Wise solutions, maths problems, exam preparation
Class 11 Maths NCERT solutions offer comprehensive explanations and step-by-step guides for each topic, making complex concepts easier to grasp. These solutions are perfect for exam preparation, helping students build a strong foundation in algebra, geometry, calculus, and more. With expert-backed recommendations, mastering Class 11 Maths becomes achievable.
Class 11 Maths NCERT Solutions: Sample Questions and Answers
Sample Question 1: Find the value of the following determinant:
det(2357)\text{det} \left( \begin{matrix} 2 & 3 \\ 5 & 7 \end{matrix} \right)det(2537)
Answer:
The determinant of a 2x2 matrix is given by ad−bcad - bcad−bc, where the matrix is:
(abcd)\left( \begin{matrix} a & b \\ c & d \end{matrix} \right)(acbd)
For the given matrix:
(2357)\left( \begin{matrix} 2 & 3 \\ 5 & 7 \end{matrix} \right)(2537)
The determinant is (2×7)−(3×5)=14−15=−1(2 \times 7) - (3 \times 5) = 14 - 15 = -1(2×7)−(3×5)=14−15=−1.
Sample Question 2: Find the distance between the points (2, 3) and (5, 7). Answer:
The distance between two points (x1,y1)(x_1, y_1)(x1,y1) and (x2,y2)(x_2, y_2)(x2,y2) is given by the formula:
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
Substituting the values:
d=(5−2)2+(7−3)2=9+16=25=5d = \sqrt{(5 - 2)^2 + (7 - 3)^2} = \sqrt{9 + 16} = \sqrt{25} = 5d=(5−2)2+(7−3)2=9+16=25=5
So, the distance is 5 units.
Sample Question 3: Solve for x: x2−5x+6=0x^2 - 5x + 6 = 0x2−5x+6=0. Answer:
This is a quadratic equation. Factorizing the equation:
x2−5x+6=(x−2)(x−3)=0x^2 - 5x + 6 = (x - 2)(x - 3) = 0x2−5x+6=(x−2)(x−3)=0
Setting each factor equal to 0:
x−2=0orx−3=0x - 2 = 0 \quad \text{or} \quad x - 3 = 0x−2=0orx−3=0
Thus, x=2x = 2x=2 or x=3x = 3x=3.
Sample Question 4: Find the slope of the line passing through the points (1, 2) and (3, 6). Answer:
The slope of a line through two points (x1,y1)(x_1, y_1)(x1,y1) and (x2,y2)(x_2, y_2)(x2,y2) is given by:
slope=y2−y1x2−x1\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}slope=x2−x1y2−y1
Substituting the given points:
slope=6−23−1=42=2\text{slope} = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2slope=3−16−2=24=2
So, the slope is 2.
Sample Question 5: What is the area of a triangle with vertices at (0, 0), (4, 0), and (0, 3)? Answer:
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 given by:
Area=12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣\text{Area} = \frac{1}{2} | x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) |Area=21∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣
Substituting the values:
Area=12∣0(0−3)+4(3−0)+0(0−0)∣=12∣0+12+0∣=122=6\text{Area} = \frac{1}{2} | 0(0 - 3) + 4(3 - 0) + 0(0 - 0) | = \frac{1}{2} | 0 + 12 + 0 | = \frac{12}{2} = 6Area=21∣0(0−3)+4(3−0)+0(0−0)∣=21∣0+12+0∣=212=6
So, the area is 6 square units.
Class 11 Chemistry NCERT Solutions: Sample Questions and Answers
Next, let's explore some sample questions from Class 11 Chemistry. These questions focus on important concepts in the subject, offering both theoretical and practical approaches.
Sample Question 1: What is the molecular mass of methane (CH₄)? Answer:
The molecular mass of methane (CH₄) is the sum of the atomic masses of its constituent atoms:
- Carbon (C) has an atomic mass of 12.
- Hydrogen (H) has an atomic mass of 1, and there are 4 hydrogen atoms.
Thus, the molecular mass of CH₄ is:
12+(4×1)=12+4=1612 + (4 \times 1) = 12 + 4 = 1612+(4×1)=12+4=16
So, the molecular mass of methane is 16 g/mol.
Sample Question 2: What is an exothermic reaction? Give an example. Answer:
An exothermic reaction is a chemical reaction that releases energy in the form of heat or light. One common example is the combustion of methane:
CH4+2O2→CO2+2H2O+heatCH₄ + 2O₂ \rightarrow CO₂ + 2H₂O + \text{heat}CH4+2O2→CO2+2H2O+heat
In this reaction, heat is released as a product.
