਀                            㰀椀渀瀀甀琀 琀礀瀀攀㴀∀挀栀攀挀欀戀漀砀∀ 椀搀㴀∀挀栀愀瀀琀攀爀㄀㌀∀ 渀愀洀攀㴀∀挀栀愀瀀琀攀爀∀ 瘀愀氀甀攀㴀∀䌀栀愀瀀琀攀爀 ㄀㌀∀㸀 13. Limits and Derivatives਀                        㰀⼀搀椀瘀㸀 ਀    㰀⼀搀椀瘀㸀 ਀    㰀搀椀瘀 挀氀愀猀猀㴀∀挀漀渀琀愀椀渀攀爀 栀椀搀搀攀渀∀ 椀搀㴀∀最攀渀攀爀愀琀攀搀倀愀瀀攀爀䌀漀渀琀愀椀渀攀爀∀㸀㰀⼀搀椀瘀㸀 ਀    㰀猀挀爀椀瀀琀㸀 // Initialize jsPDF਀        挀漀渀猀琀 笀 樀猀倀䐀䘀 紀 㴀 眀椀渀搀漀眀⸀樀猀瀀搀昀㬀 ਀        ⼀⼀ 䐀伀䴀 䔀氀攀洀攀渀琀猀 const startContainer = document.getElementById('startContainer');਀        挀漀渀猀琀 猀琀愀爀琀匀挀爀攀攀渀 㴀 搀漀挀甀洀攀渀琀⸀最攀琀䔀氀攀洀攀渀琀䈀礀䤀搀⠀✀猀琀愀爀琀匀挀爀攀攀渀✀⤀㬀 const mathOptions = document.getElementById('mathOptions');਀        挀漀渀猀琀 挀栀愀瀀琀攀爀匀攀氀攀挀琀椀漀渀 㴀 搀漀挀甀洀攀渀琀⸀最攀琀䔀氀攀洀攀渀琀䈀礀䤀搀⠀✀挀栀愀瀀琀攀爀匀攀氀攀挀琀椀漀渀✀⤀㬀 const generatedPaperContainer = document.getElementById('generatedPaperContainer');਀         // Buttons਀        挀漀渀猀琀 猀琀愀爀琀䜀攀渀攀爀愀琀椀渀最䈀琀渀 㴀 搀漀挀甀洀攀渀琀⸀最攀琀䔀氀攀洀攀渀琀䈀礀䤀搀⠀✀猀琀愀爀琀䜀攀渀攀爀愀琀椀渀最䈀琀渀✀⤀㬀 const selectMathBtn = document.getElementById('selectMathBtn');਀        挀漀渀猀琀 戀愀挀欀吀漀匀琀愀爀琀䈀琀渀 㴀 搀漀挀甀洀攀渀琀⸀最攀琀䔀氀攀洀攀渀琀䈀礀䤀搀⠀✀戀愀挀欀吀漀匀琀愀爀琀䈀琀渀✀⤀㬀 const generateAssignmentBtn = document.getElementById('generateAssignmentBtn');਀        挀漀渀猀琀 昀甀氀氀倀愀瀀攀爀䌀栀攀挀欀戀漀砀 㴀 搀漀挀甀洀攀渀琀⸀最攀琀䔀氀攀洀攀渀琀䈀礀䤀搀⠀✀昀甀氀氀倀愀瀀攀爀✀⤀㬀 const chapterCheckboxes = document.querySelectorAll('input[name="chapter"]:not(#fullPaper)');਀ // Event Listeners਀        猀琀愀爀琀䜀攀渀攀爀愀琀椀渀最䈀琀渀⸀愀搀搀䔀瘀攀渀琀䰀椀猀琀攀渀攀爀⠀✀挀氀椀挀欀✀Ⰰ 猀栀漀眀䴀愀琀栀伀瀀琀椀漀渀猀⤀㬀 selectMathBtn.addEventListener('click', showChapters);਀        戀愀挀欀吀漀匀琀愀爀琀䈀琀渀⸀愀搀搀䔀瘀攀渀琀䰀椀猀琀攀渀攀爀⠀✀挀氀椀挀欀✀Ⰰ 戀愀挀欀吀漀匀琀愀爀琀⤀㬀 generateAssignmentBtn.addEventListener('click', generateAssignment);਀        昀甀氀氀倀愀瀀攀爀䌀栀攀挀欀戀漀砀⸀愀搀搀䔀瘀攀渀琀䰀椀猀琀攀渀攀爀⠀✀挀栀愀渀最攀✀Ⰰ 琀漀最最氀攀䌀栀愀瀀琀攀爀猀⤀㬀 ਀        挀漀渀猀琀 焀甀攀猀琀椀漀渀猀 㴀 笀 "Chapter 1": {਀                洀挀焀㨀 嬀 { question: "Which of the following is a finite set?", options: ["Set of all points on a line", "Set of all natural numbers", "Set of all prime numbers less than 100", "Set of all integers"], answer: "Set of all prime numbers less than 100" },਀                    笀 焀甀攀猀琀椀漀渀㨀 ∀䤀昀 䄀 㴀 笀㄀Ⰰ ㈀Ⰰ ㌀紀Ⰰ 琀栀攀渀 琀栀攀 渀甀洀戀攀爀 漀昀 猀甀戀猀攀琀猀 漀昀 䄀 椀猀㨀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀㌀∀Ⰰ ∀㘀∀Ⰰ ∀㠀∀Ⰰ ∀㤀∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀㠀∀ 紀Ⰰ { question: "The set A = {x: x is a letter in the word 'SCHOOL'} can be written as:", options: ["{S, C, H, O, O, L}", "{S, C, H, O, L}", "{S, C, H, L}", "{S, H, O, O, L}"], answer: "{S, C, H, O, L}" }਀                崀Ⰰ assertionReason: [਀                    笀  assertion: "The empty set is a subset of every set.", ਀                        爀攀愀猀漀渀㨀 ∀吀栀攀 攀洀瀀琀礀 猀攀琀 栀愀猀 渀漀 攀氀攀洀攀渀琀猀Ⰰ 猀漀 愀氀氀 椀琀猀 攀氀攀洀攀渀琀猀 戀攀氀漀渀最 琀漀 愀渀礀 猀攀琀⸀∀Ⰰ answer: "Both A and R are true and R is the correct explanation of A."਀                    紀Ⰰ { ਀                        愀猀猀攀爀琀椀漀渀㨀 ∀䘀漀爀 愀渀礀 琀眀漀 猀攀琀猀 䄀 愀渀搀 䈀Ⰰ 䄀 ⨀•䈀 㴀 䄀 ⤀•䈀⸀∀Ⰰ  reason: "This is true only when A = B.",਀                        愀渀猀眀攀爀㨀 ∀䄀 椀猀 昀愀氀猀攀 戀甀琀 刀 椀猀 琀爀甀攀⸀∀ }਀                崀Ⰰ veryShort: [਀                    ∀圀爀椀琀攀 琀栀攀 猀攀琀 䄀 㴀 笀砀㨀 砀 椀猀 愀渀 椀渀琀攀最攀爀 愀渀搀 ⴀ㌀ 㰀 砀 㰀 㐀紀 椀渀 爀漀猀琀攀爀 昀漀爀洀⸀∀Ⰰ "If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, find A - B and B - A."਀                崀Ⰰ short: [਀                    ∀䤀渀 愀 猀甀爀瘀攀礀 漀昀 㐀　　 猀琀甀搀攀渀琀猀Ⰰ ㈀　　 琀愀欀攀 琀攀愀Ⰰ ㄀㔀　 琀愀欀攀 挀漀昀昀攀攀 愀渀搀 ㄀　　 琀愀欀攀 戀漀琀栀⸀ 䠀漀眀 洀愀渀礀 猀琀甀搀攀渀琀猀 琀愀欀攀 渀攀椀琀栀攀爀 琀攀愀 渀漀爀 挀漀昀昀攀攀㼀∀Ⰰ "Prove that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) for any three sets A, B and C."਀                崀Ⰰ long: [਀                    ∀唀猀椀渀最 嘀攀渀渀 搀椀愀最爀愀洀猀Ⰰ 瘀攀爀椀昀礀 琀栀愀琀 ⠀䄀 ⨀•䈀⤀✀ 㴀 䄀✀ ⤀•䈀✀ 愀渀搀 ⠀䄀 ⤀•䈀⤀✀ 㴀 䄀✀ ⨀•䈀✀⸀∀Ⰰ "In a class of 50 students, 30 study Mathematics, 25 study Biology, and 10 study both. Find the number of students who study (i) Mathematics only (ii) Biology only (iii) Neither Mathematics nor Biology."਀                崀Ⰰ caseStudy: [਀                    笀 context: "In a school, 60 students take Physics, 40 take Chemistry, and 30 take both. There are 20 students who take neither Physics nor Chemistry.",਀                        焀甀攀猀琀椀漀渀猀㨀 嬀 "Find the total number of students in the school.",਀                            ∀䠀漀眀 洀愀渀礀 猀琀甀搀攀渀琀猀 琀愀欀攀 漀渀氀礀 倀栀礀猀椀挀猀㼀∀ ]਀                    紀 ]਀            紀Ⰰ "Chapter 2": { ਀                洀挀焀㨀 嬀 { question: "Which of the following relations is a function?", options: ["{(1, 2), (1, 3), (2, 4)}", "{(1, 2), (2, 3), (3, 4)}", "{(1, 2), (2, 2), (2, 3)}", "{(1, 1), (2, 2), (3, 3), (1, 4)}"], answer: "{(1, 2), (2, 3), (3, 4)}" },਀                    笀 焀甀攀猀琀椀漀渀㨀 ∀吀栀攀 搀漀洀愀椀渀 漀昀 琀栀攀 昀甀渀挀琀椀漀渀 昀⠀砀⤀ 㴀 ᨀ⠢㐀 ⴀ 砀눀⤀ 椀猀㨀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀嬀ⴀ㈀Ⰰ ㈀崀∀Ⰰ ∀⠀ⴀḀⰢ ⴀ㈀崀 ⨀•嬀㈀Ⰰ Ḁ⤢∀Ⰰ ∀⠀ⴀḀⰢ ⴀ㈀⤀ ⨀•⠀㈀Ⰰ Ḁ⤢∀Ⰰ ∀⠀ⴀ㈀Ⰰ ㈀⤀∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀嬀ⴀ㈀Ⰰ ㈀崀∀ 紀Ⰰ { question: "If f(x) = x² + 1, then f(-2) is equal to:", options: ["-5", "-3", "5", "3"], answer: "5" }਀                崀Ⰰ assertionReason: [਀                    笀  assertion: "Every relation is a function.", ਀                        爀攀愀猀漀渀㨀 ∀䄀 昀甀渀挀琀椀漀渀 椀猀 愀 猀瀀攀挀椀愀氀 琀礀瀀攀 漀昀 爀攀氀愀琀椀漀渀 眀栀攀爀攀 攀愀挀栀 椀渀瀀甀琀 栀愀猀 攀砀愀挀琀氀礀 漀渀攀 漀甀琀瀀甀琀⸀∀Ⰰ answer: "A is false but R is true."਀                    紀Ⰰ { ਀                        愀猀猀攀爀琀椀漀渀㨀 ∀吀栀攀 昀甀渀挀琀椀漀渀 昀⠀砀⤀ 㴀 砀대 椀猀 漀渀攀ⴀ琀漀ⴀ漀渀攀⸀∀Ⰰ  reason: "A function is one-to-one if different inputs give different outputs.",਀                        愀渀猀眀攀爀㨀 ∀䈀漀琀栀 䄀 愀渀搀 刀 愀爀攀 琀爀甀攀 愀渀搀 刀 椀猀 琀栀攀 挀漀爀爀攀挀琀 攀砀瀀氀愀渀愀琀椀漀渀 漀昀 䄀⸀∀ }਀                崀Ⰰ veryShort: [਀                    ∀䘀椀渀搀 琀栀攀 搀漀洀愀椀渀 漀昀 琀栀攀 昀甀渀挀琀椀漀渀 昀⠀砀⤀ 㴀 ㄀⼀⠀砀 ⴀ ㈀⤀⸀∀Ⰰ "Let f = {(1, 2), (2, 3), (3, 4)} and g = {(2, 1), (3, 2), (4, 3)}. Find g o f."