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ISC Class XI Notes 2026 : Mathematics (St. Josephs College (SJC), Prayagraj / Allahabad) : maths paper

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CITY CHILDREN'S ACADEMY Chhibramau, Kannauj MATHEMATICS PROJECT Session: 2026 27 Assignment 1 (Section A Topic 1) Subsets of a Set and Verification of 2 Formula Branch: Set Theory Assignment 2 (Section C Topic 22) Statistical Significance of Percentile Branch: Statistics Submitted by: Name: ___________________________ Class: XI Section: ___________ Roll No.: ______ Subject Teacher: ___________________________ City Children's Academy, Chhibramau | Mathematics Project | Class XI | Page INDEX S.No. Topic Section Page No. 1 Acknowledgement 3 2 Certificate 4 3 Introduction to Set Theory A Topic 1 5 4 What is a Subset? A Topic 1 5 5 Types of Subsets A Topic 1 6 6 Power Set A Topic 1 6 7 Venn Diagram Representation A Topic 1 7 8 Verification of 2 Formula A Topic 1 8 9 Worked Examples & Numericals A Topic 1 9 10 Conclusion (Sets) A Topic 1 11 11 Introduction to Statistics & Percentile C Topic 22 12 12 Types of Percentiles C Topic 22 12 13 Methods to Calculate Percentile C Topic 22 13 14 Worked Examples & Inferences C Topic 22 14 15 Conclusion (Statistics) C Topic 22 16 16 Bibliography 17 City Children's Academy, Chhibramau | Mathematics Project | Class XI | Page ACKNOWLEDGEMENT I would like to express my sincere gratitude to all those who supported and guided me in completing this Mathematics project. First and foremost, I am deeply thankful to my Mathematics teacher for their constant encouragement, valuable guidance, and patient support throughout this project. Their clear explanations of Set Theory and Statistics made it easier for me to understand and present these topics effectively. I also thank the Principal and all the teachers of City Children's Academy, Chhibramau, for providing an environment that encourages learning and curiosity. I am grateful to my parents for their constant motivation and for helping me arrange the necessary resources during the summer vacation. Finally, I acknowledge the textbooks, NCERT materials, and reference books that served as the primary sources of content for this project. ______________________________ Name of Student Class XI | City Children's Academy City Children's Academy, Chhibramau | Mathematics Project | Class XI | Page CERTIFICATE This is to certify that the Mathematics Project entitled: "Assignment 1: Subsets of a Set and Verification of 2 Formula" and "Assignment 2: Statistical Significance of Percentile" has been prepared and submitted by: Name: ___________________________ Class: XI Roll No.: ______ This project has been completed under my supervision and guidance and fulfils the requirements of the ISC Curriculum for the academic session 2026 27. Teacher's Signature: Principal's Signature: ______________________ ______________________ Subject Teacher Principal School Stamp City Children's Academy, Chhibramau | Mathematics Project | Class XI | Page ASSIGNMENT 1 SECTION A, TOPIC 1 Set Theory | Class XI | ISC 2026 27 1. Introduction to Set Theory A Set is one of the most fundamental concepts in Mathematics. It forms the building block of modern mathematics and is used across almost every branch from Algebra and Geometry to Probability and Statistics. Definition: A set is a well-defined collection of distinct objects. The objects in a set are called its elements or members. Georg Cantor (1845 1918), a German mathematician, is known as the Father of Set Theory. He introduced the concept of sets in the 1870s, which revolutionised the foundations of mathematics. Sets are usually denoted by capital letters such as A, B, C, and their elements are written inside curly braces { }. Examples of Sets: A = {1, 2, 3, 4, 5} Set