Comprehensive NCERT Solutions for Class 10 Mathematics

Struggling with Class 10 Maths? Our NCERT Class 10 Maths Solutions 2025 offer clear, step-by-step explanations for every chapter, making complex problems easy to understand and solve.

Mathematics serves as the foundation for numerous academic and professional pursuits. For students in Class 10, mastering mathematical concepts is crucial not only for academic excellence but also for developing analytical and problem-solving skills. We present meticulously crafted solutions to the NCERT Class 10 Mathematics textbook, ensuring clarity and a deep understanding of each topic.

Chapter 1: Real Numbers

Real numbers encompass both rational and irrational numbers, forming the backbone of various mathematical concepts.

Key Topics:

• Euclid’s Division Lemma: A fundamental principle stating that for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b.
• Fundamental Theorem of Arithmetic: Every composite number can be expressed uniquely as a product of prime numbers, irrespective of the order of the factors.
• Revisiting Irrational Numbers: Understanding numbers that cannot be expressed as a simple fraction, such as √2 and π.
• Decimal Expansions of Rational Numbers: Exploring terminating and non-terminating repeating decimals.

Example Problem:

Problem: Prove that √3 is an irrational number.

Solution: Assume, to the contrary, that √3 is rational. Then it can be expressed in the form a/b, where a and b are coprime integers, and b ≠ 0. Squaring both sides, we get 3 = a²/b², implying a² = 3b². Therefore, a² is divisible by 3, which means a is also divisible by 3. Let a = 3k for some integer k. Substituting back, we get (3k)² = 3b², leading to 9k² = 3b², and thus b² = 3k². This implies b² is divisible by 3, and hence b is divisible by 3. This contradicts the assumption that a and b are coprime. Therefore, √3 is irrational.

Exercises Covered:

• Exercise 1.1 – Problems based on Euclid’s Division Lemma
• Exercise 1.2 – HCF and LCM using prime factorization
• Exercise 1.3 – Proving numbers as irrational
• Exercise 1.4 – Decimal expansions of rational numbers

Chapter 2: Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients, connected by addition, subtraction, and multiplication operations.

Key Topics:

• Geometrical Meaning of the Zeroes of a Polynomial: Understanding how the zeroes of a polynomial correspond to the x-intercepts of its graph.
• Relationship between Zeroes and Coefficients: For a quadratic polynomial ax² + bx + c, the sum and product of its zeroes can be derived from its coefficients.
• Division Algorithm for Polynomials: Similar to the division of integers, polynomials can be divided to obtain a quotient and a remainder.

Example Problem:

Problem: Find the zeroes of the quadratic polynomial 2x² – 8x + 6 and verify the relationship between the zeroes and the coefficients.

Solution: The given polynomial is 2x² – 8x + 6. Dividing the entire polynomial by 2, we get x² – 4x + 3. Factoring, we have (x – 1)(x – 3). Therefore, the zeroes are x = 1 and x = 3. The sum of the zeroes is 1 + 3 = 4, and the product is 1 × 3 = 3. Comparing with the standard form ax² + bx + c, we have a = 2, b = -8, and c = 6. The sum of the zeroes is -b/a = -(-8)/2 = 4, and the product is c/a = 6/2 = 3, verifying the relationship.

Exercises Covered:

• Exercise 2.1 – Finding zeros of polynomials
• Exercise 2.2 – Relationship between coefficients and zeros
• Exercise 2.3 – Polynomial division problems
• Exercise 2.4 – Advanced factorization problems

Chapter 3: Pair of Linear Equations in Two Variables

Linear equations in two variables represent straight lines in a coordinate plane. Understanding their solutions is essential for solving real-life problems involving two variables.

Key Topics:

• Graphical Method of Solution: Plotting two linear equations on a graph to find their point of intersection, representing the solution.
• Algebraic Methods of Solution: Including substitution, elimination, and cross-multiplication methods.
• Equations Reducible to Linear Form: Transforming certain nonlinear equations into linear form for easier solving.