਀                崀Ⰰ short: [਀                    ∀䰀攀琀 䄀 㴀 笀㄀Ⰰ ㈀Ⰰ ㌀Ⰰ 㐀紀 愀渀搀 䈀 㴀 笀㔀Ⰰ 㘀Ⰰ 㜀Ⰰ 㠀紀⸀ 䐀攀琀攀爀洀椀渀攀 眀栀攀琀栀攀爀 琀栀攀 爀攀氀愀琀椀漀渀 刀 㴀 笀⠀㄀Ⰰ 㔀⤀Ⰰ ⠀㈀Ⰰ 㘀⤀Ⰰ ⠀㌀Ⰰ 㜀⤀Ⰰ ⠀㐀Ⰰ 㠀⤀紀 椀猀 愀 昀甀渀挀琀椀漀渀 昀爀漀洀 䄀 琀漀 䈀⸀ 䤀昀 礀攀猀Ⰰ 昀椀渀搀 椀琀猀 搀漀洀愀椀渀 愀渀搀 爀愀渀最攀⸀∀Ⰰ "Find the domain and range of the real function f(x) = √(9 - x²)."਀                崀Ⰰ long: [਀                    ∀䰀攀琀 昀㨀 刀 鈀‡刀 戀攀 搀攀昀椀渀攀搀 戀礀 昀⠀砀⤀ 㴀 ㌀砀 ⴀ ㈀⸀ 匀栀漀眀 琀栀愀琀 昀 椀猀 椀渀瘀攀爀琀椀戀氀攀 愀渀搀 昀椀渀搀 昀笀뤠⸀∀Ⰰ "Let A = R - {3} and B = R - {1}. Consider the function f: A → B defined by f(x) = (x - 2)/(x - 3). Show that f is one-one and onto and hence find f⁻¹."਀                崀Ⰰ caseStudy: [਀                    笀 context: "A function f is defined by f(x) = 2x + 3 for x ≤ 1 and f(x) = x² + 2 for x > 1.",਀                        焀甀攀猀琀椀漀渀猀㨀 嬀 "Find f(0) and f(2).",਀                            ∀䤀猀 昀 挀漀渀琀椀渀甀漀甀猀 愀琀 砀 㴀 ㄀㼀 䨀甀猀琀椀昀礀 礀漀甀爀 愀渀猀眀攀爀⸀∀ ]਀                    紀 ]਀            紀Ⰰ "Chapter 3": { ਀                洀挀焀㨀 嬀 { question: "The value of sin 15° is:", options: ["(√3 - 1)/2√2", "(√3 + 1)/2√2", "(√3 - 1)/2", "(√3 + 1)/2"], answer: "(√3 - 1)/2√2" },਀                    笀 焀甀攀猀琀椀漀渀㨀 ∀䤀昀 琀愀渀 렀 㴀 ⴀ㐀⼀㌀ 愀渀搀 렀 椀猀 椀渀 琀栀攀 猀攀挀漀渀搀 焀甀愀搀爀愀渀琀Ⰰ 琀栀攀渀 猀椀渀 렀 椀猀㨀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀ⴀ㐀⼀㔀∀Ⰰ ∀㐀⼀㔀∀Ⰰ ∀ⴀ㌀⼀㔀∀Ⰰ ∀㌀⼀㔀∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀㐀⼀㔀∀ 紀Ⰰ { question: "The general solution of sin θ = 0 is:", options: ["θ = nπ, n ∈ Z", "θ = 2nπ, n ∈ Z", "θ = (2n + 1)π/2, n ∈ Z", "θ = nπ/2, n ∈ Z"], answer: "θ = nπ, n ∈ Z" }਀                崀Ⰰ assertionReason: [਀                    笀  assertion: "sin(A + B) = sin A + sin B for all angles A and B.", ਀                        爀攀愀猀漀渀㨀 ∀吀栀攀 猀椀渀攀 昀甀渀挀琀椀漀渀 椀猀 搀椀猀琀爀椀戀甀琀椀瘀攀 漀瘀攀爀 愀搀搀椀琀椀漀渀⸀∀Ⰰ answer: "Both A and R are false."਀                    紀Ⰰ { ਀                        愀猀猀攀爀琀椀漀渀㨀 ∀吀栀攀 洀椀渀椀洀甀洀 瘀愀氀甀攀 漀昀 ㌀ 挀漀猀 砀 ⬀ 㐀 猀椀渀 砀 ⬀ 㔀 椀猀 　⸀∀Ⰰ  reason: "The maximum value of a cos x + b sin x is √(a² + b²).",਀                        愀渀猀眀攀爀㨀 ∀䄀 椀猀 昀愀氀猀攀 戀甀琀 刀 椀猀 琀爀甀攀⸀∀ }਀                崀Ⰰ veryShort: [਀                    ∀䘀椀渀搀 琀栀攀 瘀愀氀甀攀 漀昀 琀愀渀 ㄀㌀쀀⼃㄀㈀⸀∀Ⰰ "Prove that (sin x - sin y)/(cos x + cos y) = tan((x - y)/2)."਀                崀Ⰰ short: [਀                    ∀倀爀漀瘀攀 琀栀愀琀㨀 挀漀猀 㐀砀 㴀 ㄀ ⴀ 㠀 猀椀渀눀砀 挀漀猀눀砀∀Ⰰ "Find the general solution of the equation: sin x + sin 3x + sin 5x = 0"਀                崀Ⰰ long: [਀                    ∀倀爀漀瘀攀 琀栀愀琀㨀 ⠀挀漀猀 砀 ⴀ 挀漀猀 礀⤀눀 ⬀ ⠀猀椀渀 砀 ⴀ 猀椀渀 礀⤀눀 㴀 㐀 猀椀渀눀⠀⠀砀 ⴀ 礀⤀⼀㈀⤀∀Ⰰ "In any triangle ABC, prove that: (sin(B - C))/(sin(B + C)) = (b² - c²)/a²"਀                崀Ⰰ caseStudy: [਀                    笀 context: "A ladder 5 meters long leans against a vertical wall. The foot of the ladder is 3 meters away from the wall.",਀                        焀甀攀猀琀椀漀渀猀㨀 嬀 "Find the angle the ladder makes with the wall.",਀                            ∀䤀昀 琀栀攀 昀漀漀琀 漀昀 琀栀攀 氀愀搀搀攀爀 椀猀 瀀甀氀氀攀搀 愀眀愀礀 昀甀爀琀栀攀爀 昀爀漀洀 琀栀攀 眀愀氀氀Ⰰ 眀栀愀琀 栀愀瀀瀀攀渀猀 琀漀 琀栀攀 愀渀最氀攀 椀琀 洀愀欀攀猀 眀椀琀栀 琀栀攀 眀愀氀氀㼀∀ ]਀                    紀 ]਀            紀Ⰰ "Chapter 4": { ਀                洀挀焀㨀 嬀 { question: "The principle of mathematical induction is used to prove statements related to:", options: ["Only real numbers", "Only natural numbers", "Only integers", "All real numbers"], answer: "Only natural numbers" },਀                    笀 焀甀攀猀琀椀漀渀㨀 ∀䤀昀 倀⠀渀⤀㨀 ㈀帀渀 㰀 渀℀ 椀猀 琀爀甀攀 昀漀爀㨀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀䄀氀氀 渀 攀•㄀∀Ⰰ ∀䄀氀氀 渀 攀•㐀∀Ⰰ ∀䄀氀氀 渀 攀•㔀∀Ⰰ ∀䄀氀氀 渀 攀•㄀　∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀䄀氀氀 渀 攀•㐀∀ 紀Ⰰ { question: "The first step in mathematical induction is:", options: ["Assume P(k) is true", "Prove P(1) is true", "Prove P(k+1) is true", "Find counterexamples"], answer: "Prove P(1) is true" }਀                崀Ⰰ assertionReason: [਀                    笀  assertion: "Mathematical induction can be used to prove inequalities.", ਀                        爀攀愀猀漀渀㨀 ∀䴀愀琀栀攀洀愀琀椀挀愀氀 椀渀搀甀挀琀椀漀渀 椀猀 愀瀀瀀氀椀挀愀戀氀攀 琀漀 猀琀愀琀攀洀攀渀琀猀 椀渀瘀漀氀瘀椀渀最 渀愀琀甀爀愀氀 渀甀洀戀攀爀猀⸀∀Ⰰ answer: "Both A and R are true and R is the correct explanation of A."਀                    紀Ⰰ { ਀                        愀猀猀攀爀琀椀漀渀㨀 ∀䤀昀 倀⠀渀⤀ 椀猀 琀爀甀攀 昀漀爀 渀 㴀 欀Ⰰ 琀栀攀渀 椀琀 洀甀猀琀 戀攀 琀爀甀攀 昀漀爀 渀 㴀 欀 ⬀ ㄀⸀∀Ⰰ  reason: "This is the inductive step in mathematical induction.",਀                        愀渀猀眀攀爀㨀 ∀䄀 椀猀 昀愀氀猀攀 戀甀琀 刀 椀猀 琀爀甀攀⸀∀ }਀                崀Ⰰ veryShort: [਀                    ∀倀爀漀瘀攀 戀礀 椀渀搀甀挀琀椀漀渀 琀栀愀琀 ㄀ ⬀ ㌀ ⬀ 㔀 ⬀ ⸀⸀⸀ ⬀ ⠀㈀渀 ⴀ ㄀⤀ 㴀 渀눀 昀漀爀 愀氀氀 渀 ࠀ•一⸀∀Ⰰ "Prove that 7^n - 3^n is divisible by 4 for all n ∈ N."਀                崀Ⰰ short: [਀                    ∀倀爀漀瘀攀 戀礀 椀渀搀甀挀琀椀漀渀 琀栀愀琀 ㄀눀 ⬀ ㈀눀 ⬀ ㌀눀 ⬀ ⸀⸀⸀ ⬀ 渀눀 㴀 渀⠀渀 ⬀ ㄀⤀⠀㈀渀 ⬀ ㄀⤀⼀㘀 昀漀爀 愀氀氀 渀 ࠀ•一⸀∀Ⰰ "Prove by induction that 1.2 + 2.3 + 3.4 + ... + n(n + 1) = n(n + 1)(n + 2)/3 for all n ∈ N."਀                崀Ⰰ long: [਀                    ∀倀爀漀瘀攀 戀礀 椀渀搀甀挀琀椀漀渀 琀栀愀琀 ㄀대 ⬀ ㈀대 ⬀ ㌀대 ⬀ ⸀⸀⸀ ⬀ 渀대 㴀 ⠀渀⠀渀 ⬀ ㄀⤀⼀㈀⤀눀 昀漀爀 愀氀氀 渀 ࠀ•一⸀∀Ⰰ "Prove by induction that 11^(n+2) + 12^(2n+1) is divisible by 133 for all n ∈ N."