# JEE - Algebra

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## Introduction

Algebra is a mathematical topic that covers number theory, geometry, and analysis. In the history of Mathematics, it is one of the first branches. According to the definition of algebra, the study of mathematical symbols and rules sometimes includes manipulating these mathematical symbols. From solving simple equations to analysing abstractions, algebra covers a wide range of topics. Many chapters of Mathematics that kids will study in school contain algebra equations. Algebra also has a number of formulae and identities.

### Important Terms in Algebra

• Polynomials

• Domain and Range

• Trigonometric function

• Exponential function

• Irrational expression

• Inequalities

• Linear equations

• Sequences and series

### What is Algebra

We use mathematical formulae to determine the distance and capacity of containers in our daily lives, as well as to compute sales pricing when necessary. Algebra is a mathematical language that allows you to express a mathematical equation and connection using letters or other symbols to represent things. The unknown quantities in the equation can be solved using algebra.

Algebra helps in the solution of mathematical equations and the calculation of unknown numbers such as bank interest, proportions, and percentages. We can rewrite the equations by using the variables in algebra to represent the unknown quantities that are related.

### Classification of Algebra

The branches and sub-branches of Algebra are listed below. These branches are used to find the values of two or more variables in the correct order. The following are the algebraic branches:

### Elementary Algebra

The conventional topics studied in a modern elementary algebra course are covered in Elementary Algebra. Arithmetic is the study of numbers and mathematical operations such as +, -, x, and y. However, in algebra, numbers are frequently represented by symbols and are referred to as variables, such as x, a, n, and y. It also allows for the common formulation of arithmetic laws such as a + b = b + a, and it is the first step in demonstrating the systematic exploration of all properties of a system of real numbers.

Variables, evaluating expressions and equations, properties of equalities and inequalities, solving algebraic equations and linear equations with one or two variables, and other concepts are covered in elementary algebra.

This is Algebra's intermediate level. In comparison to pre-algebra, this algebra has a lot of equations to solve. Advanced algebra will help you in understanding other topics of algebra, such as:

• Equations with inequalities

• Matrices

• Solving system of linear equations

• Graphing of functions and linear equations

• Conic sections

• Polynomial equation

• Polynomials and expressions with radicals

• Sequences and series

• Rational expressions

• Trigonometry

• Discrete Mathematics and probability

### Abstract Algebra

Abstract algebra is one of the divisions of algebra that tries to solve truths regarding algebraic systems that are independent of the nature of particular operations. In some circumstances, these operations have particular features. As a result, abstract algebra is a branch of Mathematics.

The concepts of abstract algebra are as follows:

1. Sets – Sets are defined as a collection of items whose existence is determined by a set's special property. For example, a collection of all 22 matrices, a collection of two-dimensional vectors in the plane, and many types of finite groups.

2. Binary Operations – The binary operations are created when the concept of addition is realised. Without a set, the concept of all binary operations will be useless.

3. Identity Element – The digits 0 and 1 are used to represent an identity element for a certain operation. For the addition operation, 0 is the identity element, but for the multiplication operation, 1 is the identity element.

4. Inverse Elements – The idea of inverse elements comes up with a negative number. For addition, we write “-a” as the inverse of “a” and for the multiplication, the inverse form is written as “a-1″.

5. Associativity – When adding integers, there is a property known as associativity, which states that grouping the numbers added has no effect on the result. Consider an example, (5 + 2) + 7 = 5 + (2 + 7)

### Linear Algebra

Linear algebra is an area of Mathematics that may be used in both practical and pure Mathematics. It is concerned with linear mappings between vector spaces. It also entails the investigation of planes and lines. It is the study of transformation qualities in linear sets of equations. It is nearly universally used in Mathematics. It is concerned with linear equations and their representation in vector spaces and matrices for linear functions.