Example Problem:

Problem: Solve the following pair of linear equations using the elimination method:

2x + 3y = 8
4x + 6y = 16

Solution: Observing the equations, we notice that the second equation is a multiple of the first (multiplying the first equation by 2 yields the second). This implies that the two lines are coincident, meaning they have infinitely many solutions.

Exercises Covered:

• Exercise 3.1 – Introduction to linear equations
• Exercise 3.2 – Graphical method of solving equations
• Exercise 3.3 – Substitution method
• Exercise 3.4 – Elimination method
• Exercise 3.5 – Cross-multiplication method
• Exercise 3.6 – Word problems on linear equations
• Exercise 3.7 – Advanced problems

Chapter 4: Quadratic Equations

Quadratic equations are polynomial equations of degree two and are fundamental in various mathematical analyses.

Key Topics:

• Standard Form of a Quadratic Equation: Expressed as ax² + bx + c = 0, where a, b, and c are real numbers, and a ≠ 0.
• Methods of Solving Quadratic Equations: Including factorization, completing the square, and the quadratic formula.
• Nature of Roots: Determined by the discriminant (b² – 4ac), indicating whether the roots are real and distinct, real and equal, or complex.

Example Problem:

Problem: Solve the quadratic equation x² – 5x + 6 = 0 by factorization.

Solution: The given equation is x² – 5x + 6 = 0. Factoring, we get (x – 2)(x – 4) = 0. Therefore, the solutions are x = 2 and x = 3.

Exercises Covered:

• Exercise 4.1 – Introduction to quadratic equations
• Exercise 4.2 – Solving equations by factorization
• Exercise 4.3 – Quadratic formula method
• Exercise 4.4 – Word problems based on quadratic equations

Chapter 5: Arithmetic Progressions

An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant.

Key Topics:

• General Form of an AP: Represented as a, a + d, a + 2d, …, where a is the first term and d is the common difference.
• nth Term of an AP: Given by aₙ = a + (n – 1)d.
• Sum of the First n Terms: Calculated using the formula Sₙ = n/2 [2a + (n – 1)d].

Example Problem:

Problem: Find the 10th term and the sum of the first 15 terms of the AP: 3, 8, 13, 18, …

Solution: Here, the first term a = 3, and the common difference d = 5. The 10th term is a₁₀ = a + (10 – 1)d = 3 + 9×5 = 48. The sum of the first 15 terms is *S₁₅ = 15/2 [2×3 + (15 – 1)×5] = 15/2 [6 + 70] = 15/2 × 76 = 570*.

Thus, the 10th term is 48, and the sum of the first 15 terms is 570.

Exercises Covered:

• Exercise 5.1 – Finding the nth term of an AP
• Exercise 5.2 – Sum of first n terms of an AP
• Exercise 5.3 – Word problems related to AP
• Exercise 5.4 – Application-based problems

Chapter 6: Triangles

Triangles are fundamental geometric shapes with unique properties and relationships.

Key Topics:

• Similarity of Triangles: Two triangles are similar if their corresponding angles are equal and their corresponding sides are in proportion.
• Basic Proportionality Theorem (Thales Theorem): If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.
• Criteria for Similarity of Triangles: AA (Angle-Angle), SSS (Side-Side-Side), and SAS (Side-Angle-Side) criteria.

Example Problem:

Problem: In ∆ABC, DE is parallel to BC. If AD = 3 cm, DB = 4 cm, AE = 2.25 cm, find EC.

Solution: By the Basic Proportionality Theorem:ADDB=AEEC\frac{AD}{DB} = \frac{AE}{EC}DBAD​=ECAE​

Substituting the values:34=2.25EC\frac{3}{4} = \frac{2.25}{EC}43​=EC2.25​

Cross-multiplying:EC=2.25×43=3 cmEC = \frac{2.25 \times 4}{3} = 3 \text{ cm}EC=32.25×4​=3 cm

Thus, EC = 3 cm.

