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NCERT solutions for class 9 maths are available for free. Find comprehensive explanations and step-by-step solutions to all the chapters of class 9 maths to help you study better.

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NCERT Solutions for Class 9 Maths offer comprehensive, step-by-step answers to all textbook exercises. These solutions help students grasp complex concepts and improve problem-solving skills. Expert-recommended, they serve as essential tools for mastering topics like algebra, geometry, and trigonometry. With clear explanations, these solutions ensure a solid foundation for future learning.

Sample Questions and Answers for Class 9 Maths

What is a polynomial? A polynomial is an algebraic expression made up of terms that involve variables raised to non-negative integer powers. For example, $2x^2 + 3x + 4$ is a polynomial, where the highest exponent of the variable is 2.
How do you solve a linear equation in one variable? To solve a linear equation in one variable, you simplify both sides by isolating the variable. For example, in the equation $3 x + 5 = 11$ , subtract 5 from both sides to get $3 x = 6$ , then divide both sides by 3, which gives $x = 2$ .
What is the formula to find the area of a triangle? The area of a triangle can be found using the formula: $12×base×height\frac{1}{2} \times \text{base} \times \text{height}$ .

How do you find the HCF of two numbers? To find the HCF (Highest Common Factor) of two numbers, list their factors and choose the greatest one common to both. For example, the HCF of 12 and 15 is 3 because it is the largest number that divides both 12 and 15.
What is a rational number? A rational number is any number that can be expressed as the ratio of two integers, where the denominator is not zero. For example, $23\frac{2}{3}$ is a rational number.
Explain how to add fractions with unlike denominators. To add fractions with unlike denominators, first find the least common denominator (LCD). Then, convert both fractions to equivalent fractions with this LCD and add the numerators. For example, to add $13\frac{1}{3}$ and $14\frac{1}{4}$ , the LCD is 12. So, $13=412\frac{1}{3} = \frac{4}{12}$ and $14=312\frac{1}{4} = \frac{3}{12}$ , and the sum is $412+312=712\frac{4}{12} + \frac{3}{12} = \frac{7}{12}$ .
What is the difference between surface area and volume of a cylinder? The surface area of a cylinder is the total area that covers the outside of the cylinder, including the two circular ends and the curved surface. The volume is the amount of space inside the cylinder. The surface area formula is $2πr(r+h)2\pi r(r + h)$ and the volume formula is $πr2h\pi r^2 h$ .
How do you simplify an algebraic expression? To simplify an algebraic expression, combine like terms (terms with the same variable raised to the same power). For example, in the expression $3 x + 4 x - 2 y$ , combine the $x$ -terms to get $7 x - 2 y$ .
What is a geometric progression? A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, in the sequence $\dots$ , the common ratio is 3.

How do you find the perimeter of a rectangle? The perimeter of a rectangle is the sum of all its sides. The formula is $\times (\text{length} + \text{breadth})$ . For example, if the length is 5 cm and the breadth is 3 cm, the perimeter is $\times (5 + 3) = 16 \, \text{cm}$ .
What is the formula for the volume of a sphere? The formula for the volume of a sphere is $43πr3\frac{4}{3} \pi r^3$ , where $r$ is the radius of the sphere.
Explain the difference between mean, median, and mode. The mean is the average of a set of numbers, found by dividing the sum of all numbers by the count. The median is the middle value when the numbers are arranged in order. The mode is the number that appears most frequently.
How do you subtract algebraic expressions? To subtract algebraic expressions, remove parentheses and change the signs of the terms being subtracted, then combine like terms. For example, subtracting $(3 x + 4) - (2 x - 5)$ results in $3 x + 4 - 2 x + 5 = x + 9$ .
What is a right-angled triangle? A right-angled triangle is a triangle that has one angle equal to 90 degrees. The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs.
How do you calculate the simple interest? The formula for simple interest is $SI=P×R×T100\text{SI} = \frac{P \times R \times T}{100}$ , where $P$ is the principal, $R$ is the rate of interest, and $T$ is the time period.
What is a composite number? A composite number is a positive integer that has more than two factors. For example, 6 is a composite number because it has four factors: 1, 2, 3, and 6.
How do you calculate the probability of an event? Probability is calculated as the ratio of favorable outcomes to the total number of possible outcomes. For example, if you roll a die, the probability of rolling a 4 is $16\frac{1}{6}$ , as there are 6 possible outcomes.
What is an irrational number? An irrational number is a number that cannot be written as a simple fraction. Its decimal form never repeats or terminates. For example, $π\pi$ is an irrational number.
How do you factorize a quadratic expression? To factorize a quadratic expression, you look for two numbers that multiply to give the constant term and add up to give the coefficient of the middle term. For example, to factorize $x^2 + 5x + 6$ , the numbers are 2 and 3, so the factorization is $(x + 2) (x + 3)$ .
What is the Pythagorean theorem? The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. The formula is $a^2 + b^2 = c^2$ , where $c$ is the hypotenuse.

