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NCERT Solutions for Class 10 Maths
NCERT solutions for class 10 maths comprise a detailed and step-wise explanation to all NCERT problems for CBSE 10th grade. Students can use these solutions to help them proceed if they hit a roadblock while attempting a sum. These class 10 maths NCERT solutions are prepared by a host of math experts who are wizards in this field. The solutions are presented in such a way that students not only get the answer to a specific problem but can also develop a clear understanding of the concept used in that question.
NCERT solutions for class 10 maths are extremely helpful for students preparing for the 10th board examinations. The examination paper follows the structure provided by NCERT hence, it becomes very important for students to refer to these solutions. They also build a foundation for topics that will be covered in the upcoming 11th and 12th grades. CBSE solutions for class 10 maths are useful in preparing for other competitive exams such as Olympiads.
NCERT Solutions for Class 10 Maths Updated for 2022-23 Session – Free PDF Download
NCERT solutions for class 10 maths have been written with the utmost care so that students can study all topics efficiently and effectively. NCERT is the best source to score an excellent grade in the class 10th boards. An in-depth analysis of the 10th maths CBSE solutions sums is available for free pdf download. So start your impressive educational journey today! The links to all the chapter-wise pdfs are given below.
Class 10 Maths NCERT Solutions Chapter 1 to 15
Practice is the key to acing any 10th class examination be it the units, prelims, or boards. Thus, it is imperative for students to regularly visit these links and keep revising class 10 maths NCERT solutions to master each topic. Follow the NCERT textbook given below to develop crystal clear concepts and get outstanding grades.
NCERT Solutions for Class 10 Maths Chapter 1
NCERT solutions for class 10 maths chapter 1 real numbers encompass the solutions to problems based on Euclid’s division lemma, Fundamental theorem of arithmetic, irrational numbers, rational numbers, and decimal expansions. Euclid’s division lemma is used to compute the HCF (highest common factor) of any two positive integers. The Fundamental Theorem of Arithmetic is the next chief concept introduced that is based on prime numbers. HCF (highest common factor) and LCM (least common multiple) can be found using this concept. Finally, the class 10 maths NCERT solutions teach students the different methods to prove that a number is irrational or rational and how to break down a rational number into its decimal expansion.
Given below are some of the important formulas covered in the NCERT solutions for class 10 maths chapter 1 that is required to solve problems on real numbers:
- HCF (p, q) × LCM (p, q) = p × q
- Euclid’s division lemma: a = bq + r, 0 ≤ r < b (Here, when 'a' is divided by 'b', the quotient is 'q' and the reminder is 'r'.)
|Class 10 Maths NCERT Solutions Chapter 1 Exercises|
Class 10 Maths Chapter 1 Real Numbers
Topics Covered: Problems based on Euclid’s division lemma and long division, HCF, and LCM questions that most commonly appear in board exams. Other topics included in class 10 maths NCERT solutions are - decomposing a composite number into a product of its primes, properties of positive integers, rational number into the decimal expansion.
Total Questions: Class 10 maths chapter 1 consists of a total of 18 questions out of which 10 are easy, 5 are moderate, and 3 are long answer type questions.
NCERT Solutions Class 10 Maths Chapter 2
NCERT solutions class 10 maths chapter 2 polynomials include solutions based on topics such as the degree of a polynomial, finding the value of a polynomial at a given point, and the division algorithm. students learn the importance of zeros of a polynomial and the different techniques on how to verify their relationship with the coefficients.
Linear polynomial, ax + b
- Zero of a polynomial ax + b is x = -b/a
Quadratic Polynomial, ax2 + bx + c
- Sum of zeros, α + β = -b/a
- Product of zeros, αβ = c/a
Cubic Polynomial, ax3 + bx2 + cx + d
- α + β + γ = -b/a
- αβ + βγ + γα = c/a
- α β γ = -d/a
|Class 10 Maths NCERT Solutions Chapter 2 Exercises|
Class 10 Maths Chapter 2 Polynomials
Topics Covered: Degree of a polynomial, the relationship between zeros of a polynomial, and its coefficients are topics that fall under polynomials. Another important topic included in these class 10 maths NCERT solutions is the division algorithm that helps to decompose a polynomial into polynomials of a lesser degree.
