atal tunnel case study class 10 maths

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The Constructor

Atal rohtang tunnel: construction of the world’s longest highway tunnel at 10,000 feet.

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🕑 Reading time: 1 minute

Atal tunnel (Rohtang) is a roadway tunnel constructed at an altitude of 3060 m above mean sea level, making it the highest highway tunnel in the world. At 9.02 km length, it is also the longest bi-directional single-tube roadway tunnel of India. 

The tunnel was formerly known as the Rohtang tunnel and was renamed after India's former prime minister Atal Bihari Vajpayee. It is located under the Pir Panjal mountain range of the Himalayas, near Rohtang pass in Himachal Pradesh, India.

The project is considered a key development for the Indian defense forces as it connects the Manali and Lahaul valley. The tunnel will also reduce the distance between Manali and Leh by 46 km. Earlier, the road connectivity was limited to only one season of the year as the Rohtang pass remains closed in the winters due to heavy snowfall.

It is a horse-shoe shaped tunnel with raised footpaths on both sides of the lanes. It is equipped with a semi-transverse ventilation system and a service tunnel for maintenance and emergency exit.

The tunnel was constructed using the New Austrian Tunneling Method (NATM) and is designed for a maximum speed of 80 km/h. Excavation activities were carried out every year from June to December   as construction was not possible during the rest of the year due to the accumulation of heavy snow at the tunnel portal.

1. Geology of Rohtang Tunnel

The Himalayas are full of geological uncertainties because of faults, folds, and shear zones due to ongoing tectonic activities. Therefore, it is essential to study the geology of the Himalayas before constructing any major structure in the region.

• The site has a complicated geology marked with intermixed rocks along the alignment of the tunnel. The types of rocks consist of quartzite schist, phyllite, and magmatic gneiss.
• The average dip direction and plunge of these rocks were 220° and 250°, respectively. It shows that the rocks are uniformly dipping throughout the alignment of the tunnel.
• Two main shear zones were encountered during the construction of the tunnel. Tunneling in the shear zones is difficult due to the probability of rockburst. 

Properties of Rohtang Tunnel Rocks

The presence of good quality rocks is desirable for tunneling process, and the index properties of rocks can help assess the quality of rock. The index properties of rocks present along the alignment of the Rohtang tunnel are tabled below:

Density and sonic wave velocity are highest in quartzite schist; thus, it is denser than other rocks. However, the slake durability index for all the rocks is almost the same, which indicates that the rocks are highly durable.

For half of the year, the periphery of the Rohtang tunnel is covered in snow, and the porosity of rocks is in the moderate to high range. Therefore, the possibility of water ingress during the time of construction was very high.

The compressive and tensile strengths of rock mass are very important for designing tunnels. Mostly, the wear rate of the cutter head in tunnel boring machine (TBM) and the strength of rock mass decide the amount of explosives needed for NATM. The strength properties of rock mass available along the tunnel alignment are given below:

2.  Problems Encountered during Tunneling and their Remedial Measures

Tunneling in the young Himalayan mountains is challenging due to the existing geological structures and tectonic activities. During the construction stage, many unforeseen events had occurred, which forced the team to change the excavation methods and the support system. These problems have been described below: 

• The tunnel was subjected to a maximum overburden of 2000 m. A huge overburden pressure created deformation of the crown area, bending of lattice girders, crack in shotcrete, and rockfalls. 
• Adhesion between concrete and schist rock was very poor. Therefore, the shotcrete layer was losing its adhesive properties causing it to disintegrate at the locations where the schist rock was present.
• Mainly in the crown region, two joint sets were aligned in the perpendicular direction. Thus, the wedge failure at the crown region was common. 
• During the construction of the tunnel, there was a sudden collapse of a 50 m stretch of the roof, but fortunately, it didn't cause any loss of life. The heading portion of the tunnel was 100 m ahead of the collapsed area. It took almost two months to repair the collapsed section. Repair works were carried out by reinstallation of lattice girders, reinstallation and extension of rock bolts, and shotcreting with wire-mesh.
• During construction, the primary challenge was the availability of time as the tunneling advancement was possible only from June to December. This is the main reason that the tunnel construction took more than ten years.

2.1 Major Problems  Encountered during Tunneling

Other than the issues mentioned above, some major complications were observed, which have been discussed below:

2.1.1 Rock Squeezing

The rock mass tries to get mobilized for stability after excavating an opening in it. However, during tunneling in poor rocks at higher overburden pressure, fracture of rock mass occurs before mobilization. This process is called as squeezing of rocks.

At the Rohtang tunnel site, mica-schist rock was the poorest, causing squeezing of rock mass at those locations.

2.1.2 Rock Bursting 

Rock bursting is generally observed in hard rock due to an increase in overburden pressure. In the Rohtang tunnel, mild to moderate rock bursting conditions were observed.

2.1.3 Delay in Support Installation 

After clearing the muck from the heading area, lattice girder, wire mesh, and shotcrete were applied. The rock bolts were later installed after six or seven rounds of shotcrete. This led to deformation around the tunnel periphery.

2.1.4  Stress Concentration around the Tunnel Opening

If the tunnel alignment is along the orientation of the foliation plane, then the stress concentration is observed more around the tunnel periphery. This condition was encountered at multiple locations during the Rohtang tunnel drive. An increased stress concentration created the bending of lattice girders.

2.1.5 Leaving Shotcrete Slot during Primary Support Installation

Generally, the slot in shotcrete acts as an apparent lining stress controller. Therefore, it was created during the initial years of tunnel advancement. However, this process proved to be detrimental because, in the long term, it allowed the propagation of cracks in the surrounding rock mass. 

2.2 Remedial Measures

To avoid the collapse of the tunnel, the following remedial measures were taken:

• Slots in shotcrete were no longer provided to act as apparent lining stress controller.
• Rock bolts perpendicular to the foliation plane were provided, especially in the location where stress concentration was high. 
• Continuous three-dimensional monitoring of the supports and tunnel deformation was carried out to determine the excessive deformation.
• Rock bolts and wire mesh were properly tied, and shotcrete was applied in a monolithic manner. This ensured a uniform distribution of stress concentration around the periphery of the tunnel opening.
• Ground response curves were developed to determine the allowable deformation before the installation of the main support system.
• To prevent the roof collapse, single or double pipe roofing was installed in advance. 
• Rock bolts were installed just after clearing the muck. It reduced the deformation around the tunnel periphery.

