comprehensive, easy-to-understand, solutions for all chapters, step-by-step

Class 12 Maths NCERT Solutions for all chapters with step-by-step explanations and easy-to-understand methods to help you prepare for exams effectively.

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Class 12 Maths NCERT Solutions: Questions and Answers

Question 1: What is the derivative of x2+3x+2x^2 + 3x + 2x2+3x+2?

Answer: The derivative of x2+3x+2x^2 + 3x + 2x2+3x+2 with respect to $x$ is $2x+3$.

Question 2: How do you find the area under the curve of y=x2y = x^2y=x2 between $x=0$ and $x=1$?

Answer: The area under the curve is calculated using integration. The integral of x2x^2x2 from 0 to 1 is ∫01x2dx=13\int_0^1 x^2 dx = \frac{1}{3}∫01​x2dx=31​.

Question 3: What is the integral of $\sin x$ with respect to $x$?

Answer: The integral of $\sin x$ with respect to $x$ is $-\cos x+C$, where $C$ is the constant of integration.

Question 4: If the probability of an event occurring is 0.4, what is the probability of it not occurring?

Answer: The probability of the event not occurring is $1-0.4=0.6$.

Question 5: What is the equation of the line passing through the points $(1,2)$ and $(3,4)$?

Answer: The slope of the line is 4−23−1=1\frac{4 - 2}{3 - 1} = 13−14−2​=1. Using the point-slope form, the equation is $y-2=1(x-1)$, which simplifies to $y=x+1$.

Question 6: What is the value of sin⁡45∘ \sin 45^\circsin45∘?

Answer: The value of sin⁡45∘\sin 45^\circsin45∘ is 22\frac{\sqrt{2}}{2}22​​.

Question 7: How do you calculate the mean of the following numbers: 2, 5, 7, 10, 3?

Answer: The mean is the sum of the numbers divided by the number of numbers. So, 2+5+7+10+35=275=5.4\frac{2 + 5 + 7 + 10 + 3}{5} = \frac{27}{5} = 5.452+5+7+10+3​=527​=5.4.

Question 8: What is the volume of a sphere with radius 3 units?

Answer: The volume of a sphere is given by the formula 43πr3\frac{4}{3} \pi r^334​πr3. Substituting $r=3$, we get 43π(3)3=36π cubic units\frac{4}{3} \pi (3)^3 = 36\pi \, \text{cubic units}34​π(3)3=36πcubic units.

Question 9: How do you solve the quadratic equation x2−5x+6=0x^2 - 5x + 6 = 0x2−5x+6=0?

Answer: The factors of x2−5x+6x^2 - 5x + 6x2−5x+6 are $(x-2)(x-3)=0$, so the solutions are $x=2$ and $x=3$.

Question 10: What is the derivative of exe^xex?

Answer: The derivative of exe^xex with respect to $x$ is simply exe^xex.

Question 11: What is the second derivative of x3+2x2−xx^3 + 2x^2 - xx3+2x2−x?

Answer: The first derivative is 3x2+4x−13x^2 + 4x - 13x2+4x−1. The second derivative is $6x+4$.

Question 12: How do you find the standard deviation of the following data: 1, 4, 6, 8, 10?

Answer: First, find the mean, then calculate the squared differences from the mean, and finally take the square root of the average of these squared differences.

Question 13: What is the distance between the points $(1,2)$ and $(4,6)$?

Answer: The distance between two points is given by (x2−x1)2+(y2−y1)2\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}(x2​−x1​)2+(y2​−y1​)2​. Substituting the given values: (4−1)2+(6−2)2=9+16=25=5\sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5(4−1)2+(6−2)2​=9+16​=25​=5 units.

Question 14: What is the equation of a circle with center $(2,3)$ and radius 4?

Answer: The equation of the circle is (x−2)2+(y−3)2=16(x - 2)^2 + (y - 3)^2 = 16(x−2)2+(y−3)2=16.

Question 15: How do you find the median of the following data: 2, 5, 7, 10, 12?

Answer: The median is the middle value of the data. Since there are 5 numbers, the median is the third number, which is 7.

Question 16: What is the solution to the system of equations $x+y=5$ and $x-y=1$?

Answer: Adding the two equations gives $2x=6$, so $x=3$. Substituting $x=3$ into $x+y=5$ gives $3+y=5$, so $y=2$.

Question 17: How do you simplify the expression (x2−9)/(x−3) (x^2 - 9) / (x - 3)(x2−9)/(x−3)?

Answer: Factor the numerator as $(x-3)(x+3)$, and cancel the $(x-3)$ terms to get $x+3$.

