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Questions and Answers
What is the inverse function of $y = 4 - x$?
What is the inverse function of $y = 4 - x$?
For the function $f(x) = 3x^4 - x^2 + 5$, which two options describe its critical points?
For the function $f(x) = 3x^4 - x^2 + 5$, which two options describe its critical points?
Which of the following functions is an even function?
Which of the following functions is an even function?
What is defined as the order of an equation when a function depends on several variables?
What is defined as the order of an equation when a function depends on several variables?
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What is the value of f(1) + f(2) + ... + f(33) for f(x) = x(x + 1)?
What is the value of f(1) + f(2) + ... + f(33) for f(x) = x(x + 1)?
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What is the inverse function of $y = 5x^2 + 2$, valid for $x \geq 2$?
What is the inverse function of $y = 5x^2 + 2$, valid for $x \geq 2$?
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Which statement correctly defines differential equations?
Which statement correctly defines differential equations?
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What is the domain of the function f(x) = sqrt{1 - sqrt{16 - x^2}}?
What is the domain of the function f(x) = sqrt{1 - sqrt{16 - x^2}}?
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What is the smallest positive period of the function $f(x) = 3 \tan(1.5x)$?
What is the smallest positive period of the function $f(x) = 3 \tan(1.5x)$?
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Which axis is a symmetry axis for any even function?
Which axis is a symmetry axis for any even function?
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What represents the absolute value of a complex number?
What represents the absolute value of a complex number?
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What is the value of the function $f(x) = \frac{7x - 5}{x^2 - 4}$ at the point $x = -0.2$?
What is the value of the function $f(x) = \frac{7x - 5}{x^2 - 4}$ at the point $x = -0.2$?
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What is the domain of the function $f(x) = \sqrt{(x - 1)(x - 2)}$?
What is the domain of the function $f(x) = \sqrt{(x - 1)(x - 2)}$?
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Where does the complex number 'i' correspond to in the coordinate system?
Where does the complex number 'i' correspond to in the coordinate system?
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What is the value of f(0.1) for f(x) = (x - 1)/(3x)?
What is the value of f(0.1) for f(x) = (x - 1)/(3x)?
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Which of the following represents an even function?
Which of the following represents an even function?
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Which of the following expressions evaluates to 2520?
Which of the following expressions evaluates to 2520?
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What is the modulus of the complex number z = 4 + 3i?
What is the modulus of the complex number z = 4 + 3i?
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What area is enclosed by the lines x = 2 and the graph of y = x^3?
What area is enclosed by the lines x = 2 and the graph of y = x^3?
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What is the value of the function $f(x) = \frac{7x - 14}{x^2 - 4}$ at the point $x = t - 3$?
What is the value of the function $f(x) = \frac{7x - 14}{x^2 - 4}$ at the point $x = t - 3$?
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How many ways can you arrange 3 students in 5 seats?
How many ways can you arrange 3 students in 5 seats?
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What is the number of permutations of 5 people standing in line?
What is the number of permutations of 5 people standing in line?
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How is a unit matrix defined?
How is a unit matrix defined?
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Who introduced the term 'normal numbers'?
Who introduced the term 'normal numbers'?
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How many games will be played in the first round of a tournament with 12 teams?
How many games will be played in the first round of a tournament with 12 teams?
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Calculate the value of (A_{5}^2 + C_{5}^3) * C_{5}^2.
Calculate the value of (A_{5}^2 + C_{5}^3) * C_{5}^2.
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What describes complex numbers?
What describes complex numbers?
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In the first-order differential equation dy/dx = 3x^2 - 4, what is the general solution?
In the first-order differential equation dy/dx = 3x^2 - 4, what is the general solution?
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What is the distance from point M to plane ABC in a parallelogram ABCD with AB = 20 cm and angle between MA and plane ABC as 60 degrees?
What is the distance from point M to plane ABC in a parallelogram ABCD with AB = 20 cm and angle between MA and plane ABC as 60 degrees?
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In triangle ABC, if angle C is a right angle and AC = 18 cm, SM (perpendicular from C to plane) = 12 cm, what is the distance from point M to line AB?
In triangle ABC, if angle C is a right angle and AC = 18 cm, SM (perpendicular from C to plane) = 12 cm, what is the distance from point M to line AB?
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What is the relationship between a line and the number of planes it can pass through?
What is the relationship between a line and the number of planes it can pass through?
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What can be concluded about the convergence type based on the D'Alembert criterion if it is found to be divergent?
What can be concluded about the convergence type based on the D'Alembert criterion if it is found to be divergent?
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In triangle ABC where AC = BC = 10 cm and angle B = 30°, what is the distance from point D to line AC if BD is perpendicular to the triangle's plane?
In triangle ABC where AC = BC = 10 cm and angle B = 30°, what is the distance from point D to line AC if BD is perpendicular to the triangle's plane?
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In a right triangle where CA = a, CB = a, and angle A = 30°, what can be inferred about the angles formed by the heights from points A and C?
In a right triangle where CA = a, CB = a, and angle A = 30°, what can be inferred about the angles formed by the heights from points A and C?
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What is the minimum number of regions occupied in space by three planes?
What is the minimum number of regions occupied in space by three planes?
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What angle results from the intersection of heights AA' and CC' in an acute triangle, given ∠OCA = 38°?
What angle results from the intersection of heights AA' and CC' in an acute triangle, given ∠OCA = 38°?
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Given a triangle ABC and the sides BC = 6 cm and angle ACB = 120° with the height from M perpendicular to ABC being 3 cm, what is the distance from M to line AC?
Given a triangle ABC and the sides BC = 6 cm and angle ACB = 120° with the height from M perpendicular to ABC being 3 cm, what is the distance from M to line AC?