3. Construction Process of Rohtang Tunnel 

NATM was preferred over TBM to construct the Rohtang tunnel. The main reasons for not choosing TBM are: 

• High deformations and pressures were expected on the tunnel. Thus, the TBM may have got jammed in such situations. 
• The transportation of heavy machine parts over Himalaya's bridges was risky because the Himalayan roads are not designed to bear heavy loads. Also, poor road conditions and tight hairpin bends make the transportation dangerous.  

The construction process of the Rohtang tunnel is discussed in two segments. In the first segment, the general tunnel construction sequence using NATM is discussed, and the second segment describes the construction timeline. 

3.1 Tunnel Construction Using NATM 

NATM is not a construction method for tunneling, but a strategy that streamlines all the construction activities, with safety and economy being the main principles. Flexibility in the operation and construction works is the main advantage of the NATM technique. It can be used for hard rock, soft rock, and mixed ground conditions. The general tunnel construction sequence using NATM is described below:

• Firstly, the drill holes are marked on the face of the tunnel based on the blasting pattern. Thereafter, the holes are drilled using manual excavators if the face is soft soil/rock, or with boomer machines if the face is hard rock. 
• Further, the explosives are inserted into the drilled holes and connected to detonators which are connected to one prime connection. 
• The prime connection where all the detonators are connected is detonated, which creates a chain reaction of explosions.
• After the complete removal of the dust and harmful gases suffused due to the blast, the loose materials are removed from the face of the tunnel. 
• After the removal of loose material, primary supports are installed on the tunnel periphery. Shotcrete or rock bolts are used as primary supports. 
• Water ingress inside the tunnel is a major concern as it may lead to the failure of the structure. To avoid this situation, waterproofing membranes are provided after the installation of the primary support system. 
• If the strata demand more support and structural stability, supports in the form of lattice girder, steel ribs, and a second layer of shotcrete can be provided. 
• Finally, the precast concrete linings are installed throughout the periphery of the tunnel section. 

3.2 Construction Timeline of Rohtang Tunnel 

The idea for the construction of the Rohtang tunnel was first conceived over 15 years ago. A joint venture of Afcons and Strabag won the Engineering Procurement & Construction (EPC) tender for Rs 1,458 crore from Border Road Organisation (BRO) in 2009.

The portals of the tunnel were constructed from the north and south directions as an intermediate starting point was impossible due to the topography. While it was possible to work with limitations from the south portal throughout the year, the north portal could only be reached via the pass. Therefore, only six to seven months of each year were effectively available for the work. The construction timeline of the Rohtang tunnel is discussed below:

• In 2010, the tunnel excavation work began from both sides of the portal. Two arm boomers and excavators were used for tunnel excavation. After six months of work, around 1 meter of tunnel was completed. 
• In November 2012, the construction work came to a halt for more than a year due to water ingress into the tunnel, and continuous dewatering led to the delay. 
• On 16 October 2013, the roof area of the tunnel collapsed. Before this incident, 1.95 km of the tunnel had already been excavated. It took more than two months to repair the collapsed area.
• In 2014, half of the tunnel construction work was completed. The total tunnel lengths of 2.25 km and 2.7 km were excavated from the north and south portal respectively.
• Finally, the excavation work was completed on 13th October 2017. Thereafter, the laying of concrete load, installation of electromagnetic fitting, lighting, intelligent traffic control system, ventilation system, and other miscellaneous works were executed.  

The main reasons for not choosing Tunnel boring machines (TBM) for constructing the Rohtang tunnel are given below:  1. High deformations and pressures were expected on the tunnel. Thus, the TBM may have got jammed in such situations.  2. The transportation of heavy machine parts over Himalaya's bridges was risky because the Himalayan roads are not designed to bear heavy loads. Also, poor road conditions and tight hairpin bends make the transportation dangerous.  

The Rohtang tunnel is the world's highest motorable road way tunnel of 9.02 km length.

The Rohtang tunnel has been constructed at an altitude of 3060 m above mean sea level, the highest in the world.

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CBSE Class 10 Maths Case Study

CBSE Board has introduced the case study questions for the ongoing academic session 2021-22. The board will ask the paper on the basis of a different exam pattern which has been introduced this year where 50% syllabus is occupied for MCQ for Term 1 exam. Selfstudys has provided below the chapter-wise questions for CBSE Class 10 Maths. Students must solve these case study based problems as soon as they are done with their syllabus. 

These case studies are in the form of Multiple Choice Questions where students need to answer them as asked in the exam. The MCQs are not that difficult but having a deep and thorough understanding of NCERT Maths textbooks are required to answer these. Furthermore, we have provided the PDF File of CBSE Class 10 maths case study 2021-2022.

Class 10 Maths (Formula, Case Based, MCQ, Assertion Reason Question with Solutions)

In order to score good marks in the term 1 exam students must be aware of the Important formulas, Case Based Questions, MCQ and Assertion Reasons with solutions. Solving these types of questions is important because the board will ask them in the Term 1 exam as per the changed exam pattern of CBSE Class 10th.

Important formulas should be necessarily learned by the students because the case studies are solved with the help of important formulas. Apart from that there are assertion reason based questions that are important too. 

Assertion Reasoning is a kind of question in which one statement (Assertion) is given and its reason is given (Explanation of statement). Students need to decide whether both the statement and reason are correct or not. If both are correct then they have to decide whether the given reason supports the statement or not. In such ways, assertion reasoning questions are being solved. However, for doing so and getting rid of confusions while solving. Students are advised to practice these as much as possible.

For doing so we have given the PDF that has a bunch of MCQs questions based on case based, assertion, important formulas, etc. All the Multiple Choice problems are given with detailed explanations.

CBSE Class 10th Case study Questions

Recently CBSE Board has the exam pattern and included case study questions to make the final paper a little easier. However, Many students are nervous after hearing about the case based questions. They should not be nervous because case study are easy and given in the board papers to ease the Class 10th board exam papers. However to answer them a thorough understanding of the basic concepts are important. For which students can refer to the NCERT textbook.

Basically, case study are the types of questions which are developed from the given data. In these types of problems, a paragraph or passage is given followed by the 5 questions that are given to answer . These types of problems are generally easy to answer because the data are given in the passage and students have to just analyse and find those data to answer the questions.

CBSE Class 10th Assertion Reasoning Questions

These types of questions are solved by reading the statement, and given reason. Sometimes these types of problems can make students confused. To understand the assertion and reason, students need to know that there will be one statement that is known as assertion and another one will be the reason, which is supposed to be the reason for the given statement. However, it is students duty to determine whether the statement and reason are correct or not. If both are correct then it becomes important to check, does reason support the statement? 

Moreover, to solve the problem they need to look at the given options and then answer them.