Question 18: What is the sum of the first 10 natural numbers?

Answer: The sum of the first $n$ natural numbers is n(n+1)2\frac{n(n + 1)}{2}2n(n+1)​. For $n=10$, the sum is 10(10+1)2=55\frac{10(10 + 1)}{2} = 55210(10+1)​=55.

Question 19: How do you solve the inequality $2x-3<5$?

Answer: Add 3 to both sides to get $2x<8$, then divide by 2 to get $x<4$.

Question 20: What is the area of a right triangle with base 5 and height 12?

Answer: The area of a right triangle is 12×base×height\frac{1}{2} \times \text{base} \times \text{height}21​×base×height, so the area is 12×5×12=30\frac{1}{2} \times 5 \times 12 = 3021​×5×12=30 square units.

Top Indian Books for Class 12 Mathematics NCERT Solutions

1. 1. Problems in Calculus of One Variable by I.A. Maron

      • Publication: Sharma Prakashan
      • Content: This book focuses primarily on calculus, with in-depth explanations of limits, continuity, differentiation, and integration. The problems range from basic to advanced, providing a comprehensive understanding of the subject.
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   3. Integral Calculus for JEE Main & Advanced by Amit M Agarwal

      • Publication: Arihant Publications
      • Content: This book covers all major topics in integral calculus, including definite and indefinite integrals, application of integrals, and differential equations. It's great for students aiming for competitive exams.

   4. Mathematics for Class 12 by R.D. Sharma

      • Publication: Dhanpat Rai & Co
      • Content: This book covers all the NCERT topics for Class 12 Mathematics, with an emphasis on practice problems and theoretical explanations. It is structured to help students build a solid foundation for exams.

   5. Higher Algebra by Hall and Knight

      • Publication: S. Chand Publications
      • Content: This book is an excellent resource for algebra, with detailed explanations of quadratic equations, permutations, combinations, and matrices. It is useful for both board exams and competitive exams.

   6. Objective Mathematics by R.D. Sharma

      • Publication: Dhanpat Rai
      • Content: Focused on objective type questions, this book provides plenty of practice material for students preparing for entrance exams, with solutions and a step-by-step approach to problem-solving.

   7. Mathematics Class 12 by P.L. Soni

      • Publication: A. R. Publications
      • Content: This book provides solutions to NCERT problems and additional questions that cover a wide range of topics, including coordinate geometry, probability, and vectors, making it a comprehensive tool for exam preparation.

   8. Problems in Calculus of One Variable by I.A. Maron

      • Publication: B. I. Publications
      • Content: Focuses on calculus, particularly differentiation and integration, with a collection of complex problems to sharpen problem-solving skills for competitive exams and boards.

   9. Mathematics for Class 12 by R.S. Aggarwal

      • Publication: S. Chand & Co.
      • Content: It contains all NCERT chapters, including applications of derivatives, differential equations, and vectors, accompanied by a variety of problems for practice.

   10. Trigonometry by S.L. Loney

       • Publication: S. Chand Publications
       • Content: This book focuses on trigonometric identities, equations, and applications. It provides a solid foundation in trigonometry with clear theory and problems that help in board and competitive exam preparation.

   11. Class 12 Mathematics by R. S. Aggarwal

   12.  

       • Publication: S. Chand
       • Content: A great all-rounder book covering the entire Class 12 syllabus. It includes a large number of solved and unsolved problems, covering every topic extensively.

   13. Vector Algebra by S.K. Goyal

       • Publication: Arihant Publications
       • Content: A specialized book on vector algebra with a focus on the properties of vectors, vector operations, and their applications in geometry and physics. It is ideal for deepening understanding and problem-solving skills.

   14. Advanced Problems in Mathematics for JEE Main & Advanced by Vikas Gupta

       • Publication: Cengage Learning
       • Content: A comprehensive book offering difficult-level problems, focusing on high-level topics such as calculus, coordinate geometry, and algebra, ideal for preparing for IIT JEE.

   15. Mathematics Class 12 by S.K. Goyal

       • Publication: R.S. Aggarwal
       • Content: This book is designed with a variety of questions from the NCERT syllabus and includes an extensive range of examples, along with practice sets.

   16. Problems in Mathematics by V.S. Raju

       • Publication: Popular Prakashan
       • Content: The book features challenging problems with solutions that encourage a deeper understanding of mathematical concepts such as integration, algebra, and coordinate geometry.