The following are some of the most essential subjects studied in linear algebra:

• Linear equations

• Vector Spaces

• Relations

• Matrices and matrix decomposition

• Relations and Computations

### Commutative Algebra

Commutative algebra is one of the branches of algebra that studies the commutative rings and their ideals. The algebraic number theory, as well as the algebraic geometry, depends on commutative algebra. Rings of algebraic integers, polynomial rings, and so on are examples. Commutative algebra is used in many other areas of Mathematics, including differential topology, invariant theory, order theory, and general topology. In modern pure Mathematics, it has played a significant role.

### Solved Examples

Example 1: Solve $\left|x^{2}+x-4 \right|= \left|x^{2}-4 \right|+\left|x \right|$

Solution: $\left|x^{2}+x-4 \right|= \left|x^{2}-4 \right|+\left|x \right|$

$\Rightarrow x\left ( x^{2}-4 \right )\geq 0$

$\Rightarrow x\left ( x-2 \right )\left ( x+2 \right )\geq 0$

$x\in \left [ -2, 0 \right ]\cup \left [ 2, \infty \right ]$

Example 2: Find how many of the equations $x^{4}+2x^{2}-8x+3=0$ are real.

Solution: Let $f(x)= x^{4}+2x^{2}-8x+3=0$

$\Rightarrow f'(x)= 4x^{3}+4x-8$

$\Rightarrow f'(x)= 4x^{3}+4x-8= 4(x-1)\left ( x^{2x}+4x-8 \right )$

now , $f'(x)=0\Rightarrow x =1$

Hence, the graph of y = f(x) has only one turn (maxima/minima)

Now, $f(1)=1+2-8+3<0$

Also when $x\to \pm \infty , f(x) \to \infty$

### Solved Problems of Previous Year Question

1. A value of b for which the equations $x^{2}+bx-1=0,\,\ x^{2}+x+b=0$  have one root in common is

1. $-\sqrt{2}$

2. $-i\sqrt{3}$

3. $\sqrt{2}$

4. $\sqrt{3}$

Ans: (b) here, we have

$x^{2}+bx-1=0$ ……… (i)

$x^{2}+x+b=0$ …………(ii)

Subtracting, we get $\left ( b-1 \right )x-1-b=0$

$\Rightarrow x=\frac{b+1}{b-1}$

This value of x satisfies equation (1)

$\Rightarrow \frac{(b+1)^{2}}{(b-1)^{2}} +\frac{b+1}{b-1}+b=0$

$\Rightarrow b=\sqrt{3}i, \,b=-\sqrt{3}i,\, 0$

2. Let $\alpha$ and $\beta$ be the roots of $x^{2}-6x-2=0, \,\ with ,\ ,\alpha > \beta$. If $a_{n}=\alpha ^{n}-\beta ^{n}$ for $n\geq 1,$ then the value of $\frac{a_{20}-2a_{8}}{2a_{9}}$.

1. 1

2. 3

3. 4

$a_{n}=\alpha ^{n}-\beta ^{n}$

Also $\alpha ^{2}-6\alpha -2=0$

Multiply with $\alpha^{8}$  on both sides

$\Rightarrow \alpha ^{10}-6\alpha ^{9}-2\alpha ^{8}=0$ ……….. (i)

Similarly $\beta^{10}-6\beta ^{9}-2\beta ^{8}=0$ ………. (ii)

Subtracting (2) from (1) we have

$\Rightarrow \alpha ^{10}-\beta ^{10}-6\left ( \alpha ^{9}-\beta ^{9} \right )=2\left ( \alpha ^{8}-\beta ^{8} \right )$

$\Rightarrow a_{10}-6a_{9}=2a_{8}$

$\frac{a_{20}-2a_{8}}{2a_{9}}=3$

3. The number of distinct real roots of $x^{4}-4x^{3}+12x^{2}+x-1 = 0$ is ……

Ans: b

Let $f(x) = x^{4}-4x^{3}+12x^{2}+x-1 = 0$

$\therefore f'(x)= 4x^{3}-12x^{2}+24x+1$

And $f''(x)=12x^{2}-24x+24>0$ for all real x.

So, the graph of f’(x) intersects the x - axis only once.

