Download Updated NCERT Solutions for Class 10 Maths – Free PDF for Better Exam Preparation NCERT Solutions for Math Class 10

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NCERT Solutions for Class 10 Maths (Chapters 1–15)

We provide NCERT Solutions for all the exercises from Chapters 1 to 15. These solutions are created by expert teachers to help students prepare for their exams. Students can download the chapter-wise PDFs to better understand how to solve problems.

Why NCERT Solutions are Helpful

The answers in the NCERT books are great study material for students. These solutions will help you understand the concepts more deeply. By practicing the questions in the textbook, you can check how well you're prepared and what areas you need to improve. The solutions will also help you quickly clear any doubts.

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NCERT Solutions for Class 10 Maths: Free PDF Download

The NCERT Solutions for Class 10 Maths include answers to all the questions in the NCERT book. These answers are written clearly and concisely, sticking closely to the textbook content. You can use these solutions as extra study material to help with your preparation. Practicing these solutions will certainly improve your exam readiness.

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Details of NCERT Solutions for Class 10 Maths Chapters

Chapter 1: Real Numbers

In this chapter, students will learn about real numbers and irrational numbers. The chapter begins with Euclid’s Division Lemma, which says:

• “For any two positive integers $a$ and $b$, there are unique integers $q$ (quotient) and $r$ (remainder) such that:
  $a = bq + r$
  where $0 \leq r < b$.”

Euclid’s Division Algorithm is based on this lemma and is used to calculate the HCF (Highest Common Factor) of two integers.

Next, the Fundamental Theorem of Arithmetic is explained, which helps to find the LCM (Least Common Multiple) and HCF of two numbers.

The chapter also covers irrational numbers (numbers that cannot be written as fractions, like $\sqrt{2}, \sqrt{3}, \sqrt{5}$) and rational numbers (numbers that can be written as fractions). It explains the decimal expansion of rational numbers as well.

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Topics Covered in Chapter 1: Real Numbers

• Fundamental Theorem of Arithmetic
  This is explained with examples and proofs, including the irrationality of numbers like $\sqrt{2}, \sqrt{3}, \sqrt{5}$.

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Steps to Find the HCF of Two Numbers (Using Euclid’s Division Lemma)

1. Step 1:
   Apply Euclid’s Division Lemma to the two numbers $c$ and $d$ (where $c > d$). This gives us integers $q$ and $r$ such that:
   $c = dq + r$
   where $0 \leq r < d$.

2. Step 2:
   If $r = 0$, then $d$ is the HCF of $c$ and $d$.
   If $r \neq 0$, repeat the process with $d$ and $r$.

3. Step 3:
   Continue until the remainder is 0. The divisor at that point will be the HCF.
   This works because:
   $HCF(c, d) = HCF(d, r)$

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NCERT Solutions for Class 10 Maths Chapter 2: Polynomials

In this chapter, you will learn about polynomials, which are algebraic expressions made up of terms with powers of variables. The chapter starts by explaining the degree of a polynomial and the types of polynomials:

• Linear Polynomial: Degree 1 (e.g., $ax + b$)
• Quadratic Polynomial: Degree 2 (e.g., $ax^2 + bx + c$)
• Cubic Polynomial: Degree 3 (e.g., $ax^3 + bx^2 + cx + d$)

The chapter has four exercises, including an optional one:

• Exercise 2.1: Focuses on finding the zeroes of a polynomial using graphs. You will learn the geometric meaning of the zeroes.
• Exercise 2.2: Deals with the relationship between the zeroes and coefficients of a polynomial. You will practice finding the zeroes of a quadratic polynomial and, in some cases, reconstructing the polynomial.
• Exercise 2.3: Introduces the division algorithm and requires you to apply it to polynomials.
• Exercise 2.4: Optional exercise that includes a variety of questions from the chapter.

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Topics Covered in Chapter 2: Polynomials

• Zeroes of a polynomial.
• Relationship between zeroes and coefficients of quadratic polynomials.