਀                崀Ⰰ caseStudy: [਀                    笀 context: "A student is trying to prove that 3^n > n^3 for all natural numbers n.",਀                        焀甀攀猀琀椀漀渀猀㨀 嬀 "For which base case(s) should the student verify the statement?",਀                            ∀圀栀愀琀 猀栀漀甀氀搀 戀攀 琀栀攀 椀渀搀甀挀琀椀瘀攀 栀礀瀀漀琀栀攀猀椀猀㼀∀ ]਀                    紀 ]਀            紀Ⰰ "Chapter 5": { ਀                洀挀焀㨀 嬀 { question: "The value of i⁹ + i¹⁹ is:", options: ["0", "i", "-i", "2i"], answer: "0" },਀                    笀 焀甀攀猀琀椀漀渀㨀 ∀吀栀攀 爀漀漀琀猀 漀昀 琀栀攀 攀焀甀愀琀椀漀渀 砀눀 ⬀ 砀 ⬀ ㄀ 㴀 　 愀爀攀㨀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀刀攀愀氀 愀渀搀 搀椀猀琀椀渀挀琀∀Ⰰ ∀刀攀愀氀 愀渀搀 攀焀甀愀氀∀Ⰰ ∀䤀洀愀最椀渀愀爀礀∀Ⰰ ∀一漀渀攀 漀昀 琀栀攀猀攀∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀䤀洀愀最椀渀愀爀礀∀ 紀Ⰰ { question: "The complex conjugate of (3 - 2i)(2 + 3i) is:", options: ["12 + 5i", "12 - 5i", "-12 + 5i", "-12 - 5i"], answer: "12 - 5i" }਀                崀Ⰰ assertionReason: [਀                    笀  assertion: "The sum of two complex numbers is always a complex number.", ਀                        爀攀愀猀漀渀㨀 ∀䌀漀洀瀀氀攀砀 渀甀洀戀攀爀猀 愀爀攀 挀氀漀猀攀搀 甀渀搀攀爀 愀搀搀椀琀椀漀渀⸀∀Ⰰ answer: "Both A and R are true and R is the correct explanation of A."਀                    紀Ⰰ { ਀                        愀猀猀攀爀琀椀漀渀㨀 ∀吀栀攀 攀焀甀愀琀椀漀渀 砀눀 ⬀ ㄀ 㴀 　 栀愀猀 渀漀 猀漀氀甀琀椀漀渀⸀∀Ⰰ  reason: "The discriminant of the equation is negative.",਀                        愀渀猀眀攀爀㨀 ∀䄀 椀猀 昀愀氀猀攀 戀甀琀 刀 椀猀 琀爀甀攀⸀∀ }਀                崀Ⰰ veryShort: [਀                    ∀䔀砀瀀爀攀猀猀 ⠀㔀 ⬀ ᨀⴢ㈀⤀⠀㄀ ⴀ ᨀⴢ㠀⤀ 椀渀 琀栀攀 昀漀爀洀 愀 ⬀ 椀戀⸀∀Ⰰ "Find the modulus of the complex number (3 + 4i)/(4 - 3i)."਀                崀Ⰰ short: [਀                    ∀䌀漀渀瘀攀爀琀 琀栀攀 挀漀洀瀀氀攀砀 渀甀洀戀攀爀 ⠀㄀ ⴀ 椀⤀⼀⠀㄀ ⬀ 椀⤀ 椀渀琀漀 瀀漀氀愀爀 昀漀爀洀⸀∀Ⰰ "Solve the equation x² - 2x + 3/2 = 0."਀                崀Ⰰ long: [਀                    ∀䤀昀 ⠀砀 ⬀ 椀礀⤀대 㴀 甀 ⬀ 椀瘀Ⰰ 琀栀攀渀 猀栀漀眀 琀栀愀琀 甀⼀砀 ⬀ 瘀⼀礀 㴀 㐀⠀砀눀 ⴀ 礀눀⤀⸀∀Ⰰ "Find the square roots of the complex number 7 - 24i."਀                崀Ⰰ caseStudy: [਀                    笀 context: "In an electrical circuit, the impedance Z is given by Z = R + iX, where R is resistance and X is reactance. For two impedances Z₁ = 3 + 4i and Z₂ = 1 - 2i connected in series:",਀                        焀甀攀猀琀椀漀渀猀㨀 嬀 "Find the total impedance.",਀                            ∀䘀椀渀搀 琀栀攀 洀愀最渀椀琀甀搀攀 漀昀 琀栀攀 琀漀琀愀氀 椀洀瀀攀搀愀渀挀攀⸀∀ ]਀                    紀 ]਀            紀Ⰰ "Chapter 6": { ਀                洀挀焀㨀 嬀 { question: "The solution set of the inequality 3x + 5 < x - 7 is:", options: ["x < -6", "x > -6", "x < 6", "x > 6"], answer: "x < -6" },਀                    笀 焀甀攀猀琀椀漀渀㨀 ∀吀栀攀 最爀愀瀀栀 漀昀 琀栀攀 椀渀攀焀甀愀氀椀琀礀 礀 㸀 砀 ⬀ ㈀ 椀猀㨀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀䄀 猀琀爀愀椀最栀琀 氀椀渀攀∀Ⰰ ∀吀栀攀 爀攀最椀漀渀 愀戀漀瘀攀 琀栀攀 氀椀渀攀 礀 㴀 砀 ⬀ ㈀∀Ⰰ ∀吀栀攀 爀攀最椀漀渀 戀攀氀漀眀 琀栀攀 氀椀渀攀 礀 㴀 砀 ⬀ ㈀∀Ⰰ ∀䄀 瀀愀爀愀戀漀氀愀∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀吀栀攀 爀攀最椀漀渀 愀戀漀瘀攀 琀栀攀 氀椀渀攀 礀 㴀 砀 ⬀ ㈀∀ 紀Ⰰ { question: "The solution of |x - 2| ≤ 3 is:", options: ["-1 ≤ x ≤ 5", "x ≤ -1 or x ≥ 5", "-5 ≤ x ≤ 1", "x ≤ -5 or x ≥ 1"], answer: "-1 ≤ x ≤ 5" }਀                崀Ⰰ assertionReason: [਀                    笀  assertion: "The inequality x² < 4 is equivalent to x < ±2.", ਀                        爀攀愀猀漀渀㨀 ∀䘀漀爀 愀渀礀 爀攀愀氀 渀甀洀戀攀爀 愀Ⰰ 砀눀 㰀 愀눀 椀洀瀀氀椀攀猀 ⴀ愀 㰀 砀 㰀 愀⸀∀Ⰰ answer: "A is false but R is true."਀                    紀Ⰰ { ਀                        愀猀猀攀爀琀椀漀渀㨀 ∀吀栀攀 猀漀氀甀琀椀漀渀 猀攀琀 漀昀 砀 ⬀ 礀 攀•㔀 椀猀 琀栀攀 爀攀最椀漀渀 愀戀漀瘀攀 琀栀攀 氀椀渀攀 砀 ⬀ 礀 㴀 㔀⸀∀Ⰰ  reason: "For linear inequalities, the solution set is a half-plane.",਀                        愀渀猀眀攀爀㨀 ∀䈀漀琀栀 䄀 愀渀搀 刀 愀爀攀 琀爀甀攀 愀渀搀 刀 椀猀 琀栀攀 挀漀爀爀攀挀琀 攀砀瀀氀愀渀愀琀椀漀渀 漀昀 䄀⸀∀ }਀                崀Ⰰ veryShort: [਀                    ∀匀漀氀瘀攀 琀栀攀 椀渀攀焀甀愀氀椀琀礀㨀 ㌀砀 ⴀ 㜀 㸀 砀 ⬀ ㌀∀Ⰰ "Solve: |2x - 3| < 5"਀                崀Ⰰ short: [਀                    ∀匀漀氀瘀攀 琀栀攀 猀礀猀琀攀洀 漀昀 椀渀攀焀甀愀氀椀琀椀攀猀 最爀愀瀀栀椀挀愀氀氀礀㨀 砀 ⬀ 礀 搀•㘀Ⰰ 砀 ⴀ 礀 搀•㐀Ⰰ 砀 攀•　Ⰰ 礀 攀•　∀Ⰰ "Solve the inequality: (x - 1)/(x + 2) > 2"਀                崀Ⰰ long: [਀                    ∀匀漀氀瘀攀 琀栀攀 昀漀氀氀漀眀椀渀最 猀礀猀琀攀洀 漀昀 椀渀攀焀甀愀氀椀琀椀攀猀 最爀愀瀀栀椀挀愀氀氀礀㨀 ㈀砀 ⬀ 礀 攀•㠀Ⰰ 砀 ⬀ ㈀礀 攀•㄀　Ⰰ 砀 攀•　Ⰰ 礀 攀•　 愀渀搀 昀椀渀搀 琀栀攀 挀漀爀渀攀爀 瀀漀椀渀琀猀 漀昀 琀栀攀 昀攀愀猀椀戀氀攀 爀攀最椀漀渀⸀∀Ⰰ "A manufacturer produces two products A and B. Product A requires 2 hours on machine M1 and 3 hours on machine M2. Product B requires 3 hours on machine M1 and 2 hours on machine M2. If the machines are available for at most 12 hours each day, represent this problem as a system of inequalities."਀                崀Ⰰ caseStudy: [਀                    笀 context: "A dietitian is planning a meal that provides at least 80 units of vitamin A and 100 units of vitamin B. Food F1 costs ₹4 per unit and contains 3 units of A and 4 units of B. Food F2 costs ₹6 per unit and contains 6 units of A and 3 units of B.",਀                        焀甀攀猀琀椀漀渀猀㨀 嬀 "Formulate the inequalities representing the constraints.",਀                            ∀䜀爀愀瀀栀椀挀愀氀氀礀 爀攀瀀爀攀猀攀渀琀 琀栀攀 昀攀愀猀椀戀氀攀 爀攀最椀漀渀⸀∀ ]਀                    紀 ]਀            紀Ⰰ "Chapter 7": { ਀                洀挀焀㨀 嬀 { question: "The number of 4-digit numbers that can be formed using 1, 2, 3, 4 without repetition is:", options: ["12", "16", "24", "256"], answer: "24" },਀                    笀 焀甀攀猀琀椀漀渀㨀 ∀吀栀攀 瘀愀氀甀攀 漀昀 ㄀　倀㌀ 椀猀㨀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀㄀㈀　∀Ⰰ ∀㌀㘀　∀Ⰰ ∀㜀㈀　∀Ⰰ ∀㔀　㐀　∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀㜀㈀　∀ 紀Ⰰ { question: "The number of diagonals in a hexagon is:", options: ["6", "9", "12", "15"], answer: "9" }਀                崀Ⰰ assertionReason: [਀                    笀  assertion: "The number of permutations of n distinct objects is n!.", ਀                        爀攀愀猀漀渀㨀 ∀倀攀爀洀甀琀愀琀椀漀渀 洀攀愀渀猀 愀爀爀愀渀最攀洀攀渀琀 漀昀 漀戀樀攀挀琀猀 眀栀攀爀攀 漀爀搀攀爀 洀愀琀琀攀爀猀⸀∀Ⰰ answer: "Both A and R are true and R is the correct explanation of A."