New Exam Pattern for CBSE Class 9, 10, 11, 12:

The Central Board of Secondary Education (CBSE) is making continuous efforts to enhance the education system for students. Recently, CBSE made a big update regarding the exam pattern for Class 9, 10, 11, 12 where more focus would be given to application-based questions. For every subject, the examination papers will now contain Competency-based Questions along…

The Central Board of Secondary Education (CBSE) is making continuous efforts to enhance the education system for students. Recently, CBSE made a big update regarding the exam pattern for Class 9, 10, 11, 12 where more focus would be given to application-based questions. For every subject, the examination papers will now contain Competency-based Questions along with short answers and long answer questions. Let us see what that means and why this is good news for students. 

Introduction of Competency-based Questions 

The education ministry of India wants to promote application-based learning amongst the students. It has always been seen that students of our country have only one agenda in mind – to complete the CBSE syllabus as soon as possible and score higher marks, especially in Classes 9, 10, 11 & 12. As a result, little effort is put into whether the students can apply their knowledge in real-life scenarios or not. 

So, a decision was made to include questions that can test their knowledge and skills. This set of questions are known as competency-based questions. These questions will test the problem-solving, analytical and critical thinking of the students. 

A comparison of the new CBSE exam pattern with the old exam pattern 

New Exam Pattern for CBSE Class 9 & CBSE Class 10

Exam pattern until 2020-21

Updated Exam Pattern for 2021-22

  • Case-based/ Source-based Integrated Questions: 20%
  • Objective type Questions: 20%
  • Short Answer/ Long Answer Questions: 60%
  • Competency-Based Questions (Multiple-Choice Questions, Case-Based Questions, Source-Based Integrated Questions or any other types): at least 30%
  • Objective Questions: 20%
  • Short  Answer/  Long Answer Questions: 50% 

New Exam Pattern for CBSE Class 11 & CBSE Class 12

Exam pattern until 2020-21

Updated Exam Pattern for 2021-22

  • Objective type Questions: 20%

  • Case-based/ Source-based Integrated Questions: 10%

  • Short Answer/ Long Answer Questions: 70%

  • Competency-Based Questions (Multiple-Choice Questions, Case-Based Questions, Source-Based Integrated Questions or any other types): 20%

  • Objective Questions: 20%

  • Short  Answer/  Long Answer Questions: 60% 

What does the new exam pattern indicate? 

The new exam pattern discourages rote learning and promotes application-based learning. This means students will have to understand the concepts in-depth. They must understand the objective behind every concept and how that concept is utilised in the real world, making learning more interesting. After all, that is how learning should be conducted, isn’t it? 

How to prepare yourself according to the new exam pattern? 

#1 Solve CBSE sample papers: The best way to get yourself acquainted with the changing exam pattern of CBSE is to solve sample papers. Practice as much as you can so that you gain confidence for your exams. 

#2 Focus on every concept: Your aim should not only be to complete the syllabus and solve NCERT exercises. You must understand all the concepts thoroughly so that you can answer all kinds of questions in the examination. 

#3 Try different learning techniques: Do not just focus on reading your textbooks, making notes and solving exercises. Try using flashcards, mind-maps, mnemonics, conceptual video-based lessons and other different learning techniques to memorise concepts better. 

#4 Take help from online classes: Taking guidance from an expert is the best way to remove your doubts and prepare yourself for the examination. Join an online CBSE coaching where you can find a plethora of study resources, especially question banks and test series to help you prepare as per the latest CBSE exam pattern. 

Need help in preparing for the CBSE exams? Join askIITians and prepare yourself as per the latest CBSE exam pattern and guidelines. Learn from our experts in live, interactive classes. Get chapter-wise notes, tips and tricks on how to prepare for the competency-based questions, updated question banks, sample papers, previous year CBSE papers, flashcards, mind-maps and a lot more!  

