The solution of the new math book for 9th class exercise 5.2 step by step with the help of the graph is shown below. Question 1: Maximize \( f(x, y) = 2x + 5y \) Subject to constraints: \[2y – x \leq 8, \quad x – y \leq 4, \quad x \geq 0, \quad y…
The solution of the new math book for 9th class step by step is given below. Question 1: Solve and represent the solution on a real line (i) \( 12x + 30 = -6 \) \[12x + 30 = -6\] \[12x = -6 – 30\] \[12x = -36\] \[x = -3\] Solution: \( x =…
MCQs 1. The factorization of \( 12x + 36 \) is: (a) \( 12(x + 3) \) (b) \( 12(3x) \) (c) \( 12(3x + 1) \) (d) \( x(12 + 36x) \) Correct Answer: (a) \( 12(x + 3) \) Explanation: Take 12 as the common factor, leaving \( (x + 3) \) as…
1. Find the square root of the following polynomials by factorization method (i) \( x^2 – 8x + 16 \) \[x^2 – 8x + 16 = (x – 4)^2\] \[\sqrt{x^2 – 8x + 16} = x – 4\] (ii) \( 9x^2 + 12x + 4 \) \[9x^2 + 12x + 4 = (3x + 2)^2\]…
The square root is a fundamental concept in mathematics, allowing us to simplify expressions and solve equations efficiently. When it comes to algebraic expressions, finding their square root requires understanding the structure of the terms and applying mathematical rules systematically. Square roots are widely used in geometry, algebra, and applied sciences. This article explains the…
1. Find HCF by factorization method (i) \( 21x^2y, 35xy^2 \) \[21x^2y = 3 \cdot 7 \cdot x^2 \cdot y, \quad 35xy^2 = 5 \cdot 7 \cdot x \cdot y^2\] \[\text{HCF} = 7xy\] (ii) \( 4x^2 – 9y^2, 2x^2 – 3xy \) \[4x^2 – 9y^2 = (2x + 3y)(2x – 3y), \quad 2x^2 – 3xy…
The concepts of HCF (Highest Common Factor) and LCM (Least Common Multiple) are essential tools in mathematics, particularly for simplifying fractions, solving problems with multiples, and understanding number relationships. They are widely used in number theory, algebra, and real-life applications such as scheduling and resource allocation. This article provides clear definitions, examples, and formulas for…
1. Factorize each of the following expressions: (i) \( 4x^4 + 81y^4 \) \[(2x^2)^2 + (9y^2)^2 + 2(2x^2)(9y^2) – 2(2x^2)(9y^2)\] \[(2x^2 + 9y^2)^2 – 36x^2y^2\] \[(2x^2 + 9y^2)^2 – (6xy)^2\] \[(2x^2 + 9y^2 + 6xy)(2x^2 + 9y^2 – 6xy)\] (ii) \( a^4 + 64b^4 \) \[(a^2)^2 + (8b^2)^2 + 2(a^2)(8b^2) – 2(a^2)(8b^2)\] \[(a^2 + 8b^2)^2…
Mathematics provides us with powerful formulas to simplify expressions and solve equations efficiently. Among these, square and cube formulas are widely used to expand or factorize algebraic expressions. These formulas help reduce complex calculations and are essential for solving problems in algebra, geometry, and physics. In this article, we will explore the key square and…
Question 1: Factorize by identifying common factors (i) \( 6x + 12 \) \[= 6(x + 2)\] (ii) \( 15y^2 + 20y \) \[= 5y(3y + 4)\] (iii) \( -12x^2 – 3x \) \[= -3x(4x + 1)\] (iv) \( 4a^2b + 8ab^2 \) \[= 4ab(a + 2b)\] (v) \( xy – 3x^2 + 2x \)…
