Mathematics Flashcards
1
Q
What is the formula to find the area of a rectangle?
A
Length multiplied by width (Area = Length × Width).
2
Q
How do you solve for x in the equation 3x + 4 = 19?
A
Subtract 4 from both sides, then divide by 3 (x = 5).
3
Q
What is the Pythagorean theorem used for?
A
To find the length of a side in a right triangle when the lengths of the other two sides are known (a² + b² = c²).
4
Q
How do you convert a fraction to a decimal?
A
Divide the numerator by the denominator.
5
Q
What does the slope of a line represent?
A
The rate of change of the line, showing how much the y-value changes for a unit increase in the x-value.
6
Q
How do you find the median of a set of numbers?
A
Arrange the numbers in order and select the middle number. If there is an even number of observations, the median is the average of the two middle numbers.
7
Q
What is a prime number?
A
A number greater than 1 that has no positive divisors other than 1 and itself.
8
Q
How do you calculate the perimeter of a triangle?
A
Add the lengths of all three sides.
9
Q
What is an integer?
A
A whole number that can be positive, negative, or zero, not including fractions or decimals.
10
Q
How do you simplify the expression 2(3x + 4)?
A
Distribute the 2 to get 6x + 8.
11
Q
What is the formula for the circumference of a circle?
A
Circumference = 2πr, where r is the radius of the circle.
12
Q
How do you solve the inequality 5x - 7 > 18?
A
Add 7 to both sides and then divide by 5 (x > 5).
13
Q
What is the difference between mean and median?
A
The mean is the average of a set of numbers, calculated by adding them together and dividing by the number of elements. The median is the middle value when the numbers are arranged in order.
14
Q
How do you find the mode of a data set?
A
The mode is the number that appears most frequently in the data set.
15
Q
What is a rational number?
A
A number that can be expressed as a fraction where both the numerator and the denominator are integers.
16
Q
How do you calculate the area of a parallelogram?
A
Base multiplied by height (Area = Base × Height).
17
Q
What is the formula for calculating the volume of a rectangular prism?
A
Volume = Length × Width × Height.
18
Q
How do you solve the equation 2x² = 18 for x?
A
Divide both sides by 2 to get x² = 9, then take the square root of both sides (x = ±3).
19
Q
What is a proportion in mathematics?
A
An equation stating that two ratios are equal.
20
Q
How can you determine if a number is divisible by 3?
A
A number is divisible by 3 if the sum of its digits is divisible by 3.
21
Q
What is an improper fraction, and how can it be converted into a mixed number?
A
An improper fraction has a numerator larger than its denominator. To convert it into a mixed number, divide the numerator by the denominator. The quotient becomes the whole number, and the remainder over the denominator is the fractional part.
22
Q
How do you calculate the diagonal of a square with side length s?
A
Use the formula
𝑑 = 𝑠 √ 2, where d is the diagonal.
23
Q
What is the formula for the area of a trapezoid?
A
Area = 1/2 × (𝐵𝑎𝑠𝑒1 + 𝐵𝑎𝑠𝑒2) × 𝐻𝑒𝑖𝑔ℎ𝑡.
24
Q
How do you solve the system of equations
𝑦 = 2𝑥 + 3 and 𝑦 = − 𝑥 + 5 ?
A
Set the equations equal to each other and solve for x. Substitute x back into one of the equations to find y. Solution: x = 2/3, y = 11/3.
25
How do you convert a decimal to a percent?
Multiply the decimal by 100 and add the percent sign (%).
26
What is the difference between permutations and combinations?
Permutations consider order as important, while combinations do not.
27
How do you find the range of a data set?
Subtract the smallest value from the largest value in the set.
28
What is a histogram and what information does it provide?
A histogram is a type of bar graph used to represent the distribution of a data set. Each bar represents the frequency of data within a certain range.
29
How do you simplify the expression 3𝑥 − 2𝑥 + 5𝑥
Combine like terms: 6𝑥
30
How do you solve quadratic equations using the quadratic formula?
Use the formula 𝑥 = −𝑏 ± √ 𝑏^2 − 4𝑎𝑐 / 2𝑎, , where
𝑎,𝑏 and 𝑐 are the coefficients of the quadratic equation ax^2 + bx + c = 0.
31
What does the coefficient of the x-term in a linear equation represent?
It represents the slope of the line, indicating the change in y for a one-unit increase in x.
32
How do you determine if two lines are parallel?
Two lines are parallel if their slopes are equal.
33
What is the formula for calculating simple interest?
Interest = Principal × Rate × Time.
34
How do you calculate the volume of a cylinder?
Volume = 𝜋𝑟^2ℎ, where 𝑟 is the radius of the base and ℎ is the height.
35
What is the Least Common Multiple (LCM) of two numbers, and how do you find it?
The LCM is the smallest number that is a multiple of both numbers. Find it by listing multiples or using the prime factorization method.
36
What is the complement of an angle measuring 30 degrees?
The complement is the angle that, when added to 30 degrees, equals 90 degrees. So, it is 60 degrees.
37
How do you simplify the expression (𝑥^2 𝑦)^3 ?