CBSE Class 10 Maths Case Based MCQ

CBSE Class 10 Maths Case Based MCQ are either Multiple Choice Questions or assertion reasons. To solve such types of problems it is ideal to use elimination methods. Doing so will save time and answering the questions will be much easier. Students preparing for the board exams should definitely solve these types of problems on a daily basis.

Also, the CBSE Class 10 Maths MCQ Based Questions are provided to us to download in PDF file format. All are developed as per the latest syllabus of CBSE Class Xth.

Class 10th Mathematics Multiple Choice Questions

Class 10 Mathematics Multiple Choice Questions for all the chapters helps students to quickly revise their learnings, and complete their syllabus multiple times. MCQs are in the form of objective types of questions whose 4 different options are given and one of them is a true answer to that problem. Such types of problems also aid in self assessment.

Case Study Based Questions of class 10th Maths are in the form of passage. In these types of questions the paragraphs are given and students need to find out the given data from the paragraph to answer the questions. The problems are generally in Multiple Choice Questions.

The Best Class 10 Maths Case Study Questions are available on Selfstudys.com. Click here to download for free.

To solve Class 10 Maths Case Studies Questions you need to read the passage and questions very carefully. Once you are done with reading you can begin to solve the questions one by one. While solving the problems you have to look at the data and clues mentioned in the passage.

In Class 10 Mathematics the assertion and reasoning questions are a kind of Multiple Choice Questions where a statement is given and a reason is given for that individual statement. Now, to answer the questions you need to verify the statement (assertion) and reason too. If both are true then the last step is to see whether the given reason support=rts the statement or not.

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CBSE Class 10 Maths Case Study Questions PDF

Download Case Study Questions for Class 10 Mathematics to prepare for the upcoming CBSE Class 10 Final Exam. These Case Study and Passage Based questions are published by the experts of CBSE Experts for the students of CBSE Class 10 so that they can score 100% on Boards.

CBSE Class 10 Mathematics Exam 2024  will have a set of questions based on case studies in the form of MCQs. The CBSE Class 10 Mathematics Question Bank on Case Studies, provided in this article, can be very helpful to understand the new format of questions. Share this link with your friends.

Table of Contents

Chapterwise Case Study Questions for Class 10 Mathematics

Inboard exams, students will find the questions based on assertion and reasoning. Also, there will be a few questions based on case studies. In that, a paragraph will be given, and then the MCQ questions based on it will be asked.

The above  Case studies for Class 10 Maths will help you to boost your scores as Case Study questions have been coming in your examinations. These CBSE Class 10 Mathematics Case Studies have been developed by experienced teachers of cbseexpert.com for the benefit of Class 10 students.

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Class 10 Maths Syllabus 2024

Chapter-1  real numbers.

Starting with an introduction to real numbers, properties of real numbers, Euclid’s division lemma, fundamentals of arithmetic, Euclid’s division algorithm, revisiting irrational numbers, revisiting rational numbers and their decimal expansions followed by a bunch of problems for a thorough and better understanding.

Chapter-2  Polynomials

This chapter is quite important and marks securing topics in the syllabus. As this chapter is repeated almost every year, students find this a very easy and simple subject to understand. Topics like the geometrical meaning of the zeroes of a polynomial, the relationship between zeroes and coefficients of a polynomial, division algorithm for polynomials followed with exercises and solved examples for thorough understanding.

Chapter-3  Pair of Linear Equations in Two Variables

This chapter is very intriguing and the topics covered here are explained very clearly and perfectly using examples and exercises for each topic. Starting with the introduction, pair of linear equations in two variables, graphical method of solution of a pair of linear equations, algebraic methods of solving a pair of linear equations, substitution method, elimination method, cross-multiplication method, equations reducible to a pair of linear equations in two variables, etc are a few topics that are discussed in this chapter.

Chapter-4  Quadratic Equations

The Quadratic Equations chapter is a very important and high priority subject in terms of examination, and securing as well as the problems are very simple and easy. Problems like finding the value of X from a given equation, comparing and solving two equations to find X, Y values, proving the given equation is quadratic or not by knowing the highest power, from the given statement deriving the required quadratic equation, etc are few topics covered in this chapter and also an ample set of problems are provided for better practice purposes.

Chapter-5  Arithmetic Progressions

This chapter is another interesting and simpler topic where the problems here are mostly based on a single formula and the rest are derivations of the original one. Beginning with a basic brief introduction, definitions of arithmetic progressions, nth term of an AP, the sum of first n terms of an AP are a few important and priority topics covered under this chapter. Apart from that, there are many problems and exercises followed with each topic for good understanding.

Chapter-6  Triangles

This chapter Triangle is an interesting and easy chapter and students often like this very much and a securing unit as well. Here beginning with the introduction to triangles followed by other topics like similar figures, the similarity of triangles, criteria for similarity of triangles, areas of similar triangles, Pythagoras theorem, along with a page summary for revision purposes are discussed in this chapter with examples and exercises for practice purposes.

Chapter-7  Coordinate Geometry

Here starting with a general introduction, distance formula, section formula, area of the triangle are a few topics covered in this chapter followed with examples and exercises for better and thorough practice purposes.

Chapter-8  Introduction to Trigonometry

As trigonometry is a very important and vast subject, this topic is divided into two parts where one chapter is Introduction to Trigonometry and another part is Applications of Trigonometry. This Introduction to Trigonometry chapter is started with a general introduction, trigonometric ratios, trigonometric ratios of some specific angles, trigonometric ratios of complementary angles, trigonometric identities, etc are a few important topics covered in this chapter.

Chapter-9  Applications of Trigonometry

This chapter is the continuation of the previous chapter, where the various modeled applications are discussed here with examples and exercises for better understanding. Topics like heights and distances are covered here and at the end, a summary is provided with all the important and frequently used formulas used in this chapter for solving the problems.

Chapter-10  Circle

Beginning with the introduction to circles, tangent to a circle, several tangents from a point on a circle are some of the important topics covered in this chapter. This chapter being practical, there are an ample number of problems and solved examples for better understanding and practice purposes.

Chapter-11  Constructions

This chapter has more practical problems than theory-based definitions. Beginning with a general introduction to constructions, tools used, etc, the topics like division of a line segment, construction of tangents to a circle, and followed with few solved examples that help in solving the exercises provided after each topic.

Chapter-12  Areas related to Circles

This chapter problem is exclusively formula based wherein topics like perimeter and area of a circle- A Review, areas of sector and segment of a circle, areas of combinations of plane figures, and a page summary is provided just as a revision of the topics and formulas covered in the entire chapter and also there are many exercises and solved examples for practice purposes.

