Algebra I Examination - F.Y B.Sc. Sem-1 Nov 2023

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

Match the following terms with their definitions:

Bijective function = A function that is both injective and surjective Surjective function = A function that covers every element in the codomain Injective function = A function where different inputs map to different outputs Polynomial = A mathematical expression involving a sum of powers in one or more variables

Match the following properties with their corresponding operations:

Commutative = a * b = b * a Associative = (a * b) * c = a * (b * c) Reflexive = aRa for all a Transitive = If aRb and bRc, then aRc

Match the following binary operations with their expressions:

' * ' on Z = a * b = a + 2b f(x) = 2x + 5 = Linear function gof = Composition of functions R relation = aRb iff a and b are in the same college

Match the following sets with their descriptions:

Set of all students = X Set of integers = Z Set of rational numbers = p/q where p and q are integers and q ≠ 0 Set of polynomials = Functions with integer coefficients Signup and view all the answers

Match the following concepts with their characteristics:

Root of a polynomial = Value that makes the polynomial equal to zero Set of all integers = Z Rational number = A number that can be expressed as a fraction College relation R = Defines if two students are from the same college Signup and view all the answers

Match the following properties of relations with their definitions:

Reflexive = Every element is related to itself Symmetric = If a is related to b, then b is related to a Transitive = If a is related to b and b is related to c, then a is related to c Irreflexive = No element is related to itself Signup and view all the answers

Match the following terms about polynomials with their definitions:

Polynomial = An expression consisting of variables and coefficients Degree = The highest power of the variable in the polynomial Root = A value for which the polynomial evaluates to zero Coefficient = A numerical factor in front of a term Signup and view all the answers

Match the following operations with their descriptions:

Quotient = The result of dividing one polynomial by another Remainder = The amount left over after division Multiplicity = The number of times a root appears in a polynomial Simplification = The process of reducing a mathematical expression to its simplest form Signup and view all the answers

Match the following types of roots with their characteristics:

Simple Root = A root with multiplicity of one Multiple Root = A root that is repeated in a polynomial Complex Root = A root that involves imaginary numbers Rational Root = A root that can be expressed as a fraction Signup and view all the answers

Match the following components of the polynomial $f(x) = a_nx^n + a_{n-1}x^{n-1} + .... + a_1x + a_0$ with their meanings:

$a_n$ = Leading coefficient $a_0$ = Constant term $x^n$ = Term with the highest degree $a_i$ = Coefficient of the term $x^i$ Signup and view all the answers

Match the mathematical concepts with their definitions:

G.C.D = The largest integer that divides two numbers without leaving a remainder Surjective Function = A function where every element in the codomain is mapped by at least one element in the domain Bijective Function = A function that is both injective and surjective Fermat's Little Theorem = A theorem that provides a method to determine if a number is a prime based on modular arithmetic Signup and view all the answers

Match the mathematical symbols with their meanings:

∈ = Element of ∤ = Does not divide mod = Modulus operation gcd = Greatest common divisor Signup and view all the answers

Match the mathematical statements with their related concepts:

If P is prime = It cannot be divided by any integer other than 1 and itself gof is bijective = The composition of functions f and g is a one-to-one correspondence f is surjective = Function covers the entire range of the codomain a and b in Z = Refers to integers both positive and negative Signup and view all the answers

Match the integers with their G.C.D outcomes:

G.C.D of 2210 and 357 = $1$ G.C.D of 12 and 8 = $4$ G.C.D of 30 and 12 = $6$ G.C.D of 18 and 24 = $6$ Signup and view all the answers

Match the features of functions with their types:

Injective = Each element in the domain maps to a unique element in the codomain Surjective = At least one element in the domain maps to every element in the codomain Bijective = Each element in the domain maps to a unique element in the codomain and vice-versa Non-injective = At least one element in the domain maps to more than one element in the codomain Signup and view all the answers

Study Notes

Algebra I Exam Overview

Exam conducted by Gokhale Education Society's N.B.Mehta Science College, Bordi.
Scheduled for 10/11/2023 from 10:00 to 12:30 PM.
Total marks for the exam: 75.
All questions are compulsory.

Question 1: GCD and Modular Arithmetic

Part A: Prove that any two non-zero integers ( a ) and ( b ) in ( \mathbb{Z} ) have a greatest common divisor (g.c.d), and it can be written as ( ma + nb ) for integers ( m, n \in \mathbb{Z} ).
Part B: Choose two from these:
- Compute the G.C.D. of 2210 and 357 and express it in the form ( 2210x + 357y ) for integers ( x, y \in \mathbb{N} ).
- Demonstrate ( 281 \equiv 2 \pmod{41} ) using Fermat’s Little Theorem.
- Prove: If ( P ) is a prime such that ( P \mid a ) or ( P \mid b ), then ( P \mid ab ).

Question 2: Function Properties

Part A: Show that if ( f: X \to Y ) and ( g: Y \to Z ) such that ( gof ) is bijective and ( f ) is surjective, then ( g ) must be injective.
Part B: Select two from these:
- Prove ( f: \mathbb{R} \to \mathbb{R} ) defined by ( f(x) = 2x + 5 ) is bijective.
- Define a binary operation ( a * b = a + 2b ) on ( \mathbb{Z} ) and check if it is commutative and associative.
- Analyze relation ( R ) on set ( X ) (students in a college) defined as ( aRb ) if ( a ) and ( b ) are in the same college for reflexivity, symmetry, and transitivity.

Question 3: Polynomials and Rational Roots

Part A: Prove that a polynomial ( f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 ) with integer coefficients means ( a_i \in \mathbb{Z} ).
- If ( p/q ) (in lowest terms, ( q \neq 0 )) is a root of ( f(x) ), then ( p \mid a_0 ) and ( q \mid a_n ).
Part B: Check whether relation ( R ) is reflexive, symmetric, and transitive on the set of college students.

Additional Tasks

Find the quotient and remainder when dividing ( x^4 - 3x^2 + 4x + 8 ) by ( x^2 + 2 ).
Determine the multiplicity of each root for ( f(x) = x^3 - 4x^2 + 5x - 2 ).

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

This quiz tests your understanding of Algebra I concepts covered in the F.Y B.Sc. Theory curriculum. It includes problems on the greatest common divisor (g.c.d) and requires you to prove mathematical statements and solve numerical questions. Prepare for a comprehensive assessment of your algebraic skills.