Use the power of a power property: 𝑥^6 y^3.
38
What are the steps to graph a linear equation in slope-intercept form, y = mx + b ?
Plot the y-intercept (b) on the y-axis, then use the slope (m) to find another point by rising/running from the intercept, and draw the line through these points.
39
What is a reciprocal, and how do you find the reciprocal of a fraction?
A reciprocal is found by inverting the fraction, making the numerator the denominator and vice versa.
40
How do you calculate the exterior angle of a regular polygon with n sides?
The exterior angle is 360∘/𝑛 .
41
uestion: How do you find the slope of a line given two points on the line, (x1, y1) and (x2, y2)?
Use the formula slope =𝑦^2 − 𝑦^1 / 𝑥^2 − 𝑥^1
42
What is the standard form of a linear equation?
The standard form is Ax + By = C, where A, B, and C are integers, and A should be positive.
43
How do you factor a quadratic expression like 𝑥^2 − 9 ?
Recognize it as a difference of squares: (x+3)(x−3).
44
What is a scatter plot and what does it typically show?
A scatter plot is a graph with points plotted to show a possible relationship between two sets of data.
45
What is the Absolute Value of a number, and how is it represented?
he absolute value of a number is its distance from zero on a number line, represented as ∣𝑎∣.
46
How do you convert an angle from degrees to radians?
Multiply the degree measure by π / 180.
47
How do you solve the equation 2 / 3𝑥 = 8 ?
Multiply both sides by the reciprocal of 2/3 (which is 3/2) to find 𝑥 = 12.
48
What are the steps to calculate compound interest?
Use the formula 𝐴 = 𝑃(1 + 𝑟/𝑛)^𝑛𝑡, where
𝐴 is the amount of money accumulated after n years, including interest. 𝑃 is the principal amount, 𝑟 is the annual interest rate, 𝑛 is the number of times that interest is compounded per year, and 𝑡 is the time the money is invested or borrowed for.
49
What is the discriminant of a quadratic equation and how is it used?
The discriminant is the part of the quadratic formula under the square root, b^2 - 4ac. It determines the nature of the roots (real and distinct, real and equal, or complex).
50
How do you simplify the expression (3x−4) − (x+5)?
Distribute the negative sign and combine like terms: 2𝑥 − 9.
51
What is the Interquartile Range (IQR) and how is it calculated?
The IQR is the range between the first quartile (Q1) and the third quartile (Q3) in a data set, calculated as 𝑄3 − 𝑄1 .
52
How do you calculate the surface area of a sphere?
Use the formula 4πr^2, where 𝑟 is the radius of the sphere.
53
How do you solve for y in the equation
3𝑦 + 3 = 12 ?
Subtract 3 from both sides and then divide by 3 to isolate 𝑦:𝑦 = 3.
54
What is a composite number?
A composite number is a positive integer that has at least one positive divisor other than one or itself.
55
How do you find the common denominator of fractions like 1/4 and 1/6 ?
The common denominator would be 12, the least common multiple of 4 and 6.
56
How do you graph the inequality x > 5 on a number line?
Draw a number line, place an open circle at 5, and shade the line to the right of 5.
57
What is the function of parentheses in mathematical expressions?
Parentheses indicate the order of operations, ensuring that operations within parentheses are performed first.
58
What is Euler's Formula related to polyhedra?
V−E+F=2, where 𝑉 is the number of vertices,
𝐸 is the number of edges, and 𝐹 is the number of faces in the polyhedron.
59
How do you calculate the third side of a triangle if you know the lengths of two sides and the included angle (Law of Cosines)?
Use the formula 𝑐^2 = 𝑎^2 + 𝑏^2 − 2𝑎𝑏cos(𝐶),
where 𝑎 and 𝑏 are the known sides,
𝐶 is the included angle, and 𝑐 is the side to be calculated.
60
How do you determine if a set of points are collinear?
Points are collinear if the slope between any two pairs of points is the same.
61
What is the formula for calculating the area of a circle?
Area = πr², where r is the radius of the circle.
62
How do you solve the proportion 3/4 = x/12 ?
Cross-multiply to get 3 × 12 = 4x, then divide by 4 to solve for x:𝑥 = 9.
63
What is the definition of a parallelogram?
A quadrilateral in which both pairs of opposite sides are parallel.
64
How do you find the vertex of a parabola given in vertex form y = a(x−h)^2 + k ?
The vertex of the parabola is at the point (ℎ, 𝑘)
65
How do you determine if two angles are supplementary?
Two angles are supplementary if they add up to 180 degrees.
66
What does it mean for two sets to be disjoint?
Two sets are disjoint if they have no elements in common.
67
How do you convert a percentage to a fraction?
Divide the percentage by 100 and simplify the fraction if possible.
68
What is the volume formula for a cone?
Volume = 1/3𝜋𝑟^2ℎ, where r is the radius of the base and h is the height.
69
How do you find the x-intercepts of the quadratic equation 𝑎𝑥^2 + 𝑏𝑥 + 𝑐 = 0 ?
