Varsity Geometry Questions

Q: What three dimensional shape is technically a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, which does not touch the circle? Doughnuts and inner tubes are common examples.

A: Torus (or tori)

Q: What 18th century Swiss mathematician introduced much of the modern mathematical terminology and notation? A solid geometric formula named for him is commonly written as v − e + f = 2.

A: Leonhard Euler

Q: Who first presented geometry in an ideal axiomatic form? His presentation of this form in Elements of Geometry eventually led to a branch of geometry being named for him.

A: Euclid

Q: What branch of geometry deals with three-dimensional figures and surfaces? In other contexts, then same modifier describes a common state of matter.

A: solid

Q: What regular polygon has interior angles of 108 degrees?

A: Pentagon

Q: What is the measure of an interior angle in a regular dodecagon?

A: 150 degrees

Q: What is the measure of an interior angle in a regular 18-sided polygon?

A: 160 degrees

Q: What is the measure of an exterior angle in a regular dodecagon?

A: 30 degrees

Q: How many edges does a septagonal pyramid have?

A: 14

Q: In degrees, what is the sum of the angles of all faces of a regular tetrahedron?

A: 720 degrees

Q: The angles of a quadrilateral are in a 11 to 8 to 7 to 4 ratio. What is the measure of the smallest angle?

A: 48 degrees

Q: Four of the interior angles of a hexagon are 100, 91, 124, and 87 degrees. What is the average of measure of the remaining angles?

A: 159 degrees

Q: Five of the interior angles of a hexagon are 100, 90, 120, 170, and 92 degrees. What is the measure of the fifth angle?

A: 148 degrees

Q: Determine the distance between the origin and (7, 7, 7).

A: 7 times the square root of 3

Q: Determine the distance between the origin and (7, 7, 9).

A: the square root of 179

Q: Determine the distance between the origin and (4, 5, 7).

A: 3 times the square root of 10

Q: Determine the distance between the origin and (7, 10, 12).

A: the square root of 293

Q: Determine the distance between the origin and (8, 12, 16).

A: 4 times the square root of 29

Q: Determine the distance between the origin and (8, 10, 12).

A: 2 times the square root of 77

Q: Determine the distance between the origin and (-5, -4, -3).

A: 5 times the square root of 2

Q: Determine the distance between the points (4, 8, -4) and (0, 10, 0)

A: 6

Q: Determine the distance between the points (1, -10) and (4, 8)

A: 3 times the square root of 37

Q: Determine the distance between the points (1, 2, 3) and (7, 8, 9).

A: 6 times the square root of 3

Q: Determine the distance between the points (-2, 2, 4) and (3, -3, 8).

A: s the square root of 66

Q: Determine the distance between the points (3, 6) and (4, -10). Simplify completely, as necessary.

A: the square root of 257

Q: Determine the equation of the line passing through the origin and (-3, 8). Express the equation in y = mx + b form.

A: y = -8/3x (or y = - 2 2/3x)

Q: Determine the equation of the line passing through the points (0, 4) and (8, 3). Express the equation in y = mx + b form.

A: y = -1/8x + 4

Q: Determine the equation of the line passing through the points (1, 1) and (-3, 8). Express the equation in y = mx + b form using improper fractions, if necessary.

A: y = -7/4x + 11/4

Q: Calculate the area of a square with an apothem of 9 feet.

A: 324 square feet

Q: Calculate the area of a square with an apothem of 8 feet.

A: 256 square feet

Q: A cube and a sphere have equal volumes. Determine the edge length of the cube if the radius of the sphere is 5 centimeters.

A: the cube root of 500 pi over 3

Q: A cube and a sphere have equal volumes. Determine the edge length of the cube if the surface area of the sphere is 400 pi square meters. Rationalize the denominator if needed.

A: 10 times the cube root of 12 pi over 3

Q: A cube and a cylinder have equal volumes. Determine the edge length of the cube if the height of the cylinder is 10 centimeters, and the radius is 5 centimeters.

A: 5 times the cube root of 2 pi

Q: What is the volume of a cone if its height and diameter are the same as a cylinder with a volume of 630 cubic centimeters?

