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Jan 20, 2020
Worksheet on Matrix | Solving Matrix Equations Worksheet | Answers
In Worksheet on matrix the questions are based on finding unknown elements and matrices from matrix equation. (i) Find the matrix C(B – A). (ii) Find A(B + C). (iii) Prove that A(B + C) = AB + AC. 2. Show that 6X – X^2 = 9I, where I is the unit matrix.
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Jan 19, 2020
Worksheet on Matrix Multiplication |Multiplication of Matrices|Answers
Practice the questions given in the Worksheet on Matrix Multiplication. (i) Find AB and BA if possible. (ii) Verify if AB = BA. (iii) Find A^2. (iv) Find AB^2.
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Jan 16, 2020
Problems on Classification of Matrices | Construct a Null Matrix
Here we will solve different types of Problems on classification of matrices. Indicate the class of each of the matrices. Construct a null matrix of the order 2 × 3 and a unit matrix of the order 3 × 3. Solution: A null matrix of the order 2 × 3 is
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Jan 15, 2020
Matrix | Definition of a Matrix | Examples of a Matrix | Elements
A rectangular array of mn elements aij into m rows and n columns, where the elements aij belongs to field F, is said to be a matrix of order m × n (or an m × n matrix) over the field F. Definition of a Matrix: A matrix is a rectangular arrangement or array of numbers
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Jan 15, 2020
Multiplication of Matrices | How to Multiply Matrices? |Rules|Examples
Two matrices A and B are said to be conformable for the product AB if the number of columns of A be equal to the number of rows of B. If A be an m × n matrix and B an n × p matrix then their product AB is defined to be the m × p matrix whose (ij)th element is obtained by
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Jan 15, 2020
Worksheet on Understanding Matrix | Possible Orders of the Matrix
Practice the questions given in the Worksheet on understanding matrix. (i) What is the order of the matrix A? (ii) Find the (2, 1)th, (1, 2)th and (3, 2)th elements. (iii) Is it a rectangular matrix or a square matrix? 2. (i) A matrix has 4 elements. Write the possible order
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Jan 13, 2020
Problems on Matrix Multiplication | Multiply Two Matrices
Here we will solve different types of Problems on Matrix Multiplication.write down the matrix AB. Would it be possible to find the product of BA? If so, compute it, and if not, give reasons.
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Jan 08, 2020
Problems on Understanding Matrices | Order of a Matrix | Position of
Here we will solve different types of Problems on understanding matrices. (i) What is the order of the matrix A? (ii) Find (2, 1)th and (1, 2)th elements. Solution: (i) The order is 2 × 2 because there are 2 rows and 2 columns in the matrix (ii) (2, 1)th element = the number
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Jan 07, 2020
Subtraction of Two Matrices | Matrix Subtraction | Subtract Matrices
We will learn how to find the subtraction of two matrices. If A and B two matrices of the same order then A – B is a matrix which is the addition of A and –B. The elements of A – B can also be obtained by subtracting the elements of B from the corresponding elements of A.
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Jan 06, 2020
Negative of a Matrix | Solved Examples on Negative of a Matrix
We will discuss about Negative of a Matrix. The negative of the matrix A is the matrix (-1)A, written as – A. Clearly, the negative matrix is obtained by changing the signs of each element. Solved examples on Negative of a Matrix:
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Jan 03, 2020
Multiplication of Two Matrices | Finding the Product of Two Matrices
Here we will learn the process of Multiplication of two matrices. Two matrices A and B are conformable (compatible) for multiplication (i) AB if the number of columns in A = the number of rows in B (ii) BA if the number of columns in B = the number of rows in A. To find the
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Jan 02, 2020
Multiplication of a Matrix by a Number |Scalar Multiplication|Examples
We will discuss here about the process of Multiplication of a matrix by a number. The multiplication of a matrix A by a number k gives a matrix of the same order as A, in which all the elements are k times the elements of A.