Exercises Covered:

• Exercise 6.1 – Introduction to similarity
• Exercise 6.2 – Thales theorem
• Exercise 6.3 – Criteria for triangle similarity
• Exercise 6.4 – Area theorem
• Exercise 6.5 – Pythagoras theorem
• Exercise 6.6 – Advanced similarity problems

Chapter 7: Coordinate Geometry

Coordinate geometry helps in representing geometric figures algebraically.

Key Topics:

• Distance Formula: The distance between two points (x1,y1)(x_1, y_1)(x1​,y1​) and (x2,y2)(x_2, y_2)(x2​,y2​) is:

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}d=(x2​−x1​)2+(y2​−y1​)2​

• Section Formula: The coordinates of a point dividing a line segment in the ratio m:n are:

(mx2+nx1m+n,my2+ny1m+n)\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)(m+nmx2​+nx1​​,m+nmy2​+ny1​​)

• Area of a Triangle Formula: The area of a triangle with vertices (x1,y1)(x_1, y_1)(x1​,y1​), (x2,y2)(x_2, y_2)(x2​,y2​), and (x3,y3)(x_3, y_3)(x3​,y3​) is:

A=12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣A = \frac{1}{2} \left| x_1(y_2 – y_3) + x_2(y_3 – y_1) + x_3(y_1 – y_2) \right|A=21​∣x1​(y2​−y3​)+x2​(y3​−y1​)+x3​(y1​−y2​)∣

Example Problem:

Problem: Find the distance between the points (3, 4) and (7, 1).

Solution: Using the distance formula:d=(7−3)2+(1−4)2=16+9=25=5d = \sqrt{(7 – 3)^2 + (1 – 4)^2} = \sqrt{16 + 9} = \sqrt{25} = 5d=(7−3)2+(1−4)2​=16+9​=25​=5

Thus, the distance is 5 units.

Exercises Covered:

• Exercise 7.1 – Distance formula problems
• Exercise 7.2 – Section formula
• Exercise 7.3 – Finding centroid and incenter
• Exercise 7.4 – Area of a triangle problems

Chapter 8: Introduction to Trigonometry

Trigonometry deals with the relationships between angles and sides of a right-angled triangle.

Key Topics:

• Trigonometric Ratios:
  • sin θ = Opposite / Hypotenuse
  • cos θ = Adjacent / Hypotenuse
  • tan θ = Opposite / Adjacent
• Trigonometric Identities:
  • sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1sin2θ+cos2θ=1
  • 1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta1+tan2θ=sec2θ
  • 1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta1+cot2θ=csc2θ

Example Problem:

Problem: If sin A = 3/5, find cos A.

Solution: Using the identity:sin⁡2A+cos⁡2A=1\sin^2 A + \cos^2 A = 1sin2A+cos2A=1

Substituting sin⁡A=3/5\sin A = 3/5sinA=3/5:(35)2+cos⁡2A=1\left(\frac{3}{5}\right)^2 + \cos^2 A = 1(53​)2+cos2A=1925+cos⁡2A=1\frac{9}{25} + \cos^2 A = 1259​+cos2A=1cos⁡2A=1−925=1625\cos^2 A = 1 – \frac{9}{25} = \frac{16}{25}cos2A=1−259​=2516​cos⁡A=45\cos A = \frac{4}{5}cosA=54​

Thus, cos A = 4/5.

Exercises Covered:

• Exercise 8.1 – Defining trigonometric ratios
• Exercise 8.2 – Trigonometric values for standard angles
• Exercise 8.3 – Proving trigonometric identities
• Exercise 8.4 – Problems based on trigonometric identities

Chapter 9: Some Applications of Trigonometry

Trigonometry is widely used in various real-world applications, such as measuring heights and distances.

Key Topics:

• Angle of Elevation: The angle between the line of sight and the horizontal when looking up.
• Angle of Depression: The angle between the line of sight and the horizontal when looking down.