Total Questions: Class 10 maths chapter 2 consists of 13 questions, of which 10 are long answers, 3 are moderate, and 6 are easy questions.
Class 10 Maths NCERT Solutions Chapter 3
Class 10 maths NCERT solutions chapter 3 pair of linear equations in two variables comprise of all the step-wise answers to the different topics included in this chapter such as introduction to linear equations in two variables, algebraic and graphical methods to solve linear equations such as cross multiplication.
Given below is a list of tips that can help you understand class 10 maths NCERT solutions chapter 3 faster.
- Make notes on the important theories such as consistent and inconsistent lines.
- Practice the NCERT solutions regularly.
- For graphical methods, draw proper graphs for correct analysis.
|Class 10 Maths NCERT Solutions Chapter 3 Exercises|
Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables
Topics Covered: Class 10 maths NCERT solutions chapter 3 include an introduction to linear equations in two variables, different algebraic techniques such as substitution and elimination used to solve them. The chapter has a total of 29 questions.
NCERT Solutions for Class 10 Maths Chapter 4
NCERT solutions for class 10 maths chapter 4 quadratic equations consist of an in-depth explanation of solutions to topics such as solving quadratic equations using different methods like factorizing, completing the squares, finding the roots of the equation, and splitting the middle term.
- Quadratic formula: x = (-b ± √(b2 - 4ac))/(2a)
- Discriminant: Δ = b2 - 4ac
|Class 10 Maths NCERT Solutions Chapter 4 Exercises|
Class 10 Maths Chapter 4 Quadratic Equations
Topics Covered: Finding the discriminant and drawing conclusions from it, solving quadratic equations by completing the squares, factorizing, and using the quadratic formula to find the roots of the equation.
Total Questions: NCERT solutions for class 10 maths chapter 4 have a total of 24 well-structured questions. They are further divided into 8 long answer questions, 10 easy sums, and 6 moderate problems.
NCERT Solutions Class 10 Maths Chapter 5
NCERT solutions class 10 maths chapter 5 arithmetic progressions help students get an insight into what arithmetic progressions actually are. Students learn how to apply the formula to different questions such as finding the nth term, finding the sum of the first n terms, and associated word problems.
- nth term of AP: an = a + (n – 1) d
- Sum of n terms: S = n/2 [2a + (n – 1) d]
|Class 10 Maths NCERT Solutions Chapter 5 Exercises|
Class 10 Maths Chapter 5 Arithmetic Progressions
Topics Covered: The topics covered in NCERT class 10 maths solutions chapter 5 are finding an arithmetic progression if the first term and common difference are given, calculating the sum of the first n terms using the formula, and determining the nth term of a given AP.
Total Questions: Chapter 5 arithmetic progressions have a total of 49 questions. They can be further divided into 20 simple ones, 15 moderately difficult questions, and 14 tough sums.
Class 10 Maths NCERT Solutions Chapter 6
Class 10 maths NCERT solutions chapter 6 triangles comprise a step-wise and detailed approach to attempting problems based on proving the similarity as well as congruence between two triangles. They learn about the AAA, AA, SSS, SAS criterion, and the Pythagoras Theorem.
|Class 10 Maths NCERT Solutions Chapter 6 Exercises|
Class 10 Maths Chapter 6 Triangles
Total Questions: There are a total of 65 questions in class 10 maths chapter 6. These include 30 complex questions, 25 moderate, and 10 easy sums.
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NCERT Solutions for Class 10 Maths Chapter 7
NCERT solutions for class 10 maths chapter 7 coordinate geometry include problems on the distance formula, section formula, collinearity of points, area of a triangle using coordinates, and several real-life questions based on the same.