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Jan 01, 2020
Addition of Two Matrices | Matrix Addition | Sum of Two Matrices
We will learn how to find the addition of two matrices. Two matrices A and B are conformable (compatible) for addition if A and B are of the same order. The sum of A and B is a matrix of the same order and the elements of the matrix A + are obtained by adding the
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Dec 30, 2019
Classification of Matrices | Row Matrix | Column Matrix | Null Matrix
Matrices are classified according to the number of rows and columns, and the specific elements therein. (i) Row matrix: A matrix which has exactly one row is called a row matrix. For example, [45 -8], [8 9 10] are row matrices because each has only one row.
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Dec 28, 2019
Definition of Equal Matrices | Examples of Equal Matrices
Equality of two matrix: Two matrices [aij] and [bij] are said to be equal when they have the same number of rows and columns and aij = bij for all admissible values of i and j. Definition of Equal Matrices: Two matrices A and B are said to be equal if A and B have the same
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Dec 27, 2019
Position of an Element in a Matrix | 10th Grade Matrix | Examples
We will learn how to find the position of an element in a matrix? 1. Consider the matrixHere, the element 5 falls on row number 1 and column 1. We say, 5 is the (1, 1)th element. Similarly, (1, 2)th element = -6 (2, 1)th element = 1 (2, 2)th element = 7 (3, 1)th element
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Dec 26, 2019
Order of a Matrix | Determine the Order of Matrix | Solved Examples
How to determine the order of matrix? If a matrix has m rows and n columns, its order is said to be m × n (read as ‘m by n’). a matrix of the order m × n has mn elements. Hence, if the number of elements in a matrix be prime, it must have one row or one column.
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Dec 25, 2019
nth Root of a | Meaning of \(\sqrt[n]{a}\) | Solved Examples
We will discuss here about the meaning of \(\sqrt[n]{a}\). The expression \(\sqrt[n]{a}\) means ‘nth rrot of a’. So, (\(\sqrt[n]{a}\))^n = a. Also, (a^1/a)^n = a^n*1/n = a^1 = a. So, \(\sqrt[n]{a}\) = a^1/n. Examples: \(\sqrt[3]{8}\) = 8^1/3 = (2^3)^1/3 = 2^3 * 1/3 = 2^1
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Dec 22, 2019
The laws of exponents are explained here along with their examples.
The laws of exponents are explained here along with their examples. In multiplication of exponents if the bases are same then we need to add the exponents.
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Dec 22, 2019
Laws of Indices | Laws of Exponents| Rules of Indices |Solved Examples
We will discuss here about the different Laws of Indices. If a, b are real numbers (>0, ≠ 1) and m, n are real numbers, following properties hold true. (i) am × an = am + n (ii) a-m = \(\frac{1}{a^{m}}\) (iii) \(\frac{a^{m}}{a^{n}}\) = am – n = \(\frac{1}{a^{m - n}}\)
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Dec 20, 2019
Power of a Number | Exponent | Index | Negative Exponents | Examples
Here we will learn the Power of a Number. We know a × a = a^2, a × a × a = a^3, etc., and a × a × a × ... n times = a^n, where n is a positive integer. a^n is a power of a whose base is a and the index of power is n. a^p/q is the qth root of a^p if p, q are positive integers
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Dec 19, 2019
