Class XII Mathematics

਀                            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㄀　∀ 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 ㄀　∀㸀 10. Vector Algebra਀                        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 ਀        氀攀琀 猀挀栀漀漀氀一愀洀攀 㴀 ✀✀㬀 let questionCounter = 1;਀        氀攀琀 猀攀猀猀椀漀渀夀攀愀爀 㴀 ✀✀㬀 ਀        挀漀渀猀琀 焀甀攀猀琀椀漀渀猀 㴀 笀 mathematics: {਀                ∀䌀栀愀瀀琀攀爀 ㄀∀㨀 笀 mcq: [ ਀                        笀 焀甀攀猀琀椀漀渀㨀 ∀䰀攀琀 刀 戀攀 愀 爀攀氀愀琀椀漀渀 漀渀 琀栀攀 猀攀琀 一 漀昀 渀愀琀甀爀愀氀 渀甀洀戀攀爀猀 搀攀昀椀渀攀搀 戀礀 渀刀洀 椀昀 渀 搀椀瘀椀搀攀猀 洀⸀ 吀栀攀渀 刀 椀猀㨀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀刀攀昀氀攀砀椀瘀攀 愀渀搀 猀礀洀洀攀琀爀椀挀∀Ⰰ ∀吀爀愀渀猀椀琀椀瘀攀 愀渀搀 猀礀洀洀攀琀爀椀挀∀Ⰰ ∀䔀焀甀椀瘀愀氀攀渀挀攀∀Ⰰ ∀刀攀昀氀攀砀椀瘀攀Ⰰ 琀爀愀渀猀椀琀椀瘀攀 戀甀琀 渀漀琀 猀礀洀洀攀琀爀椀挀∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀刀攀昀氀攀砀椀瘀攀Ⰰ 琀爀愀渀猀椀琀椀瘀攀 戀甀琀 渀漀琀 猀礀洀洀攀琀爀椀挀∀ 紀Ⰰ { question: "The number of bijective functions from set A to itself when A contains 106 elements is:", options: ["106", "106²", "106!", "2¹⁰⁶"], answer: "106!" },਀                        笀 焀甀攀猀琀椀漀渀㨀 ∀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 㐀⸀ 吀栀攀渀 昀笀뤠⠀砀⤀ 椀猀㨀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀⠀砀⬀㐀⤀⼀㌀∀Ⰰ ∀砀⼀㌀ ⬀ 㐀∀Ⰰ ∀㌀砀 ⬀ 㐀∀Ⰰ ∀一漀渀攀 漀昀 琀栀攀猀攀∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀⠀砀⬀㐀⤀⼀㌀∀ 紀Ⰰ { question: "Let f : N → N be defined by f(n) = the highest prime factor of n. Then f is:", options: ["Injective", "Surjective", "Bijective", "Neither injective nor surjective"], answer: "Neither injective nor surjective" },਀                        笀 焀甀攀猀琀椀漀渀㨀 ∀吀栀攀 爀攀氀愀琀椀漀渀 刀 椀渀 琀栀攀 猀攀琀 娀 漀昀 椀渀琀攀最攀爀猀 最椀瘀攀渀 戀礀 刀 㴀 笀⠀愀Ⰰ戀⤀㨀 ㈀ 搀椀瘀椀搀攀猀 愀ⴀ戀紀 椀猀㨀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀刀攀昀氀攀砀椀瘀攀 漀渀氀礀∀Ⰰ ∀匀礀洀洀攀琀爀椀挀 漀渀氀礀∀Ⰰ ∀吀爀愀渀猀椀琀椀瘀攀 漀渀氀礀∀Ⰰ ∀䔀焀甀椀瘀愀氀攀渀挀攀 爀攀氀愀琀椀漀渀∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀䔀焀甀椀瘀愀氀攀渀挀攀 爀攀氀愀琀椀漀渀∀ 紀 ],਀                    愀猀猀攀爀琀椀漀渀㨀 嬀  { statement1: "Every relation which is symmetric and transitive is also reflexive.", statement2: "An empty relation is symmetric and transitive.", options: ["Both true", "Both false", "Only statement 1 is true", "Only statement 2 is true"], answer: "Only statement 2 is true" },਀                        笀 猀琀愀琀攀洀攀渀琀㄀㨀 ∀吀栀攀 挀漀洀瀀漀猀椀琀椀漀渀 漀昀 琀眀漀 戀椀樀攀挀琀椀漀渀猀 椀猀 愀 戀椀樀攀挀琀椀漀渀⸀∀Ⰰ 猀琀愀琀攀洀攀渀琀㈀㨀 ∀吀栀攀 椀渀瘀攀爀猀攀 漀昀 愀 戀椀樀攀挀琀椀漀渀 椀猀 愀 戀椀樀攀挀琀椀漀渀⸀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀䈀漀琀栀 琀爀甀攀∀Ⰰ ∀䈀漀琀栀 昀愀氀猀攀∀Ⰰ ∀伀渀氀礀 猀琀愀琀攀洀攀渀琀 ㄀ 椀猀 琀爀甀攀∀Ⰰ ∀伀渀氀礀 猀琀愀琀攀洀攀渀琀 ㈀ 椀猀 琀爀甀攀∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀䈀漀琀栀 琀爀甀攀∀ 紀 ],਀                    琀眀漀䴀愀爀欀㨀 嬀  "Check whether the relation R in R defined as R = {(a, b): a ≤ b³} is reflexive, symmetric or transitive.",਀                        ∀匀栀漀眀 琀栀愀琀 琀栀攀 昀甀渀挀琀椀漀渀 昀㨀 刀 鈀‡刀 最椀瘀攀渀 戀礀 昀⠀砀⤀ 㴀 砀대 ⬀ 砀 椀猀 愀 戀椀樀攀挀琀椀漀渀⸀∀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 bijective.",਀                        ∀䰀攀琀 昀㨀 一 鈀‡一 戀攀 搀攀昀椀渀攀搀 戀礀 昀⠀渀⤀ 㴀 渀 ⬀ ㄀ 椀昀 渀 椀猀 漀搀搀 愀渀搀 昀⠀渀⤀ 㴀 渀 ⴀ ㄀ 椀昀 渀 椀猀 攀瘀攀渀⸀ 匀栀漀眀 琀栀愀琀 昀 椀猀 戀椀樀攀挀琀椀瘀攀⸀∀ ],਀                    琀栀爀攀攀䴀愀爀欀㨀 嬀  "Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b): |a - b| is even} is an equivalence relation.",਀                        ∀䰀攀琀 昀㨀 刀 鈀‡刀 戀攀 搀攀昀椀渀攀搀 愀猀 昀⠀砀⤀ 㴀 ㄀　砀 ⬀ 㜀⸀ 䘀椀渀搀 琀栀攀 昀甀渀挀琀椀漀渀 最㨀 刀 鈀‡刀 猀甀挀栀 琀栀愀琀 最 漀 昀 㴀 昀 漀 最 㴀 䤀开刀⸀∀Ⰰ "Given a non-empty set X, consider the binary operation *: P(X) × P(X) → P(X) given by A * B = A ∩ B for all A, B ∈ P(X). Show that X is the identity element for this operation."