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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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