Worksheet on Factorization | Hints | Miscellaneous Factorization
Practice the questions given in the Worksheet on Factorization. Factorization of expressions of the form a^3 ± b^3 1. Factorize: (i) 8x^3 + 27y^3 (ii) 216a^3 + 1 (iii) a^6 + 1 (iv) x^3 + \(\frac{1}{x^{3}}\) (v) a^3 + 8b^6
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Dec 09, 2019
Factorization of expressions of the Form a^3 + b^3 + c^3 – 3abc
Here we will learn the process of On Factorizations of expressions of the Form a^3 + b^3 + c^3 – 3abc. We have, a^3 + b^3 + c^3 – 3abc = (a + b + c)(a^2 + b^2 + c^2 – bc – ca – ab). [Verify by actual multiplication.] Example: Factorize: x^3 + y^3 – 3xy + 1. Solution: Here
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Dec 09, 2019
Factorization of Expressions of the Form a^3 + b^3 + c^3, a + b + c=0
Here we will learn the process of On Factorization of expressions of the Form a^3 + b^3 + c^3 , where a + b + c = 0. We have, a^3 + b^3 + c^3 = a^3 + b^3 – (-c)^3 = a^3 + b^3 – (a + b)^3, [Since, a + b + c = 0] = a^3 + b^3 – {a^3 + b^3 + 3ab(a + b)} = -3ab(a + b) = -3ab(-c
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Dec 09, 2019
Miscellaneous Problems on Factorization | Application Problems
Here we will solve different types of Miscellaneous Problems on Factorization. 1. Factorize: x(2x + 5) – 3 Solution: Given expression = x(2x + 5) – 3 = 2x^2 + 5x – 3 = 2x^2 + 6x – x – 3, [Since, 2(-3) = - 6 = 6 × (-1), and 6 + (-1) = 5] = 2x(x + 3) – 1(x + 3)
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Nov 24, 2019
Factorization of Expressions of the Form a^3 - b^3 | Solved Examples
Here we will learn the process of Factorization of Expressions of the Form a^3 - b^3. We know that (a - b)^3 = a^3 - b^3 - 3ab(a - b), and so a^3 - b^3 = (a - b)^3 + 3ab(a - b) = (a - b){(a - b)^2 + 3ab} Therefore, a^3 - b^3 = (a - b)(a^2 + ab + b^2) Example: 1. Factorize:
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Nov 24, 2019
Factorization of Expressions of the Form a^3 + b^3 | Solved Examples
Here we will learn the process of Factorization of Expressions of the Form a^3 + b^3. We know that (a + b)^3 = a^3 + b^3 + 3ab(a + b), and so a^3 + b^3 = (a + b)^3 – 3ab(a + b) = (a + b){(a + b)^2 – 3ab} Therefore, a^3 + b^3 = (a + b)(a^2 – ab + b^2). Example: 1. Factorize:
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Nov 23, 2019
Worksheet on Factorization of the Trinomial ax^2 + bx + c | Answers
Practice the questions given in the worksheet on factorization of the trinomial ax^2 + bx + c. 1. Factorization of a perfect-square trinomial. (i) a^2 + 6a + 9 (ii) a^2 + a + \(\frac{1}{4}\) (iii) 25x^2 – 10x + 1 (iv) 4x^2 – 4xy + y^2 2. Factorization of expressions of the
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Nov 18, 2019
Problems on Factorization of Expressions of the Form x^2 +(a + b)x +ab
Here we will solve different types of Problems on Factorization of Expressions of the Form x^2 + (a + b)x + ab. 1. Factorize: a^2 + 25a - 54 Solution: Here, constant term = -54 = (27) × (-2), and 27 + (-2) = 25 (= coefficient of a). Therefore, a^2 + 25a – 54
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Nov 17, 2019
Factorization of Expressions of the Form ax^2 + bx + c, a ≠ 1|Examples
The below examples show that the method of factorizing ax^2 + bx + c by breaking the middle term involves the following steps. Steps: 1.Take the product of the constant term and the coefficient of x^2, i.e., ac. 2. Break ac into two factors p, q whose sum is b,
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Nov 13, 2019