A: 210 cubic centimeters

Q: What is the total surface area of a cube if a face diagonal has a length of 4 times the square root of 3 inches?

A: 144 square inches

Q: The two legs of a right triangle have lengths equal to the two smallest prime numbers greater than 10. Determine the length of the hypotenuse.

A: square root of 290

Q: The two legs of a right triangle have lengths equal to the two smallest prime numbers greater than 15. Determine the length of the hypotenuse.

A: 5 times the square root of 26

Q: Two legs of a right triangle measure 9 and 18 inches. Determine the length of the hypotenuse.

A: 9 times the square root of 5 inches

Q: Two legs of a right triangle measure 15 and 17 inches. Determine the length of the hypotenuse.

A: the square root of 514 inches

Q: Given a right triangle with a height of 8 units, what is the length of the hypotenuse if the area of the triangle is 60 square units?

A: 17 (units)

Q: Given a right triangle with a height of 8 units, what is the length of the hypotenuse if the area of the triangle is 64 square units?

A: 8 times the square root of 2 (units)

Q: The hypotenuse of a 30-60-90 right triangle is 20 feet. What is the area of the triangle?

A: 50 times the square root of 3 feet

Q: The area of a circle is 48 square inches. What is the area within a 15 degree arc?

A: 2 square inches

Q: Determine the lateral area of a cylinder with a height of 7 inches and a radius of 40 inches. Leave your answer in terms of pi.

A: 560 pi square inches

Q: What is the lateral area of a cube with an edge length of square root of 7 inches?

A: 28 square inches

Q: What is the surface area of a cube if its volume is 343 cubic inches?

A: 294 square inches

Q: Determine the radius of a sphere if the numeric values for the surface area and volume are the same.

A: 3

Q: The height of a trapezoid is 2 times the lower base plus 4 centimeters, and the upper base is 3 times the lower base. What is the area of the trapezoid if the upper base is 15 centimeters?

A: 140 square centimeters

Q: Determine the equation of a hyperbola with center at the origin, one vertex at (7, 0) and a focus at (12, 0).

A: x squared over 49 – y squared over 95 = 1

Q: What is the compliment of an angle pi over 8 radians?

A: 3 pi over 8 radians

Q: If the compliment of the supplement of an angle is 65 degrees, what is the original angle?

A: 155 degrees

Q: If the compliment of the supplement of an angle is 12 degrees, what is the original angle?

A: 102 degrees

Q: If the Supplement of the compliment of an angle is 148 degrees, what is the original angle?

A: 58 degrees

Q: The compliment of the supplement of 32 degrees is found in what quadrant?

A: 4th

Q: If the supplement of the compliment of an angle is 158 degrees, what is the original angle?

A: 68 degrees

Q: What is the length of the altitude drawn to the hypotenuse in an isosceles right triangle with a hypotenuse of 10 inches?

A: 5 inches

Q: Give the ratio of radii of two spheres is 7 to 4. What is the ratio of their volumes?

A: 343 to 64

Q: Given that the hypotenuse of a 30-60-90 triangle is 15, find the following, expressing answers as improper fractions, and rationalizing where appropriate.

1. Length of the side across from the 30 degree angle.2. Length of the other side3. Area of the triangle4. Perimeter of the triangle

1. 15/22. 15 times the square root of 3 over 23. 225 times the square root of 3 over 8.4. 45 + 15 times the square root of 3 all over 2 or

45/2 + 15 times the square root of 3 over 2

Q: Given that the hypotenuse of a 30-60-90 triangle is 16, find the following:

1. Length of the side across from the 30 degree angle.2. Length of the other side3. Area of the triangle4. Perimeter of the triangle

1. 82. 8 times the square root of 3 3. 32 times the square root of 3 4. 24 + 8 times the square root of 3

Q: A right triangle with legs of 4 and 7 is inscribed in a circle. Express answers as improper fractions, as necessary:

1. What is the hypotenuse of the right triangle?2. What is the radius of the circle?3. What is the area of the circle?4. What is the area inside the circle but outside the triangle? Express your answer to

the nearest tenth.