NCERT Solutions for Class 9 Maths

1. Understanding Polynomials

A polynomial is an expression made up of variables and coefficients, involving sums, differences, and products. For example, the expression $x^2 + 3x + 2$ is a polynomial.

Sample Questions:

What is a polynomial?
Write an example of a polynomial of degree 2.
How can you factor the expression $x^2 + 5x + 6$ ?
What do we mean by the degree of a polynomial?
What is the value of $x$ when $x^2 - 4 = 0$ ?
Explain how to find the roots of a quadratic polynomial.
Solve for $x$ in the equation $x^2 + 7x + 12 = 0$ .
How can you determine if a polynomial is factorable?
Can a polynomial have negative exponents? Why or why not?
What is the remainder when $x^3 - 3x^2 + 2x - 5$ is divided by $x - 1$ ?

2. Linear Equations in Two Variables

A linear equation in two variables represents a straight line when plotted on a graph.

Sample Questions:

What is a linear equation in two variables?
Solve the system of equations: $x + y = 5$ and $2 x - y = 3$ .
What is the graphical representation of the equation $x - 3 y = 6$ ?
How can we find the slope of the line represented by the equation $2 x + 3 y = 12$ ?
What is the solution to the equation $4 x + 2 y = 10$ ?
Solve for $x$ and $y$ in the equations $3 x + 4 y = 7$ and $x - 2 y = 3$ .
Explain how to graph the equation $x + 2 y = 8$ .
What does it mean if two lines are parallel in the context of linear equations?
How do you solve the system of equations $x + y = 8$ and $3 x - y = 10$ ?
What is the point of intersection for the equations $2 x - y = 4$ and $x + y = 3$ ?

3. Coordinate Geometry

Coordinate geometry helps in analyzing and solving geometric problems using algebraic methods.

Sample Questions:

What is the distance formula in coordinate geometry?
Find the distance between the points $(3, 4)$ and $(7, 1)$ .
What does the midpoint formula calculate?
Calculate the midpoint of the points $(1, 2)$ and $(3, 8)$ .
How do you find the slope of the line joining two points?
Find the slope of the line joining $(1, 2)$ and $(3, 4)$ .
What is the equation of a line with slope 2 passing through the point $(1, 3)$ ?
What is the area of a triangle with vertices at $(0, 0)$ , $(4, 0)$ , and $(0, 3)$ ?
How do you determine if three points are collinear?
What is the equation of a line parallel to the line $y = 2 x + 3$ ?

4. Introduction to Euclid's Geometry

Euclidean geometry studies points, lines, and planes and their properties.

Sample Questions:

What is the difference between a line and a line segment?
Define a ray in Euclidean geometry.
How many points are needed to define a plane in Euclidean geometry?
What are the postulates of Euclid's geometry?
What does Euclid's first postulate state?
Explain Euclid's fifth postulate.
What is the definition of a parallel line?
If two angles are complementary, what is the sum of their measures?
Define a right angle.
How do you construct the perpendicular bisector of a line segment?

5. Introduction to Trigonometry

Trigonometry studies the relationships between the sides and angles of triangles.

