Unit 5 Class 10 Math New Book Solutions
Unit 5 Class 10 Math New Book Solutions are available here in PDF format with complete step-by-step answers for Exercise 5.1, Exercise 5.2, Exercise 5.3, Exercise 5.4, Exercise 5.5, and Review Exercise 5. This unit covers algebraic fractions, simplification, multiplication and division of algebraic fractions, equations reducible to quadratic form, exponential equations, reciprocal equations, and practical applications.
Exercise wise Solutions of Unit 5
Quick Overview of Unit 5 Class 10 Math New Book Solutions
What Is Unit 5 Class 10 Math New Book About?
Unit 5 is titled Algebraic Fractions. This unit extends the idea of ordinary fractions to algebraic expressions containing variables.
An algebraic fraction is a fraction in which the numerator, denominator, or both contain algebraic expressions.
Example:
\[
\frac{x+2}{x-3}
\]
Students first learn how to reduce algebraic fractions to their simplest form by factorizing the numerator and denominator and cancelling common factors.
After that, they learn addition and subtraction of algebraic fractions. These questions require students to find a common denominator before combining the fractions.
The unit then explains multiplication and division of algebraic fractions. Factorization plays an important role because many expressions become easier after common factors are cancelled.
A major part of this chapter focuses on equations reducible to quadratic form. Students use substitution techniques to convert complicated equations into quadratic equations and then solve them by factorization or formula methods.
The final exercise applies algebraic fractions to real-life situations involving travelling speed, work rates, pipes filling tanks, and similar practical problems.
Solutions of Exercise 5.1 Class 10 Math New Book
Solutions of Exercise 5.1 Class 10 Math New Book are available above in PDF format. This exercise introduces reducing algebraic fractions to their lowest form through factorization and cancellation.
Main Topics Covered
- Factorization of numerator and denominator
- Cancellation of common factors
- Difference of squares
- Sum and difference of cubes
- Rational expressions in lowest form
Reducing Algebraic Fractions
Just like ordinary fractions, algebraic fractions can be simplified by cancelling common factors.
Example:
\[
\frac{x^2-5x}{x^2-4x-5}
\]
First factorize both expressions:
\[
\frac{x(x-5)}{(x-5)(x+1)}
\]
Cancel the common factor:
\[
\frac{x}{x+1}
\]
Important Rule
Only common factors can be cancelled.
Terms connected by addition or subtraction cannot be cancelled directly.
Why Exercise 5.1 Is Important
Exercise 5.1 forms the foundation of the entire chapter. Students who understand factorization properly will find the remaining exercises much easier.
Solutions of Exercise 5.2 Class 10 Math New Book
Solutions of Exercise 5.2 Class 10 Math New Book are available above in PDF format. This exercise explains simplification of algebraic fractions involving addition and subtraction.
Main Topics Covered
- Finding LCM of algebraic denominators
- Addition of algebraic fractions
- Subtraction of algebraic fractions
- Simplification after combining fractions
- Factorization and cancellation
Simplifying Algebraic Fractions
Before adding or subtracting algebraic fractions, students must find a common denominator.
Example:
\[
\frac{3}{x-y}+\frac{1}{y-x}
\]
Since:
\[
y-x=-(x-y)
\]
we have:
\[\frac{1}{y-x}-\frac{1}{x-y}
\]
Therefore:
\[ \frac{3}{x-y}+\frac{1}{y-x}
=\frac{3}{x-y}-\frac{1}{x-y}
=\frac{2}{x-y}
\]
Common Denominator Method
When denominators are different, students must find the least common denominator and rewrite each fraction before combining them.
Why Exercise 5.2 Is Important
Many examination questions require students to simplify long algebraic expressions. Strong skills in finding common denominators help students solve such questions accurately.
Solutions of Exercise 5.3 Class 10 Math New Book
Solutions of Exercise 5.3 Class 10 Math New Book are available above in PDF format. This exercise focuses on multiplication and division of algebraic fractions.
Main Topics Covered
- Multiplication of algebraic fractions
- Division of algebraic fractions
- Reciprocal method
- Factorization before multiplication
- Cancellation of common factors
Multiplication of Algebraic Fractions
To multiply algebraic fractions, multiply the numerators together and multiply the denominators together.
Example:
\[
\frac{a^2-4b^2}{a^2+2ab}
\times
\frac{2a^2+10ab}{a^2+3ab-10b^2}
\]
After factorization and cancellation, the expression becomes much simpler.