Chapter-13  Surface Areas and Volumes

Starting with the introduction, the surface area of a combination of solids, the volume of a combination of solids, conversion of solid from one shape to another, frustum of a cone, etc are to name a few topics explained in detail provided with a set of examples for a better comprehension of the concepts.

Chapter-14  Statistics

In this chapter starting with an introduction, topics like mean of grouped data, mode of grouped data, a median of grouped, graphical representation of cumulative frequency distribution are explained in detail with exercises for practice purposes. This chapter being a simple and easy subject, securing the marks is not difficult for students.

Chapter-15  Probability

Probability is another simple and important chapter in examination point of view and as seeking knowledge purposes as well. Beginning with an introduction to probability, an important topic called A theoretical approach is explained here. Since this chapter is one of the smallest in the syllabus and problems are also quite easy, students often like this chapter

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CBSE 10th Standard Maths Subject Case Study Questions with Solutions

By QB365 on 20 May, 2021

QB365 Provides the updated CASE Study Questions for Class 10 , and also provide the detail solution for each and every case study questions . Case study questions are latest updated question pattern from NCERT, QB365 will helps to get  more marks in Exams

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CBSE Class 10 Maths Case Study Questions for Chapter 4 Quadratic Equations (Published by CBSE)

Cbse class 10 maths case study questions for chapter 4 - quadratic equations are released by the board. solve all these questions to perform well in your cbse class 10 maths exam 2021-22..

Check here the case study questions for CBSE Class 10 Maths Chapter 4 - Quadratic Equations. The board has published these questions to help class 10 students to understand the new format of questions. All the questions are provided with answers. Students must practice all the case study questions to prepare well for their Maths exam 2021-2022.

Case Study Questions for Class 10 Maths Chapter 4 - Quadratic Equations

CASE STUDY 1:

Raj and Ajay are very close friends. Both the families decide to go to Ranikhet by their own cars. Raj’s car travels at a speed of x km/h while Ajay’s car travels 5 km/h faster than Raj’s car. Raj took 4 hours more than Ajay to complete the journey of 400 km.

1. What will be the distance covered by Ajay’s car in two hours?

 a) 2(x + 5)km

b) (x – 5)km

c) 2(x + 10)km

d) (2x + 5)km

Answer: a) 2(x + 5)km

2. Which of the following quadratic equation describe the speed of Raj’s car?

a) x 2 – 5x – 500 = 0

b) x 2 + 4x – 400 = 0

c) x 2 + 5x – 500 = 0

d) x 2 – 4x + 400 = 0

Answer: c) x 2 + 5x – 500 = 0

3. What is the speed of Raj’s car?

a) 20 km/hour

b) 15 km/hour

c) 25 km/hour

d) 10 km/hour

Answer: a) 20 km/hour

4. How much time took Ajay to travel 400 km?

Answer: d) 16 hour

CASE STUDY 2:

The speed of a motor boat is 20 km/hr. For covering the distance of 15 km the boat took 1 hour more for upstream than downstream.

1. Let speed of the stream be x km/hr. then speed of the motorboat in upstream will be

a) 20 km/hr

b) (20 + x) km/hr

c) (20 – x) km/hr

Answer: c) (20 – x)km/hr

2. What is the relation between speed ,distance and time?

a) speed = (distance )/time

b) distance = (speed )/time

c) time = speed x distance

d) speed = distance x time

Answer: b) distance = (speed )/time

3. Which is the correct quadratic equation for the speed of the current?

a) x 2 + 30x − 200 = 0

b) x 2 + 20x − 400 = 0

c) x 2 + 30x − 400 = 0

d) x 2 − 20x − 400 = 0

Answer: c) x 2 + 30x − 400 = 0

4. What is the speed of current ?

b) 10 km/hour

c) 15 km/hour

d) 25 km/hour

Answer: b) 10 km/hour

5. How much time boat took in downstream?

a) 90 minute

b) 15 minute

c) 30 minute

d) 45 minute

Answer: d) 45 minute

Also Check:

CBSE Case Study Questions for Class 10 Maths - All Chapters

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• NCERT Solutions
• NCERT Solutions for Class 10

NCERT Solutions for Class 10 Maths

Ncert solutions for class 10 maths updated for 2023-24 session – free pdf download.

The answers to the questions present in the NCERT books are undoubtedly the best study material a student can get hold of. These CBSE NCERT Solutions for Class 10 Maths 2023-24 will also help students to build a deeper understanding of concepts covered in the textbook. Practising the textbook questions will help students analyse their level of preparation and knowledge of concepts. The solutions to these questions present in the NCERT books can help students to clear their doubts quickly.

NCERT Solutions Class 10 Maths Chapters NCERT Solutions Class 10 Maths Chapter Details CBSE Class 10 Maths Exam Pattern 2023-2024 CBSE Class 10 Chapter-wise Marks Weightage CBSE Question Paper Design

NCERT Solutions Class 10 Maths Chapters

• Chapter 1 Real Numbers
• Chapter 2 Polynomials
• Chapter 3 Pair of Linear Equations in Two Variables
• Chapter 4 Quadratic Equations
• Chapter 5 Arithmetic Progressions
• Chapter 6 Triangles
• Chapter 7 Coordinate Geometry
• Chapter 8 Introduction to Trigonometry
• Chapter 9 Some Applications of Trigonometry
• Chapter 10 Circles
• Chapter 11 Areas Related to Circles
• Chapter 12 Surface Areas and Volumes
• Chapter 13 Statistics
• Chapter 14 Probability

The following are the chapters that have been removed from the NCERT Class 10 Maths textbook 2023-24.

Constructions

NCERT Solutions for Class 10 Maths Free PDF Download

NCERT Solutions of Class 10 Maths list comprises all the chapter-wise answers to the questions present in the NCERT Book for Class 10 Maths, written in a very precise and lucid manner, maintaining the objective of textbooks. The students can refer to the  NCERT Solutions for Class 10 as their additional references and study materials. Practising NCERT textbook exercise solutions will surely help the students in their preparation for the examination.

NCERT Solutions Class 10 Maths Chapter Details and Exercises

Ncert solutions for class 10 maths chapter 1 real numbers.

In Chapter 1 of Class 10, students will explore real numbers and irrational numbers. The chapter starts with the Euclid’s Division Lemma, which states that “Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0≤r<b”. The Euclid’s Division algorithm is based on this lemma and is used to calculate the HCF of two positive integers. Then, the Fundamental Theorem of Arithmetic is defined, which is used to find the LCM and HCF of two positive integers. After that, the concept of an irrational number, a rational number and the decimal expansion of rational numbers are explained with the help of the theorem.

