Algebra

Question 1

What is the formula for the area of a rectangle?

Answer 1

Area = length × width

Question 2

True or False: The formula for the perimeter of a square is P = 4s, where s is the length of a side.

Answer 2

True

Question 3

Fill in the blank: The formula for the volume of a cylinder is V = _____, where r is the radius and h is the height.

Answer 3

πr²h

Question 4

What is the quadratic formula used to find the roots of a quadratic equation?

Answer 4

x = (-b ± √(b² - 4ac)) / 2a

Question 5

Which formula represents the slope of a line given two points (x1, y1) and (x2, y2)?

Answer 5

Slope = (y2 - y1) / (x2 - x1)

Question 6

True or False: The formula for the area of a triangle is A = 1/2 × base × height.

Answer 6

True

Question 7

What is the formula for the distance between two points (x1, y1) and (x2, y2)?

Answer 7

Distance = √((x2 - x1)² + (y2 - y1)²)

Question 8

Fill in the blank: The formula for the circumference of a circle is C = _____, where r is the radius.

Answer 8

2πr

Question 9

What is the formula for the sum of the interior angles of a polygon with n sides?

Answer 9

Sum = (n - 2) × 180°

Question 10

What is the formula for converting Celsius to Fahrenheit?

Answer 10

F = (C × 9/5) + 32

Question 11

True or False: The formula for the Pythagorean theorem is a² + b² = c².

Answer 11

True

Question 12

What is the formula for the exponential growth model?

Answer 12

y = a(1 + r)ⁿ

Question 13

Fill in the blank: The formula for the area of a circle is A = _____, where r is the radius.

Answer 13

πr²

Question 14

What is the formula for the roots of a cubic equation?

Answer 14

There is no simple formula; it generally requires numerical methods or Cardano’s method.

Question 15

What is the formula for the sum of an arithmetic series?

Answer 15

S_n = n/2 × (a + l), where n is the number of terms, a is the first term, and l is the last term.

Question 16

True or False: The formula for the surface area of a sphere is 4πr².

Answer 16

True

Question 17

What is the formula for the compound interest?

Answer 17

A = P(1 + r/n)^(nt)

Question 18

Fill in the blank: The formula for the difference of squares is a² - b² = _____ .

Answer 18

(a - b)(a + b)

Question 19

What is the formula for the slope-intercept form of a line?

Answer 19

y = mx + b, where m is the slope and b is the y-intercept.

Question 20

What is the formula for calculating the mean of a set of numbers?

Answer 20

Mean = (Sum of all values) / (Number of values)

Question 21

True or False: The formula for the harmonic mean is 1/H = (1/a + 1/b + 1/c + … + 1/n) / n.

Answer 21

True

Question 22

What is the formula for the binomial theorem?

Answer 22

(a + b)ⁿ = Σ (n choose k) a^(n-k) b^k

Question 23

Fill in the blank: The formula for the factorial of n is n! = _____ .

Answer 23

n × (n - 1) × (n - 2) × … × 1

Question 24

What is the formula for the standard deviation of a population?

Answer 24

σ = √(Σ(x - μ)² / N), where μ is the mean and N is the number of data points.

Question 25

True or False: The formula for the law of cosines is c² = a² + b² - 2ab cos(C).

Answer 25

True

Question 26

What is the formula for the area of a trapezoid?

Answer 26

Area = 1/2 × (base1 + base2) × height

Question 27

What is a set in set theory?

Answer 27

A collection of distinct objects, considered as an object in its own right.

Question 28

True or False: The empty set is a subset of every set.

Answer 28

True

Question 29

What symbol is used to denote the union of two sets?

Answer 29

∪

Question 30

Fill in the blank: The intersection of sets A and B is denoted as ___ .

Answer 30

A ∩ B

Question 31

What does the complement of a set A refer to?

Answer 31

The set of all elements not in A.

Question 32

Which of the following is an example of a finite set? (a) The set of all integers (b) The set of days in a week (c) The set of all real numbers

Answer 32

b) The set of days in a week

Question 33

True or False: A set can contain itself as an element.

Answer 33

False

Question 34

If A = {1, 2, 3} and B = {3, 4, 5}, what is A ∪ B?

Answer 34

{1, 2, 3, 4, 5}

Question 35

What is the power set of a set?

Answer 35

The set of all possible subsets of a set.

Question 36

How many subsets does a set with n elements have?

Answer 36

2^n

Question 37

What is the difference between two sets A and B denoted as A - B?

Answer 37

The set of elements in A that are not in B.

Question 38

Fill in the blank: The symbol for 'is a subset of' is ___ .

Answer 38

⊆

Question 39

True or False: Two sets are equal if they have the same elements.

Answer 39

True

Question 40

What is the notation for the Cartesian product of sets A and B?

Answer 40

A × B

Question 41

If U is the universal set and A = {1, 2}, what is the complement of A?

Answer 41

U - A

Question 42

What is a disjoint set?

Answer 42

Two sets that have no elements in common.

Question 43

If A = {x | x is an even number} and B = {x | x is a prime number}, are A and B disjoint?

Answer 43

Yes

Question 44

What is the formula for the number of elements in the union of two sets A and B?

Answer 44

|A ∪ B| = |A| + |B| - |A ∩ B|

Question 45

True or False: The intersection of two sets can be the empty set.

Answer 45

True

Question 46

What does it mean for a set to be countably infinite?

Answer 46

It can be put into a one-to-one correspondence with the natural numbers.

Question 47

Fill in the blank: A set that contains all subsets of a set S is called the ___ of S.

Answer 47

power set

Question 48

If A = {2, 4, 6} and B = {1, 2, 3}, what is A ∩ B?

Answer 48

{2}

Question 49

What is the definition of a proper subset?

Answer 49

A subset that is not equal to the set it is a subset of.

Question 50

What symbol is used to denote the difference between two sets?

Answer 50

−

Question 51

True or False: The union of two sets can be equal to one of the original sets.

Answer 51

True

Question 52

What is the result of the Cartesian product of sets A = {1, 2} and B = {a, b}?

Answer 52

{(1, a), (1, b), (2, a), (2, b)}