਀                    紀Ⰰ { ਀                        愀猀猀攀爀琀椀漀渀㨀 ∀吀栀攀 渀甀洀戀攀爀 漀昀 挀漀洀戀椀渀愀琀椀漀渀猀 漀昀 渀 漀戀樀攀挀琀猀 琀愀欀攀渀 爀 愀琀 愀 琀椀洀攀 椀猀 最爀攀愀琀攀爀 琀栀愀渀 琀栀攀 渀甀洀戀攀爀 漀昀 瀀攀爀洀甀琀愀琀椀漀渀猀⸀∀Ⰰ  reason: "Combinations do not consider the order of selection.",਀                        愀渀猀眀攀爀㨀 ∀䄀 椀猀 昀愀氀猀攀 戀甀琀 刀 椀猀 琀爀甀攀⸀∀ }਀                崀Ⰰ veryShort: [਀                    ∀䘀椀渀搀 琀栀攀 渀甀洀戀攀爀 漀昀 眀漀爀搀猀 琀栀愀琀 挀愀渀 戀攀 昀漀爀洀攀搀 昀爀漀洀 琀栀攀 氀攀琀琀攀爀猀 漀昀 琀栀攀 眀漀爀搀 ✀䌀伀䴀䴀䤀吀吀䔀䔀✀⸀∀Ⰰ "How many different committees of 3 men and 2 women can be formed from 5 men and 4 women?"਀                崀Ⰰ short: [਀                    ∀䤀渀 栀漀眀 洀愀渀礀 眀愀礀猀 挀愀渀 㔀 最椀爀氀猀 愀渀搀 ㌀ 戀漀礀猀 戀攀 猀攀愀琀攀搀 椀渀 愀 爀漀眀 猀漀 琀栀愀琀 渀漀 琀眀漀 戀漀礀猀 愀爀攀 琀漀最攀琀栀攀爀㼀∀Ⰰ "Find the number of 4-letter words that can be formed from the letters of the word 'NUMBER'."਀                崀Ⰰ long: [਀                    ∀䄀 挀漀洀洀椀琀琀攀攀 漀昀 㔀 椀猀 琀漀 戀攀 昀漀爀洀攀搀 昀爀漀洀 㘀 洀攀渀 愀渀搀 㐀 眀漀洀攀渀⸀ 䤀渀 栀漀眀 洀愀渀礀 眀愀礀猀 挀愀渀 琀栀椀猀 戀攀 搀漀渀攀 眀栀攀渀 ⠀椀⤀ 愀琀 氀攀愀猀琀 ㈀ 眀漀洀攀渀 愀爀攀 椀渀挀氀甀搀攀搀 ⠀椀椀⤀ 愀琀 洀漀猀琀 ㈀ 眀漀洀攀渀 愀爀攀 椀渀挀氀甀搀攀搀㼀∀Ⰰ "Find the number of arrangements of the letters of the word 'INDEPENDENCE'. In how many of these arrangements do all the vowels always occur together?"਀                崀Ⰰ caseStudy: [਀                    笀 context: "A student has 5 books on Mathematics, 4 on Physics and 3 on Chemistry. The student wants to arrange these books on a shelf.",਀                        焀甀攀猀琀椀漀渀猀㨀 嬀 "In how many ways can the books be arranged if books on each subject must be kept together?",਀                            ∀䠀漀眀 洀愀渀礀 愀爀爀愀渀最攀洀攀渀琀猀 愀爀攀 瀀漀猀猀椀戀氀攀 椀昀 漀渀氀礀 琀栀攀 䴀愀琀栀攀洀愀琀椀挀猀 戀漀漀欀猀 洀甀猀琀 戀攀 欀攀瀀琀 琀漀最攀琀栀攀爀㼀∀ ]਀                    紀 ]਀            紀Ⰰ "Chapter 8": { ਀                洀挀焀㨀 嬀 { question: "The middle term in the expansion of (x + y)⁶ is:", options: ["15x³y³", "20x³y³", "15x⁴y²", "20x²y⁴"], answer: "20x³y³" },਀                    笀 焀甀攀猀琀椀漀渀㨀 ∀吀栀攀 挀漀攀昀昀椀挀椀攀渀琀 漀昀 砀甀†椀渀 ⠀㄀ ⬀ 砀⤀砀†椀猀㨀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀㠀∀Ⰰ ∀㈀㠀∀Ⰰ ∀㔀㘀∀Ⰰ ∀㜀　∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀㔀㘀∀ 紀Ⰰ { question: "The number of terms in the expansion of (a + b + c)³ is:", options: ["6", "9", "10", "12"], answer: "10" }਀                崀Ⰰ assertionReason: [਀                    笀  assertion: "The sum of the coefficients in the expansion of (1 + x)ⁿ is 2ⁿ.", ਀                        爀攀愀猀漀渀㨀 ∀吀栀椀猀 椀猀 漀戀琀愀椀渀攀搀 戀礀 瀀甀琀琀椀渀最 砀 㴀 ㄀ 椀渀 琀栀攀 攀砀瀀愀渀猀椀漀渀⸀∀Ⰰ answer: "Both A and R are true and R is the correct explanation of A."਀                    紀Ⰰ { ਀                        愀猀猀攀爀琀椀漀渀㨀 ∀吀栀攀 最攀渀攀爀愀氀 琀攀爀洀 椀渀 琀栀攀 攀砀瀀愀渀猀椀漀渀 漀昀 ⠀砀 ⬀ 礀⤀缀†椀猀 渀䌀爀 砀帀⠀渀ⴀ爀⤀ 礀帀爀⸀∀Ⰰ  reason: "This is the (r+1)th term in the binomial expansion.",਀                        愀渀猀眀攀爀㨀 ∀䈀漀琀栀 䄀 愀渀搀 刀 愀爀攀 琀爀甀攀 愀渀搀 刀 椀猀 琀栀攀 挀漀爀爀攀挀琀 攀砀瀀氀愀渀愀琀椀漀渀 漀昀 䄀⸀∀ }਀                崀Ⰰ veryShort: [਀                    ∀䘀椀渀搀 琀栀攀 㐀琀栀 琀攀爀洀 椀渀 琀栀攀 攀砀瀀愀渀猀椀漀渀 漀昀 ⠀㈀砀 ⴀ ㌀礀⤀眀⸠∀Ⰰ "Find the coefficient of x⁵ in (1 + x)¹⁰."਀                崀Ⰰ short: [਀                    ∀䘀椀渀搀 琀栀攀 洀椀搀搀氀攀 琀攀爀洀⠀猀⤀ 椀渀 琀栀攀 攀砀瀀愀渀猀椀漀渀 漀昀 ⠀砀⼀㌀ ⬀ 㤀礀⤀뤀瀀⸠∀Ⰰ "Show that the coefficient of xⁿ in (1 + x)²ⁿ is twice the coefficient of xⁿ in (1 + x)²ⁿ⁻¹."਀                崀Ⰰ long: [਀                    ∀䘀椀渀搀 琀栀攀 琀攀爀洀 椀渀搀攀瀀攀渀搀攀渀琀 漀昀 砀 椀渀 琀栀攀 攀砀瀀愀渀猀椀漀渀 漀昀 ⠀ᨀ砢 ⬀ ㄀⼀⠀㌀砀눀⤀⤀뤀瀀⸠∀Ⰰ "If the coefficients of (r - 1)th, rth and (r + 1)th terms in the expansion of (1 + x)ⁿ are in the ratio 1:3:5, find n and r."਀                崀Ⰰ caseStudy: [਀                    笀 context: "A student is studying the binomial expansion of (1 + x)ⁿ. The student observes that the coefficients of certain terms form a pattern.",਀                        焀甀攀猀琀椀漀渀猀㨀 嬀 "What is the sum of the coefficients of the terms in the expansion of (1 + x)⁵?",਀                            ∀䤀昀 琀栀攀 挀漀攀昀昀椀挀椀攀渀琀猀 漀昀 琀栀攀 㐀琀栀 愀渀搀 ㄀㌀琀栀 琀攀爀洀猀 椀渀 琀栀攀 攀砀瀀愀渀猀椀漀渀 漀昀 ⠀㄀ ⬀ 砀⤀缀†愀爀攀 攀焀甀愀氀Ⰰ 眀栀愀琀 椀猀 琀栀攀 瘀愀氀甀攀 漀昀 渀㼀∀ ]਀                    紀 ]਀            紀Ⰰ "Chapter 9": { ਀                洀挀焀㨀 嬀 { question: "The 10th term of the sequence 2, 6, 18, 54, ... is:", options: ["2 × 3⁹", "2 × 3¹⁰", "2 × 3⁸", "2 × 3¹¹"], answer: "2 × 3⁹" },਀                    笀 焀甀攀猀琀椀漀渀㨀 ∀吀栀攀 猀甀洀 漀昀 琀栀攀 猀攀爀椀攀猀 ㄀ ⬀ ㌀ ⬀ 㔀 ⬀ ⸀⸀⸀ ⬀ ㄀㤀 椀猀㨀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀㠀㄀∀Ⰰ ∀㄀　　∀Ⰰ ∀㄀㈀㄀∀Ⰰ ∀㄀㐀㐀∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀㄀　　∀ 紀Ⰰ { question: "The sum to infinity of the series 1 + 1/2 + 1/4 + 1/8 + ... is:", options: ["1", "2", "4", "∞"], answer: "2" }਀                崀Ⰰ assertionReason: [਀                    笀  assertion: "The sequence 1, 4, 9, 16, ... is an arithmetic progression.", ਀                        爀攀愀猀漀渀㨀 ∀吀栀攀 搀椀昀昀攀爀攀渀挀攀 戀攀琀眀攀攀渀 挀漀渀猀攀挀甀琀椀瘀攀 琀攀爀洀猀 椀猀 挀漀渀猀琀愀渀琀⸀∀Ⰰ answer: "Both A and R are false."਀                    紀Ⰰ { ਀                        愀猀猀攀爀琀椀漀渀㨀 ∀吀栀攀 猀甀洀 漀昀 琀栀攀 昀椀爀猀琀 渀 漀搀搀 渀甀洀戀攀爀猀 椀猀 渀눀⸀∀Ⰰ  reason: "This can be proved by mathematical induction.",਀                        愀渀猀眀攀爀㨀 ∀䈀漀琀栀 䄀 愀渀搀 刀 愀爀攀 琀爀甀攀 愀渀搀 刀 椀猀 琀栀攀 挀漀爀爀攀挀琀 攀砀瀀氀愀渀愀琀椀漀渀 漀昀 䄀⸀∀ }਀                崀Ⰰ veryShort: [਀                    ∀䘀椀渀搀 琀栀攀 猀甀洀 漀昀 琀栀攀 猀攀爀椀攀猀 ㄀ ⬀ 㐀 ⬀ 㜀 ⬀ ⸀⸀⸀ 琀漀 ㈀　 琀攀爀洀猀⸀∀Ⰰ "Find the sum to infinity of the series 1 - 1/3 + 1/9 - 1/27 + ..."