Factorization of Expressions of the Form x^2 + (a + b)x + ab |Examples
Here we will learn the process of Factorization of Expressions of the Form x^2 + (a + b)x + ab. We know, (x + a)(x + b) = x^2 + (a + b)x + ab. Therefore, x^2 + (a + b)x + ab = (x + a)(x + b). 1. Factorize: a^2 + 7a + 12. Solution: Here, constant term = 12 = 3 × 4, and 3 + 4
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Nov 10, 2019
Factorization of a Perfect-square Trinomial | Solved Examples
Here we will learn the process of Factorization of a Perfect-square Trinomial. A trinomial of the form a^2 ± 2ab + b^2 = (a ± b)^2 = (a ± b)(a ± b) Solved examples on Factorization of a Perfect-square Trinomial 1. Factorize: x^2 + 6x + 9 Solution: Here, given expression
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Nov 04, 2019
Problems on Factorization Using a^2 – b^2 = (a + b)(a – b)
Problems on Factorization using a^2 – b^2 = (a + b)(a – b) Here we will solve different types of Problems on Factorization using a^2 – b^2 = (a + b)(a – b). 1. Factorize: 4a^2 – b^2 + 2a + b Solution: Given expression = 4a^2 – b^2 + 2a + b = (4a^2 – b^2) + 2a + b
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Nov 02, 2019
Problems on Factorization of Expressions of the Form a^2 – b^2
Here we will solve different types of Problems on Factorization of expressions of the form a^2 – b^2. 1. Resolve into factors: 49a^2 – 81b^2 Solution: Given expression = 49a^2 – 81b^2 = (7a)^2 – (9b)^2 = (7a + 9b)(7a – 9b). 2.Factorize: (x + y)^2 – 4(x - y)^2 Solution: Given
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Nov 01, 2019
Problems on Factorization by Grouping of Terms | Find the Factors
Here we will solve different types of Problems on Factorization by grouping of terms. 1. Factorize: a^2 – (b – 5)a – 5b. Solution: Given expression = a^2 – (b – 5)a – 5b = a^2 – ba + 5a – 5b = a(a - b) + 5(a - b) = (a – b)(a + 5). 2. Factorize: a^2 + b^2 + a + b + 2ab
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Oct 31, 2019
Introduction to Factorization | Different of Two Squares | Examples
We will discuss here about the introduction to factorization. The method of expressing a given polynomial as a product of two or more polynomials is called factorization. The polynomials whose product is the given polynomial are called its factors. You are already familiar
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Oct 25, 2019
Problems on Expanding of (a ± b)\(^{3}\) and its Corollaries |Examples
Here we will solve different types of application problems on expanding of (a ± b)\(^{3}\) and its corollaries. 1. Expanding the following: (i) (1 + x)\(^{3}\) (ii) (2a – 3b)\(^{3}\) (iii) (x + \(\frac{1}{x}\))\(^{3}\) Solution: (i) (1 + x)\(^{3}\) = 1\(^{3}\) +
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Oct 22, 2019
Worksheet on Application Problems on Expansion of Powers of Binomials
Practice the questions given in the worksheet on application problems on expansion of powers of binomials and trinomials. 1. Use (a ± b)^2 = a^2 ± 2ab + b2 to evaluate the following: (i) (3.001)^2 (ii) (5.99)^2 (iii) 1001 × 999 (iv) 5.63 × 5.63 + 11.26 × 2.37 + 2.37 × 2.37
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Oct 22, 2019
Application Problems on Expansion of Powers of Binomials & Trinomials
Here we will solve different types of application problems on expansion of powers of binomials and trinomials. 1. Use (x ± y)^2 = x^2 ± 2xy + y^2 to evaluate (2.05)^2. Solution: (2.05)^2 = (2 + 0.05)^2 = 2^2 + 2 × 2 × 0.05 + (0.05)^2 = 4 + 0.20 + 0.0025 = 4.2025.