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NCERT Solutions for Class 12 Maths Chapter 3

NCERT Solutions for Class 12 Maths Chapter 312th Maths Chapter 3 Solutions in English MediumClass: 12Maths (English and Hindi Medium)Chapter 3:Matrices12th Maths Chapter 3 SolutionsDownload NCERT Solutions Class 12 Maths chapter 3 exercise 3.1, 3.2, 3.3, 3.4 and miscellaneous exercises in PDF form English and Hindi Medium for CBSE Board. UP Board Students are using…

NCERT Solutions for Class 12 Maths Chapter 3

  • 12th Maths Chapter 3 Solutions in English Medium

Class: 12Maths (English and Hindi Medium)
Chapter 3:Matrices

12th Maths Chapter 3 Solutions

Download NCERT Solutions Class 12 Maths chapter 3 exercise 3.1, 3.2, 3.3, 3.4 and miscellaneous exercises in PDF form English and Hindi Medium for CBSE Board. UP Board Students are using the same NCERT Books, so they can also use these solutions as UP Board Solutions 12 Maths Chapter 3 for the academic session 2021-22. We have done everything perfectly to provide solutions of Class 12 Maths for CBSE and UP Board, if you feel some problem, inform us. We will definitely rectify.

  • Class 12 Math Chapter 3 Solutions in Hindi Medium

  • Class 12 Maths Chapter 3 Solutions in PDF

Class 12 Maths Exercise 3.1 Video Solutions

Class 12 Maths Exercise 3.1 Question 1, 2, 3Class 12 Maths Exercise 3.1 Question 1, 2 in Hindi

Class 12 Maths Exercise 3.2 Video Solutions

Class 12 Maths Exercise 3.2 Question 1, 2Class 12 Maths Exercise 3.2 Question 1 in Hindi

Class 12 Maths Exercise 3.3 Video Solutions

Class 12 Maths Exercise 3.3 Question 1, 2Class 12 Maths Exercise 3.3 Question 1, 2 in Hindi

Class 12 Maths Exercise 3.4 Video Solutions

Class 12 Maths Exercise 3.4 Question 1Class 12 Maths Exercise 3.4 Question 1, 2 in Hindi

Class 12 Maths Miscellaneous Exercise 3 Video Solutions

Class 12 Maths Miscellaneous Exercise 3 Question 1Class 12 Maths Miscellaneous Exercise 3 Question 1 in Hindi

Matrix

The arrangement of real numbers in a rectangular array enclosed in brackets as [] or () is known as a Matrix(Matrices is plural of matrix). Matrix operations are used in electronic physics, computers, budgeting, cost estimation, analysis and experiments. They are also used in cryptography, modern psychology, genetics, industrial management etc. In general an m x n matrix is matrix having m rows and n columns. it can be written as follows:

class 12 chapter 3 matrixclass 12 chapter 3 matrix

Important Terms related to 12th Maths Chapter 3

Order of a Matrix

There may be any number of rows and any number of columns in a matrix. If there are m rows and n columns in matrix A, its order is m x n and it is read as an m x n matrix.

Transpose of a Matrix

The transpose of a given matrix A is formed by interchanging its rows and columns and is denoted by A’.

Symmetric Matrix

A square matrix A is said to be a symmetric matrix if A’ = A.

Skew-Symmetric Matrix

A square matrix A is said to be a skew symmetric if A’ = – A. all elements in the principal diagonal of a skew symmetric matrix are zeroes.

Addition of Matrix

If A and B are any two given matrices of the same order, then their sum is defined to be a matrix C whose respective elements are the sum of the corresponding elements of the matrices A and B and we write this as C = A + B.