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Important Steps for Dividing Polynomials (Division Algorithm)

1. Step 1:
   Arrange the terms of both the dividend and divisor in decreasing order of their degrees (i.e., highest to lowest power of the variable). This is called writing the polynomials in standard form.

2. Step 2:
   To find the first term of the quotient, divide the highest degree term of the dividend by the highest degree term of the divisor. Then, proceed with the division process.

3. Step 3:
   Now, to find the next term in the quotient, divide the highest degree term of the new dividend by the highest degree term of the divisor, and continue the division.

4. Step 4:
   Eventually, the degree of the remainder will be lower than the degree of the divisor, and the division will stop.

   This method is similar to Euclid’s Division Algorithm you learned in Chapter 1, where we have:
   $\text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder}$

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NCERT Solutions for Class 10 Maths Chapter 3: Pair of Linear Equations in Two Variables

This chapter focuses on solving pairs of linear equations with two variables (x and y). There are 7 exercises, where different methods of solving these equations are introduced:

• Exercise 3.1: Explains how to represent situations using algebraic and graphical methods.
• Exercise 3.2: Shows how to solve linear equations using the Graphical Method.
• Exercise 3.3, 3.4, 3.5, 3.6: These exercises explain various Algebraic Methods, such as:
  • Substitution Method
  • Elimination Method
  • Cross-Multiplication Method
• Exercise 3.7: Optional exercise that covers all the methods and types of questions from the chapter.

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Topics Covered in Chapter 3: Pair of Linear Equations in Two Variables

• Pair of linear equations in two variables.
• Graphical Method of finding the solution of these equations.
• Consistency/Inconsistency of the system of equations (whether a solution exists or not).
• Algebraic conditions for the number of solutions.
• Solving the system of linear equations using:
  • Substitution Method
  • Elimination Method
  • Cross-Multiplication Method
• Solving simple situational problems involving linear equations.

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Important Formula

The general form of a pair of linear equations in two variables, $x$ and $y$, is:

$$a_1x + b_1y + c_1 = 0$$ $$a_2x + b_2y + c_2 = 0$$

Where $a_1, b_1, c_1, a_2, b_2, c_2$ are real numbers and $a_1^2 + b_1^2 \neq 0$, $a_2^2 + b_2^2 \neq 0$.

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Here’s the article rewritten in simpler language:

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NCERT Solutions for Class 10 Maths Chapter 4: Quadratic Equations

In this chapter, you will learn about quadratic equations and their standard form, which is:

$$ax^2 + bx + c = 0$$

Where $a \neq 0$.

You will study two methods to solve quadratic equations:

1. Factorization Method – Breaking the equation into factors to find the solution.
2. Completing the Square Method – A method to solve the equation by making a perfect square.

The chapter also covers how to determine the nature of the roots of the quadratic equation. Based on the discriminant ($b^2 - 4ac$), we can say:

• Two distinct real roots: if $b^2 - 4ac > 0$
• Two equal roots: if $b^2 - 4ac = 0$
• No real roots: if $b^2 - 4ac < 0$

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Topics Covered in Chapter 4: Quadratic Equations

• The standard form of a quadratic equation.
• Solving quadratic equations by factorization and using the quadratic formula.
• The relationship between the discriminant and the nature of the roots.
• Solving situational problems based on quadratic equations related to real-life scenarios.

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NCERT Solutions for Class 10 Maths Chapter 5: Arithmetic Progressions

This chapter introduces Arithmetic Progressions (AP), which is a sequence of numbers where the difference between each term is constant. The chapter includes four exercises:

• Exercise 5.1: Focuses on representing real-life situations as an AP, finding the first term and common difference, and checking if a sequence is an AP.
• Exercise 5.2: Teaches how to find the nth term of an AP using the formula: $$a_n = a + (n-1) d$$ where $a$ is the first term and $d$ is the common difference.
• Exercise 5.3: Deals with finding the sum of the first $n$ terms of an AP.
• Exercise 5.4: Contains higher-level problems to improve your understanding and problem-solving skills related to AP.