਀                崀Ⰰ short: [਀                    ∀䘀椀渀搀 琀栀攀 猀甀洀 漀昀 琀栀攀 猀攀爀椀攀猀 ㄀ 휀 ㈀ ⬀ ㈀ 휀 ㌀ ⬀ ㌀ 휀 㐀 ⬀ ⸀⸀⸀ 琀漀 渀 琀攀爀洀猀⸀∀Ⰰ "If the sum of n terms of an AP is 3n² + 5n, find its 10th term."਀                崀Ⰰ long: [਀                    ∀吀栀攀 猀甀洀 漀昀 琀栀攀 昀椀爀猀琀 琀栀爀攀攀 琀攀爀洀猀 漀昀 愀 䜀倀 椀猀 ㄀㌀⼀㄀㈀ 愀渀搀 琀栀攀椀爀 瀀爀漀搀甀挀琀 椀猀 ⴀ㄀⸀ 䘀椀渀搀 琀栀攀 挀漀洀洀漀渀 爀愀琀椀漀 愀渀搀 琀栀攀 琀攀爀洀猀⸀∀Ⰰ "If the pth term of an AP is q and the qth term is p, prove that its nth term is p + q - n."਀                崀Ⰰ caseStudy: [਀                    笀 context: "A ball is dropped from a height of 100 meters. Each time it hits the ground, it rebounds to 3/4 of the height from which it fell.",਀                        焀甀攀猀琀椀漀渀猀㨀 嬀 "Find the total distance traveled by the ball before coming to rest.",਀                            ∀䘀椀渀搀 琀栀攀 琀漀琀愀氀 琀椀洀攀 琀愀欀攀渀 戀礀 琀栀攀 戀愀氀氀 琀漀 挀漀洀攀 琀漀 爀攀猀琀 椀昀 琀栀攀 琀椀洀攀 琀漀 昀愀氀氀 昀爀漀洀 栀攀椀最栀琀 栀 椀猀 ᨀ⠢㈀栀⼀最⤀ 眀栀攀爀攀 最 㴀 㤀⸀㠀 洀⼀猀눀⸀∀ ]਀                    紀 ]਀            紀Ⰰ "Chapter 10": { ਀                洀挀焀㨀 嬀 { question: "The slope of the line joining the points (2, -3) and (4, 5) is:", options: ["1/4", "4", "-4", "1"], answer: "4" },਀                    笀 焀甀攀猀琀椀漀渀㨀 ∀吀栀攀 攀焀甀愀琀椀漀渀 漀昀 琀栀攀 氀椀渀攀 眀椀琀栀 猀氀漀瀀攀 ⴀ㈀ 愀渀搀 礀ⴀ椀渀琀攀爀挀攀瀀琀 ㌀ 椀猀㨀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀礀 㴀 ⴀ㈀砀 ⬀ ㌀∀Ⰰ ∀礀 㴀 ㈀砀 ⬀ ㌀∀Ⰰ ∀礀 㴀 ⴀ㈀砀 ⴀ ㌀∀Ⰰ ∀礀 㴀 ㈀砀 ⴀ ㌀∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀礀 㴀 ⴀ㈀砀 ⬀ ㌀∀ 紀Ⰰ { question: "The distance between the parallel lines 3x + 4y = 5 and 6x + 8y = 15 is:", options: ["1/2", "1", "5/2", "5"], answer: "1/2" }਀                崀Ⰰ assertionReason: [਀                    笀  assertion: "Two lines with slopes m1 and m2 are perpendicular if m1 × m2 = -1.", ਀                        爀攀愀猀漀渀㨀 ∀吀栀攀 瀀爀漀搀甀挀琀 漀昀 猀氀漀瀀攀猀 漀昀 瀀攀爀瀀攀渀搀椀挀甀氀愀爀 氀椀渀攀猀 椀猀 愀氀眀愀礀猀 ⴀ㄀⸀∀Ⰰ answer: "Both A and R are true and R is the correct explanation of A."਀                    紀Ⰰ { ਀                        愀猀猀攀爀琀椀漀渀㨀 ∀吀栀攀 氀椀渀攀猀 ㈀砀 ⬀ ㌀礀 㴀 㔀 愀渀搀 㐀砀 ⬀ 㘀礀 㴀 ㄀　 愀爀攀 挀漀椀渀挀椀搀攀渀琀⸀∀Ⰰ  reason: "One equation is a multiple of the other.",਀                        愀渀猀眀攀爀㨀 ∀䈀漀琀栀 䄀 愀渀搀 刀 愀爀攀 琀爀甀攀 愀渀搀 刀 椀猀 琀栀攀 挀漀爀爀攀挀琀 攀砀瀀氀愀渀愀琀椀漀渀 漀昀 䄀⸀∀ }਀                崀Ⰰ veryShort: [਀                    ∀䘀椀渀搀 琀栀攀 攀焀甀愀琀椀漀渀 漀昀 琀栀攀 氀椀渀攀 瀀愀猀猀椀渀最 琀栀爀漀甀最栀 ⠀㄀Ⰰ ㈀⤀ 愀渀搀 瀀愀爀愀氀氀攀氀 琀漀 ㌀砀 ⴀ 㐀礀 㴀 㜀⸀∀Ⰰ "Find the distance of the point (3, 4) from the line 4x + 3y = 10."਀                崀Ⰰ short: [਀                    ∀䘀椀渀搀 琀栀攀 攀焀甀愀琀椀漀渀 漀昀 琀栀攀 氀椀渀攀 眀栀椀挀栀 挀甀琀猀 漀昀昀 椀渀琀攀爀挀攀瀀琀猀 漀渀 琀栀攀 愀砀攀猀 眀栀漀猀攀 猀甀洀 椀猀 㤀 愀渀搀 瀀爀漀搀甀挀琀 椀猀 ㈀　⸀∀Ⰰ "Find the coordinates of the foot of the perpendicular from the point (2, 3) to the line 3x - 4y + 8 = 0."਀                崀Ⰰ long: [਀                    ∀䘀椀渀搀 琀栀攀 攀焀甀愀琀椀漀渀 漀昀 琀栀攀 氀椀渀攀 瀀愀猀猀椀渀最 琀栀爀漀甀最栀 琀栀攀 瀀漀椀渀琀 漀昀 椀渀琀攀爀猀攀挀琀椀漀渀 漀昀 ㈀砀 ⬀ 礀 㴀 㔀 愀渀搀 砀 ⬀ ㌀礀 ⬀ 㠀 㴀 　 愀渀搀 瀀愀爀愀氀氀攀氀 琀漀 琀栀攀 氀椀渀攀 ㌀砀 ⬀ 㐀礀 㴀 㜀⸀∀Ⰰ "A line is such that its segment between the lines 5x - y + 4 = 0 and 3x + 4y - 4 = 0 is bisected at the point (1, 5). Find its equation."਀                崀Ⰰ caseStudy: [਀                    笀 context: "A company manufactures two products A and B. The profit per unit is ₹5 for A and ₹6 for B. The production constraints are given by the inequalities x + 2y ≤ 10, 3x + 2y ≤ 18, x ≥ 0, y ≥ 0 where x and y are quantities of A and B respectively.",਀                        焀甀攀猀琀椀漀渀猀㨀 嬀 "Find the coordinates of the corner points of the feasible region.",਀                            ∀䘀椀渀搀 琀栀攀 洀愀砀椀洀甀洀 瀀爀漀昀椀琀⸀∀ ]਀                    紀 ]਀            紀Ⰰ "Chapter 11": { ਀                洀挀焀㨀 嬀 { question: "The eccentricity of the ellipse x²/25 + y²/9 = 1 is:", options: ["2/5", "3/5", "4/5", "1/5"], answer: "4/5" },਀                    笀 焀甀攀猀琀椀漀渀㨀 ∀吀栀攀 攀焀甀愀琀椀漀渀 漀昀 琀栀攀 瀀愀爀愀戀漀氀愀 眀椀琀栀 昀漀挀甀猀 ⠀㈀Ⰰ 　⤀ 愀渀搀 搀椀爀攀挀琀爀椀砀 砀 㴀 ⴀ㈀ 椀猀㨀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀礀눀 㴀 㠀砀∀Ⰰ ∀礀눀 㴀 ⴀ㠀砀∀Ⰰ ∀砀눀 㴀 㠀礀∀Ⰰ ∀砀눀 㴀 ⴀ㠀礀∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀礀눀 㴀 㠀砀∀ 紀Ⰰ { question: "The equation of the hyperbola with vertices at (±2, 0) and foci at (±3, 0) is:", options: ["x²/4 - y²/5 = 1", "x²/5 - y²/4 = 1", "x²/4 - y²/9 = 1", "x²/9 - y²/4 = 1"], answer: "x²/4 - y²/5 = 1" }਀                崀Ⰰ assertionReason: [਀                    笀  assertion: "A circle is a special case of an ellipse.", ਀                        爀攀愀猀漀渀㨀 ∀圀栀攀渀 琀栀攀 洀愀樀漀爀 愀渀搀 洀椀渀漀爀 愀砀攀猀 漀昀 愀渀 攀氀氀椀瀀猀攀 愀爀攀 攀焀甀愀氀Ⰰ 椀琀 戀攀挀漀洀攀猀 愀 挀椀爀挀氀攀⸀∀Ⰰ answer: "Both A and R are true and R is the correct explanation of A."਀                    紀Ⰰ { ਀                        愀猀猀攀爀琀椀漀渀㨀 ∀吀栀攀 氀愀琀甀猀 爀攀挀琀甀洀 漀昀 琀栀攀 瀀愀爀愀戀漀氀愀 礀눀 㴀 㐀愀砀 椀猀 㐀愀⸀∀Ⰰ  reason: "The length of the latus rectum is the distance between the points where the parabola intersects the line through the focus parallel to the directrix.",਀                        愀渀猀眀攀爀㨀 ∀䈀漀琀栀 䄀 愀渀搀 刀 愀爀攀 琀爀甀攀 愀渀搀 刀 椀猀 琀栀攀 挀漀爀爀攀挀琀 攀砀瀀氀愀渀愀琀椀漀渀 漀昀 䄀⸀∀ }਀                崀Ⰰ veryShort: [਀                    ∀䘀椀渀搀 琀栀攀 挀漀漀爀搀椀渀愀琀攀猀 漀昀 琀栀攀 昀漀挀椀 漀昀 琀栀攀 栀礀瀀攀爀戀漀氀愀 砀눀⼀㤀 ⴀ 礀눀⼀㄀㘀 㴀 ㄀⸀∀Ⰰ "Find the equation of the circle with center (2, -3) and radius 5."