Use the quadratic formula or factor the equation, and set each factor equal to zero.
70
What is an isosceles triangle, and what are its properties?
An isosceles triangle has at least two sides of equal length. The angles opposite these sides are also equal.
71
How do you calculate the sum of the interior angles of a polygon?
The sum of the interior angles of a polygon with n sides is (𝑛 − 2) × 180 degrees.
72
What is the difference between direct and inverse variation?
In direct variation, the variables change in the same way (as one increases, so does the other). In inverse variation, the variables change in opposite ways (as one increases, the other decreases).
73
How do you calculate the diagonal of a cube with side length s?
Use the formula 𝑠√3, where s is the side length of the cube.
74
What is a sector in a circle, and how do you find its area?
sector is a portion of a circle defined by two radii and an arc. The area of a sector is 1/2 r^2 θ, where r is the radius and
𝜃 is the angle in radians.
75
How do you solve the linear equation
5(x − 2) = 3x + 6 ?
Expand and simplify the equation to 5𝑥 − 10 = 3𝑥 + 6, then solve for x by isolating x on one side.
76
What is a binomial, and how is it different from a trinomial?
A binomial is a polynomial with two terms. A trinomial has three terms.
77
How do you graph the equation
𝑦 = −2𝑥 + 3 using the slope-intercept method?
Plot the y-intercept (3) on the y-axis, then from this point, use the slope (-2) to find another point (down 2 units, right 1 unit), and draw the line.
78
What is a unit circle, and why is it important in trigonometry?
A unit circle is a circle with a radius of one unit, centered at the origin of a coordinate plane. It is used to define the trigonometric functions sine, cosine, and tangent.
79
How do you simplify the expression
4𝑥 − 3𝑦 + 2𝑥 + 5𝑦 ?
Combine like terms: 6𝑥 + 2𝑦.
80
What are coprime numbers?
Coprime numbers are two or more numbers whose greatest common divisor is 1; they have no other common factors besides 1.
81
What is a hypotenuse, and how do you find its length in a right triangle?
The hypotenuse is the longest side of a right triangle, opposite the right angle. Use the Pythagorean theorem,
𝑎^2 + 𝑏^2 = 𝑐^2, where 𝑐 is the hypotenuse.
82
How do you calculate the lateral area of a cylinder?
Lateral area = 2𝜋𝑟ℎ where 𝑟 is the radius and ℎ is the height of the cylinder.
83
What is a quadratic inequality, and how do you solve it?
A quadratic inequality looks like a quadratic equation but involves inequalities ( <, >, ≤ or
≥). To solve it, find the roots using the quadratic formula and test the intervals.
84
How do you find the greatest common divisor (GCD) of two numbers?
The GCD of two numbers can be found using the Euclidean algorithm, which involves repeated division.
85
What does the term 'rationalize the denominator' mean?
To rationalize the denominator means to eliminate any radicals (like square roots) from the denominator of a fraction by multiplying both the numerator and the denominator by an appropriate value.
86
What is an asymptote in graphing?
An asymptote is a line that a graph approaches but never touches. Horizontal and vertical asymptotes occur in rational functions and some algebraic expressions.
87
What is meant by the term 'order of operations' in mathematics?
The order of operations is the rule that defines the sequence in which operations should be performed in an expression: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
88
How do you calculate the area of an equilateral triangle with side length 𝑠 ?
Area = √3/4s^2.
89
How do you find the average rate of change of a function between two points?
The average rate of change = Change in function value (Δy) / Change in independent variable (Δx) .
.
90
What is the Binomial Theorem and how is it used?
The Binomial Theorem provides a formula for expanding powers of binomials,
(𝑎 + 𝑏)^𝑛, involving binomial coefficients.
91
How do you determine if a number is a perfect square?
A number is a perfect square if its square root is an integer.
92
What is the difference between an arithmetic and a geometric sequence?
In an arithmetic sequence, the difference between consecutive terms is constant. In a geometric sequence, the ratio between consecutive terms is constant.
93
How do you graph an absolute value function?
Graph the function 𝑦 = ∣𝑥∣ as two rays: one ray goes up and to the right, and the other goes up and to the left, both starting from the vertex, typically at the origin.
94
Question: How do you find the zeros of a polynomial function?
Find the values of 𝑥 for which the polynomial equals zero, either by factoring, using the Rational Root Theorem, or applying synthetic division.
95
What are inverse functions, and how do you find one?
Inverse functions reverse the effect of the original function. To find an inverse function, swap the and 𝑦 in the original equation and solve for 𝑦.
96
How do you find the probability of a single event occurring?
Probability =
Number of favorable outcomes /
Total number of possible outcomes.
97
How do you simplify the expression √12?
Simplify by factoring:
√12 = √4 × 3 = 2√3.
98
What is the angle sum property of a triangle?
The sum of the interior angles of any triangle is always 180 degrees.
99
What is a reflex angle?
A reflex angle is an angle greater than 180 degrees but less than 360 degrees.
100
How do you solve a linear equation in one variable involving fractions?
Clear the fractions by multiplying every term by the least common denominator, then solve the resulting equation as usual.
