Objective Section Subjective Section Q.2. Write detailed answers to the given questions: (10 Marks) (Questions based on poetry and prose texts.) Q.3. Explain the following verses in detail: (10 Marks) (Extracts from poetry are provided.) Q.4. Answer the following questions in short: (5 Marks) (Based on grammar and prose comprehension.) Q.5. Write an essay on…
All Punjab Boards: Lahore, Rawalpindi, Faisalabad, Sargodha, Gujranwala, Sahiwal, Multan, Bahawalpur, Dera Ghazi Khan, Azad Kashmir Mathematics (Science Group) MCQs (15 Marks) Chapter No. MCQs Chapter No. MCQs 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 10 1 11 1 12 1 13 0 14…
1. Questions, Options, Correct Answers, and Reasons (i) The set builder form of the set \( \left\{1, \frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \dots \right\}\) is: (a) \(\{x \, | \, x = \frac{1}{n}, \, n \in W\}\) (b) \(\{x \, | \, x = \frac{1}{2n + 1}, \, n \in W\}\) (c) \(\{x \, | \, x…
1. State the domain and range of each relation. For \(A = \{1, 2, 3, 4\}\), find \(A \times A\) first, then solve other parts. Find \(A \times A\) \[A \times A = \{(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3),…
Binary relations and functions are fundamental concepts in mathematics that describe relationships between elements of sets. They are widely used in various fields, from computer science to economics. Functions, in particular, serve as mathematical models for real-world phenomena. This article covers binary relations, the definition and types of functions, their domain and range, function notation,…
1. Consider the universal set \( U = \{x : x \text{ is multiple of 2 and } 0 < x \leq 30\} \) \( A = \{x : x \text{ is a multiple of 6}\} \) \( B = \{x : x \text{ is a multiple of 8}\} \) (i) List all elements of…
Sets are a fundamental concept in mathematics, and operations on sets allow us to combine, compare, and analyze their elements. These operations help solve problems in various fields, including computer science, probability, and logic. This article covers the union, intersection, difference, and complement of sets, explores disjoint and overlapping sets, and delves into operations on…
1. Write the following sets in set builder notation: (i) \(\{1, 4, 9, 16, 25, 36, \dots, 484\}\) \[\{x \, | \, x = n^2, \, n \in \mathbb{N}, \, 1 \leq n \leq 22\}\] (ii) \(\{2, 4, 8, 16, 32, 64, \dots, 150\}\) \[\{x \, | \, x = 2^n, \, n \in \mathbb{N},…
Sets and functions form the foundation of modern mathematics, playing a key role in various branches such as algebra, geometry, and computer science. Sets are used to group related elements, while functions define relationships between elements. In this article, we will explore the basics of sets, their different representations, types, and subsets, providing clear definitions,…
1. Questions, Correct Options, and Reasons (i) The standard form of \(5.2 \times 10^6\) is: (a) 52,000 (b) 520,000 (c) 5,200,000 (d) 52,000,000 Correct Answer: (c) 5,200,000 Reason: Multiplying \(5.2\) by \(10^6\) gives \(5,200,000\). (ii) Scientific notation of \(0.00034\) is: (a) \(3.4 \times 10^3\) (b) \(3.4 \times 10^{-4}\) (c) \(3.4 \times 10^4\) (d) \(3.4 \times…