Topics Covered in Class 10 Maths Chapter 1 Real Numbers :

Fundamental Theorem of Arithmetic – statements after reviewing work done earlier and after illustrating and motivating through examples, Proofs of irrationality of √2, √3, √5

Important Steps –

To obtain the HCF of two positive integers, say c and d, with c > d, follow the steps below:

Step 1: Apply Euclid’s division lemma to c and d. So, we find whole numbers, q and r, such that c = dq + r, 0 ≤ r < d.

Step 2: If r = 0, d is the HCF of c and d. If r ≠ 0, apply the division lemma to d and r.

Step 3: Continue the process till the remainder is zero. The divisor at this stage will be the required HCF. This algorithm works because HCF (c, d) = HCF (d, r), where the symbol HCF (c, d) denotes the HCF of c and d, etc.

Also, access the following resources for NCERT Class 10 Chapter 1 Real Numbers, at BYJU’S:

• NCERT Real Numbers Class 10 Notes
• Real Numbers Class 10 Important Questions
• NCERT Exemplar Solutions Class 10 Real Numbers
• RD Sharma Solutions Real Numbers Class 10

NCERT Solutions for Class 10 Maths Chapter 2 Polynomials

In Polynomials , the chapter begins with the definition of the degree of the polynomial, linear polynomial, quadratic polynomial and cubic polynomial. This chapter has a total of 4 exercises, including an optional exercise. Exercise 2.1 includes questions on finding the number of zeroes through a graph. It requires an understanding of the Geometrical Meaning of the Zeroes of a Polynomial. Exercise 2.2 is based on the Relationship between Zeroes and Coefficients of a Polynomial, where students have to find the zeros of a quadratic polynomial and in some of the questions, they have to find the quadratic polynomial. In Exercise 2.3, the concept of the division algorithm is defined, and students will find the questions related to it. The optional exercise, 2.4, consists of the questions from all the concepts of Chapter 2.

Topics Covered in Class 10 Maths Chapter 2 Polynomials :

Zeros of a polynomial. Relationship between zeros and coefficients of quadratic polynomials.

We first arrange the terms of the dividend and the divisor in the decreasing order of their degrees. Recall that arranging the terms in this order is called writing the polynomials in standard form.

Step 1: To obtain the first term of the quotient, divide the highest degree term of the dividend by the highest degree term of the divisor. Then carry out the division process.

Step 2: Now, to obtain the second term of the quotient, divide the highest degree term of the new dividend by the highest degree term of the divisor. Again, carry out the division process.

Step 3: Now, the degree of the remainder is less than the degree of the divisor. So, we cannot continue the division any further.

Here again, we see that Dividend = Divisor × Quotient + Remainder. What we are applying here is an algorithm which is similar to Euclid’s division algorithm that you studied in Chapter 1.

This says that

If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that

p(x) = g(x) × q(x) + r(x),

where r(x) = 0 or degree of r(x) < degree of g(x).

This result is known as the Division Algorithm for polynomials.

Also, access the following resources for NCERT Class 10 Chapter 2 Polynomials, at BYJU’S:

• NCERT Polynomials Class 10 Notes
• Polynomials Class 10 Important Questions
• NCERT Exemplar Solutions Class 10 Polynomials
• RD Sharma Solutions Polynomials Class 10

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

This chapter explains the concept of Pair of Linear Equations in Two Variables . This chapter has a total of 7 exercises, and in these exercises, different methods of solving the pair of linear equations are described. Exercise 3.1 describes how to represent a situation algebraically and graphically. Exercise 3.2 explains the methods of solving the pair of the linear equation through the Graphical Method. Exercises 3.3, 3.4, 3.5 and 3.6 describe the Algebraic Method, Elimination Method, Cross-Multiplication Method, and Substitution Method, respectively. Exercise 3.7 is an optional exercise which contains all types of questions. Students must practise these exercises to master the method of solving linear equations.

Topics Covered in Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables:

Pair of linear equations in two variables and graphical method of their solution, consistency/inconsistency. Algebraic conditions for number of solutions. Solution of a pair of linear equations in two variables algebraically – by substitution, by elimination. Simple situational problems.

Important Formulas –

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

a 1 x + b 1 y + c 1 = 0

and a 2 x + b 2 y + c 2 = 0,

where a 1 , b 1 , c 1 , a 2 , b 2 , c 2 are all real numbers and a 1 2 + b 1 2 ≠ 0, a 2 2 + b 2 2 ≠ 0.

Also, access the following resources for NCERT Class 10 Chapter 3 Pair of Linear Equations in Two Variables, at BYJU’S:

• NCERT Pair of Linear Equations in Two Variables Class 10 Notes
• Pair of Linear Equations in Two Variables Class 10 Important Questions ,
• NCERT Exemplar Solutions Class 10 Pair of Linear Equations in Two Variables
• RD Sharma Solutions Pair of Linear Equations in Two Variables Class 10

NCERT Solutions of Class 10 Maths Chapter 4 Quadratic Equations

In this chapter, students will get to know the standard form of writing a Quadratic Equation . The chapter goes on to explain the method of solving the quadratic equation through the factorization method and completing the square method. The chapter ends with the topic on finding the nature of roots, which states that a quadratic equation ax² + bx + c = 0 has

• Two distinct real roots, if b² – 4ac > 0
• Two equal roots, if b² – 4ac = 0
• No real roots, if b² – 4ac < 0

Topics Covered in Class 10 Maths Chapter 4 Quadratic Equations :

The standard form of a quadratic equation ax 2 + bx + c = 0, (a ≠ 0). Solutions of quadratic equations (only real roots) by factorization and by using the quadratic formula. Relationship between discriminant and nature of roots. Situational problems based on quadratic equations related to day-to-day activities are to be incorporated.

If b 2 – 4ac > 0, we get two distinct real roots

If b 2 – 4ac = 0, then

So, the roots of the equation ax 2 + bx + c = 0 are both -b/2a.

Therefore, we say that the quadratic equation ax 2 + bx + c = 0 has two equal real roots in this case.

If b 2 – 4ac < 0, then there is no real number whose square is b 2 – 4ac. Therefore, there are no real roots for the given quadratic equation in this case.

Since b 2 – 4ac determines whether the quadratic equation ax 2 + bx + c = 0 has real roots or not, b 2 – 4ac is called the discriminant of this quadratic equation.

So, a quadratic equation ax 2 + bx + c = 0 has

(i) two distinct real roots, if b 2 – 4ac > 0,

(ii) two equal real roots, if b 2 – 4ac = 0,

(iii) no real roots, if b 2 – 4ac < 0.