- Distance formula: √[(x2 - x1)2 + (y2 - y1)2]
- Area of a Triangle: 1/2 × [x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)]
- Midpoint Formula: [(x1 + x2)/2 , (y1 + y2)/2]
|Class 10 Maths NCERT Solutions Chapter 7 Exercises|
Class 10 Maths Chapter 7 Coordinate Geometry
Topics Covered: Finding the area of a triangle when all three coordinates are given, determining the distance between two points or a point from the origin, and applications of the section formula are the topics encompassed in the NCERT solutions for class 10 maths.
Total Questions: Chapter 7 coordinate geometry includes a total of 33 questions out of which 15 are of a medium level, 10 are easy and 8 are difficult problems.
NCERT Solutions Class 10 Maths Chapter 8
NCERT solutions class 10 maths chapter 8 introduction to trigonometry gives students an insight into the different trigonometric ratios such as sine, cosine, tangent, etc. Values of these ratios at particular angles, how to solve complementary angles of trigonometric ratios, and trigonometric identities are other concepts that students get a grasp of by attempting questions.
- sin A = (side opposite to A)/hypotenuse; cos A = (side adjacent to A)/hypotenuse; tan A = (side opposite to A)/(side adjacent to A)
- cosec A = 1/sin A; sec A = 1/cos A; cot A = 1/tan A
- tan A = (sin A) / (cos A)
- sin2A + cos2A = 1; cosec2A - cot2A = 1; sec2A - tan2A = 1
- sin (90 - A) = cos A; cos (90 - A) = sin A
cosec (90 - A) = sec A; sec (90 - A) = cosec A
tan (90 - A) = cot A; cot (90 - A) = tan A
|Class 10 Maths NCERT Solutions Chapter 8 Exercises|
NCERT Solutions Class 10 Maths Chapter 8 Preparation Tips
Trigonometry can be a complicated topic, especially when first introduced to students. Following are some tips that can help students to assimilate the chapter in a better manner.
- Make mnemonic devices to learn the correct sequence of the ratios and their formulas.
- Find patterns wherever possible.
- Understand the Pythagoras Theorem thoroughly.
Class 10 Maths Chapter 8 Introduction to Trigonometry
Topics Covered: NCERT solutions class 10 maths chapter 8 encompass the trigonometric ratios (sine, cosine, tangent, secant, cosec, and cotangent), problems on the value of these ratios at a particular angle, and several other sums on trigonometric identities. They also include questions, to prove if an equation is true, that appear frequently on board exams.
Total Questions: Chapter 8 introduction to trigonometry has 27 questions. 5 are easy, 10 require some manipulation, and 12 are difficult sums.
Class 10 Maths NCERT Solutions Chapter 9
Class 10 maths NCERT solutions chapter 9 some applications of trigonometry give in-depth insight to students into how to find the height and distance of various objects using trigonometry. It focuses on helping students to understand how trigonometry can be applied to real-life situations.
|Class 10 Maths NCERT Solutions Chapter 9 Exercises|
Class 10 Maths Chapter 9 Some Applications of Trigonometry
Topics Covered: There are a variety of problems on the practical applications of trigonometry using the angle of elevation, angle of depression, and line of sight that are covered under class 10 maths NCERT solutions chapter 9.
Total Questions: Chapter 9 has a total of 16 questions that deal with different scenarios cropping up in daily life. 7 are easy questions, 5 are tough and 4 are medium-level problems. The chapter clear concepts related to applying trigonometry to real-life examples for calculating the height and distance between two objects that are far away from each other.
NCERT Solutions for Class 10 Maths Chapter 10
NCERT solutions for class 10 maths chapter 10 circles develop a strong foundation on the concepts of circles. These include problems on the tangent of a circle, concentric circles, chords, and various construction questions.
|Class 10 Maths NCERT Solutions Chapter 10 Exercises|
Class 10 Maths Chapter 10 Circles
Topics Covered: Class 10 maths chapter 10 covers topics that help students to grasp the meaning of a tangent and associated properties that can be applied to solve different levels of problems.
Total Questions: There are 17 questions in class 10 maths chapter 10 out of which 4 are of an intermediate level, 3 are easy and 10 are difficult problems that challenge a student’s conceptual understanding.
