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Questions and Answers
Which of the following statements about the greatest common divisor (G.C.D) is true?
If P is a prime number and P divides the product of two integers a and b, it necessarily follows that P divides at least one of those integers.
True
What does it mean for a function f: X→Y to be surjective?
A function f is surjective if every element in the codomain Y is mapped to by at least one element in the domain X.
If A and B are reflexive relations, then their union is also ______.
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Match the following functions with their properties:
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Study Notes
Exam Information
- Subject: Algebra I (USMT102)
- Date: November 10, 2023
- Exam Duration: 2 hours, from 10:00 to 12:30 PM
- Maximum Marks: 75
- All questions are mandatory
Question Breakdown
Q1: Number Theory
- G.C.D of Integers: Any two non-zero integers a and b have a greatest common divisor (g.c.d), expressible as ( ma + nb ) for integers m and n.
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Example Problems:
- Calculate G.C.D of 2210 and 357, express as ( 2210x + 357y ).
- Prove ( 281 \equiv 2 \mod 41 ) using Fermat's Little Theorem.
- If P is a prime and divides the product ab, it also divides a or b.
Q2: Functions and Relations
- Function Properties: If ( g \circ f ) is bijective and f is surjective, function g is injective.
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Example Problems:
- Prove f: R→R defined by ( f(x) = 2x + 5 ) is a bijective function.
- Define binary operation ( a * b = a + 2b ) on Z, check for commutativity and associativity.
- Analyze the relation R among students in college for reflexivity, symmetry, and transitivity.
Q3: Polynomials
- Polynomial Definition: Polynomial ( f(x) = a_n x^n + a_{n-1}x^{n-1} +...+ a_1x + a_0 ) has integer coefficients, where each ( a_i \in Z ).
- Rational Roots Theorem: If a rational number ( \frac{p}{q} ) is a root of f(x), then ( \frac{p}{a_n} ) and ( \frac{q}{a_0} ) must also hold.
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Example Problems:
- Find quotient and remainder of ( x^3 - 3x^2 + 4x + 8 ) when divided by ( x^2 + 2 ).
- Determine the multiplicity of each root for ( f(x) = x^3 - 4x^2 + 5x - 2 ).
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Description
Test your knowledge with this Algebra I theory exam based on the syllabus for F.Y. B.Sc. The quiz includes questions on the fundamental concepts of integers and their greatest common divisor. All questions are compulsory, so prepare thoroughly and showcase your understanding!