Types of Matrices
    1. Row matrix: A row matrix has only one row but any number of columns.
    2. Column matrix: A column matrix has only one column but any number of rows.
    3. Square matrix: A square matrix has the number of column equal to the number of rows.
    4. Rectangular Matrix: A matrix is said to be a rectangular matrix if the number of rows is not equal to the number of columns.
    5. Diagonal matrix: If in a square matrix has all elements 0 except principal diagonal elements, it is called diagonal matrix.
    6. Scalar Matrix: A diagonal matrix is said to be a scalar matrix if all the elements in its principal diagonal are equal to some non-zero constant.
    7. Zero or Null matrix: If all elements of a matrix are zero, then the matrix is known as zero matrix and denoted by O.
    8. Unit or Identity matrix: If in a square matrix has all elements 0 and each diagonal elements are non-zero, it is called identity matrix and denoted by I.
    9. Equal Matrices: Two matrices are said to be equal if they are of the same order and if their corresponding elements are equal.
    Properties of Matrices
      • When a matrix is multiplied by a scalar, then each of its element is multiplied by the same scalar.
      • If A and B are any two given matrices of the same order, then their sum is defined to be a matrix C whose respective elements are the sum of the corresponding elements of the matrices A and B and we write this as C = A + B.
      • For any two matrices A and B of the same order, A + B = B + A. i.e. matrix addition is commutative.
      • For any three matrices A, B and C of the same order, A + (B + C) = (A + B) + C i.e., matrix addition is associative.
      • Additive identity is a zero matrix, which when added to a given matrix, gives the same given matrix, i.e., A + O = A = O + A.
      • If A + B = O, then the matrix B is called the additive inverse of the matrix of A.
      • If A and B are two matrices of order m x p and p x n respectively, then their product will be a matrix C of order m x n.
      Invertible Matrix

      A square matrix of order n is invertible if there exists a square matrix B of the same order such that AB = I = BA, Where I is identify matrix of order n.
      Theorems of invertible matrices

        • Theorem 1: Every invertible matrix possesses a unique inverse.
        • Theorem 2: A square matrix is invertible iff it is non-singular.
        Historical Facts!

        Matrix is a latin word. Originally matrices are used for solutions of simultaneous linear equations. An important Chinese Text between 300 BC and 200 AD, nine chapters of Mathematical Art(Chiu Chang Suan Shu), give the use of matrix methods to solve simultaneous equations. Carl Friedrich Gauss(1777 – 1855) also gave the method to solve simultaneous equations by matrix method.

        What is the matrix as per chapter 3 of class 12th Maths?

        A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix. We denote matrices by capital letters.
        The following are some examples of matrices: –
        In the above examples, the horizontal lines of elements are said to constitute, rows of the matrix and the vertical lines of elements are said to constitute, columns of the matrix. Thus A has 3 rows and 2 columns and B has 3 rows and 3 columns.

        What are the practical implimentations of matrices chapter 3 of 12th Maths in different fields?

        The knowledge of matrices is necessary in various branches of mathematics. Matrices are one of the most powerful tools in mathematics. The evolution of the concept of matrices is the result of an attempt to obtain compact and simple methods of solving a system of linear equations. Matrix notation and operations are used in electronic spreadsheet programs for personal computer, which in turn is used in different areas of business and science like budgeting, sales projection, cost estimation, analyzing the results of an experiment, etc.
        Also, many physical operations such as magnification, rotation, and reflection through a plane can be represented mathematically by matrices. Matrices are also used in cryptography. This mathematical tool is not only used in certain branches of sciences but also in genetics, economics, sociology, modern psychology, and industrial management.

        How many exercises are there in chapter 3 of class 12th Maths for 1st term exam?

        There are 5 exercises in chapter 3 of class 12th Maths. The first exercise (Ex 3.1) has 15 sums (5 examples and 10 questions). The second exercise (Ex 3.2) contains 36 sums (14 examples and 22 questions). In the third exercise (Ex 3.3), there are 15 sums (3 examples and 12 questions). The fourth exercise has 21 sums (3 examples and 18 questions). There are 18 sums (3 examples and 15 questions) in the last (Miscellaneous) exercise.

        Are there any theorems in chapter 3 of class 12th Maths?

        Yes, there are four theorems in chapter 3 of class 12th Maths. All the theorems are nice, easy, and important. Proofs of these theorems are short and easy. Students can easily understand the theorems and proofs of these theorems.

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        Made Easy Aptitude Book PDF Free Download 2021

        Made easy aptitude book pdf: In this article we will share with you the made easy aptitude book pdf for the GATE and other competitive exams aspirants. The book is highly useful for the GATE aspirants, it covers previous years questions with solutions.Made easy aptitude and reasoning book includes comprehensive theory with solved examples and previous…

        Made easy aptitude book pdf: In this article we will share with you the made easy aptitude book pdf for the GATE and other competitive exams aspirants. The book is highly useful for the GATE aspirants, it covers previous years questions with solutions.

        Made easy aptitude and reasoning book includes comprehensive theory with solved examples and previous solved questions of GATE and ESE prelims.