Division of Algebraic Fractions
Division is performed by multiplying by the reciprocal.
Example: \[
\frac{A}{B}
\div
\frac{C}{D}
=
\frac{A}{B}
\times
\frac{D}{C}
=
\frac{AD}{BC}
\]
Students should always factorize first before cancelling common factors.
Reciprocal Method
The reciprocal method is one of the most important skills in this exercise. A small mistake while taking the reciprocal can change the entire answer.
Why Exercise 5.3 Is Important
This exercise combines factorization and fraction operations. It prepares students for more advanced algebraic questions and provides important practice for examinations.
Solutions of Exercise 5.4 Class 10 Math New Book
Solutions of Exercise 5.4 Class 10 Math New Book are available above in PDF format. This exercise introduces equations reducible to quadratic form, exponential equations, and reciprocal equations.
Main Topics Covered
- Equations reducible to quadratic form
- Substitution method
- Exponential equations
- Reciprocal equations
- Factorization
- Quadratic formula
Equations Reducible to Quadratic Form
Some equations appear complicated because the variable has a higher power. However, they can be converted into quadratic equations by substitution.
Example:
\[
5x^4-19x^2+12=0
\]
Let:
\[
y=x^2
\]
Then:
\[
5y^2-19y+12=0
\]
After solving for y, substitute back to find the values of x.
Exponential Equations
An exponential equation contains the variable in the exponent.
Example:
\[
3^x+3^{3-x}-12=0
\]
Such questions are solved by introducing a new variable and reducing the equation to quadratic form.
Reciprocal Equations
Some questions contain expressions such as:
\[
x+\frac1x
\]
or
\[
x^2+\frac1{x^2}
\]
Students use substitution to convert these equations into a simpler quadratic equation.
Why Exercise 5.4 Is Important
Exercise 5.4 combines algebraic techniques learned in previous chapters. Students must understand substitution, factorization, and quadratic equations to solve these questions successfully.
Solutions of Exercise 5.5 Class 10 Math New Book
Solutions of Exercise 5.5 Class 10 Math New Book are available above in PDF format. This exercise applies algebraic fractions to real-life situations involving speed, work, time, and rates.
Main Topics Covered
- Speed and distance problems
- Work and time problems
- Pipe and cistern questions
- Fractional equations
- Application of algebraic fractions
Speed Problems
Students form equations using:
\[
\text{Speed}=\frac{\text{Distance}}{\text{Time}}
\]
Example situations include trains, vehicles, and travelling at different speeds.
Work Problems
Work-rate questions use the idea that:
\[
\text{Work Rate}=\frac1{\text{Time}}
\]
If two people work together, their rates are added.
Example:
\[
\frac16+\frac1x=\frac14
\]
Pipe Problems
Pipe questions use the same principle as work-rate questions.
A filling pipe contributes positive work, while a draining pipe contributes negative work.
Students learn how to calculate the total time required to fill a tank when multiple pipes are involved.
Why Exercise 5.5 Is Important
This exercise shows how algebra is used in real-life situations. Many students find word problems challenging, so regular practice is necessary.
Solutions of Review Exercise 5 Class 10 Math New Book
Solutions of Review Exercise 5 Class 10 Math New Book are available above in PDF format. This review exercise combines all major concepts from Unit 5.
Topics Included
- MCQs
- Reducing algebraic fractions
- Simplification
- Multiplication and division
- Exponential equations
- Reciprocal equations
- Equations reducible to quadratic form
- Application problems
Complete Revision
Review Exercise 5 is designed to help students revise the entire chapter before examinations.
Students should attempt this exercise only after completing Exercise 5.1 to Exercise 5.5.
Exam Preparation Value
The review exercise contains a mixture of conceptual and calculation-based questions. It is very useful for final revision and self-assessment.
Important Formulas of Unit 5
Algebraic Fraction
\(\frac{P(x)}{Q(x)} where, Q(x)\neq0\)
Difference of Squares
\[
a^2-b^2=(a-b)(a+b)
\]
Sum of Cubes
\[
a^3+b^3=(a+b)(a^2-ab+b^2)
\]
Difference of Cubes
\[
a^3-b^3=(a-b)(a^2+ab+b^2)
\]
Multiplication of Algebraic Fractions
\[
\frac ab\times\frac cd=\frac{ac}{bd}
\]
Division of Algebraic Fractions
\[
\frac ab\div\frac cd=\frac ab\times\frac dc
\]
Speed Formula
\[
\text{Speed}=\frac{\text{Distance}}{\text{Time}}
\]
Work Formula
\[
\text{Work Rate}=\frac1{\text{Time}}
\]
Quadratic Formula
\[
x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
\]
Substitution Method
If: \( x^4 \)
appears in an equation, use: \( y=x^2\)
to reduce the equation to quadratic form.