Sample Questions:

What is the sine of an angle in a right triangle?
What is the cosine of $30∘30^\circ$ ?
Define the tangent of an angle.
How do you find the height of an object using trigonometry?
What is the formula for finding the length of the hypotenuse?
If $sin⁡θ=1/2\sin \theta = 1/2$ , what is the value of $θ\theta$ ?
What is the relationship between the sides and angles in a right-angled triangle?
How do you use trigonometry to calculate distances in real life?
What is the value of $cos⁡90∘\cos 90^\circ$ ?
Define the term 'angle of elevation.'

6. Areas of Parallelograms and Triangles

Understanding the calculation of areas in different shapes helps in many practical applications.

Sample Questions:

What is the area of a parallelogram?
Calculate the area of a triangle with a base of 5 units and a height of 3 units.
How do you find the area of a trapezium?
What is the formula for the area of a rectangle?
How do you calculate the area of a rhombus?
If the area of a triangle is 24 square units, and the base is 6 units, what is the height?
What is the area of a square with a side length of 4 units?
Explain how to calculate the area of a sector of a circle.
How do you find the area of a quadrilateral?
What is the area of a circle with a radius of 7 units?

7. Surface Areas and Volumes

This section helps in understanding the surface area and volume of three-dimensional shapes.

Sample Questions:

What is the formula for the surface area of a cube?
How do you find the volume of a cylinder?
Calculate the surface area of a cone with a radius of 3 units and a height of 5 units.
What is the volume of a sphere with a radius of 4 units?
How do you calculate the surface area of a sphere?
Find the volume of a rectangular prism with dimensions 3x4x5 units.
What is the surface area of a cylinder?
How do you calculate the volume of a cone?
What is the surface area of a pyramid with a square base?
How do you find the volume of a hemisphere?

Top Indian Books for Class 9 Maths with Author Details and Content Insights

Mathematics Class 9 by R.D. Sharma
Publisher: Dhanpat Rai
Content: This book covers all topics of Class 9 Maths in detail. It provides a solid theoretical understanding, followed by numerous practice problems to help reinforce learning. The content includes topics like algebra, geometry, trigonometry, and statistics.
Mathematics for Class 9 by R.S. Aggarwal
Publisher: S. Chand
Content: It is known for its systematic approach, where each chapter begins with theory and ends with various exercise sets, ensuring ample practice. The book is ideal for improving problem-solving techniques.
Understanding Mathematics Class 9 by R.S. Aggarwal
Publisher: S. Chand
Content: This book offers a blend of theory and problems, with the problems ranging from easy to challenging. It helps develop a clear conceptual understanding of key mathematical concepts.
NCERT Solutions for Class 9 Mathematics
Publisher: NCERT
Content: This book contains solutions to all the exercises from the NCERT Class 9 Mathematics textbook. It is helpful for students who need step-by-step solutions to problems and aids in preparing for exams.
Exemplar Problems in Mathematics Class 9
Publisher: NCERT
Content: This book focuses on problem-solving and critical thinking. It contains advanced questions to enhance logical reasoning and provides in-depth explanations for various concepts.
Mathematics Class 9 by S.K. Gupta
Publisher: Laxmi Publications
Content: The book provides a comprehensive set of problems to practice. The theory is well-explained, and the book is structured to prepare students for competitive exams and school-level tests.
Guide to Class 9 Mathematics by M.L. Aggarwal
Publisher: Arya Publishing
Content: M.L. Aggarwal’s guide is a must-have for thorough preparation. The book contains an extensive collection of practice questions, including objective, subjective, and application-based problems.
Trigonometry and Geometry Class 9 by A. R. Shastri
Publisher: Goyal Publishers
Content: Focusing on geometry and trigonometry, this book is great for building a strong foundation in these important mathematical areas, with detailed examples and exercises.
Mathematics Class 9 by S.K. Singh
Publisher: New Saraswati House
Content: The book offers a perfect mix of theory and practice questions that cater to students who prefer to build their skills gradually, moving from basic concepts to complex applications.
NCERT Mathematics Class 9
Publisher: NCERT
Content: The official NCERT textbook for Class 9 Mathematics covers essential topics and provides a clear understanding of the subject. It includes a variety of problems, ranging from simple to challenging, helping students develop conceptual clarity.
Mathematics Class 9 by M. R. Sethi
Publisher: Deep & Deep Publications
Content: Known for its practical problem-solving approach, this book focuses on all the important areas of Class 9 Maths with clear explanations and a large number of practice problems.
Mathematics for Class 9 by P.S. Verma
Publisher: Pearson
Content: This book aims at strengthening the mathematical foundation. It includes well-structured theory, solved examples, and exercises that help reinforce learning through consistent practice.
Practice Book on Mathematics Class 9 by A. K. Sharma
Publisher: New Age International
Content: It is a practical guide for practice and revision, with clear explanations of all chapters and enough exercises for mastering each concept.
Mathematics Class 9 by H.K. Dass
Publisher: Laxmi Publications
Content: The book focuses on building a conceptual understanding of maths and presents a range of problems that gradually build the student’s confidence in mathematics.
Mathematics for Class 9 by M.K. Soni
Publisher: Ramesh Publishing House
Content: This book contains exercises designed to help students refine their skills, with a focus on application-based problems, ensuring they are well-prepared for various exams.
Objective Mathematics Class 9 by R.D. Sharma
Publisher: Dhanpat Rai
Content: Known for its objective-style questions, this book is great for students looking for practice in MCQs, short-answer, and long-answer questions that are frequently asked in exams.
Mathematics Class 9 by V.K. Agarwal
Publisher: S. Chand
Content: This book combines theoretical explanations with a wide variety of problems. The exercises are progressively structured, making it suitable for both beginners and advanced learners.
Advance Problems in Mathematics Class 9 by V.K. Kapoor
Publisher: New Age International
Content: Designed to push students to higher levels of understanding, this book presents a variety of difficult problems, ideal for those looking to hone their problem-solving skills.
Mathematics Class 9 by K.C. Sinha
Publisher: Goyal Publishers
Content: The book includes both theory and exercises that are clearly explained, helping students to strengthen their conceptual foundation.
Maths Magic Class 9 by R. S. Aggarwal
Publisher: S. Chand
Content: This book offers an innovative approach to learning, with a focus on creative problem-solving techniques. The exercises range from basic to complex, offering a balanced learning experience.