਀                崀Ⰰ short: [਀                    ∀䘀椀渀搀 琀栀攀 攀焀甀愀琀椀漀渀 漀昀 琀栀攀 瀀愀爀愀戀漀氀愀 眀椀琀栀 瘀攀爀琀攀砀 愀琀 ⠀　Ⰰ 　⤀Ⰰ 愀砀椀猀 愀氀漀渀最 琀栀攀 砀ⴀ愀砀椀猀 愀渀搀 瀀愀猀猀椀渀最 琀栀爀漀甀最栀 ⠀㈀Ⰰ ㌀⤀⸀∀Ⰰ "Find the equation of the ellipse with foci at (±3, 0) and vertices at (±5, 0)."਀                崀Ⰰ long: [਀                    ∀䘀椀渀搀 琀栀攀 攀焀甀愀琀椀漀渀 漀昀 琀栀攀 栀礀瀀攀爀戀漀氀愀 眀椀琀栀 昀漀挀椀 愀琀 ⠀　Ⰰ 넀ᨀㄢ　⤀ 愀渀搀 瀀愀猀猀椀渀最 琀栀爀漀甀最栀 ⠀㈀Ⰰ ㌀⤀⸀∀Ⰰ "A rod of length 12 cm moves with its ends always touching the coordinate axes. Find the equation of the locus of a point P on the rod which is 3 cm from the end in contact with the x-axis."਀                崀Ⰰ caseStudy: [਀                    笀 context: "A satellite dish has a parabolic shape. The dish is 2 meters wide and 0.5 meters deep at its center.",਀                        焀甀攀猀琀椀漀渀猀㨀 嬀 "Find the equation of the parabola assuming the vertex is at the origin and the parabola opens upwards.",਀                            ∀䘀椀渀搀 琀栀攀 瀀漀猀椀琀椀漀渀 漀昀 琀栀攀 爀攀挀攀椀瘀攀爀 眀栀椀挀栀 猀栀漀甀氀搀 戀攀 瀀氀愀挀攀搀 愀琀 琀栀攀 昀漀挀甀猀⸀∀ ]਀                    紀 ]਀            紀Ⰰ "Chapter 12": { ਀                洀挀焀㨀 嬀 { question: "The distance between the points (2, 3, 5) and (4, 3, 1) is:", options: ["2", "√20", "√24", "√28"], answer: "√20" },਀                    笀 焀甀攀猀琀椀漀渀㨀 ∀吀栀攀 挀漀漀爀搀椀渀愀琀攀猀 漀昀 琀栀攀 挀攀渀琀爀漀椀搀 漀昀 琀栀攀 琀爀椀愀渀最氀攀 眀椀琀栀 瘀攀爀琀椀挀攀猀 ⠀㄀Ⰰ ㈀Ⰰ ㌀⤀Ⰰ ⠀㈀Ⰰ ㌀Ⰰ ㄀⤀ 愀渀搀 ⠀㌀Ⰰ ㄀Ⰰ ㈀⤀ 愀爀攀㨀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀⠀㈀Ⰰ ㈀Ⰰ ㈀⤀∀Ⰰ ∀⠀㌀Ⰰ ㌀Ⰰ ㌀⤀∀Ⰰ ∀⠀㄀Ⰰ ㄀Ⰰ ㄀⤀∀Ⰰ ∀⠀㘀Ⰰ 㘀Ⰰ 㘀⤀∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀⠀㈀Ⰰ ㈀Ⰰ ㈀⤀∀ 紀Ⰰ { question: "The direction ratios of the line joining the points (1, 0, 0) and (0, 1, 1) are:", options: ["(1, 1, 1)", "(1, -1, 1)", "(-1, 1, 1)", "(1, 1, -1)"], answer: "(-1, 1, 1)" }਀                崀Ⰰ assertionReason: [਀                    笀  assertion: "The distance between two points in space is always positive.", ਀                        爀攀愀猀漀渀㨀 ∀䐀椀猀琀愀渀挀攀 椀猀 琀栀攀 洀漀搀甀氀甀猀 漀昀 琀栀攀 搀椀猀瀀氀愀挀攀洀攀渀琀 瘀攀挀琀漀爀⸀∀Ⰰ answer: "Both A and R are true and R is the correct explanation of A."਀                    紀Ⰰ { ਀                        愀猀猀攀爀琀椀漀渀㨀 ∀吀栀攀 搀椀爀攀挀琀椀漀渀 挀漀猀椀渀攀猀 漀昀 愀 氀椀渀攀 愀爀攀 甀渀椀焀甀攀⸀∀Ⰰ  reason: "The direction cosines are the cosines of the angles made by the line with the positive directions of the coordinate axes.",਀                        愀渀猀眀攀爀㨀 ∀䈀漀琀栀 䄀 愀渀搀 刀 愀爀攀 琀爀甀攀 愀渀搀 刀 椀猀 琀栀攀 挀漀爀爀攀挀琀 攀砀瀀氀愀渀愀琀椀漀渀 漀昀 䄀⸀∀ }਀                崀Ⰰ veryShort: [਀                    ∀䘀椀渀搀 琀栀攀 搀椀猀琀愀渀挀攀 戀攀琀眀攀攀渀 琀栀攀 瀀漀椀渀琀猀 ⠀㄀Ⰰ ㈀Ⰰ ㌀⤀ 愀渀搀 ⠀㐀Ⰰ 㔀Ⰰ 㘀⤀⸀∀Ⰰ "Find the direction cosines of the line joining (1, 2, -3) and (3, -1, 2)."਀                崀Ⰰ short: [਀                    ∀䘀椀渀搀 琀栀攀 爀愀琀椀漀 椀渀 眀栀椀挀栀 琀栀攀 礀稀ⴀ瀀氀愀渀攀 搀椀瘀椀搀攀猀 琀栀攀 氀椀渀攀 猀攀最洀攀渀琀 樀漀椀渀椀渀最 琀栀攀 瀀漀椀渀琀猀 ⠀㈀Ⰰ 㐀Ⰰ 㔀⤀ 愀渀搀 ⠀ⴀ㐀Ⰰ ㌀Ⰰ ⴀ㈀⤀⸀∀Ⰰ "Find the coordinates of the point which divides the line segment joining (1, -2, 3) and (3, 4, -5) in the ratio 2:3 internally."਀                崀Ⰰ long: [਀                    ∀䘀椀渀搀 琀栀攀 攀焀甀愀琀椀漀渀 漀昀 琀栀攀 猀攀琀 漀昀 瀀漀椀渀琀猀 倀 猀甀挀栀 琀栀愀琀 倀䄀눀 ⬀ 倀䈀눀 㴀 ㈀欀눀Ⰰ 眀栀攀爀攀 䄀 愀渀搀 䈀 愀爀攀 琀栀攀 瀀漀椀渀琀猀 ⠀㌀Ⰰ 㐀Ⰰ 㔀⤀ 愀渀搀 ⠀ⴀ㄀Ⰰ ㌀Ⰰ ⴀ㜀⤀ 爀攀猀瀀攀挀琀椀瘀攀氀礀⸀∀Ⰰ "Find the coordinates of the foot of the perpendicular drawn from the point (1, 2, 3) to the plane x + 2y + 3z = 0."਀                崀Ⰰ caseStudy: [਀                    笀 context: "A room is 5 meters long, 4 meters wide and 3 meters high. A spider is at one corner of the floor and a fly is at the diagonally opposite corner of the ceiling.",਀                        焀甀攀猀琀椀漀渀猀㨀 嬀 "Find the straight-line distance between the spider and the fly.",਀                            ∀䤀昀 琀栀攀 猀瀀椀搀攀爀 眀愀氀欀猀 愀氀漀渀最 琀栀攀 攀搀最攀猀 漀昀 琀栀攀 爀漀漀洀Ⰰ 眀栀愀琀 椀猀 琀栀攀 猀栀漀爀琀攀猀琀 搀椀猀琀愀渀挀攀 椀琀 洀甀猀琀 琀爀愀瘀攀氀 琀漀 爀攀愀挀栀 琀栀攀 昀氀礀㼀∀ ]਀                    紀 ]਀            紀Ⰰ "Chapter 13": { ਀                洀挀焀㨀 嬀 { question: "The derivative of x² + 1 with respect to x is:", options: ["x", "2x", "x²", "2x + 1"], answer: "2x" },਀                    笀 焀甀攀猀琀椀漀渀㨀 ∀吀栀攀 搀攀爀椀瘀愀琀椀瘀攀 漀昀 猀椀渀 砀 愀琀 砀 㴀 쀀⼃㈀ 椀猀㨀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀㄀∀Ⰰ ∀　∀Ⰰ ∀ⴀ㄀∀Ⰰ ∀㄀⼀㈀∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀　∀ 紀Ⰰ { question: "lim(x→0) (sin x)/x is equal to:", options: ["0", "1", "-1", "∞"], answer: "1" }਀                崀Ⰰ assertionReason: [਀                    笀  assertion: "The derivative of a constant function is zero.", ਀                        爀攀愀猀漀渀㨀 ∀吀栀攀 爀愀琀攀 漀昀 挀栀愀渀最攀 漀昀 愀 挀漀渀猀琀愀渀琀 昀甀渀挀琀椀漀渀 椀猀 稀攀爀漀⸀∀Ⰰ answer: "Both A and R are true and R is the correct explanation of A."਀                    紀Ⰰ { ਀                        愀猀猀攀爀琀椀漀渀㨀 ∀吀栀攀 昀甀渀挀琀椀漀渀 昀⠀砀⤀ 㴀 簀砀簀 椀猀 搀椀昀昀攀爀攀渀琀椀愀戀氀攀 愀琀 砀 㴀 　⸀∀Ⰰ  reason: "The left-hand and right-hand derivatives at x = 0 are equal.",਀                        愀渀猀眀攀爀㨀 ∀䄀 椀猀 昀愀氀猀攀 戀甀琀 刀 椀猀 琀爀甀攀⸀∀ }਀                崀Ⰰ veryShort: [਀                    ∀䘀椀渀搀 琀栀攀 搀攀爀椀瘀愀琀椀瘀攀 漀昀 昀⠀砀⤀ 㴀 ㌀砀눀 ⴀ ㈀砀 ⬀ ㄀⸀∀Ⰰ "Evaluate lim(x→3) (x² - 9)/(x - 3)."਀                崀Ⰰ short: [਀                    ∀䘀椀渀搀 琀栀攀 搀攀爀椀瘀愀琀椀瘀攀 漀昀 昀⠀砀⤀ 㴀 ⠀砀 ⬀ ㄀⤀⼀砀 昀爀漀洀 昀椀爀猀琀 瀀爀椀渀挀椀瀀氀攀猀⸀∀Ⰰ "Find the derivative of tan x using the quotient rule."਀                崀Ⰰ long: [਀                    ∀䘀椀渀搀 琀栀攀 搀攀爀椀瘀愀琀椀瘀攀 漀昀 ⠀砀 ⬀ 愀⤀⠀砀눀 ⬀ 愀눀⤀⠀砀琀†⬀ 愀琀⤠⠀砀砀†⬀ 愀砀⤠⸀∀Ⰰ "If y = (tan⁻¹x)², show that (1 + x²)² y'' + 2x(1 + x²) y' = 2."