Sample Question 3: What is the difference between an ionic bond and a covalent bond? Answer:
An ionic bond is formed when one atom donates an electron to another atom, resulting in the formation of oppositely charged ions that are held together by electrostatic attraction. For example, in NaCl (sodium chloride), sodium donates an electron to chlorine.
A covalent bond, on the other hand, is formed when two atoms share electrons. For example, in H₂O (water), hydrogen and oxygen atoms share electrons to form a stable molecule.
Sample Question 4: Define the term 'molarity'. Answer:
Molarity (M) is defined as the number of moles of solute present in one liter of solution. It is given by the formula:
Molarity=Moles of soluteVolume of solution in liters\text{Molarity} = \frac{\text{Moles of solute}}{\text{Volume of solution in liters}}Molarity=Volume of solution in litersMoles of solute
Sample Question 5: What is the pH of pure water? Answer:
The pH of pure water is 7. This is because pure water is neutral, meaning the concentration of hydrogen ions (H⁺) is equal to the concentration of hydroxide ions (OH⁻), both being 1×10−71 \times 10^{-7}1×10−7 mol/L.
Class 11 Physics NCERT Solutions: Sample Questions and Answers
Finally, let's look at some sample questions and answers from Class 11 Physics, designed to help you grasp key concepts in mechanics, electricity, and more.
Sample Question 1: What is Newton's First Law of Motion? Answer:
Newton's First Law of Motion states that an object at rest will remain at rest, and an object in motion will continue to move with a constant velocity in a straight line, unless acted upon by an external force. This is also known as the law of inertia.
Sample Question 2: What is the SI unit of force? Answer:
The SI unit of force is the Newton (N), which is defined as the force required to accelerate a one-kilogram mass by one meter per second squared.
Sample Question 3: What is the law of conservation of energy? Answer:
The law of conservation of energy states that energy cannot be created or destroyed; it can only be transformed from one form to another. The total energy of an isolated system remains constant.
Sample Question 4: Explain the concept of acceleration. Answer:
Acceleration is the rate of change of velocity of an object with respect to time. It is a vector quantity, meaning it has both magnitude and direction. The SI unit of acceleration is meters per second squared (m/s²).
Sample Question 5: What is the relationship between work and energy? Answer:
Work is said to be done when a force causes displacement of an object. The amount of work done is equal to the change in the energy of the object. The SI unit of work is the joule (J), and it is numerically equal to the energy transferred.
Best Indian Books for Class 11 Maths NCERT Solutions
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R.D. Sharma - Mathematics for Class 11
- Publisher: Dhanpat Rai Publications
- Content: This book provides a clear explanation of basic concepts, step-by-step solutions to problems, and multiple practice exercises for every chapter, enhancing problem-solving skills.
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S. Chand – Class 11 Mathematics
- Publisher: S. Chand Publishing
- Content: The book covers all NCERT topics with additional questions to practice, including various difficulty levels. It also includes solved examples that offer a deeper understanding of concepts.
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M.L. Khanna – Higher Secondary Mathematics for Class 11
- Publisher: K.K. Gupta & Co.
- Content: Known for its detailed explanations, this book is ideal for students aiming to build a thorough understanding of each concept. It features a variety of problems, ranging from simple to complex.
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Laxmi Publications – Mathematics for Class 11
- Publisher: Laxmi Publications
- Content: The book is designed to strengthen problem-solving skills and includes both solved and unsolved questions with detailed explanations. It also offers additional practice material.
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I.A. Maron – Problems in Calculus of One Variable
- Publisher: Arihant Publications
- Content: This book is a comprehensive guide for calculus, one of the key areas in Class 11 Maths. It offers practice problems for a deeper understanding and mastering of calculus concepts.
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Arihant - Mathematics for Class 11
- Publisher: Arihant Publications
- Content: Known for its wide array of problems and detailed solutions, this book offers chapter-wise explanation along with additional important questions. It also provides a well-structured approach to solving problems.
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R.S. Aggarwal – Mathematics for Class 11
- Publisher: S. Chand Publishing
- Content: This book is well-structured, with clear explanations and a variety of problems. It features a variety of types of questions, such as multiple-choice, long-answer, and short-answer.
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Cengage Learning - Mathematics for Class 11
- Publisher: Cengage Learning
- Content: This book is comprehensive and detailed, offering students practice on a wide range of topics. It’s perfect for those who seek advanced problems in addition to NCERT solutions.
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K.C. Sinha – Mathematics for Class 11
- Publisher: S. Chand Publishing
- Content: A great book for self-study, it covers the NCERT syllabus thoroughly. It features practice exercises, previous years’ questions, and hints for solving problems.