1. the square root of 652. the square root of 65 over 23. 65 pi over 44. 37.1

Q: An equilateral triangle with sides of 7 is inscribed in a circle. Find the following:

1. The triangle’s altitude.2. The circle’s radius.3. The triangle’s area4. The circle’s area.

1. 7 times the square root of 3 over 22. 7 times the square root of 3 over 33. 49 times the square root of 3 over 44. 49 pi over 3

Q: An equilateral triangle with sides of 6 is inscribed in a circle. Find the following:

1. the triangle’s altitude2. the circle’s radius3. the triangle’s area4. the circle’s area

1. 3 times the square root of 32. 2 times the square root of 33. 9 times the square root of 34. 12 pi

Q: Name the conic section associated with the following terms:

1. Focus, directrix 2. Transverse axis, asymptotes 3. Center, radius 4. Vertices, major axis

1. Parabola 2. Hyperbola 3. Circle 4. Ellipse

Q: Identify the following terms:

1. a mathematical object having a location, but no size2. a mathematical object with length only, and no endpoints3. Angle formed where two planes meet4. two angles which have a common vertex, and one common side

1. Point2. Line3. Dihedral4. Adjacent

Q: Given a rhombus ABCD with diagonals that intersect at E. Angle ECB is 70 degrees. Find the measures of the following:

1. ACD2. DEC3. EDC4. ABC

1. 702. 903. 204. 40

Q: The length of a rectangle is 13 cm more than twice the width. The perimeter is 62 cm.

1. Find the length.2. Find the width.3. Find the area.4. Find the volume of a rectangular prism using this rectangle and a depth 3 less than

double the length.

1. 25 cm2. 6 cm3. 150 square cm4. 7050 cubic cm

Q: The length of a rectangle is 10 cm less than the square of the width. The perimeter is 40 cm.

1. Find the length.2. Find the width.3. Find the area.4. Find the volume of a rectangular prism using this rectangle and a depth equal to

double the width.

1. 15 cm2. 5 cm3. 75 square cm4. 750 cubic cm

Q: Determine the distances between the following points in spaces of various dimensions.

1. (6, 22) and (14, 37)2. (1, 0, 3) and (1, 0, 13)3. (3, 0, 13, 7) and (1, 4, 18, 1)4. (4, -2, 3, 9, 5) and (5, -2, 8, -4, -4)5. (4, 8) and (-5, 30)6. (0, 3, 0) and (3, 0, 3)7. (1, 1, 2, 2) and (2, 4, 6, 8)8. (2, 3, 5, 7, 11) and (4, 6, 8, 10, 12)

1. 172. 103. 94. 2 times the square root of 695. square root of 5656. 3 times the square root of 37. square root of 628. 4 times the square root of 2

Q: Given a circle whose center is (4,5), and passes through (0,7), find the following, leaving answers in terms of pi where applicable:

1. Find the radius2. Find the area3. Find the circumference4. Find the question

1. 2 times the square root of 52. 20 pi3. 4 pi times the square root of 54. (x – 4) squared + (y – 5) squared = 20

Q: Determine the equation of the lines passing through these pairs of points. Express your answers in y = mx + b form, leaving fractions in improper form, as necessary.

1. (-1, 2) and (4, 5)2. (-6, 4) and (3, 8)3. (0, 3) and ( 5, -2)4. (1.2, -1/4) and (3/5, 1/3)

1. y = 3/5 x + 13/52. y = 4/9 x + 70/93. y = -x + 34. y = 35/6 x - 19/6

Q: Given a cone with a base diameter of 60 units and a height of 40 units. Find the following, leaving answers in terms of pi, where necessary:

1. Slant height2. Lateral area3. Volume4. Surface Area

1. 50 units2. 1500 pi square units3. 12000 pi cubic units4. 2400 pi square units

Q: A square is inscribed in a circle with an area of 1089 pi square inches. Find the following:

1. The circle’s radius2. Measure of the square’s sides3. Area of the square4. Area of the region inside the circle, but outside the square, expressed to the nearest

whole number

1. 33 inches2. 33 times the square root of 2 inches3. 2178 square inches4. 1243 square inches

Q: A square is inscribed in a circle with an area of 625 pi. Find the following:

1. The circle’s radius2. Measure of the square’s side3. Area of the square4. Area of the circle outside of the square. Express your answer to the nearest whole

number.

