W . Suppose we also know the coordinates ( x 0 , y 0 , z 0 ) of point P . Now, imagine that we multiply all the values a, b , and c by −1, obtaining the vector − m = (− a ,− b ,− c ). Let the resulting plane, which is determined by the point P and which is normal to the vector − m , be called plane X . Which of the following is true?

(a) Planes X and W are perpendicular

(b) Planes X and W are distinct, but parallel

(c) Planes X and W coincide

(d) Plane X cannot be defined because there are infinitely many possibilities

(e) Plane X cannot exist

15. Suppose two planes intersect at an angle of 140°. This is the less common of two ways the intersection angle can be expressed. The more common value for the intersection angle of these two planes is

(a) 40°

(b) −40°

(c) −140°

(d) 220°

(e) none of the above

16. The dot product of the vectors a = (2,4,−1) and b = (−5,1,2) in Cartesian xyz -space is equal to

(a) the scalar quantity −8

(b) the vector (−10,4,−2)

(c) the scalar quantity 80

(d) a vector perpendicular to the plane containing a and b

(e) a vector in the plane containing a and b

17. Suppose the spherical coordinates of a certain object in the sky are specified as ( θ,φ,r ), where θ is its elevation with respect to the plane of the horizon, φ is its azimuth, and r is its radius (distance from us). Imagine that the object flies horizontally away from us, traveling at the same heading as its azimuth as we see it (so φ remains constant). What happens to θ and r ?

(a) The value of θ approaches 0, while r increases without limit

(b) The values of θ and r both approach 0

(c) The value of θ increases without limit, while r approaches 0

(d) The values of θ and r both increase without limit

(e) None of the above

18. Suppose the spherical coordinates of a certain object in the sky are specified as ( θ,φ,r ), where θ is its elevation with respect to the plane of the horizon, φ is its azimuth, and r is its radius (distance from us). Imagine that the object flies straight away from us. What happens to θ and r ?

(a) The value of θ approaches 0, while r increases without limit

(b) The values of θ and r both approach 0

(c) The value of θ increases without limit, while r approaches 0

(d) The values of θ and r both increase without limit

(e) None of the above

19. Considered with respect to the speed of light, one minute-equivalent represents a distance of about

(a) 18,000 kilometers

(b) 180,000 kilometers

(c) 1,800,000 kilometers

(d) 18,000,000 kilometers

(e) 180,000,000 kilometers

20. The faces (including the base) of a rectangular pyramid are all

(a) triangles

(b) squares

(c) rectangles

(d) rhombuses

(e) plane polygons

21. Suppose there are two planes X and Y such that, for all lines L passing through both X and Y , the acute angle between L and X has the same measure as the acute angle between L and Y . From this information, it is reasonable to suppose that

(a) planes X and Y are perpendicular

(b) planes X and Y are non-Euclidean

(c) planes X and Y are parallel

(d) there are no lines parallel to either plane X or plane Y

(e) there are no lines perpendicular to either plane X or plane Y

22. How many straight line-segment edges does a tesseract (or four-cube) have?

(a) 16

(b) 24

(c) 32

(d) 48

(e) 96

23. What is the 4D hypervolume of a tesseract (or four-cube) measuring 1 meter-equivalent on each edge? (Call the standard unit of 4D hypervolume a quartic meter-equivalent .)

(a) 1 quartic meter-equivalent

(b) 4 quartic meter-equivalents

(c) 16 quartic meter-equivalents

(d) 64 quartic meter-equivalents

24. What is the 5D hypervolume of a five-cube measuring 1 meter-equivalent on each edge? (Call the standard unit of 5D hypervolume a quintic meter-equivalent .)

(a) 1 quintic meter-equivalent

(b) 5 quintic meter-equivalents

(c) 25 quintic meter-equivalents

(d) 125 quintic meter-equivalents

25. Given a slant circular cone whose base has radius r and whose height, expressed between the apex and the plane containing the base, is h , the volume V is given by the following formula:

V = πr 2 h /3

Let K 1 and K 2 be slant circular cones. Suppose the radius of K 2 is twice as great as the radius of K 1 , but the height of K 2 is only half the height of K 1 . Which of the following statements is true?

(a) The volume of K 2 is four times the volume of K 1

(b) The volume of K 2 is twice the volume of K 1

(c) The volume of K 1 is four times the volume of K 2

(d) The volume of K 1 is twice the volume of K 2

(e) The volumes of K 1 and K 2 are the same

26. Refer to Fig. Test 2-3. If the rectangular coordinates x 0 and y 0 of point P are both doubled, what happens to the value of r 0 ?