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For a rhombus ABCD with side lengths of 8 cm and angle A = 45°, if a perpendicular BE is dropped from point E located 4 cm from line AD, what is the distance from E to plane ABC?
For a rhombus ABCD with side lengths of 8 cm and angle A = 45°, if a perpendicular BE is dropped from point E located 4 cm from line AD, what is the distance from E to plane ABC?
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What is the volume of the cylinder that is circumscribed around a triangular prism with edges measuring 3 cm each?
What is the volume of the cylinder that is circumscribed around a triangular prism with edges measuring 3 cm each?
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What is the angle between the line SB and the plane ABC in a square pyramid with edges measuring 1 cm?
What is the angle between the line SB and the plane ABC in a square pyramid with edges measuring 1 cm?
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What is the distance from point B to the line AD₁ in a unit cube?
What is the distance from point B to the line AD₁ in a unit cube?
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What is the radius of the inscribed sphere in a unit cube?
What is the radius of the inscribed sphere in a unit cube?
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What is the radius of the circumscribed sphere around a unit cube?
What is the radius of the circumscribed sphere around a unit cube?
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What is the surface area of a rectangular parallelepiped with dimensions 1 cm, 2 cm, and 3 cm?
What is the surface area of a rectangular parallelepiped with dimensions 1 cm, 2 cm, and 3 cm?
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What is the critical point of the function f(x) = 1/(x² - 3x + 2)?
What is the critical point of the function f(x) = 1/(x² - 3x + 2)?
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What is the value of the expression ( \frac{x^{2} - 2x}{4x^{2}} \cdot \frac{2x}{2 - x} )?
What is the value of the expression ( \frac{x^{2} - 2x}{4x^{2}} \cdot \frac{2x}{2 - x} )?
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How far is point B from the line AD₁ in a unit cube configuration?
How far is point B from the line AD₁ in a unit cube configuration?
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What is the area of triangle ΔAОC given that AA1 = 15 cm, BB1 = 9 cm and the medians AA1 and CC1 are perpendicular?
What is the area of triangle ΔAОC given that AA1 = 15 cm, BB1 = 9 cm and the medians AA1 and CC1 are perpendicular?
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If the perimeter of a rectangle is 80 cm and the ratio of its sides is 2:3, what is its area?
If the perimeter of a rectangle is 80 cm and the ratio of its sides is 2:3, what is its area?
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Given a rhombus with diagonals of lengths 6 cm and 10 cm, what is its area?
Given a rhombus with diagonals of lengths 6 cm and 10 cm, what is its area?
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What is the length of the base of an isosceles triangle with an area of 60 cm² and height of 8 cm?
What is the length of the base of an isosceles triangle with an area of 60 cm² and height of 8 cm?
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If a triangle's sides are in the ratio 7:8:9 and the perimeter of a triangle formed by its midsegments is 12 cm, what are the lengths of the original triangle's sides?
If a triangle's sides are in the ratio 7:8:9 and the perimeter of a triangle formed by its midsegments is 12 cm, what are the lengths of the original triangle's sides?
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What is the opposite angle to the longest side in a triangle with sides measuring 8 cm, 15 cm, and 17 cm?
What is the opposite angle to the longest side in a triangle with sides measuring 8 cm, 15 cm, and 17 cm?
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What was the original edge length of a cube if increasing its edge by 2 cm resulted in a volume increase of 98 cm³?
What was the original edge length of a cube if increasing its edge by 2 cm resulted in a volume increase of 98 cm³?
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In a regular triangular pyramid, if the angle between the height and the base is 90°, with a base area of 15√3, what is the lateral area?
In a regular triangular pyramid, if the angle between the height and the base is 90°, with a base area of 15√3, what is the lateral area?
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Given an inclined plane making a 30° angle and a length of 20 cm, what is the perpendicular length to the inclined plane?
Given an inclined plane making a 30° angle and a length of 20 cm, what is the perpendicular length to the inclined plane?
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In a cube, what is the angle between the lines AB and CB1?
In a cube, what is the angle between the lines AB and CB1?
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Study Notes
Mathematical Functions
- Inverse functions: A function's inverse reverses the input-output relationship. To find the inverse, swap x and y, then solve for y.
- Even functions: A function is even if f(-x) = f(x). The graph is symmetrical about the y-axis.
- Odd functions: A function is odd if f(-x) = -f(x). The graph is symmetrical about the origin.
Derivatives
- Derivative of a constant: The derivative of a constant is zero (e.g., d/dx(5) = 0).
- Power rule: The derivative of xn is nxn-1.
- Chain rule: The derivative of a composite function is the derivative of the outer function times the derivative of the inner function.
- Product rule: The derivative of the product of two functions is the first function times the derivative of the second function, plus the second function times the derivative of the first function
- Quotient rule: The derivative of a quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.
- Trigonometric derivatives: The derivatives of trigonometric functions (sin x, cos x, tan x, etc.) are well-known and should be memorized.
Integrals
- Power rule of integration: ∫ xn dx = (xn+1)/(n+1) + C (where C is the constant of integration)
- Constant multiple rule: ∫ cf(x)dx = c∫f(x)dx
- Sum/difference rule: ∫ (f(x) ± g(x)) dx = ∫ f(x) dx ± ∫ g(x) dx.
- Trigonometric integrals: Memorizing the integrals of trigonometric functions is crucial for calculating definite integrals.
- Integration by parts: A technique for integrating products of functions, involving the product rule in reverse.
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Description
Test your knowledge on various algebraic topics including inverse functions, critical points, and function domains. This quiz covers essential concepts related to even functions, periodic functions, and complex numbers, providing a comprehensive overview of algebra and functions.