Also, access the following resources for NCERT Class 10 Chapter 4 Quadratic Equations, at BYJU’S:

• NCERT Quadratic Equations Class 10 Notes
• Quadratic Equations Class 10 Important Questions
• NCERT Exemplar Solutions Class 10 Quadratic Equations
• RD Sharma Solutions Quadratic Equations Class 10

NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions

This chapter introduces students to a new topic, Arithmetic Progression , i.e. AP. The chapter constitutes a total of 4 exercises. In Exercise 5.1, students will find the questions related to representing a situation in the form of AP, finding the first term and difference of an AP, finding out whether a series is AP or not. Exercise 5.2 includes the questions on finding out the nth term of an AP by using the following formula:

a n  = a + (n-1) d

The next exercise, i.e. 5.3, contains the questions on finding the sum of the first n terms of an AP. The last exercise includes higher-level questions based on AP to enhance students’ analytical and problem-solving skills.

Topics Covered in Class 10 Maths Chapter 5 Arithmetic Progressions :

Motivation for studying Arithmetic Progression Derivation of the n th term and sum of the first n terms of A.P. and their application in solving daily life problems.

If a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , …  are the terms of AP and d is the common difference between each term, then we can write the sequence as; a ,  a+d, a+2d, a+3d, a+4d, a+5d,…., nth term… where a is the first term. Now, n th  term for arithmetic progression is given as;

n th  term = a + (n-1) d

Sum of the first n terms in Arithmetic Progression;

S n  = n/2 [2a + (n-1) d]

Also, access the following resources for NCERT Class 10 Chapter 5 Arithmetic Progressions, at BYJU’S:

• NCERT Arithmetic Progressions Class 10 Notes
• Arithmetic Progressions Class 10 Important Questions
• NCERT Exemplar Solutions Class 10 Arithmetic Progressions
• RD Sharma Solutions Arithmetic Progressions Class 10

NCERT Solutions for Class 10 Maths Chapter 6 Triangles

In Chapter 6 of Class 10 CBSE Maths, students will study those figures which have the same shape, but not necessarily the same size. The chapter Triangles starts with the concept of a similar and congruent figure. It further explains the condition for the similarity of two triangles and theorems related to the similarity of triangles. After that, the areas of similar triangles have been explained with a theorem. At the end of this chapter, the Pythagoras Theorem and the Converse of Pythagoras Theorem are described.

Topics Covered in Class 10 Maths Chapter 6 Triangles :

Definitions, examples, and counter examples of similar triangles. 1. (Prove) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. 2. (Motivate) If a line divides two sides of a triangle in the same ratio, the line is parallel to the third side. 3. (Motivate) If in two triangles, the corresponding angles are equal, their corresponding sides are proportional, and the triangles are similar. 4. (Motivate) If the corresponding sides of two triangles are proportional, their corresponding angles are equal, and the two triangles are similar. 5. (Motivate) If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the two triangles are similar.

Important Theorems –

Theorem 6.1: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

Theorem 6.2: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

Theorem 6.3: If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion), and hence the two triangles are similar.

Theorem 6.4: If in two triangles, the sides of one triangle are proportional to (i.e., in the same ratio of ) the sides of the other triangle, then their corresponding angles are equal, and hence the two triangles are similar.

Theorem 6.5: If one angle of a triangle is equal to one angle of the other 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 equal to the square of the ratio of their corresponding sides.

Theorem 6.7: If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then 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.

Theorem 6.9: In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

Also, access the following resources for NCERT Class 10 Chapter 6 Triangles, at BYJU’S:

• NCERT Triangles Class 10 Notes
• Triangles Class 10 Important Questions
• NCERT Exemplar Solutions Class 10 Triangles
• RD Sharma Solutions Triangles Class 10

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

In this chapter, students will learn how to find the distance between two points whose coordinates are given, and to find the area of the triangle formed by three given points. Along with this, students will also study how to find the coordinates of the point which divides a line segment joining two given points in a given ratio. For this purpose, students will get introduced to the Distance Formula, Section Formula and Area of a Triangle in this chapter on Coordinate Geometry .

Topics Covered in Class 10 Maths Chapter 7 Coordinate Geometry :

Review: Concepts of coordinate geometry, graphs of linear equations. Distance formula. Section formula (internal division).

Distance Formula

Section Formula

Also, access the following resources for NCERT Class 10 Chapter 7 Coordinate Geometry, at BYJU’S:

• NCERT Coordinate Geometry Class 10 Notes
• Coordinate Geometry Class 10 Important Questions
• NCERT Exemplar Solutions Class 10 Coordinate Geometry
• RD Sharma Solutions Coordinate Geometry Class 10

NCERT Solutions for Class 10 Maths Chapter 8 Introduction to Trigonometry

This chapter will introduce students to Trigonometry . They will study some ratios of a right triangle with respect to its acute angles, called trigonometric ratios of the angles. The chapter also defines the trigonometric ratios for angles of 0 0 and 90 0 . Further, students will also know how to calculate trigonometric ratios for some specific angles and establish some identities involving these ratios, called trigonometric identities.

Topics Covered in Class 10 Maths Chapter 8 Introduction to Trigonometry :

Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined); motivate the ratios, whichever are defined at 0 o and 90 o . Values of the trigonometric ratios of 30 0 , 45 0 and 60 0 . Relationships between the ratios. TRIGONOMETRIC IDENTITIES Proof and applications of the identity sin 2 A + cos 2 A = 1. Only simple identities to be given

Trigonometry Maths formulas for Class 10 cover three major functions Sine, Cosine and Tangent for a right-angle triangle. Let a right-angled triangle ABC is right-angled at point B and have ∠θ.

Sin θ= \(\begin{array}{l}\frac{Side\, opposite\, to\, angle\, \theta}{Hypotenuse}\end{array} \) = \(\begin{array}{l}\frac{Perpendicular}{Hypotenuse}\end{array} \) = P/H

Cos θ = \(\begin{array}{l}\frac{Adjacent\, side\, to\, angle\, \theta}{Hypotenuse}\end{array} \) = \(\begin{array}{l}\frac{Base}{Hypotenuse}\end{array} \) = B/H

Tan θ = \(\begin{array}{l}\frac{Side\, opposite\, to\, angle\, \theta}{Adjacent\, side\, to\, angle\, \theta}\end{array} \) = P/B

Sec θ = \(\begin{array}{l}\frac{1}{cos\, \theta }\end{array} \)

Cot θ = \(\begin{array}{l}\frac{1}{tan\, \theta }\end{array} \)

Cosec θ = \(\begin{array}{l}\frac{1}{sin\, \theta }\end{array} \)

Tan θ = \(\begin{array}{l}\frac{Sin\, \theta }{Cos\, \theta }\end{array} \)