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Topics Covered in Chapter 5: Arithmetic Progressions

• Why we study Arithmetic Progressions.
• How to derive the nth term and sum of the first $n$ terms of an AP.
• Solving real-life problems using the formulas for AP.

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Important Formulas

• If the terms of an AP are:
  $a_1, a_2, a_3, a_4, a_5, a_6, \dots$, with the common difference $d$, the sequence is:
  $a, a+d, a+2d, a+3d, a+4d, \dots$, where $a$ is the first term.

• The nth term of an AP is:

  $$a_n = a + (n-1) d$$

• The sum of the first $n$ terms of an AP is:

  $$S_n = \frac{n}{2} [2a + (n-1) d]$$

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Here’s the article rewritten in simpler language:

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NCERT Solutions for Class 10 Maths Chapter 6: Triangles

In Chapter 6, students learn about similar triangles, which are triangles that have the same shape but may differ in size. This chapter introduces the concepts of similar and congruent figures and provides the conditions for when two triangles are similar. It also covers the areas of similar triangles using a theorem and explains the Pythagoras Theorem and its converse.

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Topics Covered in Chapter 6: Triangles

• Similar Triangles: Understanding what makes two triangles similar.
• Key Theorems and Conditions:
  1. Theorem 1: If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio.
  2. Theorem 2: If a line divides two sides of a triangle in the same ratio, the line is parallel to the third side.
  3. Theorem 3: If two triangles have equal corresponding angles and their sides are in proportion, the triangles are similar.
  4. Theorem 4: If the sides of two triangles are proportional, their corresponding angles are equal, making the triangles similar.
  5. Theorem 5: If one angle of a triangle equals one angle of another triangle and the sides forming these angles are proportional, the triangles are similar.

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Important Theorems

• Theorem 6.1: If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio.
• Theorem 6.2: If a line divides two sides of a triangle in the same ratio, it is parallel to the third side.
• Theorem 6.3: If in two triangles, the corresponding angles are equal, the triangles are similar, and their corresponding sides are proportional.
• Theorem 6.4: If the corresponding sides of two triangles are proportional, their corresponding angles are equal, and the triangles are similar.
• Theorem 6.5: If one angle of a triangle is equal to one angle of another triangle and the sides, including these angles, are proportional, then the two triangles are similar.
• Theorem 6.6: The ratio of the areas of two similar triangles is the square of the ratio of their corresponding sides.
• Theorem 6.7: In a right triangle, if a perpendicular is drawn from the right angle to the hypotenuse, the triangles on both sides of the perpendicular are similar to the whole triangle and to each other.
• Theorem 6.8: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. (This is the Pythagoras Theorem.)
• Theorem 6.9: In a triangle, if the square of one side equals the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

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NCERT Solutions for Class 10 Maths Chapter 7: Coordinate Geometry

In this chapter, students learn how to:

• Find the distance between two points using their coordinates.
• Find the area of a triangle formed by three points.
• Find the coordinates of a point that divides a line segment joining two points in a specific ratio.

Students are introduced to the Distance Formula, Section Formula, and how to calculate the Area of a Triangle using coordinates.

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NCERT Solutions for Class 10 Maths Chapter 8: Introduction to Trigonometry

This chapter introduces Trigonometry, which deals with the relationships between the sides and angles of right-angled triangles. Students will study:

• Trigonometric ratios for the angles of a right triangle.
• The ratios for angles of 0° and 90°.
• Trigonometric ratios for special angles like 30°, 45°, and 60°.
• Trigonometric identities, such as sin²A + cos²A = 1.

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NCERT Solutions for Class 10 Maths Chapter 9: Some Applications of Trigonometry

In this chapter, students learn how Trigonometry is used in real-life applications, such as:

• Geography, navigation, and map-making.
• Determining the position of objects using angles and distances.