NCERT Solutions for Class 9 Maths: A Complete Guide for Success

Class 9 Maths is an important milestone in a student's academic journey, and NCERT solutions serve as a comprehensive guide to mastering the subject. These solutions, based on the National Council of Educational Research and Training's textbook, provide step-by-step answers to all the exercises in the book. Whether you're looking to clarify concepts or practice problems, the NCERT solutions are the perfect companion for your studies.

The key to excelling in Class 9 Maths lies in understanding the fundamental concepts laid out in the NCERT textbook. From algebra to geometry, every topic is broken down methodically, ensuring students grasp even the most complex concepts. The solutions are designed to offer clarity, starting with easier problems and gradually progressing to more challenging ones.

One of the most significant advantages of NCERT solutions is that they provide detailed explanations for each problem. These explanations guide students through the logical steps required to solve each question. By following the methodical approach, students can understand the reasoning behind each answer and apply the same principles to similar problems.

Another crucial aspect of these solutions is the wide range of question types they cover. From multiple-choice questions (MCQs) to long-answer questions, the solutions include everything. The variety ensures that students are prepared for any question format they might encounter in their exams.

Additionally, NCERT solutions help improve problem-solving skills. With practice exercises, students can test their knowledge and enhance their ability to solve complex problems. The more problems a student practices, the better their chances of scoring well in the final exam.

For students aiming to perform well in competitive exams, the NCERT solutions also offer a solid foundation. The in-depth coverage of topics, along with the numerous examples and exercises, makes it an excellent resource for revision and preparation for higher-level exams.

Moreover, the use of NCERT solutions promotes self-study. Students can use the solutions independently to clear doubts and review concepts without the need for additional resources. This encourages students to take control of their learning, boosting confidence and autonomy.

NCERT solutions also emphasize the importance of regular revision. By consistently referring to the solutions and practicing regularly, students can solidify their understanding and retain important concepts.

The layout of the NCERT solutions is user-friendly. The solutions are easy to navigate, with clear headings, subheadings, and step-by-step breakdowns. This structure ensures that students can find what they need quickly and easily, making their study sessions more efficient.