(a) It increases by a factor of the square root of 2

(b) It doubles

(d) It does not change

Fig. Test 2-3 . Illustration for Questions 26, 27, and 28 in the test for Part Two.

27. Refer to Fig. Test 2-3. If the rectangular coordinates x 0 and y 0 of point P are both doubled, what happens to the value of θ 0 ?

(a) It increases by a factor of the square root of 2

(b) It doubles

(c) It is multiplied by −1

(d) It does not change

(e) It increases by π rad

28. Refer to Fig. Test 2-3. If the rectangular coordinates x 0 and y 0 of point P are both multiplied by −1, what happens to the value of θ 0 ?

(a) It increases by a factor of the square root of 2

(b) It doubles

(c) It is multiplied by −1

(d) It does not change

(e) It increases by π rad

29. What is the volume of a rectangular prism that is 200 millimeters high, 700 millimeters wide, and 500 millimeters deep?

(a) 1.4 square meters

(b) 1.4 cubic meters

(c) 0.07 square meters

(d) 0.07 cubic meters

(e) None of the above

30. A donut-shaped geometric solid is called a

(a) right circular cylinder

(b) frustum of a cylinder

(c) bent circular cylinder

(d) truncated ellipsoid

(e) none of the above

31. A single, specific line can be contained within

(a) one and only one plane

(b) parallel planes

(c) at most two planes

(d) at most three planes

(e) infinitely many planes

32. The entire polar coordinate plane, showing all possible points with angular values from 0° to 360° and radial values corresponding to any non-negative real number, can be portrayed within a finite circular region by

(a) using a logarithmic angular scale

(b) using a logarithmic radial scale

(c) geometric compression of the angular scale

(d) geometric compression of the radial scale

(e) no known means

33. The height, or altitude, in cylindrical coordinates is expressed

(a) in linear units, perpendicular to the plane in which the direction angle is expressed

(b) in linear units, in the plane in which the direction angle is expressed

(c) in linear units, in a direction parallel to the direction in which the radius is expressed.

(d) in degrees, as an angle relative to the horizon

(e) in radians, as an angle relative to the horizon

34. The direction angle in the mathematician’s polar coordinate system is expressed

(a) in a clockwise sense

(b) in a counterclockwise sense

(c) in either sense

(e) only in degrees

35. An example of a negatively curved 2D surface is

(a) the surface of a sphere

(b) the surface of an ellipsoid

(c) the surface of a tesseract

(d) the surface of a four-sphere

(e) none of the above

36. The product of the vectors a = (−3,0,4) and b = (2,1,−5) in Cartesian xyz -space is equal to

(a) the scalar quantity −26

(b) the scalar quantity −1

(c) the vector (−6,0,−20)

(d) the vector (−1,1,−1)

37. The sum of the vectors a = (−3,0,4) and b = (2,1,−5) in Cartesian xyz -space is equal to

(a) the scalar quantity −26

(b) the scalar quantity −1

(c) the vector (−6,0,−20)

(d) the vector (−1,1,−1)

38. Given a slant circular cylinder whose base has radius r and whose height, expressed between the top and the plane containing the base, is h , the volume V is given by the following formula:

V = πr 2 h

Let C 1 and C 2 be slant circular cylinders. Suppose the radius of C 2 is four times as great as the radius of C 1 , but the height of C 2 is only 1/16 the height of C 1 . Which of the following statements is true?

(a) The volume of C 2 is four times the volume of C 1

(b) The volume of C 2 is twice the volume of C 1

(c) The volume of C 1 is four times the volume of C 2

(d) The volume of C 1 is twice the volume of C 2

(e) The volumes of C 1 and C 2 are the same

39. Let X be a plane. Suppose a line O , which is not normal to plane X , intersects plane X at some point S as shown in Fig. Test 2-4. Let N be a line normal to plane X , passing through point S . Let Y be the plane determined by the intersecting lines N and O . Let L be the line formed by the intersection of planes X and Y . Let M be a line in plane X that passes through point S , but is different from line L . The angle between line O and plane X is the same as

(a) the angle between line O and line L

(b) the angle between line O and line M

(c) the angle between line O and plane Y

(d) the angle between line L and line M

(e) the angle between line L and line N

Fig. Test 2-4 . Illustration for Questions 39 and 40 in the test for Part Two.