Common Mistakes Students Make in Unit 5
Cancelling Terms Instead of Factors
Students often cancel terms connected by addition or subtraction.
Incorrect:
\[
\frac{x+2}{x}
\]
The x cannot be cancelled.
Only common factors may be cancelled.
Not Factorizing Completely
Many questions become easy only after complete factorization.
Students should always factorize the numerator and denominator before simplifying.
Using the Wrong Common Denominator
In Exercise 5.2, students sometimes combine fractions without finding a common denominator first.
Forgetting the Reciprocal in Division
When dividing algebraic fractions, the second fraction must be inverted before multiplication.
Errors in Substitution
In Exercise 5.4, students sometimes forget to replace the substituted variable back with x.
Ignoring Restrictions
Values that make a denominator equal to zero are not allowed.
Students should keep these restrictions in mind throughout the chapter.
Misreading Word Problems
In Exercise 5.5, students often choose the wrong variable or equation.
Read the question carefully before forming the equation.
How to Prepare Unit 5 for Exams
Start with Exercise 5.1 and master factorization techniques.
After that, practise Exercise 5.2 until finding common denominators becomes easy.
Exercise 5.3 should be practised carefully because multiplication and division questions frequently appear in examinations.
Exercise 5.4 is one of the most important exercises of the chapter. Students should revise substitution methods, quadratic equations, and factorization before attempting these questions.
Exercise 5.5 requires careful reading and equation formation. Practise speed, work, and pipe problems several times.
Finally, solve Review Exercise 5 without looking at the solutions. This will help identify weak areas before examinations.
FAQs About Unit 5 Class 10 Math New Book Solutions
What is Unit 5 of Class 10 Math New Book about?
Unit 5 is about Algebraic Fractions. It includes simplification, multiplication, division, equations reducible to quadratic form, exponential equations, reciprocal equations, and practical applications.
How many exercises are included in Unit 5?
Unit 5 includes Exercise 5.1, Exercise 5.2, Exercise 5.3, Exercise 5.4, Exercise 5.5, and Review Exercise 5.
Which exercise is about reducing algebraic fractions?
Exercise 5.1 focuses on reducing algebraic fractions to their lowest form.
Which exercise is about simplification of algebraic fractions?
Exercise 5.2 explains simplification using common denominators.
Which exercise covers multiplication and division?
Exercise 5.3 covers multiplication and division of algebraic fractions.
Which exercise contains exponential equations?
Exercise 5.4 contains exponential equations, reciprocal equations, and equations reducible to quadratic form.
Which exercise contains word problems?
Exercise 5.5 contains practical problems involving speed, work, time, and pipes.
Why is factorization important in Unit 5?
Factorization is used in almost every exercise. It helps simplify fractions, solve equations, and reduce expressions to their lowest form.
What is an algebraic fraction?
An algebraic fraction is a fraction that contains algebraic expressions in the numerator, denominator, or both.
Which exercise is most important for exams?
Exercise 5.4 and Review Exercise 5 are especially important because they contain higher-level algebraic techniques and mixed revision questions.
Disclaimer
These Unit 5 Class 10 Math New Book Solutions are prepared for educational, learning, and revision purposes. Every effort has been made to provide correct, clear, and easy solutions.
Students should also consult their official mathematics textbook and follow the instructions of their teachers. Notes of Math is not affiliated with the textbook publisher or any educational board.
Final Words
Unit 5 is one of the most important algebra chapters in the new Class 10 Mathematics book. It combines factorization, fractions, equation solving, and real-life applications into a single unit.
Students who understand algebraic fractions properly will find many later mathematics topics easier. The key to success in this chapter is regular practice of factorization and simplification techniques.
Exercise 5.1 to Exercise 5.3 build the basic skills needed for algebraic fractions. Exercise 5.4 introduces more advanced equations, while Exercise 5.5 shows how mathematics can be applied to practical situations involving speed, work, and rates.
Students should practise every exercise carefully, revise the important formulas regularly, and complete Review Exercise 5 before examinations. The PDF solutions provided above can be used for step-by-step learning, homework support, revision, and exam preparation.