NCERT solutions for class 10 maths chapter 10 help work to develop a good understanding of the concept of a tangent, applying certain properties of it such as the tangent will always be perpendicular to the radius at a point of contact and finding the length of a tangent from a specific point.
NCERT Solutions Class 10 Maths Chapter 11
NCERT solutions class 10 maths chapter 11 constructions will teach students how to use instruments such as a compass and ruler to construct various figures. They learn how to divide a line segment into the desired ratio, and construct a tangent to a circle using different techniques.
|Class 10 Maths NCERT Solutions Chapter 11 Exercises|
Class 10 Maths Chapter 11 Constructions
Topics Covered: NCERT class 10 maths solutions chapter 11 cover the construction of lines given a certain ratio using a ruler and compass, drawing a tangent to a circle from a given point, as well as constructing a triangle when a scale factor is mentioned.
Total Questions: Chapter 11 include 14 questions, 8 of which challenge a student’s critical thinking, 3 are easy and 3 belong to the moderate category of sums.
Practicing NCERT solutions class 10 maths chapter 11 students get an in-depth understanding of how to use different geometrical instruments to divide a line into a given ratio, drawing tangents to a given point on a circle, and constructing a triangle with certain restraints.
Class 10 Maths NCERT Solutions Chapter 12
Class 10 maths NCERT solutions chapter 12 areas related to circles encompass a detailed explanation of solutions based on direct application of the area and circumference formula. Additionally, students learn about finding the area of a sector, segment, and combinations of figures that include circles.
- Circumference of a circle = 2πr
- Area of a circle = πr2
- Area of sector = θ/360 × πr2
- Length of an arc = θ/360 × 2πr
|Class 10 Maths NCERT Solutions Chapter 12 Exercises|
Class 10 Maths Chapter 12 Areas Related to Circles
Topics Covered: Finding the area and circumference of a circle using standard formulas. It also includes questions on finding the area of composite figures, along with the area enclosed within a sector, and a segment.
Total Questions: Class 10 maths chapter 12 has a total of 35 questions. 8 are easy questions based on the direct application of formulas, 15 are moderate sums and 12 are difficult problems.
NCERT Solutions for Class 10 Maths Chapter 13
NCERT solutions for class 10 maths chapter 13 surface areas and volumes deal with finding the TSA (total surface area), LSA/CSA (lateral or curved surface area), and volume of different composite figures. Additionally, the questions involve converting a solid shape from one form to another and drawing inferences.
- TSA of frustum of a cone = πl (r1 + r2) + π(r12 + r22 )
- Slant height of frustum of a cone = √[h2 + (r1 - r2 )2]
- Volume of frustum of a cone = ⅓ πh (r12 + r22 + r1r2)
|Class 10 Maths NCERT Solutions Chapter 13 Exercises|
Class 10 Maths Chapter 13 Surface Areas and Volumes
Topics Covered: Frustum of a cone, converting a solid shape to another figure, finding the resultant’s area and volume are topics covered in class 10 NCERT solutions chapter 13. Solving questions on composite figures by combining cones, cubes, cuboids, etc., is also encompassed. There are 38 well-curated questions in this chapter.
Class 10 Maths NCERT Solutions Chapter 14
Class 10 maths NCERT solutions chapter 14 statistics help students understand the use of various statistical concepts such as mean, mode, standard deviation, and the graphical representation of cumulative frequency distribution.
|Class 10 Maths NCERT Solutions Chapter 14 Exercises|
Class 10 Maths Chapter 14 Statistics
Topics Covered: Finding the mean, median, and mode of grouped data using statistical techniques such as direct mean, assumed mean, and step deviation methods. Problems based on plotting graphs and drawing conclusions are frequently asked as a long answer question in board examinations.
Total Questions: Chapter 14 statistics has a total of 28 questions. 5 are easy, 15 belong to an intermediate level and 8 are tough questions that challenge a student’s speed and accuracy in calculations.