40. Examine Fig. Test 2-4. Suppose lines L, M , and N are all mutually perpendicular, like the axes of Cartesian xyz-space, and they all intersect at point S . Also suppose line L is common to both planes X and Y . If line M is in plane X and line N is in plane Y , then

(a) planes X and Y are parallel

(b) planes X and Y are obtuse

(c) planes X and Y are skew

(d) planes X and Y are acute

(e) planes X and Y are perpendicular

41. Two intersecting lines define a single

(a) line

(b) ray

(c) plane

(d) triangle

42. The faces of a parallelepiped are all

(a) triangles

(b) squares

(c) rectangles

(d) rhombuses

(e) parallelograms

43. If a straight line in Cartesian xyz -space has direction defined by m = 2 i + 2 j + 2 k , we can surmise

(a) that the line is parallel to the x axis

(b) that the line is parallel to the y axis

(c) that the line is parallel to the z axis

(d) that the line is not parallel to the x axis, the y axis, or the z axis

(e) that the line passes through the origin

44. The smaller of the two definable angles between a line and a plane has a measure that can range anywhere between 0 rad and

45. Imagine two vectors in Cartesian xyz -space. Suppose vector a begins at the origin and ends at the point ( x a , y a , z a ) = (2,3,5). Suppose vector b begins at point (1,1,1) and ends at the point ( x b , y b , z b ) = (3,4,6). What can we say about these two vectors?

(a) They are parallel to each other, but they are not equivalent

(b) They are equivalent, but they are not parallel to each other

(c) They are equivalent, and they are parallel to each other

(d) They are parallel to each other, but they point in opposite directions

(e) They are skewed with respect to each other

46. A significant difference between the mathematician’s polar coordinate plane and the navigator’s polar coordinate plane is

(a) that the radii are measured in opposite directions

(b) that the direction angles are expressed in opposite senses

(c) that one scheme uses linear units, and the other does not

(d) that the radii are expressed in degrees in one scheme, and in radians in the other scheme

(e) nothing; there is no difference between the two systems

47. In the dimensionally reduced illustration Fig. Test 2-5 showing the earth’s path through time-space, each vertical division represents 1/4 of a year, or approximately 91.3 days. Suppose the vertical scale is changed so that the pitch of the helix becomes only half as great (as if it were a spring compressed by a factor of 2). Further suppose that the horizontal scales remain unchanged. Then each vertical division represents

(a) approximately 22.8 days

(b) approximately 45.7 days

(c) approximately 91.3 days

(d) approximately 183 days

(e) approximately 365 days

Fig. Test 2-5 . Illustration for Questions 47 and 48 in the test for Part Two.

48. In the dimensionally reduced illustration Fig. Test 2-5 showing the earth’s path through time-space, each vertical division represents 1/4 of a year, or approximately 91.3 days. Suppose the x and y scales are both changed so that the radius of the helix becomes twice as great. Further suppose that the vertical scale remains unchanged. Then each vertical division represents

(a) approximately 22.8 days

(b) approximately 45.7 days

(c) approximately 91.3 days

(d) approximately 183 days

(e) approximately 365 days

49. Suppose there are two ellipsoids E 1 and E 2 that have identical proportions, but the radii of E 2 are all exactly four times the radii of E 1 . Suppose V 1 is the volume of E 1 , and V 2 is the volume of E 2 . Which, if any, of the following equations (a), (b), (c), or (d) is true?

(a) V 2 = 4 V 1

(b) V 2 = 8 V 1

(c) V 2 = 16 V 1

(d) V 2 = 32 V 1

(e) None of the above equations (a), (b), (c), or (d) is true

50. The distance between a point and a plane is expressed along

(a) a line passing through the point and parallel to the plane

(b) a line passing through the point and skewed relative to the plane

(c) a line passing through the point and normal to the plane

(d) a line passing through the point and contained in the plane

(e) none of the above

1. e

2. b

3. c

4. e

5. d

6. c

7. b

8. a

9. d

10. c

11. c

12. b

13. d

14. c

15. a

16. a

17. a

18. e

19. d

20. e

21. c

22. c

23. a

24. a

25. b

26. b

27. d

28. e

29. d

30. e

31. e

32. d

33. a

34. b

35. e

36. e

37. d

38. e

39. a

40. e

41. c

42. e

43. d

44. b

45. c

46. b

47. d

48. c

49. e

50. c

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