Hence, f(x) has only one turning point.

Also, f(0)= -1.

So, the graph of f(x) cuts the x axis at two points.

Hence, f(x) = 0 has two real roots.

### Practice Question:

1. The sum of the non - real roots of $\left ( x^{2}+x-2 \right )\left ( x^{2} +x - 3\right )=12$ is

1. -1

2. 1

3. -6

4. 6

2. The number of irrational roots of the equation $\frac{4x}{x^{2}+x+3}-\frac{5x}{x^{2}-5x+3}=- \frac{3}{2}$

1. 4

2. 1

3. 2

### Conclusion

In this chapter, we have elaborated on concepts and solutions to questions on the topic Algebra. Everything you're looking for is available in a single location. Students can carefully read through the concepts, Definitions, and questions in the PDFs, which are also free to download and understand the concepts used to solve these questions. This will be extremely beneficial to the students in their exams.

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IIT Bombay has announced the JEE Advanced 2022 application form release date on the official website https://jeeadv.ac.in/. JEE Advanced 2022 Application Form is available on the official website for online registration. Besides JEE Advanced 2022 application form release date, learn about the application process, steps to fill the form, how to submit, exam date sheet etc online. Check our website for more details.

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It is crucial for the the engineering aspirants to know and download the JEE Advanced 2022 syllabus PDF for Maths, Physics and Chemistry. Check JEE Advanced 2022 syllabus here along with the best books and strategies to prepare for the entrance exam. Download the JEE Advanced 2022 syllabus consolidated as per the latest NTA guidelines from Vedantu for free.
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## JEE Advanced 2022 Study Material

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JEE Advanced 2022 Study Materials: Strengthen your fundamentals with exhaustive JEE Advanced Study Materials. It covers the entire JEE Advanced syllabus, DPP, PYP with ample objective and subjective solved problems. Free download of JEE Advanced study material for Physics, Chemistry and Maths are available on our website so that students can gear up their preparation for JEE Advanced exam 2022 with Vedantu right on time.
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In order to prepare for JEE Advanced 2022, candidates should know the list of important books i.e. RD Sharma Solutions, NCERT Solutions, HC Verma books and RS Aggarwal Solutions. They will find the high quality readymade solutions of these books on Vedantu. These books will help them in order to prepare well for the JEE Advanced 2022 exam so that they can grab the top rank in the all India entrance exam.
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IIT Bombay is responsible for the release of the JEE Advanced 2022 cut off score. The qualifying percentile score might remain the same for different categories. According to the latest trends, the expected cut off mark for JEE Advanced 2022 is 50% for general category candidates, 45% for physically challenged candidates, and 40% for candidates from reserved categories. For the general category, JEE Advanced qualifying marks for 2021 ranged from 17.50%, while for OBC/SC/ST categories, they ranged from 15.75% for OBC, 8.75% for SC and 8.75% for ST category.
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## FAQs on JEE - Algebra

FAQ

1. What are the different branches or types of algebra?

There are five different branches or types of algebra. They are elementary algebra, abstract algebra, advanced algebra, commutative algebra, and linear algebra. All these branches have different formulas, different applications, and different uses in finding out the values of variables.

• Pre-Algebra: Helps in transforming real-life problems into algebraic expressions.

• Elementary Algebra: Helps in solving the algebraic expression into a viable answer.

• Abstract Algebra: uses advanced levels of algebra with abstract concepts other than mathematical ones like rings and vectors.

• Universal Algebra: uses mathematical concepts involving trigonometry, calculus, and coordinate geometry.

2. What are Polynomials in Algebra?

Polynomials are algebraic expressions that consist of variables as well as coefficients. The variables here can also be termed as indeterminate. Polynomials only include addition, subtraction, and multiplication of the variables. It also includes non-negative or positive integer exponentiation of the required variables. Students also learn factoring polynomials in geometry. A polynomial expression includes at least one variable and also includes constants and positive exponents as well. They are the building blocks in rational expressions and are used to express numbers as a result of mathematical expressions.