        Made easy aptitude book pdf

        The book also consists of lots of questions given to practice which is useful to practice and make the topic clear and it also helps to speed up the time and accuracy to attempt questions.

        Here, in this post we will try to provide you the free pdf of gate aptitude books pdf free download. Interested candidates can easily download the book from the below available link. It is highly recommended to the students to refer to the hard copy of the book, below we will also provide you the affiliated link to buy this book from amazon.

        About made easy aptitude book pdf

        Book Name – made easy aptitude and reasoning book 

        Author Name- ME Editorial Board

        Format- PDF

        Size-  mb 

        Pages- 344

        Language- English

        Publication- Made Easy Publications

        Contents of made easy aptitude book for gate pdf

        Section A: Arithmetic

        • Number system
        • Percentage
        • Profit and loss
        • Simple interest and compound interest
        • Ratio and proportion
        • Averages, mixture and alligation
        • Time and work
        • Time, speed and distance

        Section B: Algebra and Geometry

        • Surds, indices and logarithms
        • Progressions
        • Permutations and combinations
        • Probability
        • Set theory

        Section C: Reasoning and Data Interpretations

        • Blood relations
        • Coding and decoding
        • Cubes and dices
        • Direction sense test
        • Line graph
        • Tables
        • Bar diagrams
        • Pie-charts
        • Miscellaneous puzzles
        • Logical venn diagram
        • Analytical reasoning

         Section D: Previous GATE and ESE solved questions

        • Previous solved questions
        • Previous ESE prelims solved questions

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        Other quantitative aptitude book 

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        Friends, if you need any E-Book PDF related to any topic or subjects and need any assistance and inquiry related to exams you can comment below. We will respond as soon as possible. And please don’t forget to share this post with your friends and on social media platforms.

        Disclaimer: Sarkari Rush does not own books pdf, neither created nor scanned. We just provide the link already available on the internet and in google drive. If any way it violates the law or has any issues then kindly mail us [email protected] to request removal of the link.

         

         

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        NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2

        NCERT Solutions for class 10 Maths Chapter 2 Exercise 2.2Class 10 Maths Exercise 2.2 SolutionsClass: 10Maths (English and Hindi Medium)Chapter 2:Exercise 2.210 Maths Chapter 2 Exercise 2.2 SolutionsUpdated and simplified form of solutions fit for academic years 2021-22 as per CBSE Curriculum for CBSE board, Gujrat Board, MP Board and UP Board (High School –…

        NCERT Solutions for class 10 Maths Chapter 2 Exercise 2.2

        • Class 10 Maths Exercise 2.2 Solutions

        Class: 10Maths (English and Hindi Medium)
        Chapter 2:Exercise 2.2

        10 Maths Chapter 2 Exercise 2.2 Solutions

        Updated and simplified form of solutions fit for academic years 2021-22 as per CBSE Curriculum for CBSE board, Gujrat Board, MP Board and UP Board (High School – 2021-22) students using NCERT Books 2021-22. NCERT Solutions for class 10 Maths Chapter 2 Exercise 2.2 Polynomials in English and Hindi Medium are given below. If you find any difficulty to understand these solutions, please specify us without any hesitation.

        Class 10 Maths Exercise 2.2 Solution in Hindi Medium Video

        Class 10 Maths Exercise 2.2 Question 1 Solutions in Video

        Class 10 Maths Exercise 2.2 Question 1 Part 1, 2, 3 SolutionsClass 10 Maths Exercise 2.2 Question 1 Part 4, 5, 6 Solutions

        Class 10 Maths Exercise 2.2 Question 2 Solutions in Video

        Class 10 Maths Exercise 2.2 Question 2 Solution in VideoClass 10 Maths Exercise 2.2 all Questions Solutions in Video