1. 252. 25 times the square root of 23. 12504. 713

General Math

Q: Compute the difference between the geometric mean of 16 and 19, and the arithmetic mean of the same numbers. Express your answers as a mixed number , if necessary.

A: 1/2

Q: Compute the product between the geometric mean of 12 and 10 multiplied by the arithmetic mean of the same two numbers.

A: 22 times the square root of 30

Q: What type of number is defined as one which equals the sum of its proper divisors, excluding itself?

A: Perfect

Q: Compute the sum of the first four perfect cubes.

A: 100

Q: Give the prime factorization of 405.

A: 3 to the 4th times 5 or 3 x 3 x 3 x 3 x 5

Q: Give the prime factorization of 1620.

A: 3 to the 4th times 2 squared times 5 or 3 x 3 x 3 x 3 x 2 x 2 x 5

Q: Give the prime factorization of 1575.

A: 3 squared times 5 squared times 7 or 3 x 3 x 5 x 5 x 7

Q: Determine the next term in the progression: 0, 1, 4, 15, 48, …

A: 147

Q: Expressing your answer as a decimal to the nearest thousandth, compute the square root of 1/9.

A: 0.333

Q: Expressing your answer as a decimal, compute the square root of 40.96?

A: 6.4

Q: Calculate the square root of 7.5 to the nearest hundreth.

A: 2.74

Q: Expressing 28 quintillion in scientific notation.

A: 2.8 times 10 to the 19th

Q: Expressing your answer as a decimal, compute the square root of 0.3844.

A: 0.62

Q: What is the base ten equivalent of the base seven number six four?

A: 46

Q: Expressed as a decimal number, what is the product of the hexadecimal A times B?

A: 110

Q: What is the decimal equivalent of the hexadecimal number A 0?

A: 160

Q: Express the hexadecimal number 3 A in decimal form?

A: 58

2. Math/General Math

Q: Express the octal number three four in hexadecimal form.

A: 1 C

Q: Sum the base eight integers four four and two two and convert the answer to base 10.

A: 54

Q: If a person is two zero years old in hexadecimal, how old is he in decimal form?

A: 32 (years)

Q: Sum the hexadecimal number 1 A with the binary number 1 0 0 and express your answer in decimal form.

A: 30

Q: Subtract the value of the Roman numeral C from the hexadecimal digit C, giving your answer in decimal form.

A: -88

Q: Determine the value of the quantity x cubed times x to the -3 power, close quantity, factorial.

A: 1

Q: Sum 3 cubed plus 4 to the 4th plus 5 to the first plus 5 to the zero power.

A: 289

Q: If a series of 96 objects is divided into three portions in a ratio of eight to three to one. How many objects make up the largest portion?

A: 64

Q: What is the probability of rolling a sum of 4 in three rolls of a standard die?

A: 1/72

Q: What is the probability of rolling a sum of 4 in five rolls of a standard die?

A: 0

Q: What is the probability of getting at least 4 heads in 5 tosses of a fair coin?

A: 3/16

Q: What is the probability of rolling a sum of 6 in five rolls of a standard die?

A: 5/7776

Q: What is the probability of rolling a sum of 5 in four rolls of a standard die?

A: 1/324

Q: Given that 5 consecutive flips of a fair coin turn up tails, what is the probability that the next 3 tosses will be tails?

A: 1/8

Q: Express 34 quadrillion in scientific notation?

A: 3.4 times 10 to the 16th

Q: What is the simplest name of the number that can be represented by 10 to the 13th power?

A: 10 trillion

Q: What fraction, reduced to lowest terms, is 50 percent greater than 5/13

A: 15/26

Q: What digit appears in the ten-thousandths place for pi?