   17. A Problem Book in Mathematical Analysis by G.N. Berman

       • Publication: Tata McGraw-Hill
       • Content: Concentrates on analysis, providing a detailed set of problems with solutions for deepening the understanding of real analysis, sequences, and series.

   18. Differential Calculus for JEE Main & Advanced by Amit M Agarwal

       • Publication: Arihant Publications
       • Content: This book delves into differential calculus, including tangents, normal, and rate of change problems, with varied difficulty levels catering to JEE preparation.

   19. Mathematics for Class 12 by S.K. Mapa

       • Publication: Himalaya Publishing House
       • Content: Covers topics such as matrices, determinants, and calculus with explanations and ample solved problems to help students understand and apply concepts effectively.

   20. Concise Objective Mathematics by R.S. Aggarwal

       • Publication: S. Chand Publications
       • Content: Focuses on objective-type questions for competitive exams like JEE and NEET. It provides concise theory, important formulas, and large sets of practice problems.

   21. Mathematical Analysis by Shanti Narayan

       • Publication: S. Chand
       • Content: Focuses on mathematical analysis, providing a solid grounding in sequences, series, and real analysis. The book is ideal for deep theoretical study and practice.

   22. Mathematics for Class 12 by C.B. Sharma

       • Publication: Vikas Publishing
       • Content: Contains solved examples and unsolved problems, covering all key topics such as calculus, differential equations, and probability, which are essential for both board exams and competitive exams.
    

   Class 12 Maths NCERT Solutions: A Key to Mastering Mathematics

   Mathematics in Class 12 is a crucial step in shaping a student’s academic and professional future. With the curriculum designed to cover an extensive range of topics, mastering Class 12 Maths can be a daunting task. However, the right guidance and resources can make all the difference, and NCERT Solutions for Class 12 Maths stand out as one of the best tools for students aiming for excellence in their exams.

   NCERT (National Council of Educational Research and Training) solutions provide a comprehensive guide to mastering mathematical concepts. They are meticulously crafted to align with the syllabus prescribed by CBSE, offering a detailed breakdown of every chapter. From calculus and algebra to geometry and statistics, these solutions cover all essential topics, ensuring that no concept is left unexplored.

   Why NCERT Solutions for Class 12 Maths Matter

   The importance of NCERT Solutions lies in their structure and approach. Each solution is written in a simple, step-by-step format, making it easy for students to follow and understand complex problems. This approach helps in developing problem-solving skills and building a strong foundation in mathematics.

   One of the key benefits of using NCERT Solutions is their alignment with the CBSE exam pattern. By practicing the questions provided in these solutions, students become familiar with the types of problems that are likely to appear in exams. This ensures that students are not only prepared for the theory but also for the practical application of mathematical concepts.

   Expert-Recommended Strategies for Using NCERT Solutions

   To truly benefit from NCERT Solutions, students must approach them with a structured strategy. Experts recommend starting with a thorough understanding of each concept before diving into problem-solving. Reading through the chapter summary and examples before attempting the exercises will help students grasp the core principles.

   It’s essential to practice regularly. Repetition of problems not only boosts confidence but also improves speed and accuracy in solving mathematical questions. Solving different variations of problems helps in understanding the application of the same concept in various contexts.

   Additionally, students should not hesitate to seek help if they encounter difficulties. While NCERT Solutions are a self-help tool, consulting teachers or peers for clarification can further strengthen a student’s grasp on tricky topics. Many students find that group study sessions can be highly effective in clearing doubts and exchanging problem-solving strategies.

   How NCERT Solutions Improve Exam Performance

   NCERT Solutions have been designed keeping in mind the diverse learning needs of students. The clear explanations help break down complex problems into smaller, more manageable steps. This not only boosts a student’s problem-solving abilities but also enhances their overall understanding of the subject.

   Moreover, the solutions cover a wide range of difficulty levels, from basic questions to challenging ones, ensuring that students are prepared for any kind of problem they may encounter during the exam. Regular practice of these solutions can lead to improved speed and accuracy, which is critical for scoring well in the CBSE Class 12 Maths exam.

   The Role of NCERT Solutions in Building Confidence

   Class 12 Maths can often be overwhelming for students due to its depth and complexity. However, with consistent practice using NCERT Solutions, students gradually build their confidence in tackling even the most difficult problems. When students feel more confident in their mathematical abilities, they approach exams with a positive mindset, which can make a huge difference in their performance.

   The clarity and structure provided by NCERT Solutions help reduce exam anxiety and allow students to focus on solving problems rather than stressing over what might appear in the exam. With a solid understanding of all topics, students can confidently face their Class 12 Maths exam and achieve their desired results.

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