NCERT Solutions for Class 10 Maths Chapter 15
NCERT solutions for class 10 maths chapter 15 probability help students build an understanding of the concept behind experimental and theoretical probability. These solutions give a practical approach to students that can also be applied to daily life problems when finding if an event is likely to occur or not.
|Class 10 Maths NCERT Solutions Chapter 15 Exercises|
Class 10 Maths Chapter 15 Study Tips
Probability can prove to be a tough subject as it involves using abstract concepts that can be hard to visualize. Thus, students can follow the tips given below to help them in this process.
- Read the problem carefully and break it down into parts to solve it.
- Create easy scenarios and try to solve them yourself.
- Practice till you do not feel confident.
Class 10 Maths Chapter 15 Probability
Topics Covered: The probability of an event happening, elementary event concepts, and the complement of events are the topics covered under NCERT solutions for class 10 maths chapter 15. The chapter has a total of 30 questions. 7 can be easily solved by using the given formula, 10 are of intermediate level, and 13 require careful concentration as they are of a higher level.
Class 10 Maths NCERT Solutions Important Formulas
Class 10 maths covers a lot of important concepts crucial for understanding higher-grade mathematical topics. Math formulas are the key to solving problems with speed and accuracy. The first step is to understand how a formula came into existence and the underlying concept behind it. Then you can proceed to memorise them and apply the formulas to questions. Some of the essential formulas covered in NCERT solutions for class 10 maths are given below.
- (a + b)2 = a2 + b2 + 2ab
- (a + b) (a - b) = a2 - b2
- (a - b)2 = a2 + b2 - 2ab
- nth term = a + (n-1) d
- Sum upto nth term = n/2 [2a + (n - 1) d]
- sin2θ + cos2θ = 1
- cosec2θ = 1 + cot2θ
- sec2θ = 1 + tan2θ
- Circumference = 2πr
- Area = πr2
Importance of NCERT Solutions for Class 10 Maths
Class 10 Maths NCERT solutions are extremely helpful for CBSE students for several reasons. The CBSE board and school exams follow the NCERT coursebook and hence, it becomes very important for students to be well-versed with it. Given below are the reasons that outline the vitality of NCERT solutions.
- Understanding Concepts: By referring to these solutions students get the opportunity to build strong and reliable concepts. While attempting a problem there is a very high possibility that a student might not be able to complete it due to doubts. The NCERT solutions for class 10th maths are written in a way that makes it easy to understand even the toughest theories thereby, enabling students to successfully solve all levels of sums.
- Revision: NCERT Class 10 Maths Solutions give a quick overview of how to solve any sum. As students do not have enough time to solve all problems during examination days they can skim through these solutions and recall how to attempt different types of questions. This helps in saving a lot of time, making room for students to focus on building other skills necessary to crack a test.
- Board Exam Preparation: The ncert maths class 10 solutions give students an idea regarding what type of questions can be asked in the paper. Additionally, they also advise students on how to solve the exam sums ensuring that there are no missed steps and the solution is very clear. Writing a well-organized examination increases the chances of getting better marks.
FAQs on NCERT Solutions for Class 10 Maths
Why are NCERT Solutions for Class 10 Maths Important for Students?
NCERT Solutions for class 10 maths help students to prepare for board exams and instill a very strong mathematical foundation that can be used in higher classes. It also provides students with a way to reliably cross-check their answers. Additionally, these class 10th maths NCERT solutions help students with time management so that they can complete their syllabus well within the given limit.
How CBSE Students can utilize Class 10 Maths NCERT Solutions Effectively?
To get the most out of class 10 maths NCERT solutions students must first build a robust knowledge base of the theoretical concepts. The next step is to attempt the questions without consulting the solutions. Finally, students should refer to these solutions for checking their answers and analyzing the various steps that go into reaching the result.
Do I Need to Practice all Questions Provided in NCERT Solutions Class 10 Maths?
It is of utmost importance that students practice all questions provided in NCERT solutions for class 10 maths. Practice makes a person perfect and there can never be too much of it. Each question will help a student learn some different skill or concept that is vital when appearing for home and board exams.
How are NCERT Solutions Class 10 Maths Promoting Problem-Solving in Students?