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Oct 21, 2019
Expansion of (x + a)(x + b)(x + c) | Solved Examples | Problems | Hint
We will discuss here about the expansion of (x + a)(x + b)(x + c). (x + a)(x + b)(x + c) = (x + a){(x + b)(x + c)} = (x + a){x\(^{2}\) + (b + c)x + bc} = x{x\(^{2}\) + (b + c)x + bc} + a{x\(^{2}\) + (b + c)x + bc} = x\(^{3}\) + (b + c)x\(^{2}\) + bcx + ax\(^{2}\) + a(b + c)x
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Oct 20, 2019
Simplification of (a + b + c)(a\(^{2}\)+b\(^{2}\)+c\(^{2}\)–ab–bc– ca)
We will discuss here about the expansion of (a + b + c)(a\(^{2}\) + b\(^{2}\) + c\(^{2}\) – ab – bc – ca). (a + b + c)(a\(^{2}\) + b\(^{2}\) + c\(^{2}\) – ab – bc – ca) = a(a\(^{2}\) + b\(^{2}\) + c\(^{2}\) – ab – bc – ca) + b(a\(^{2}\) + b\(^{2}\) + c\(^{2}\) –ab – bc – ca)
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Oct 17, 2019
Simplification of (a ± b)(a^2 ∓ ab + b^2) | Sum or Difference of Cubes
We will discuss here about the expansion of (a ± b)(a\(^{2}\) ∓ ab + b\(^{2}\)). (a + b)(a\(^{2}\) - ab + b\(^{2}\)) = a(a\(^{2}\) - ab + b\(^{2}\)) + b(a\(^{2}\) - ab + b\(^{2}\)) = a\(^{3}\) - a\(^{2}\)b + ab\(^{2}\) + ba\(^{2}\) - ab\(^{2}\) + b\(^{3}\) =
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Oct 15, 2019
Expansion of (a ± b)\(^{3}\) | Algebraic Expressions and Formulas
We will discuss here about the expansion of (a ± b)\(^{3}\). (a + b)\(^{3}\) = (a + b) ∙ (a + b)\(^{2}\) = (a + b)(a\(^{2}\) + 2ab + b\(^{2}\)) = a(a\(^{2}\) + 2ab + b\(^{2}\)) + b(a\(^{2}\) + 2ab + b\(^{2}\))=a\(^{3}\)+2a\(^{2}\)b+ab\(^{2}\)+ba\(^{2}\)+2ab\(^{2}\)+b\(^{3}\)
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Oct 14, 2019
Express a^2 + b^2 + c^2 – ab – bc – ca as Sum of Squares
Here we will express a^2 + b^2 + c^2 – ab – bc – ca as sum of squares. If a, b, c are real numbers then (a – b)^2, (b – c)^2 and (c – a)^2 are positive as square of every real number is positive. So, a^2 + b^2 + c^2 – ab – bc – ca is always positive.
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Oct 11, 2019
Worksheet on Simplification of (a + b)(a – b) | Hint | Answers
Practice the questions given in the worksheet on simplification of (a + b)(a – b). 1. Simplify by applying standard formula. (i) (5x – 9)(5x + 9) (ii) (2x + 3y)(2x – 3y) (iii) (a + b – c)(a – b + c) (iv) (x + y – 3)(x + y + 3) (v) (1 + a)(1 – a)(1 + a^2)
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Oct 10, 2019
Worksheet on Completing Square |Find the Missing Term| Perfect Squares
Practice the questions given in the worksheet on completing square. Write the following as a perfect square. (i) 4X^2 + 4X + 1 (ii) 9a^2 – 12ab + 4b^2 (iii) 1 + 6/a + 9/a^2 2. Indicate the perfect squares among the following. Express each of the perfect squares as the square
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Oct 10, 2019
Completing a Square | Solved Examples on Completing a Square
Here we will learn how to completing a square.Problems on completing a square 1. What should be added to the polynomial 4m^2 + 8m so that it becomes perfect square? Solution: 4m^2 + 8m = (2m)^2 + 2 ∙ (2m) ∙ 2
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Oct 09, 2019
Worksheet on Expansion of (x ± a)(x ± b) | Find the Product | Answers
Practice the questions given in the worksheet on expansion of (x ± a)(x ± b). 1. (i) Find the product using standard formula. (i) (x + 2)(x + 5) (ii) (a – 4)(a – 7) (iii) (x + 1)(x – 8) (iv) (a – 3)(a + 2) (v) (3x + 1)(3x + 2) (vi) (4x – y)(4x + 2y) 2. Find the product.
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Oct 01, 2019
Worksheet on Expanding of (a ± b ± c)^2 and its Corollaries | Answers
Practice the questions given in the worksheet on expanding of (a ± b ± c)^2 and its corollaries. 1. Expand the squares of the following trinomials. (i) a + 2b + 3c (ii) 2x + 3y + 4z (iii) x + 2y – 3z (iv) 3a – 4b – c (v) 1 – x - \(\frac{1}{x}\) (vi) 1 – a – a^2. 2. Simplify:
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