Example Problem:

Problem: A tower is 50 m high. The angle of elevation of its top from a point 30 m away from the base is θ. Find θ.

Solution: Using tan θ:tan⁡θ=Height of TowerDistance from Observer\tan θ = \frac{\text{Height of Tower}}{\text{Distance from Observer}}tanθ=Distance from ObserverHeight of Tower​tan⁡θ=5030=53\tan θ = \frac{50}{30} = \frac{5}{3}tanθ=3050​=35​

Thus, θ = tan⁻¹(5/3).

Exercises Covered:

• Exercise 9.1 – Angle of elevation and depression problems

Chapter 10: Circles

A circle is a geometric shape where all points are equidistant from a central point.

Key Topics:

• Tangent to a Circle: A line that touches the circle at exactly one point.
• Theorem on Tangents: The tangents drawn from an external point to a circle are equal in length.

Example Problem:

Problem: Two tangents are drawn from a point P to a circle with center O. If the radius of the circle is 6 cm and OP = 10 cm, find the length of each tangent.

Solution: Using the Pythagorean theorem:OP2=OT2+PT2OP^2 = OT^2 + PT^2OP2=OT2+PT2102=62+PT210^2 = 6^2 + PT^2102=62+PT2100=36+PT2100 = 36 + PT^2100=36+PT2PT2=64PT^2 = 64PT2=64PT=8 cmPT = 8 \text{ cm}PT=8 cm

Thus, each tangent is 8 cm long.

Exercises Covered:

• Exercise 10.1 – Definition of tangents and secants
• Exercise 10.2 – Theorems on tangents

Chapter 11: Constructions

Constructions involve drawing geometric figures with precision using a compass, ruler, and protractor.

Key Topics:

• Division of a Line Segment: Constructing a line segment in a given ratio.
• Constructing Tangents: Drawing tangents from a point outside a given circle.

Example Problem:

Problem: Construct a triangle similar to a given triangle ABC with a scale factor of 3:2.

Solution:

1. Draw the given triangle ABC.
2. Draw a ray BX making an acute angle with BC.
3. Mark 3 equal divisions on BX (since 3 is the greater ratio).
4. Join the 2nd division point to C (as per the 3:2 ratio) and draw a parallel line to BC.
5. Extend AB accordingly to get the required triangle.

Exercises Covered:

• Exercise 11.1 – Basic constructions
• Exercise 11.2 – Constructing similar triangles and tangents

Chapter 12: Areas Related to Circles

This chapter deals with finding the perimeter and area of circles and combinations of plane figures.

Key Topics:

• Perimeter and Area of a Circle: Formulas and applications.
• Sector and Segment of a Circle: Calculating their respective areas.
• Combinations of Figures: Finding areas of figures made up of circles and other shapes.

Example Problem:

Problem: Find the area of a sector of a circle with radius 14 cm and a central angle of 60°.

Solution:

Using the formula:

Exercises Covered:

• Exercise 12.1 – Perimeter and area of circles
• Exercise 12.2 – Areas of sector and segment
• Exercise 12.3 – Areas of combined figures

Chapter 13: Surface Areas and Volumes

This chapter focuses on the measurement of 3D shapes such as cubes, cylinders, and spheres.

Key Topics:

• Surface Areas: Finding surface areas of solids.
• Volumes: Calculating volumes of different 3D shapes.
• Combination of Solids: Determining surface area and volume when solids are combined or subtracted.

Example Problem:

Problem: Find the volume of a cylinder with radius 7 cm and height 10 cm.

Solution:

Using the formula:

Exercises Covered:

• Exercise 13.1 – Surface areas of solids
• Exercise 13.2 – Volumes of solids
• Exercise 13.3 – Combination of solids
• Exercise 13.4 – Frustum of a cone

Chapter 14: Statistics

Statistics involves the study of data, its collection, analysis, and interpretation.

