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) \( xy – 3x^2 + 2x \)…
Factorization is a fundamental concept in mathematics, enabling us to break down complex expressions into simpler, more manageable components. It plays a vital role in solving equations, simplifying fractions, and understanding the structure of algebraic expressions. By mastering factorization, you can tackle mathematical problems with greater confidence and efficiency. In this article, we will define…
Applicable for All Punjab Boards: Lahore, Rawalpindi, Faisalabad, Sargodha, Gujranwala, Sahiwal, Multan, Bahawalpur, Dera Ghazi Khan, Azad Kashmir MCQs (Total: 15 Marks) Chapter No. MCQs Chapter No. 1 2 Chapter No. 2 2 Chapter No. 3 2 Chapter No. 4 3 Chapter No. 5 3 Chapter No. 6 3 Part – I (Short Questions) Attempt…
Objective Type Maximum Marks: 19Time Allowed: 20 Minutes Objective Part Sr. No. Description Marks 1 (a) Choose the correct form of verb and fill up the bubbles. PTB Page No. 163 to 168 5 1 (b) Choose the word with correct spelling and fill up the bubbles. 4 1 (c) Choose the correct meanings (synonyms/antonyms)….
MULTIPLE CHOICE QUESTIONS (9 from Exercise, 3 Additional) Follow Physics Postmortem Series for MCQs Preparation. Chapter # MCQs 1 1 2 1 3 2 4 1 5 1 6 2 7 1 8 2 9 1 SHORT QUESTIONS 2 Analytical Questions in each portion. Question Q#2 Chapter # Short Questions (S.Q) 1 3 3 5…
