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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?
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?
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?
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
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
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
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
Equivalence Relation
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Indefinite Integral
Indefinite Integral
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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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