of first five natural numbers B = {a, e, i, o, u} Set of vowels in English C = {Red, Blue, Green} Set of primary colours D = { } Empty set (no elements) 2. What is a Subset? If every element of a set A is also an element of set B, then A is called a Subset of B. Notation: A B (read as 'A is a subset of B') Condition: For every element x, if x A then x B Important: Every set is a subset of itself. The empty set { } is a subset of every set. These are called Improper subsets. Example: Let A = {1, 2} and B = {1, 2, 3} A = {1, 2} B = {1, 2, 3} (A is a subset of B) City Children's Academy, Chhibramau | Mathematics Project | Class XI | Page B is NOT a subset of A, because 3 B but 3 A 3. Types of Subsets Type Definition Example (A = {1,2}) Proper Subset A B, and A B (A is subset but not equal) {1} {1,2}, {2} {1,2}, { } {1,2} Improper Subset A B and A = B, or A = { } (empty set) {1,2} {1,2}, { } {1,2} 4. Power Set The Power Set of a set A is the collection of ALL possible subsets of A, including the empty set and A itself. Notation: P(A) or 2^A Formula: If n(A) = n, then the number of subsets = 2 Example Let A = {a, b, c}, so n(A) = 3: No. of elements Subsets Count 0 elements {} 1 1 element {a}, {b}, {c} 3 2 elements {a,b}, {b,c}, {a,c} 3 3 elements {a,b,c} 1 TOTAL P(A) has 8 subsets 2 = 8 5. Venn Diagram Representation A Venn Diagram is a visual tool that uses overlapping circles or rectangles to show the relationship between sets. It was introduced by John Venn (1834 1923). Venn Diagram Subset Relationship (A B) City Children's Academy, Chhibramau | Mathematics Project | Class XI | Page Draw: A large rectangle (Universal Set U). Inside it, draw a large circle for B = {1, 2, 3, 4}. Inside the circle B, draw a smaller circle for A = {1, 2}. Label each element inside the correct circle. This shows A B visually. In the Venn diagram: A = {1, 2} is drawn as a smaller circle completely inside B = {1, 2, 3, 4} Elements 1 and 2 are inside circle A (and also inside B) Elements 3 and 4 are inside circle B but outside circle A Since A lies entirely within B, we confirm A B Venn Diagram Subsets of A = {p, q} Draw: Universal Set U as rectangle. Draw 4 small circles/regions representing the 4 subsets: { } (empty), {p}, {q}, {p,q}. All are shown as being contained in or equal to A. This visually demonstrates that n=2 gives 2 = 4 subsets. The four subsets of A = {p, q} shown in the Venn diagram are: Subset Subset Subset Subset 1: 2: 3: 4: { } the empty set (always a subset of any set) {p} contains only element p {q} contains only element q {p, q} the set A itself (improper subset) Total subsets: 4 = 2 = 2 where n = 2 City Children's Academy, Chhibramau | Mathematics Project | Class XI | Page 6. Verification of the Formula: Number of Subsets = 2 We will now systematically verify this formula for sets of different sizes. Set n(A) All Subsets 2 Count {} 0 { } only 1 2 = 1 {x} 1 { }, {x} 2 2 = 2 {1,2} 2 { }, {1}, {2}, {1,2} 4 2 = 4 {a,b,c} 3 { },{a},{b},{c},{a,b},{b,c},{a,c},{a,b,c} 8 2 = 8 {1,2,3,4} 4 All 16 subsets (listed below) 16 2 = 16 Complete list of 16 subsets of A = {1, 2, 3, 4}: 0-element subsets (1): { } 1-element subsets (4): {1}, {2}, {3}, {4} 2-element subsets (6): {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4} 3-element subsets (4): {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4} 4-element subsets (1): {1,2,3,4} Total: 1 + 4 + 6 + 4 + 1 = 16 = 2 Formula verified! Observation: The number of subsets with each size (1, 4, 6, 4, 1) follows Pascal's Triangle, which is another beautiful connection in Mathematics! 