਀                崀Ⰰ caseStudy: [਀                    笀 context: "The distance s (in meters) traveled by a particle in time t (in seconds) is given by s = t³ - 6t² + 9t + 4.",਀                        焀甀攀猀琀椀漀渀猀㨀 嬀 "Find the velocity of the particle at time t.",਀                            ∀䘀椀渀搀 琀栀攀 愀挀挀攀氀攀爀愀琀椀漀渀 眀栀攀渀 琀栀攀 瘀攀氀漀挀椀琀礀 椀猀 稀攀爀漀⸀∀ ]਀                    紀 ]਀            紀Ⰰ "Chapter 14": { ਀                洀挀焀㨀 嬀 { question: "The negation of the statement 'All birds can fly' is:", options: ["No birds can fly", "Some birds can fly", "Some birds cannot fly", "All birds cannot fly"], answer: "Some birds cannot fly" },਀                    笀 焀甀攀猀琀椀漀渀㨀 ∀吀栀攀 挀漀渀琀爀愀瀀漀猀椀琀椀瘀攀 漀昀 ✀䤀昀 愀 渀甀洀戀攀爀 椀猀 搀椀瘀椀猀椀戀氀攀 戀礀 㘀Ⰰ 琀栀攀渀 椀琀 椀猀 搀椀瘀椀猀椀戀氀攀 戀礀 ㌀✀ 椀猀㨀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀䤀昀 愀 渀甀洀戀攀爀 椀猀 渀漀琀 搀椀瘀椀猀椀戀氀攀 戀礀 ㌀Ⰰ 琀栀攀渀 椀琀 椀猀 渀漀琀 搀椀瘀椀猀椀戀氀攀 戀礀 㘀∀Ⰰ ∀䤀昀 愀 渀甀洀戀攀爀 椀猀 搀椀瘀椀猀椀戀氀攀 戀礀 ㌀Ⰰ 琀栀攀渀 椀琀 椀猀 搀椀瘀椀猀椀戀氀攀 戀礀 㘀∀Ⰰ ∀䤀昀 愀 渀甀洀戀攀爀 椀猀 渀漀琀 搀椀瘀椀猀椀戀氀攀 戀礀 㘀Ⰰ 琀栀攀渀 椀琀 椀猀 渀漀琀 搀椀瘀椀猀椀戀氀攀 戀礀 ㌀∀Ⰰ ∀䄀 渀甀洀戀攀爀 椀猀 搀椀瘀椀猀椀戀氀攀 戀礀 㘀 椀昀 愀渀搀 漀渀氀礀 椀昀 椀琀 椀猀 搀椀瘀椀猀椀戀氀攀 戀礀 ㌀∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀䤀昀 愀 渀甀洀戀攀爀 椀猀 渀漀琀 搀椀瘀椀猀椀戀氀攀 戀礀 ㌀Ⰰ 琀栀攀渀 椀琀 椀猀 渀漀琀 搀椀瘀椀猀椀戀氀攀 戀礀 㘀∀ 紀Ⰰ { question: "Which of the following is a tautology?", options: ["p ∨ ¬p", "p ∧ ¬p", "p → ¬p", "¬p ∧ p"], answer: "p ∨ ¬p" }਀                崀Ⰰ assertionReason: [਀                    笀  assertion: "The converse of a statement is logically equivalent to the original statement.", ਀                        爀攀愀猀漀渀㨀 ∀吀栀攀 挀漀渀瘀攀爀猀攀 椀猀 漀戀琀愀椀渀攀搀 戀礀 爀攀瘀攀爀猀椀渀最 琀栀攀 挀漀洀瀀漀渀攀渀琀猀 漀昀 愀 挀漀渀搀椀琀椀漀渀愀氀 猀琀愀琀攀洀攀渀琀⸀∀Ⰰ answer: "A is false but R is true."਀                    紀Ⰰ { ਀                        愀猀猀攀爀琀椀漀渀㨀 ∀吀栀攀 猀琀愀琀攀洀攀渀琀 ✀䤀昀 渀 椀猀 攀瘀攀渀Ⰰ 琀栀攀渀 渀눀 椀猀 攀瘀攀渀✀ 椀猀 琀爀甀攀⸀∀Ⰰ  reason: "The square of an even number is always even.",਀                        愀渀猀眀攀爀㨀 ∀䈀漀琀栀 䄀 愀渀搀 刀 愀爀攀 琀爀甀攀 愀渀搀 刀 椀猀 琀栀攀 挀漀爀爀攀挀琀 攀砀瀀氀愀渀愀琀椀漀渀 漀昀 䄀⸀∀ }਀                崀Ⰰ veryShort: [਀                    ∀圀爀椀琀攀 琀栀攀 渀攀最愀琀椀漀渀 漀昀 琀栀攀 猀琀愀琀攀洀攀渀琀㨀 ✀䔀瘀攀爀礀 渀愀琀甀爀愀氀 渀甀洀戀攀爀 椀猀 最爀攀愀琀攀爀 琀栀愀渀 稀攀爀漀⸀✀∀Ⰰ "Write the contrapositive of: 'If a triangle is equilateral, then it is isosceles.'"਀                崀Ⰰ short: [਀                    ∀嘀攀爀椀昀礀 戀礀 琀爀甀琀栀 琀愀戀氀攀 琀栀愀琀 瀀 鈀‡焀 椀猀 攀焀甀椀瘀愀氀攀渀琀 琀漀 가焀 鈀‡가瀀⸀∀Ⰰ "Check whether the following statement is a tautology or contradiction: (p ∧ q) ∨ (¬p ∨ ¬q)"਀                崀Ⰰ long: [਀                    ∀倀爀漀瘀攀 琀栀愀琀 琀栀攀 昀漀氀氀漀眀椀渀最 猀琀愀琀攀洀攀渀琀 椀猀 愀 琀愀甀琀漀氀漀最礀㨀 ⠀瀀 鈀‡焀⤀ 鐀‡⠀가焀 鈀‡가瀀⤀∀Ⰰ "Using truth tables, show that p → q ≡ ¬p ∨ q."਀                崀Ⰰ caseStudy: [਀                    笀 context: "A student is analyzing logical statements and their truth values. The student considers the statement: 'If it rains, then the ground will be wet.'",਀                        焀甀攀猀琀椀漀渀猀㨀 嬀 "What is the converse of this statement?",਀                            ∀䤀猀 琀栀攀 漀爀椀最椀渀愀氀 猀琀愀琀攀洀攀渀琀 氀漀最椀挀愀氀氀礀 攀焀甀椀瘀愀氀攀渀琀 琀漀 椀琀猀 挀漀渀琀爀愀瀀漀猀椀琀椀瘀攀㼀∀ ]਀                    紀 ]਀            紀Ⰰ "Chapter 15": { ਀                洀挀焀㨀 嬀 { question: "The mean deviation about the median for the data 3, 9, 5, 3, 12, 10, 18, 4, 7, 19, 21 is:", options: ["5.27", "4.27", "6.27", "7.27"], answer: "5.27" },਀                    笀 焀甀攀猀琀椀漀渀㨀 ∀吀栀攀 瘀愀爀椀愀渀挀攀 漀昀 琀栀攀 搀愀琀愀 㘀Ⰰ 㠀Ⰰ ㄀　Ⰰ ㄀㈀Ⰰ ㄀㐀Ⰰ ㄀㘀Ⰰ ㄀㠀Ⰰ ㈀　Ⰰ ㈀㈀Ⰰ ㈀㐀 椀猀㨀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀㌀㌀∀Ⰰ ∀㌀　∀Ⰰ ∀㌀㘀∀Ⰰ ∀㌀㤀∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀㌀㌀∀ 紀Ⰰ { question: "The standard deviation of the first n natural numbers is:", options: ["√(n² - 1)/12", "√(n² + 1)/12", "√(n² - 1)/6", "√(n² + 1)/6"], answer: "√(n² - 1)/12" }਀                崀Ⰰ assertionReason: [਀                    笀  assertion: "The mean deviation is minimum when calculated about the median.", ਀                        爀攀愀猀漀渀㨀 ∀吀栀攀 洀攀搀椀愀渀 洀椀渀椀洀椀稀攀猀 琀栀攀 猀甀洀 漀昀 愀戀猀漀氀甀琀攀 搀攀瘀椀愀琀椀漀渀猀⸀∀Ⰰ answer: "Both A and R are true and R is the correct explanation of A."਀                    紀Ⰰ { ਀                        愀猀猀攀爀琀椀漀渀㨀 ∀吀栀攀 瘀愀爀椀愀渀挀攀 漀昀 愀 搀愀琀愀 猀攀琀 椀猀 愀氀眀愀礀猀 渀漀渀ⴀ渀攀最愀琀椀瘀攀⸀∀Ⰰ  reason: "Variance is the average of squared deviations from the mean.",਀                        愀渀猀眀攀爀㨀 ∀䈀漀琀栀 䄀 愀渀搀 刀 愀爀攀 琀爀甀攀 愀渀搀 刀 椀猀 琀栀攀 挀漀爀爀攀挀琀 攀砀瀀氀愀渀愀琀椀漀渀 漀昀 䄀⸀∀ }਀                崀Ⰰ veryShort: [਀                    ∀䘀椀渀搀 琀栀攀 洀攀愀渀 搀攀瘀椀愀琀椀漀渀 愀戀漀甀琀 琀栀攀 洀攀愀渀 昀漀爀 琀栀攀 搀愀琀愀 㐀Ⰰ 㜀Ⰰ 㠀Ⰰ 㤀Ⰰ ㄀　Ⰰ ㄀㈀Ⰰ ㄀㌀Ⰰ ㄀㜀⸀∀Ⰰ "Find the variance of the first 10 multiples of 3."਀                崀Ⰰ short: [਀                    ∀䌀愀氀挀甀氀愀琀攀 琀栀攀 洀攀愀渀 愀渀搀 瘀愀爀椀愀渀挀攀 昀漀爀 琀栀攀 昀漀氀氀漀眀椀渀最 搀愀琀愀㨀 㐀Ⰰ 㠀Ⰰ ㄀㈀Ⰰ ㄀㘀Ⰰ ㈀　Ⰰ ㈀㐀Ⰰ ㈀㠀Ⰰ ㌀㈀∀Ⰰ "Find the standard deviation of the following distribution: x: 5, 10, 15, 20, 25; f: 3, 5, 7, 3, 2"਀                崀Ⰰ long: [਀                    ∀吀栀攀 洀攀愀渀 愀渀搀 猀琀愀渀搀愀爀搀 搀攀瘀椀愀琀椀漀渀 漀昀 ㈀　 漀戀猀攀爀瘀愀琀椀漀渀猀 愀爀攀 昀漀甀渀搀 琀漀 戀攀 ㄀　 愀渀搀 ㈀ 爀攀猀瀀攀挀琀椀瘀攀氀礀⸀ 伀渀 爀攀挀栀攀挀欀椀渀最Ⰰ 椀琀 眀愀猀 昀漀甀渀搀 琀栀愀琀 愀渀 漀戀猀攀爀瘀愀琀椀漀渀 㠀 眀愀猀 椀渀挀漀爀爀攀挀琀⸀ 䌀愀氀挀甀氀愀琀攀 琀栀攀 挀漀爀爀攀挀琀 洀攀愀渀 愀渀搀 猀琀愀渀搀愀爀搀 搀攀瘀椀愀琀椀漀渀 椀昀 琀栀攀 眀爀漀渀最 漀戀猀攀爀瘀愀琀椀漀渀 椀猀 爀攀瀀氀愀挀攀搀 戀礀 ㄀㈀⸀∀Ⰰ "Find the mean and variance for the following frequency distribution: Class: 0-10, 10-20, 20-30, 30-40, 40-50; Frequency: 5, 8, 15, 16, 6"਀                崀Ⰰ caseStudy: [਀                    笀 context: "The runs scored by two batsmen in 10 innings are as follows: Batsman A: 38, 70, 48, 34, 42, 55, 63, 46, 54, 44; Batsman B: 28, 29, 110, 0, 64, 94, 72, 4, 33, 66",਀                        焀甀攀猀琀椀漀渀猀㨀 嬀 "Find the mean and standard deviation for each batsman.",਀                            ∀圀栀椀挀栀 戀愀琀猀洀愀渀 椀猀 洀漀爀攀 