Trigonometry Table

Trigonometric Ratios of Complementary Angles

sin (90° – A) = cos A,

cos (90° – A) = sin A,

tan (90° – A) = cot A,

cot (90° – A) = tan A,

sec (90° – A) = cosec A,

cosec (90° – A) = sec A

sin 2 A + cos 2 A = 1,

sec 2 A – tan 2 A = 1 for 0° ≤ A < 90°,

cosec 2 A = 1 + cot 2 A for 0° < A ≤ 90°

Also, access the following resources for NCERT Class 10 Chapter 8 Trigonometry, at BYJU’S: 

• NCERT Introduction to Trigonometry Class 10 Notes
• Introduction to Trigonometry Class 10 Important Questions
• NCERT Exemplar Solutions Class 10 Introduction to Trigonometry
• RD Sharma Solutions Introduction to Trigonometry Class 10

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

This chapter is the continuation of the previous chapter; here, the students will study the applications of trigonometry . It is used in geography, navigation, construction of maps, and determining the position of an island in relation to the longitudes and latitudes. In this chapter, students will see how trigonometry is used for finding the heights and distances of various objects without actually measuring them. They will get introduced to the term line of sight, angle of elevation, and angle of depression.

Topics Covered in Class 10 Maths Chapter 9 Some Applications of Trigonometry:

HEIGHTS AND DISTANCES – Angle of elevation, Angle of Depression. Simple problems on heights and distances. Problems should not involve more than two right triangles. Angles of elevation/depression should be only 30°, 45°, and 60°.

Important Points –

The line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer.

The angle of elevation of the point viewed is the angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level, i.e. the case when we raise our head to look at the object.

The angle of depression of a point on the object being viewed is the angle formed by the line of sight with the horizontal when the point is below the horizontal level, i.e. the case when we lower our head to look at the point being viewed.

You would need to know the following:

(i) The distance DE at which the student is standing from the foot of the minar

(ii) The angle of elevation, ∠ BAC, of the top of the minar

(iii) The height AE of the student.

Assuming that the above three conditions are known, how can we determine the height of the minar?

In the figure, CD = CB + BD. Here, BD = AE, which is the height of the student.

To find BC, we will use trigonometric ratios of ∠ BAC or ∠ A.

In ∆ ABC, the side BC is the opposite side in relation to the known ∠ A. Our search narrows down to using either tan A or cot A, as these ratios involve AB and BC.

Therefore, tan A = BC/AB or cot A = AB/BC, which on solving, would give us BC.

By adding AE to BC, you will get the height of the minar.

Also, access the following resources for NCERT Class 10 Chapter 9, Some Applications of Trigonometry, at BYJU’S: 

• NCERT Some Applications of Trigonometry Class 10 Notes
• Some Applications of Trigonometry Class 10 Important Questions
• NCERT Exemplar Solutions Class 10 Some Applications of Trigonometry
• RD Sharma Solutions Some Applications of Trigonometry Class 10

NCERT Solutions for Class 10 Maths Chapter 10 Circles

In earlier classes, students have studied about a circle and various terms related to a circle, such as a chord, segment, arc, etc. In this chapter, students will study the different situations that arise when a circle and a line are given in a plane. So, they will get thorough with the concept of Tangent to a Circle and Number of Tangents from a Point on a Circle.

Topics Covered in Class 10 Maths Chapter 10 Circles:

Tangent to a circle at point of contact 1. (Prove) The tangent at any point of a circle is perpendicular to the radius through the point of contact. 2. (Prove) The lengths of tangents drawn from an external point to a circle are equal.

Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Theorem 10.2: The lengths of tangents drawn from an external point to a circle are equal.

Number of Tangents from a Point on a Circle

Case 1: There is no tangent to a circle passing through a point lying inside the circle.

Case 2: There is one and only one tangent to a circle passing through a point lying on the circle.

Case 3: There are exactly two tangents to a circle through a point lying outside the circle.

Also, access the following resources for NCERT Class 10 Chapter 10 Circles at BYJU’S: 

• NCERT Circles Class 10 Notes
• Circles Class 10 Important Questions
• NCERT Exemplar Solutions Class 10 Circles
• RD Sharma Solutions Circles Class 10

NCERT Solutions for Class 10 Maths Chapter 11 Constructions

This chapter consists of a total of 2 exercises. Whatever students have learned about construction in earlier classes will also help them. In Exercise 11.1, students will study how to divide a line segment, whereas in Exercise 11.2, they will study the construction of tangents to a circle. Methods and steps for construction are explained, and also some examples are additionally given to make it clearer to the students.

Topics Covered in Class 10 Maths Chapter 11 Constructions :

1. Division of a line segment in a given ratio (internally). 2. Tangent to a circle from a point outside it. 3. Construction of a triangle similar to a given triangle.

Construction 11.1: To divide a line segment in a given ratio.

Construction 11.2: To construct a triangle similar to a given triangle as per the given scale factor.

Construction 11.3: To construct the tangents to a circle from a point outside it.

Also, access the following resources for NCERT Class 10 Chapter 11 Constructions, at BYJU’S:

• NCERT Constructions Class 10 Notes
• Constructions Class 10 Important Questions
• NCERT Exemplar Solutions Class 10 Constructions
• RD Sharma Solutions Constructions Class 10

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

This chapter begins with the concepts of the perimeter and area of a circle. Using this concept, the chapter further explains how to find the area of sector and segment of a circular region. Moreover, students will get clarity on finding  the areas of some combinations of plane figures involving circles or their parts.

Topics Covered in Class 10 Maths Chapter 12 Areas Related to Circles :

Area of sectors and segments of a circle. Problems based on areas and perimeter/circumference of the above-said plane figures. (In calculating the area of segment of a circle, problems should be restricted to the central angle of 60°, 90° and 120° only.

circumference = 2πr

area of the circle = πr 2

Area of the sector of angle θ = (θ/360) × π r 2

Length of an arc of a sector of angle θ = (θ/360) × 2 π r where r is the radius of the circle

Also access the following resources for NCERT Class 10 Chapter 12 Areas Related to Circles at BYJU’S: 

• NCERT Areas Related to Circles Class 10 Notes
• Areas Related to Circles Class 10 Important Questions
• NCERT Exemplar Solutions Class 10 Areas Related to Circles
• RD Sharma Solutions Areas Related to Circles Class 10

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

In Chapter 13, there are a total of 5 exercises. The first exercise consists of questions based on finding the surface area of an object formed by combining any two of the basic solids, i.e. cuboid, cone, cylinder , sphere and hemisphere. In exercise 13.2, questions are based on finding the volume of objects formed by combining any two of a cuboid, cone, cylinder, sphere and hemisphere. Exercise 13.3 deals with the questions in which a solid is converted from one shape to another. Exercise 13.4 is based on finding the volume, curved surface area and total surface area of a frustum of a cone. The last exercise is optional and has high-level questions based on all the topics of this chapter.