7. Worked Examples and Numerical Problems Numerical 1 If A = {2, 4, 6, 8, 10}, find: (a) n(A) (b) Total number of subsets (c) Number of proper subsets Solution: (a) n(A) = 5 (the set has 5 elements) (b) Total subsets = 2 = 2 = 32 (c) Proper subsets = 2 1 = 32 1 = 31 (excluding the set itself) Numerical 2 City Children's Academy, Chhibramau | Mathematics Project | Class XI | Page A set has 64 subsets. Find the number of elements in the set. Solution: 2 = 64 = 2 n = 6 The set has 6 elements. Numerical 3 Find all subsets of A = {x, y} and verify the formula. Solution: n(A) = 2, so expected subsets = 2 = 4 Subsets: { }, {x}, {y}, {x, y} Count: 4 subsets = 2 Verified Numerical 4 If A = {1, 2} and B = {1, 2, 3, 4}, verify that A B using a Venn diagram and check whether B A. Solution: Every element of A (i.e., 1 and 2) is present in B. So A B B has elements 3 and 4 which are NOT in A. So B A Since A B but A B, we say A is a Proper Subset of B: A B Numerical 5 How many subsets does a set with n = 0, 1, 2, 3, 4, 5 elements have? Represent using a table. n (No. of elements) Number of Subsets (2 ) 0 2 = 1 1 2 = 2 2 2 = 4 3 2 = 8 4 2 = 16 5 2 = 32 Numerical 6 If a set has 128 subsets, find n. If it has 1 subset, what does that mean? Solution: 128 subsets: 2 = 128 = 2 n = 7 City Children's Academy, Chhibramau | Mathematics Project | Class XI | Page 1 subset: 2 = 1 = 2 n = 0. This means the set is the Empty Set { } 8. Conclusion Set Theory Through this assignment, the following key conclusions were drawn: A set is a well-defined collection of distinct objects. A subset A of B means every element of A is in B, written as A B. The empty set { } is a subset of every set; every set is a subset of itself. The Power Set P(A) contains all subsets of A, including { } and A itself. If a set has n elements, it has exactly 2 subsets this formula was verified using Venn diagrams and by listing all subsets for sets with 0 to 4 elements. The number of proper subsets = 2 1. Formula: Number of Subsets of a set with n elements = 2 VERIFIED City Children's Academy, Chhibramau | Mathematics Project | Class XI | Page ASSIGNMENT 2 SECTION C, TOPIC 22 Statistics | Class XI | ISC 2026 27 1. Introduction to Statistics Statistics is the branch of Mathematics that deals with the collection, organisation, analysis, interpretation, and presentation of data. It helps us make sense of large amounts of information by summarising it meaningfully. One important area of Statistics is Measures of Position, which tell us where a particular value stands relative to the rest of the data. Percentile is one such measure. 2. What is a Percentile? Definition: A Percentile is a value below which a certain percentage of data falls. The pth percentile (Pp) is the value such that p% of the data lies below it. Example: If a student's score is at the 80th percentile, it means the student scored higher than 80% of all other students. Percentiles divide a dataset into 100 equal parts. They are denoted as P1, P2, P3, ..., P99. P25 = First Quartile (Q1) 25% of data falls below this value P50 = Median (Q2) 50% of data falls below this value P75 = Third Quartile (Q3) 75% of data falls below this value 3. Types of Percentiles Percentile Also Known As Significance P25 Lower Quartile / Q1 25% of data lies below this P50 Median / Q2 50% of data lies below this P75 Upper Quartile / Q3 75% of data lies below this P90 Top Decile 90% of data lies below this 4. Formula for Percentile City Children's Academy, Chhibramau | Mathematics Project | Class XI | Page 4.1 For Ungrouped (Raw) Data Step 1: Arrange data in ascending order. Step 2: Use the formula: Location of P = L = (p/100) (n + 1) Where p = percentile required, n = total number of data values If L is a whole number, P = value at position L If L is a decimal, interpolate between the two nearest values. 