挀漀渀猀椀猀琀攀渀琀 椀渀 栀椀猀 瀀攀爀昀漀爀洀愀渀挀攀㼀∀ ]਀                    紀 ]਀            紀Ⰰ "Chapter 16": { ਀                洀挀焀㨀 嬀 { question: "A die is thrown once. The probability of getting a prime number is:", options: ["1/2", "1/3", "1/4", "1/6"], answer: "1/2" },਀                    笀 焀甀攀猀琀椀漀渀㨀 ∀吀眀漀 挀漀椀渀猀 愀爀攀 琀漀猀猀攀搀 猀椀洀甀氀琀愀渀攀漀甀猀氀礀⸀ 吀栀攀 瀀爀漀戀愀戀椀氀椀琀礀 漀昀 最攀琀琀椀渀最 愀琀 氀攀愀猀琀 漀渀攀 栀攀愀搀 椀猀㨀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀㄀⼀㐀∀Ⰰ ∀㄀⼀㈀∀Ⰰ ∀㌀⼀㐀∀Ⰰ ∀㄀∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀㌀⼀㐀∀ 紀Ⰰ { question: "If P(A) = 0.4, P(B) = 0.3 and P(A ∩ B) = 0.1, then P(A ∪ B) is:", options: ["0.6", "0.7", "0.8", "0.9"], answer: "0.6" }਀                崀Ⰰ assertionReason: [਀                    笀  assertion: "The probability of an impossible event is 0.", ਀                        爀攀愀猀漀渀㨀 ∀䄀渀 椀洀瀀漀猀猀椀戀氀攀 攀瘀攀渀琀 渀攀瘀攀爀 漀挀挀甀爀猀⸀∀Ⰰ answer: "Both A and R are true and R is the correct explanation of A."਀                    紀Ⰰ { ਀                        愀猀猀攀爀琀椀漀渀㨀 ∀䤀昀 䄀 愀渀搀 䈀 愀爀攀 椀渀搀攀瀀攀渀搀攀渀琀 攀瘀攀渀琀猀Ⰰ 琀栀攀渀 倀⠀䄀 ⤀•䈀⤀ 㴀 倀⠀䄀⤀ ⬀ 倀⠀䈀⤀⸀∀Ⰰ  reason: "For independent events, P(A ∩ B) = P(A) × P(B).",਀                        愀渀猀眀攀爀㨀 ∀䄀 椀猀 昀愀氀猀攀 戀甀琀 刀 椀猀 琀爀甀攀⸀∀ }਀                崀Ⰰ veryShort: [਀                    ∀䄀 挀愀爀搀 椀猀 搀爀愀眀渀 昀爀漀洀 愀 眀攀氀氀ⴀ猀栀甀昀昀氀攀搀 搀攀挀欀 漀昀 㔀㈀ 挀愀爀搀猀⸀ 䘀椀渀搀 琀栀攀 瀀爀漀戀愀戀椀氀椀琀礀 琀栀愀琀 椀琀 椀猀 愀 爀攀搀 欀椀渀最⸀∀Ⰰ "Find the probability that a leap year selected at random will contain 53 Sundays."਀                崀Ⰰ short: [਀                    ∀䄀 戀愀最 挀漀渀琀愀椀渀猀 㔀 爀攀搀 愀渀搀 㐀 戀氀愀挀欀 戀愀氀氀猀⸀ 吀眀漀 戀愀氀氀猀 愀爀攀 搀爀愀眀渀 愀琀 爀愀渀搀漀洀⸀ 䘀椀渀搀 琀栀攀 瀀爀漀戀愀戀椀氀椀琀礀 琀栀愀琀 琀栀攀礀 愀爀攀 漀昀 搀椀昀昀攀爀攀渀琀 挀漀氀漀爀猀⸀∀Ⰰ "Three coins are tossed simultaneously. Find the probability of getting exactly two heads."਀                崀Ⰰ long: [਀                    ∀䤀渀 愀 挀氀愀猀猀 漀昀 ㌀　 猀琀甀搀攀渀琀猀Ⰰ ㄀㔀 琀愀欀攀 䴀愀琀栀攀洀愀琀椀挀猀Ⰰ ㄀　 琀愀欀攀 倀栀礀猀椀挀猀Ⰰ 愀渀搀 㔀 琀愀欀攀 戀漀琀栀⸀ 䤀昀 愀 猀琀甀搀攀渀琀 椀猀 猀攀氀攀挀琀攀搀 愀琀 爀愀渀搀漀洀Ⰰ 昀椀渀搀 琀栀攀 瀀爀漀戀愀戀椀氀椀琀礀 琀栀愀琀 ⠀椀⤀ 琀栀攀 猀琀甀搀攀渀琀 琀愀欀攀猀 䴀愀琀栀攀洀愀琀椀挀猀 漀爀 倀栀礀猀椀挀猀 ⠀椀椀⤀ 琀栀攀 猀琀甀搀攀渀琀 琀愀欀攀猀 漀渀氀礀 䴀愀琀栀攀洀愀琀椀挀猀 ⠀椀椀椀⤀ 琀栀攀 猀琀甀搀攀渀琀 琀愀欀攀猀 渀攀椀琀栀攀爀 䴀愀琀栀攀洀愀琀椀挀猀 渀漀爀 倀栀礀猀椀挀猀⸀∀Ⰰ "A pair of dice is thrown. Find the probability of getting (i) a doublet (ii) a sum of 9 (iii) a sum greater than 9."਀                崀Ⰰ caseStudy: [਀                    笀 context: "In a housing society, 70% of the residents own cars, 50% own bikes, and 20% own both cars and bikes.",਀                        焀甀攀猀琀椀漀渀猀㨀 嬀 "Find the probability that a randomly selected resident owns a car or a bike.",਀                            ∀䘀椀渀搀 琀栀攀 瀀爀漀戀愀戀椀氀椀琀礀 琀栀愀琀 愀 爀攀猀椀搀攀渀琀 漀眀渀猀 渀攀椀琀栀攀爀 愀 挀愀爀 渀漀爀 愀 戀椀欀攀⸀∀ ]਀                    紀 ]਀            紀 };਀ function getRandomQuestions(chapters, type, count) {਀            琀爀礀 笀 let allQuestions = [];਀                挀栀愀瀀琀攀爀猀⸀昀漀爀䔀愀挀栀⠀挀栀愀瀀琀攀爀 㴀㸀 笀 if (questions[chapter] && questions[chapter][type]) {਀                        愀氀氀儀甀攀猀琀椀漀渀猀 㴀 愀氀氀儀甀攀猀琀椀漀渀猀⸀挀漀渀挀愀琀⠀焀甀攀猀琀椀漀渀猀嬀挀栀愀瀀琀攀爀崀嬀琀礀瀀攀崀⤀㬀 }਀                紀⤀㬀 allQuestions = allQuestions.sort(() => Math.random() - 0.5);਀                爀攀琀甀爀渀 愀氀氀儀甀攀猀琀椀漀渀猀⸀猀氀椀挀攀⠀　Ⰰ 挀漀甀渀琀⤀㬀 } catch (error) {਀                挀漀渀猀漀氀攀⸀攀爀爀漀爀⠀∀䔀爀爀漀爀 椀渀 最攀琀刀愀渀搀漀洀儀甀攀猀琀椀漀渀猀㨀∀Ⰰ 攀爀爀漀爀⤀㬀 return [];਀            紀 }਀ function showMathOptions() {਀            猀琀愀爀琀匀挀爀攀攀渀⸀挀氀愀猀猀䰀椀猀琀⸀愀搀搀⠀✀栀椀搀搀攀渀✀⤀㬀 mathOptions.classList.remove('hidden');਀        紀 ਀        昀甀渀挀琀椀漀渀 猀栀漀眀䌀栀愀瀀琀攀爀猀⠀⤀ 笀 mathOptions.classList.add('hidden');਀            挀栀愀瀀琀攀爀匀攀氀攀挀琀椀漀渀⸀挀氀愀猀猀䰀椀猀琀⸀爀攀洀漀瘀攀⠀✀栀椀搀搀攀渀✀⤀㬀 }਀ function toggleChapters() {਀            挀栀愀瀀琀攀爀䌀栀攀挀欀戀漀砀攀猀⸀昀漀爀䔀愀挀栀⠀挀戀 㴀㸀 挀戀⸀搀椀猀愀戀氀攀搀 㴀 昀甀氀氀倀愀瀀攀爀䌀栀攀挀欀戀漀砀⸀挀栀攀挀欀攀搀⤀㬀 }਀ function backToStart() {਀            最攀渀攀爀愀琀攀搀倀愀瀀攀爀䌀漀渀琀愀椀渀攀爀⸀挀氀愀猀猀䰀椀猀琀⸀愀搀搀⠀✀栀椀搀搀攀渀✀⤀㬀 startContainer.classList.remove('hidden');਀            挀栀愀瀀琀攀爀匀攀氀攀挀琀椀漀渀⸀挀氀愀猀猀䰀椀猀琀⸀愀搀搀⠀✀栀椀搀搀攀渀✀⤀㬀 startScreen.classList.remove('hidden');਀        紀 ਀        昀甀渀挀琀椀漀渀 最攀渀攀爀愀琀攀䄀猀猀椀最渀洀攀渀琀⠀⤀ 笀 try {਀                挀漀渀猀漀氀攀⸀氀漀最⠀∀䜀攀渀攀爀愀琀椀渀最 愀猀猀椀最渀洀攀渀琀⸀⸀⸀∀⤀㬀 ਀                ⼀⼀ 䠀椀搀攀 猀琀愀爀琀 挀漀渀琀愀椀渀攀爀 愀渀搀 猀栀漀眀 愀猀猀椀最渀洀攀渀琀 挀漀渀琀愀椀渀攀爀 startContainer.classList.add('hidden');਀                最攀渀攀爀愀琀攀搀倀愀瀀攀爀䌀漀渀琀愀椀渀攀爀⸀挀氀愀猀猀䰀椀猀琀⸀爀攀洀漀瘀攀⠀✀栀椀搀搀攀渀✀⤀㬀 generatedPaperContainer.innerHTML = '';਀                 // Get selected chapters਀                挀漀渀猀琀 猀攀氀攀挀琀攀搀䌀栀愀瀀琀攀爀猀 㴀 昀甀氀氀倀愀瀀攀爀䌀栀攀挀欀戀漀砀⸀挀栀攀挀欀攀搀  ? Object.keys(questions) ਀                    㨀 䄀爀爀愀礀⸀昀爀漀洀⠀搀漀挀甀洀攀渀琀⸀焀甀攀爀礀匀攀氀攀挀琀漀爀䄀氀氀⠀✀椀渀瀀甀琀嬀渀愀洀攀㴀∀挀栀愀瀀琀攀爀∀崀㨀挀栀攀挀欀攀搀✀⤀⤀⸀洀愀瀀⠀挀戀 㴀㸀 挀戀⸀瘀愀氀甀攀⤀㬀 ਀                椀昀 ⠀猀攀氀攀挀琀攀搀䌀栀愀瀀琀攀爀猀⸀氀攀渀最琀栀 㴀㴀㴀 　⤀ 笀 alert("Please select at least one chapter!");਀                    猀琀愀爀琀䌀漀渀琀愀椀渀攀爀⸀挀氀愀猀猀䰀椀猀琀⸀爀攀洀漀瘀攀⠀✀栀椀搀搀攀渀✀⤀㬀 return;਀                紀 ਀                挀漀渀猀漀氀攀⸀氀漀最⠀∀匀攀氀攀挀琀攀搀 挀栀愀瀀琀攀爀猀㨀∀Ⰰ 猀攀氀攀挀琀攀搀䌀栀愀瀀琀攀爀猀⤀㬀 ਀                挀漀渀猀琀 挀栀愀瀀琀攀爀一愀洀攀猀 㴀 猀攀氀攀挀琀攀搀䌀栀愀瀀琀攀爀猀⸀洀愀瀀⠀挀栀 㴀㸀 笀 if (ch === "Full Paper") return "Full Syllabus";਀                    挀漀渀猀琀 氀愀戀攀氀 㴀 搀漀挀甀洀攀渀琀⸀焀甀攀爀礀匀攀氀攀挀琀漀爀⠀怀氀愀戀攀氀嬀昀漀爀㴀∀␀笀挀栀⸀琀漀䰀漀眀攀爀䌀愀猀攀⠀⤀紀∀崀怀⤀㬀 return label ? ch + ". " + label.textContent.split(". ")[1] : ch;਀                紀⤀⸀樀漀椀渀⠀∀Ⰰ ∀⤀㬀 ਀                挀漀渀猀漀氀攀⸀氀漀最⠀∀䌀栀愀瀀琀攀爀 渀愀洀攀猀㨀∀Ⰰ 挀栀愀瀀琀攀爀一愀洀攀猀⤀㬀 ਀                ⼀⼀ 䌀爀攀愀琀攀 愀猀猀椀最渀洀攀渀琀 挀漀渀琀攀渀琀 let paperContent = `਀                    㰀搀椀瘀 挀氀愀猀猀㴀∀栀攀愀搀攀爀∀㸀

Class XI Mathematics Assignment

਀                        㰀栀㌀㸀␀笀猀攀氀攀挀琀攀搀䌀栀愀瀀琀攀爀猀⸀氀攀渀最琀栀 㴀㴀㴀 伀戀樀攀挀琀⸀欀攀礀猀⠀焀甀攀猀琀椀漀渀猀⤀⸀氀攀渀最琀栀 㼀 ∀䘀甀氀氀 匀礀氀氀愀戀甀猀∀ 㨀 挀栀愀瀀琀攀爀一愀洀攀猀紀㰀⼀栀㌀㸀