Students will also learn how trigonometry helps find the height and distance of objects without physically measuring them. Key terms introduced include:

• Line of sight: The straight line from the observer’s eye to the point being viewed.
• Angle of elevation: The angle formed when looking up at an object.
• Angle of depression: The angle formed when looking down at an object.

The chapter includes simple problems involving angles of elevation and depression of 30°, 45°, and 60°.

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NCERT Solutions for Class 10 Maths Chapter 10: Circles

In this chapter, students will revisit the concept of circles and study the relationships between a circle and a line. They will focus on:

• Tangents to a Circle: A tangent is a line that touches the circle at exactly one point.
• Number of Tangents from a Point on a Circle:
  1. No tangents pass through a point inside the circle.
  2. One tangent touches the circle at a point on the circle itself.
  3. Two tangents can be drawn from a point outside the circle.

Important Theorems:

1. The tangent at any point of a circle is perpendicular to the radius at that point.
2. The lengths of tangents drawn from an external point to a circle are equal.

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NCERT Solutions for Class 10 Maths Chapter 11: Constructions

This chapter focuses on geometric constructions. Students will learn how to:

• Divide a line segment in a given ratio.
• Construct tangents to a circle from a point outside the circle.
• Construct a triangle similar to a given triangle.

Important Points:

1. Construction 11.1: Dividing a line segment in a given ratio.
2. Construction 11.2: Constructing a triangle similar to another triangle with a given scale factor.
3. Construction 11.3: Drawing tangents to a circle from a point outside the circle.

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NCERT Solutions for Class 10 Maths Chapter 12: Areas Related to Circles

In this chapter, students will learn about the area and perimeter of a circle. The chapter also explains how to calculate the area of sectors and segments of a circle. Students will also learn how to find the areas of combinations of plane figures that include circles or parts of circles.

Important Formulas:

1. Circumference of a circle = $2\pi r$
2. Area of a circle = $\pi r^2$
3. Area of a sector (with angle θ) = $\frac{\theta}{360} \times \pi r^2$
4. Length of an arc (with angle θ) = $\frac{\theta}{360} \times 2\pi r$

In this chapter, the central angle for calculating the area of a segment is limited to 60°, 90°, and 120°.

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NCERT Solutions for Class 10 Maths Chapter 13: Surface Areas and Volumes

This chapter focuses on surface areas and volumes of solids, particularly when combining different 3D shapes like cuboids, cones, cylinders, spheres, and hemispheres. There are 5 exercises, covering:

• Surface area of objects formed by combining basic solids.
• Volume of objects formed by combining basic solids.
• Converting one solid into another.
• Finding the surface area and volume of a frustum of a cone.
• High-level questions based on all topics of the chapter.

Important Formulas:

1. Surface area of a sphere = $4\pi r^2$
2. Volume of a sphere = $\frac{4}{3}\pi r^3$
3. Curved surface area (CSA) of a cylinder = $2\pi rh$
4. Total surface area (TSA) of a cylinder = $2\pi r (h + r)$
5. Volume of a cylinder = $\pi r^2 h$
6. Curved surface area of a cone = $\pi rl$ (where $l$ is the slant height)
7. Volume of a cone = $\frac{1}{3}\pi r^2 h$
8. Surface area of a cuboid = $2(l \times b + b \times h + l \times h)$
9. Volume of a cuboid = $l \times b \times h$

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NCERT Solutions for Class 10 Maths Chapter 14: Statistics

In this chapter, students will learn how to represent data numerically, starting from ungrouped data to grouped data. They will also learn how to find the mean, mode, and median of grouped data. The chapter also covers cumulative frequency, cumulative frequency distribution, and how to draw cumulative frequency curves.

Key Topics:

• Finding Mean, Median, and Mode for grouped data (avoiding the bimodal situation).