        Important Questions with Answers on Polynomials

          1. If one zero of the polynomial (a² + 9)x² + 13x + 6a is the reciprocal of the other, find the value of a. [Answer: a = 3]
          2. How many (i) maximum (ii) minimum number of zeroes can a quadratic polynomial have? [Answer: (i) 2, (ii) 0]
          3. What will be the number of real zeroes of the polynomial x² + 1? [Answer: 0]
          4. If α and β are zeroes of polynomial 6x² – 7x – 3, then form a quadratic polynomial where zeroes are 2α and 2β. [Answer: 3x² – 7x – 6]
          5. If α and 1/α are zeroes of 4x² – 17x + k – 4, find value of k. [Answer: k = 8]
          6. What will be the number of zeroes of the polynomials whose graphs are parallel to (i) y-axis (ii) x-axis. [Answer: (i) 1, (ii) 0]
          7. Divide 2x² + x – 20 by x + 3 and verify division algorithm.
          8. What will be number of zeroes of the polynomials whose graphs are either touching or intersecting the axis only at the points: (i) (–3, 0), (0, 2) & (3, 0) (ii) (0, 4), (0, 0) and (0, –4). [Answer: (i) 2, (ii) 1]
          9. If –3 is one of the zeroes of the polynomial (k– 1)x² + k x + 1, find the value of k. [Answer: 4/3]
          10. If α and β are the zeroes of the polynomial p(x) = x² + x + 1, find the value of α² + β². [Answer: -1]

          Questions from Board Papers

          1. If the product of zeroes of ax² – 6x – 6 is 4, find the value of a. Hence find the sum of its zeroes. [Answer: a = -3/2, sum of zeroes = – 4]
          2. If α and β are zeroes of the polynomial x² – a(x + 1) – b such that (α + 1) (β + 1) = 0, find the value of b. [Answer: 1]
          3. It is given that 1 is one of the zeroes of the polynomial 7x – x³ – 6. Find its other zeroes. [Answer: -3, 2]
          4. If zeroes of x² – kx + 6 are in the ratio 3 : 2, find k. [Answer: -5, 5]

          5. If one zero of the quadratic polynomial (k² + k)x² + 68x + 6k is reciprocal of the other, find k. [Answer: 5]
          Download NCERT Books 2021-22 for offline use and ask your questions in Discussion Forum.

          Important Questions of 10th Maths Exercise 2.2

          What do you understand by value of p(x) at k?

          If p(x) is a polynomial in x, and if k is any real number, then the value obtained by replacing x by k in p(x), is called the value of p(x) at x = k, and is denoted by p(k).

          What is zero of a polynomial?

          A real number k is said to be a zero of a polynomial p(x), if p(k) = 0.

          How do we find the zeros of a quadratic polynomial graphically?

          The zeroes of a quadratic polynomial ax² + bx + c, a ≠ 0, are precisely the x-coordinates of the points where the parabola representing y = ax² + bx + c intersects the x-axis.

          How many zeros are there of a quadratic polynomial?

          A polynomial of degree 2 has at most two zeroes.

          What are the main topics in exercise 2.2 class 10th Maths to learn?

          In exercise 2.2 (chapter 2 Polynomials) of class 10th mathematics students will learn the following things:

            1. How to find zeroes of quadratic polynomial.
            2. Relationship between zeroes and coefficients of quadratic polynomial.
            3. Relationship between zeroes and coefficients of cubic polynomial.
            4. How to find a quadratic polynomial, the sum and product of whose zeroes are given.
            How many questions are there in exercise 2.2 of class 10th Mathematics?

            There are in all 2 questions (each question have 6 parts) in exercise 2.2 of class 10th mathematics chapter 2 (Polynomials) and both the questions are important.

            How many examples are there in exercise 2.2 of 10th Maths?

            4 examples (example 2, 3, 4, 5) are based on exercise 2.2 (chapter 2 Polynomials) of class 10th mathematics. All examples are of different type and example 2, 3, 4 are important. If we compare examples with questions of exercise 2.2 then we can observe that example 2, 3 and question 1 are of same type. Example 4 and question 2 are of same type.

            Is exercise 2.2 of 10th Maths easy?

            Exercise 2.2 (chapter 2 Polynomials) of class 10th mathematics is not that easy and not that difficult it somewhere in middle of easy and difficult. But difficulty level of anything varies from student to student. So, Exercise 2.2 (chapter 2 Polynomials) of class 10th mathematics is easy or not depends on students also. Some students find it difficult some find it easy or some find it in middle of easy and difficult.

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