Key Topics:

• Mean, Median, and Mode: Methods to find the central tendency of grouped data.
• Cumulative Frequency Distribution: Organizing data into a cumulative format.

Example Problem:

Problem: Find the mean of the following data:

Solution:

Using the formula:

(Where f = frequency and x = class midpoint)

After calculating, we find the mean to be 23.75.

Exercises Covered:

• Exercise 14.1 – Mean of grouped data
• Exercise 14.2 – Median and mode
• Exercise 14.3 – Graphical representation

Chapter 15: Probability

Probability deals with the likelihood of an event occurring.

Key Topics:

• Theoretical Probability: Basic probability concepts and calculations.
• Experimental Probability: Probability based on experiments and observations.

Example Problem:

Problem: What is the probability of rolling an even number on a six-sided die?

Solution:

The even numbers are {2, 4, 6}. Since there are 3 favorable outcomes and 6 possible outcomes:

Thus, the probability is 0.5 or 50%.

Exercises Covered:

• Exercise 15.1 – Basic probability concepts
• Exercise 15.2 – Experimental probability

CBSE Class 10 Maths Unit-wise Weightage (2011-2024)

Conclusion

Our comprehensive solutions for NCERT Class 10 Mathematics Solutions cover every essential concept, ensuring students grasp the fundamentals and excel in their exams. We provide step-by-step solutions with clear explanations, real-world applications, and solved examples to aid understanding.

For further detailed explanations and additional practice, we encourage students to explore our in-depth exercises and topic-wise problem sets. With diligent practice and conceptual clarity, mastering Class 10 Mathematics becomes an achievable goal.

Frequently Asked Questions (FAQs)

1. How can I master Class 10 Maths for CBSE?

Students can master Class 10 Maths by consistently practicing problems from the NCERT textbook, understanding concepts thoroughly, and solving sample papers. Using NCERT Solutions for reference will help in clarifying doubts and strengthening problem-solving skills.

2. How do I solve Class 10 Mathematics problems easily?

To solve problems efficiently, focus on understanding concepts rather than memorization. Regularly practice different types of problems and refer to NCERT Solutions for step-by-step explanations.

3. Where can I download CBSE Class 10 NCERT Mathematics PDF Solutions?

You can download NCERT Solutions for Class 10 Mathematics from educational websites that provide free access to PDF solutions for all chapters.

4. How will NCERT Solutions for Class 10 Maths benefit CBSE students?

NCERT Solutions help students build a strong mathematical foundation, improve problem-solving abilities, and boost confidence for board exams. They also serve as a helpful guide for completing homework and revision.

5. What concepts are covered in NCERT Class 10 Maths Solutions?

The NCERT Class 10 Maths syllabus consists of 15 chapters, covering topics such as Algebra, Trigonometry, Coordinate Geometry, Probability, and Mensuration. Each chapter includes exercises to reinforce learning.

6. How can I score high marks in Class 10 Maths CBSE Board Exams?

To score high, focus on:

• Completing all NCERT textbook exercises
• Revising formulas and theorems regularly
• Practicing previous years’ question papers
• Solving additional problems from reference books like RD Sharma or RS Aggarwal

7. Is NCERT enough for Class 10 Maths Board Exams?

Yes, NCERT textbooks provide comprehensive coverage of all concepts required for CBSE exams. However, students aiming for extra practice can use additional reference books for more challenging problems.

8. Which reference books are best for Class 10 CBSE Maths?

The best reference books for Class 10 Maths include:

• RD Sharma – Ideal for in-depth concept clarity and higher-level problems
• RS Aggarwal – Suitable for additional practice and conceptual understanding
• NCERT Exemplar – Contains challenging questions for competitive exam preparation

9. What are the most scoring topics in Class 10 Maths?

Algebra (20 marks), Trigonometry (12 marks), and Statistics & Probability (11 marks) are some of the highest-scoring topics in the CBSE Class 10 Maths syllabus.