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NCERT Solutions for Class 10 Maths Chapter 15: Probability

The final chapter introduces the concept of Probability. It begins with the theoretical approach to probability and then explains the difference between experimental probability (based on actual experiments) and theoretical probability (based on calculations or theory). The chapter provides examples to help explain these concepts before students start solving exercise problems.

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Benefits of NCERT Solutions for Class 10 Maths

The NCERT Solutions for Class 10 Maths available in PDFs have many benefits, including:

• Easy to understand: The solutions are simple and clear.
• Step-by-step guidance: Each solution is broken down into easy steps for better understanding.
• Visual aids: Diagrams are included to help students visualize the solutions.
• Complete coverage: All questions from each chapter are covered.
• In-depth learning: By preparing all the chapters, students will deepen their understanding of the concepts. It is important to follow each step in the solutions for effective learning.

How NCERT Solutions for Class 10 Maths Help with Board Exams

Mathematics is an important subject for Class 10 students, and preparing well is crucial for success. The NCERT Class 10 Maths Solutions offer complete support to help students prepare effectively for their exams.

Class 10 marks a key milestone in a student’s academic journey, and good performance here will help set a strong foundation for future achievements. The NCERT books are recommended by CBSE for exam preparation, as they present concepts in a simple and easy-to-understand way.

The preparation process for exams requires a deep understanding of each chapter. NCERT Solutions play a major role in breaking down complex problems and guiding students step by step. They are also very helpful in preparing for competitive exams later on.

To get all the chapter-wise NCERT Solutions for Class 10 Maths in PDF format, you can download them for free. Students can also access NCERT Solutions for Class 10 Science on https://www.learnscbse.in/. For a more personalized learning experience, students can download the https://www.learnscbse.in/ App and prepare more effectively for exams.

Frequently Asked Questions on NCERT Solutions for Class 10 Maths

Q1: How can I understand the important concepts in the NCERT Solutions for Class 10 Maths?

To score well in Class 10 exams, students should download the NCERT Solutions. These solutions are created by experienced teachers and explain every detail in a clear, interactive way. The solutions are organized step-by-step, keeping in mind the marks weightage based on CBSE guidelines.

Q2: Will NCERT Solutions for Class 10 Maths help me solve problems easily?

Class 10 exams are an important milestone for students, and Maths is a subject with many numerical problems. Understanding the concepts will help you score better. It’s recommended to learn the syllabus and understand the concepts first. Then, solving textbook problems with the help of the solutions will improve your problem-solving and analytical skills.

Q3: Where can I find the NCERT Solutions for Class 10 Maths PDF?

You can find the NCERT Solutions for Class 10 Maths PDF online. The solutions are created by experts and are designed to help students understand complex concepts easily. You can use them to check your answers and learn other methods to solve problems.

Q4: Are NCERT Solutions for Class 10 Maths helpful for CBSE students?

Here are some benefits of using NCERT Maths Solutions for Class 10:

1. It builds a strong understanding of the basic concepts and boosts confidence for exams.
2. You can easily learn how to solve complex problems.
3. It’s a great study resource to complete assignments and score well in exams.

Q5: How can I understand the important concepts in the NCERT Solutions for Class 10 Maths?

Before starting the year, students should download the CBSE syllabus to get an idea of the important topics. Then, practice the NCERT textbook problems using the solutions. Regular practice will help you improve your problem-solving skills.

Q6: Are NCERT Solutions for Class 10 Maths useful?

Yes, the NCERT Solutions are designed to help students understand concepts easily and clear any doubts. The solutions are detailed, making the subject more engaging. The PDF can be downloaded and used to strengthen your knowledge.

Q7: Are NCERT Solutions for Class 10 Maths the best reference guide for CBSE students?

There are many study materials available, but the NCERT Solutions are considered one of the best for CBSE students. The solutions are organized by chapter and exercise, so you can download and use them anytime without any limitations.

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