Topics Covered in Class 10 Maths Chapter 13 Surface Areas and Volumes:

Surface areas and volumes of combinations of any two of the following: cubes, cuboids, spheres, hemispheres and right circular cylinders/cones.

TSA of new solid = CSA of one hemisphere + CSA of cylinder + CSA of other hemisphere

Diameter of sphere = 2r

Surface area of sphere = 4 π r 2

Volume of Sphere = 4/3 π r 3

Curved surface area of Cylinder = 2 πrh

Area of two circular bases = 2 πr 2

Total surface area of Cylinder = Circumference of Cylinder + Curved surface area of Cylinder = 2 πrh + 2 πr 2

Volume of Cylinder = π r 2  h

Slant height of cone = l = √(r 2  + h 2 )

Curved surface area of cone = πrl

Total surface area of cone = πr (l + r)

Volume of cone = ⅓ π r 2  h

Perimeter of cuboid = 4(l + b +h)

Length of the longest diagonal of a cuboid = √(l 2  + b 2  + h 2 )

Total surface area of cuboid = 2(l×b + b×h + l×h)

Volume of Cuboid = l × b × h

Also, access the following resources for NCERT Class 10 Chapter 13 Surface Areas and Volumes, at BYJU’S: 

• NCERT Surface Areas and Volumes Class 10 Notes
• Surface Areas and Volumes Class 10 Important Questions
• NCERT Exemplar Solutions Class 10 Surface Areas and Volumes
• RD Sharma Solutions Surface Areas and Volumes Class 10

NCERT Solutions for Class 10 Maths Chapter 14 Statistics

Here, students will learn the numerical representation of ungrouped data to grouped data and find the Mean, Mode and Median. Also, the concept of cumulative frequency, cumulative frequency distribution and how to draw cumulative frequency curves will be explained.

Topics Covered in Class 10 Maths Chapter 14 Statistics :

Mean, median and mode of grouped data (bimodal situation to be avoided).

The mean of the grouped data  can be found by 3 methods.

• Direct Method: x̅ = \(\begin{array}{l}\frac{\sum_{i=1}^{n}f_i x_i}{\sum_{i=1}^{n}f_i}\end{array} \) , where ∑f i x i is the sum of observations from value i = 1 to n And ∑f i is the number of observations from value i = 1 to n
• Assumed mean method : x̅ = \(\begin{array}{l}a+\frac{\sum_{i=1}^{n}f_i d_i}{\sum_{i=1}^{n}f_i}\end{array} \)
• Step deviation method : x̅ = \(\begin{array}{l}a+\frac{\sum_{i=1}^{n}f_i u_i}{\sum_{i=1}^{n}f_i}\times h\end{array} \)

The mode of grouped data:

Mode = \(\begin{array}{l}l+\frac{f_1 – f_0}{2f_1 – f_0 – f_2} \times h\end{array} \)

The median for a grouped data:

Median = \(\begin{array}{l}l+\frac{\frac{n}{2} – cf}{f} \times h\end{array} \)

Also, access the following resources for NCERT Class 10 Chapter 14 Statistics, at BYJU’S: 

• NCERT Statistics Class 10 Notes
• Statistics Class 10 Important Questions
• NCERT Exemplar Solutions Class 10 Statistics
• RD Sharma Solutions Statistics Class 10

NCERT Solutions for Class 10 Maths Chapter 15 Probability

The last chapter deals with Probability. The chapter starts with the theoretical approach of probability. Subsequently, the chapter explains the difference between  experimental probability and theoretical probability. There are various examples given to  explain it in a n effective way . So, before going through the exercise problems, students must solve the examples of CBSE Maths first.

Topics Covered in Class 10 Maths Chapter 15 Probability:

Classical definition of probability. Simple problems on finding the probability of an event.

• The theoretical probability (also called classical probability) of an event E, written as P(E), is defined as

where we assume that the outcomes of the experiment are equally likely.

• The probability of a sure event (or certain event) is 1.
• The probability of an impossible event is 0.
• The probability of an event E is a number P(E) such that 0 ≤ P (E) ≤ 1
• An event having only one outcome is called an elementary event. The sum of the probabilities of all the elementary events of an experiment is 1.

Also, access the following resources for NCERT Class 10 Chapter 15 Probability, at BYJU’S: 

• NCERT Probability Class 10 Notes
• Probability Class 10 Important Questions
• NCERT Exemplar Solutions Class 10 Probability
• RD Sharma Solutions Probability Class 10

NCERT Solutions for Class 10 Maths PDF in English Medium, as well as Hindi Medium (हिंदी मीडियम) for the academic year 2023-24, are followed by not only CBSE but also UP Board, Uttarakhand board and all other boards following NCERT Textbooks.

CBSE Class 10 Maths Chapter-wise Marks Weightage for 2023-24

 internal assessment class 10, benefits of ncert solutions class 10 maths:.

The Class 10 NCERT Solutions of Maths given here in PDFs have several benefits, which include:

• The solutions given here are very easy-to-understand.
• Solutions are provided in steps for better understanding.
• Diagrams are given to help the students to visualise the solutions.
• All the questions from each chapter are covered.

Students are advised to prepare all the chapters covered in the solution modules, which will eventually help them to gain a deeper knowledge of concepts. For effective preparations, it is essential for students to understand all the steps provided in the solutions.

How are CBSE Class 10 Maths Solutions of NCERT Helpful for the Board Exams?

CBSE Class 10 Maths is an important subject for students. Here, we have provided complete assistance to students for their preparation. Class 10 is the first benchmark for any student that will reflect in future records of achievement. CBSE always prescribes the NCERT books for exam preparation.

Exam preparation is a rigorous process that requires an overall understanding of individual chapters. This process demands hard work towards studies and an effective approach to getting through the solutions. This significant role is played by NCERT 10 Class Maths Solutions to equip students in preparation for competitive entrance exams. NCERT books are best known for putting forth the concepts in a simple way for better understanding. NCERT Class 10 Books for Mathematics are written in the most lucid and clear manner that helps to break the complex problems in the most efficient way.

Keep visiting to get complete chapter-wise NCERT Solutions for Class 10 Maths PDF free download for all the classes. Students can also find the NCERT Solutions for Class 10 Science , at BYJU’S. Students can download BYJU’S App to get a personalised learning experience and prepare for the exams more effectively.

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