4.2 For Grouped Data P = L + [(pn/100 cf) / f] h Symbol Meaning L Lower boundary of the percentile class p Desired percentile (e.g., 25, 50, 75) n Total frequency cf Cumulative frequency before the percentile class f Frequency of the percentile class h Class width (class size) 5. Worked Examples and Numerical Problems Numerical 1 Ungrouped Data Find the 25th and 75th percentile of the following data: 12, 18, 22, 25, 30, 35, 40, 45, 50, 60 Solution: Step 1: Data is already in ascending order. n = 10 For P25: L = (25/100) (10+1) = 0.25 11 = 2.75 P25 lies between 2nd value (18) and 3rd value (22) City Children's Academy, Chhibramau | Mathematics Project | Class XI | Page P25 = 18 + 0.75 (22 18) = 18 + 3 = 21 For P75: L = (75/100) (10+1) = 0.75 11 = 8.25 P75 lies between 8th value (45) and 9th value (50) P75 = 45 + 0.25 (50 45) = 45 + 1.25 = 46.25 Inference: 25% of the data values are below 21, and 75% of the data values are below 46.25. The Interquartile Range (IQR) = P75 P25 = 46.25 21 = 25.25, which shows the spread of the middle 50% of the data. Numerical 2 Grouped Frequency Distribution Find P50 (the Median percentile) from the following data: Class Interval Frequency (f) Cumulative Freq (cf) 0 10 5 5 10 20 8 13 20 30 12 25 30 40 10 35 40 50 5 40 Solution: n = 40, p = 50 Step 1: pn/100 = 50 40 / 100 = 20 Step 2: Find class where cf first exceeds 20. cf = 25 at class 20 30. So percentile class = 20 30 Step 3: L = 20, cf = 13 (before class), f = 12, h = 10 Step 4: P50 = 20 + [(20 13) / 12] 10 = 20 + [7/12] 10 = 20 + 5.83 = 25.83 Inference: The median (P50) of this data is approximately 25.83. This means 50% of the observations lie below 25.83 and 50% lie above it. This is the central tendency of the distribution. Numerical 3 Real-Life Application In a class of 50 students, a student scored 42 marks. If 35 students scored below 42, find the student's percentile rank. Formula: Percentile Rank = (Number of values below score / Total values) 100 Solution: Percentile Rank = (35/50) 100 = 70th Percentile City Children's Academy, Chhibramau | Mathematics Project | Class XI | Page Inference: The student is at the 70th percentile, meaning they scored better than 70% of the class. This is a very useful measure in education, competitive exams, and performance analysis. 6. Conclusion Statistics (Percentile) Through this assignment on Percentiles, the following conclusions were drawn: A percentile indicates the position of a value relative to the rest of the data. The pth percentile means p% of values lie below that point. Special percentiles P25, P50, P75 are the Quartiles, which divide data into four equal parts. P50 is the Median the most commonly used measure of central position. Percentiles are extremely useful in real life: NEET/JEE rank analysis, school marksheets, salary comparisons, BMI charts, and more. For ungrouped data, the formula L = (p/100)(n+1) is used. For grouped data, the interpolation formula P = L + [(pn/100 cf)/f] h is used. Key Insight: Percentiles allow us to compare individual performance against a group making them one of the most powerful tools in applied statistics. City Children's Academy, Chhibramau | Mathematics Project | Class XI | Page BIBLIOGRAPHY Books & Textbooks Mathematics Class XI, NCERT (National Council of Educational Research and Training) ISC Mathematics Class XI, S. Chand Publications R.D. Sharma Mathematics Class XI, Dhanpat Rai Publications Understanding ISC Mathematics (Vol. 1) M.L. Aggarwal Online References www.ncert.nic.in NCERT official textbooks and solutions www.byjus.com Concept explanations and solved examples www.khanacademy.org Video tutorials on Sets and Statistics www.mathsisfun.com Interactive examples on subsets and power sets Other Sources Class notes and teacher explanations City Children's Academy, Chhibramau ISC Board Previous Year Question Papers (2022 2025) ______________________________ Signature of Student City Children's Academy, Chhibramau | Mathematics Project | Class XI | Page

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