A: 5

Q: Compute the mean of this set of numbers: -3, 0, 4, 8, 11, 16, and 20.

A: 8

Q: How many points are required to define a plane, assuming that not all are collinear?

A: 3

Q: The modulus of a complex number is a synonym for what two word term? For real numbers, the same term refers to the numerical value of a number without regard toits sign.

A: Absolute value

Q: Calculate the square roots of the following to the nearest whole number:

1. 1369 1. 372. 1370 2. 373. 9000 3. 954. 0.0009 4. 0.03

Q: Compute the square roots of the following numbers to the nearest hundredth:

1. 98 1. 9.902. 108 2. 10.393. 115 3. 10.724. 133 4. 11.53

Q: Identify the following numbers:

1. 10 to the 6th 1. 1 million2. 10 to the 15th 2. 1 quadrillion3. 10 to the 21st 3. 1 sextrillion4. 10 to the 30th 4. 1 nonillion

Q: Give the names for the following numbers:

1. 10 to the 27th power 1. Octillion2. 10 to the 30th power 2. Nonillion3. 10 to the 33rd power 3. Decillion4. 10 to the 36th power 4. Undecillion

Q: Given the following set of numbers: 1, 6, 3, 9, 9, 15, 10, 11. Find the following:

1. mean 1. 82. mode 2. 93. median 3. 94. standard deviation 4. square root of 71 over 2

Q: Given the following set of numbers: 1, 2, 3, 8, 8, 17, 18, 23. Find the following:

1. mean 1. 102. mode 2. 83. median 3. 84. range 4. 22

Q: Convert the following decimal numbers to hexadecimal:

1. 13 1. D2. 40 2. 2 83. 100 3. 6 44. 512 4. 2 0 0

Q: Evaluate the following to the nearest whole number:

1. The principal square root of 1166 1. 342. The principal square root of 940 2. 313. The cube root of 4096 3. 164. The cube root of 6000 4. 185. The principal square root of 345 5. 196. The principal square root of 2304 6. 487. The cube root of 3375 7. 158. The tenth root of 59049 8. 39. the principal square root of 22 9. 4.710. the principal square root of 22.9 10. 4.811. the principal square root of 229 11. 15.112. the principal square root of 2299 12. 47.913. the principle square root of 5184 13. 7214. the principle square root of 1111 14. 3315. the principle square root of 864 15. 2916. the principle square root of 40352 16. 20117. the principle square root of 6241 17. 7918. the principle square root of 1100 18. 3319. the principle square root of .15251425 19. 020. the principle square root of 1 quintillion 20. 1 billion or 10 to the 9th

Q: Evaluate the following:

1. i to the 95th power 1. Negative i2. i to the 120th power 2. 13. 2 + 6i the quantity squared 3. -32 + 24i4. 2 + 6i the quantity cubed 4. -208 + 144i

Q: Identify the following mathematicians:

1. Formulated the fundamentals of geometry2. Considered the father of Algebra3. First European to use mathematics to predict eclipses4. Co-founded theory of probability and was the leader in number theory

1. Euclid2. Diophantus3. Thales4. Pierre de Fermat

Computer Science

Q: What type of number is a digital representation for a rational number, often used to approximate an arbitrary number on a computer? The slang term “flops” is a acronym for operations on these numbers per second.

A: floating point

Q: What prominent object-oriented programming language was initially developed by Sun Microsystems? It was originally named “oak”, but now shares a name with an island and a type of coffee.

A: Java

Q: With regard to computer databases, what does SQL stand for?

A: Structured Query Language

Q: What discipline focuses on information security and related issues?

A: cryptography

Q: What does XML stand for?

A: Extensible Markup Language

Q: What is the name of the most recent Microsoft Windows operating system, which was released in January 2007?

A: Vista

Q: Give the meaning of the following computer related acronyms or abbreviations:

1. ABEND2. DOS3. Kbps, with only the “K” capitalized4. KBps, with both the “K” and “B” capitalized

1. Abnormal End2. Disk Operating System or Denial Of Service3. Kilobits per second4. Kilobytes per second