The 10th maths CBSE solutions gives a step-wise explanation of all problems by breaking down the sum into smaller chunks in an attempt to present it in a more organized manner. It helps students to approach the question with a critical thinking and problem-solving mindset. Hence, they can apply this skill to other problems as well.
Do I Need to Make Notes while Referring to NCERT Solutions for Class 10 Maths?
Making notes is key while working with class 10 maths NCERT solutions. Notes help students relate to a topic as by writing down pointers and tricks they can recall the associated concepts faster. These also help students comprehend the solution to a question more efficiently.
What are the Best Ways to Learn Concepts Covered in NCERT Solutions Class 10th Maths?
The best way to learn NCERT Solutions Class 10th Maths is by regular practice and continuous revision. While attempting a question if a student has doubts, she must immediately clear them by referring to these solutions so that there is no gap in learning. Additionally, a student must go through all the tips and tricks included with the solutions to get the most out of them.
What are the Important Topics Covered in Class 10 Maths NCERT Solutions?
All topics such as quadratic equations, circles, surface areas, and volumes, trigonometry, etc are interlinked wherein concepts under different headings are used throughout the scope of the subject. Thus, all topics covered in NCERT Solutions for Class 10 Maths must be given equal weightage. However, a student should spend more time perfecting topics that he might not be confident with.
Where Can I Get Chapter-wise Class 10 Maths NCERT Solutions?
Students can download the chapter-wise class 10 maths NCERT solutions by the links mentioned above. These solutions are available in a scrollable pdf format that is also compatible with mobile screens. Exercise-wise pdfs are provided that also include techniques to help improve the understanding of the concept at hand.
Can We Use Different Methods to Solve Problems Apart from Those Mentioned in CBSE Solutions Class 10 Maths?
Many methods can be used to solve problems apart from those mentioned in the NCERT solutions for class 10 maths. However, the techniques used must be correct with the proper steps being mentioned clearly. In addition to this, shortcuts that might be useful for competitive exams cannot be used to solve questions appearing in the board exams.
How Many Chapters are there in NCERT Class 10 Maths Solutions?
There are a total of 15 chapters covered in the NCERT solutions for class 10 maths. Also, these solutions cover every exercise problem giving a detailed explanation and an in-depth analysis of all questions.
What are the Benefits of Class 10 Maths NCERT Solutions for CBSE Students?
The NCERT class 10 maths solutions help students belonging to not only CBSE but other boards get an overview of how to solve problems based on trigonometry, coordinate geometry, arithmetic progressions, numbers, and so on. The techniques listed and the meticulous explanation aids students to build a very sturdy mathematical foundation.
What are the Important Formulas in NCERT Solutions for Class 10 Maths?
A student must be proficient with formulas throughout all chapters in the ncert maths class 10 as they form the basis for topics taught in higher grades. However, quadratic formula, Pythagoras theorem, TSA/LSA/CSA formulas, nth term in an AP, statistical mean, are formulas required to solve questions that frequently appear in board exams.
Why Should I Practice CBSE Solutions for Class 10 Maths Regularly?
NCERT class 10 maths solutions must be practiced regularly to master all chapters within the given period of time. Regular revision help students build the confidence required to ace any examination. It helps them recall formulas faster and deciphers the methods to solve a problem thus, increasing their speed and precision. Additionally, students can use these class 10 maths NCERT solutions to cross-check their answers.
Why Choose Cuemath for Class 10 Maths NCERT Solutions?
Cuemath is the best website for class 10 Maths NCERT solutions as the solutions provided by them are prepared by IITians who are geniuses in this subject. All solutions are thoroughly checked to ensure that they give the best and easiest approach to resolving any problem. This makes these class 10 maths NCERT solutions very reliable.
Are NCERT Solutions for Class 10 Maths Sufficient for CBSE Board Exams?
NCERT solutions for class 10 maths are sufficient for CBSE board exams. However, it is always best to consult some supplementary NCERT books that give you a broader understanding of the kind of questions that can appear on the exams. As the questions in the exam paper are based on NCERT concepts hence, it is advisable to include more matter in your studying curriculum.