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P.K. Gupta – Mathematics for Class 11
- Publisher: Arihant Publications
- Content: This book focuses on enhancing problem-solving techniques and understanding key concepts. It includes a wide variety of questions, from basic to advanced, to cater to different learning needs.
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B.D. Sharma – Mathematics for Class 11
- Publisher: Dhanpat Rai Publications
- Content: Known for its clear explanations, the book focuses on developing problem-solving skills. It is designed for students to master both basic and advanced topics in Class 11 Maths.
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K. R. S. N. – Maths for Class 11
- Publisher: Pradeep Publications
- Content: This book is suitable for practice and self-assessment, offering a wide range of exercises based on the NCERT syllabus.
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V.K. Kapoor – Mathematics for Class 11
- Publisher: Sultan Chand & Sons
- Content: This book provides ample problems for each topic, allowing students to develop their skills progressively. It is highly recommended for both beginners and advanced learners.
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V. V. M. Sharma – Class 11 Mathematics
- Publisher: Vikas Publications
- Content: This book offers clear explanations along with practice questions for each concept. It’s a great supplementary resource to tackle complex questions.
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R.K. Gupta – Mathematics for Class 11
- Publisher: New Age International Publishers
- Content: This book features numerous solved and unsolved problems, helping students to refine their problem-solving abilities. It is great for revising concepts and practicing.
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C. L. Arora – Mathematics for Class 11
- Publisher: S. Chand Publishing
- Content: C.L. Arora’s book is ideal for students who wish to strengthen their basic concepts. It provides easy-to-understand solutions and numerous examples to guide students.
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K.C. Das – Class 11 Mathematics
- Publisher: R.Lall Publications
- Content: The book focuses on breaking down complex mathematical problems into simple steps. It covers all the major topics included in the NCERT syllabus.
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D.C. Pandey – Problems in Calculus for Class 11
- Publisher: Arihant Publications
- Content: This book offers a great way to practice calculus, a crucial topic in Class 11 Maths. It provides ample problems with solutions for self-assessment.
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R.S. Aggarwal – Objective Mathematics for Class 11
- Publisher: S. Chand Publishing
- Content: This book is packed with objective-type questions, offering a unique approach to practicing maths for competitive exams and school exams alike.
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V.K. Suri – Problems in Trigonometry
- Publisher: Laxmi Publications
- Content: Focused on trigonometry, this book provides a series of problems ranging from simple to advanced, designed to build proficiency in the subject.
Class 11 Maths NCERT Solutions: Expert Recommendations for Mastering the Subject
Mathematics in Class 11 is pivotal, setting the tone for not only the following academic years but also for various competitive exams like JEE and NEET. The NCERT Solutions for Class 11 Maths serve as the cornerstone for students' preparation. These solutions guide students through the rigorous and detailed study required to understand complex mathematical concepts.
The primary benefit of working with NCERT Solutions is that they are meticulously aligned with the CBSE syllabus. This ensures that students learn the fundamental principles that are both integral to their academic growth and key for exams. The NCERT Solutions are structured in a step-by-step approach, which makes it easier for students to grasp each concept, no matter how difficult it may initially seem.
Students should focus on mastering the basics first. The best way to approach Class 11 Maths is by first solving the NCERT exercises thoroughly, and then practicing additional questions from reference books. This method ensures that the core concepts are solidified before moving on to complex problems.
While working on problems, students must focus on understanding the logic behind each solution rather than memorizing the steps. This understanding will not only help in solving similar problems but also in developing a deeper analytical mindset that will aid in solving competitive exam questions. Practice is key. Working through varied problem types—from basic to advanced—will enhance speed and accuracy.
Additionally, when using reference books, students must carefully choose those that provide detailed explanations and ample practice questions. Books like R.D. Sharma and R.S. Aggarwal complement the NCERT syllabus perfectly and offer clear step-by-step solutions, ensuring thorough concept clarity.
Another essential element in mastering Class 11 Maths is self-assessment. After completing each chapter, students should attempt to solve previous year’s question papers and sample papers. This will not only boost confidence but also highlight areas that need more attention.
For concepts like calculus, trigonometry, and algebra, focused practice with a variety of problems is essential. Books by authors like I.A. Maron and M.L. Khanna provide comprehensive coverage of these advanced topics, helping students to tackle higher difficulty-level problems effectively.
In conclusion, Class 11 Maths requires consistent effort and smart study techniques. By focusing on NCERT Solutions and supplementing learning with the right reference books, students can build a strong mathematical foundation. The goal should always be a deep understanding of the subject, as this will help excel not only in school exams but also in competitive tests.
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