Mathematics Class Quiz

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Questions and Answers

Which property does the relation R defined by xRy if and only if 2x - 2y + 3√5 is an irrational number possess?

  • Equivalence
  • Transitive
  • Reflexive
  • Symmetric (correct)

What is the result of the indefinite integral ∫(x² - 6x + 9)√(x² - 6x + 4) dx?

  • √(x² - 6x + 4) / 5(x - 3) + C
  • 2√(x² - 6x + 4) / (x - 3) + C (correct)
  • 5√(x² - 6x + 4) / (x - 3) + C
  • √(x² - 6x + 4) / (x - 3) + C

What is the value of the definite integral ∫₀^π x sin x / (1 + cos² x) dx?

  • π²/3
  • π²/4 (correct)
  • π²/6
  • π²/2

Flashcards

Reflexive Relation

A relation R on a set X is reflexive if every element of X is related to itself. Formally, for every element x in X, xRx must be true. Imagine a mirror reflecting everything back to itself.

Symmetric Relation

A relation R on a set X is symmetric if whenever x is related to y, then y is also related to x. Formally, if xRy is true, then yRx must also be true. Think of two-way streets where relationships go both ways.

Transitive Relation

A relation R on a set X is transitive if whenever x is related to y and y is related to z, then x is related to z. Think of a chain reaction where the relationship flows from one element to another.

Equivalence Relation

A relation R on a set X is an equivalence relation if it satisfies all three properties: reflexivity, symmetry, and transitivity. Equivalence relations group elements with similar characteristics into equivalence classes, just like sorting objects into categories based on shared features.

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Indefinite Integral

The indefinite integral of a function f(x) is a family of functions whose derivative is f(x). The constant of integration 'C' represents a family of functions that differ by a constant. You can think of it as adding a constant to the antiderivative.

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Study Notes

Relation R and Properties

  • A relation R is defined for x, y ∈ℝ, where xRy if and only if 2x - 2y + 3√5 is an irrational number
  • The relation R is not symmetric, transitive, or reflexive

Integration

  • ∫(x² - 6x + 9)⁻¹/² dx = √(x² - 6x + 4) / (x - 3) + C
  • Another correct integral is 2√(x² - 6x + 4) / (x - 3) + C (Note that the denominator remains the same, consistent throughout)
  • Another correct integral is 5√(x² - 6x + 4) / (x - 3) + C
  • Another correct integral is √(x² - 6x + 4) / 5(x-3) + C

Definite Integral

  • ∫₀^π xsin x / (1 + cos² x) dx = π²/4
  • Another correct answer is π²/2
  • Another correct answer is π²/3
  • Another correct answer is π²/6

Equation of Image of a Circle

  • The image of the circle x² + y² + 16x - 24y + 183 = 0 reflected across the line 4x + 7y + 13 = 0 is x² + y² + 2gx + 2fy + c = 0
  • The value of g + f + c is 221

Permutations of Letters

  • The letters of the word OLUVME are permutated and arranged in a dictionary
  • The serial number of the word VOLUME is 724

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