਀                    崀Ⰰ fiveMark: [ ਀                        ∀䰀攀琀 䄀 㴀 一 휀 一 愀渀搀 ⨀ 戀攀 琀栀攀 戀椀渀愀爀礀 漀瀀攀爀愀琀椀漀渀 漀渀 䄀 搀攀昀椀渀攀搀 戀礀 ⠀愀Ⰰ 戀⤀ ⨀ ⠀挀Ⰰ 搀⤀ 㴀 ⠀愀 ⬀ 挀Ⰰ 戀 ⬀ 搀⤀⸀ 匀栀漀眀 琀栀愀琀 ⨀ 椀猀 挀漀洀洀甀琀愀琀椀瘀攀 愀渀搀 愀猀猀漀挀椀愀琀椀瘀攀⸀ 䘀椀渀搀 琀栀攀 椀搀攀渀琀椀琀礀 攀氀攀洀攀渀琀 昀漀爀 ⨀ 漀渀 䄀Ⰰ 椀昀 愀渀礀⸀∀Ⰰ "Consider f: R+ → [-5, ∞) given by f(x) = 9x² + 6x - 5. Show that f is invertible and find f⁻¹.",਀                        ∀䰀攀琀 䄀 㴀 刀 ⴀ 笀㌀紀 愀渀搀 䈀 㴀 刀 ⴀ 笀㄀紀⸀ 䌀漀渀猀椀搀攀爀 琀栀攀 昀甀渀挀琀椀漀渀 昀㨀 䄀 鈀‡䈀 搀攀昀椀渀攀搀 戀礀 昀⠀砀⤀ 㴀 ⠀砀ⴀ㈀⤀⼀⠀砀ⴀ㌀⤀⸀ 匀栀漀眀 琀栀愀琀 昀 椀猀 戀椀樀攀挀琀椀瘀攀 愀渀搀 昀椀渀搀 昀笀뤠⸀∀ ],਀                    挀愀猀攀匀琀甀搀礀㨀 嬀  { case: "A general election of Lok Sabha is a gigantic exercise. About 911 million people were eligible to vote and voter turnout was about 67%, the highest ever. Let I be the set of all citizens of India who were eligible to exercise their voting right in the general election held in 2019. A relation R is defined on I as follows: R = {(V1, V2): V1, V2 ∈ I and both use their voting right in the general election-2019}.", questions: ["Determine whether R is reflexive, symmetric or transitive.", "Three friends F1, F2 and F3 exercised their voting right in general election-2019, then examine whether (F1, F2) ∈ R and (F2, F3) ∈ R implies (F1, F3) ∈ R.", "The above defined relation R is an equivalence relation?"] }਀                    崀 },਀                ∀䌀栀愀瀀琀攀爀 ㈀∀㨀 笀 mcq: [ ਀                        笀 焀甀攀猀琀椀漀渀㨀 ∀吀栀攀 瀀爀椀渀挀椀瀀愀氀 瘀愀氀甀攀 漀昀 猀椀渀笀뤠⠀ⴀ㄀⼀㈀⤀ 椀猀㨀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀쀀⼃㘀∀Ⰰ ∀쀀⼃㌀∀Ⰰ ∀ⴀ쀀⼃㘀∀Ⰰ ∀ⴀ쀀⼃㌀∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀ⴀ쀀⼃㘀∀ 紀Ⰰ { question: "The value of tan⁻¹√3 - sec⁻¹(-2) is:", options: ["π", "-π/3", "π/3", "2π/3"], answer: "-π/3" },਀                        笀 焀甀攀猀琀椀漀渀㨀 ∀吀栀攀 瀀爀椀渀挀椀瀀愀氀 瘀愀氀甀攀 漀昀 挀漀猀笀뤠⠀挀漀猀⠀㜀쀀⼃㘀⤀⤀ 椀猀㨀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀㜀쀀⼃㘀∀Ⰰ ∀㔀쀀⼃㘀∀Ⰰ ∀쀀⼃㌀∀Ⰰ ∀쀀⼃㘀∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀㔀쀀⼃㘀∀ 紀Ⰰ { question: "The value of sin(2tan⁻¹(0.75)) is:", options: ["0.75", "1.5", "0.96", "1.25"], answer: "0.96" },਀                        笀 焀甀攀猀琀椀漀渀㨀 ∀䤀昀 琀愀渀笀뤠砀 ⬀ 琀愀渀笀뤠礀 㴀 㐀쀀⼃㔀Ⰰ 琀栀攀渀 挀漀琀笀뤠砀 ⬀ 挀漀琀笀뤠礀 攀焀甀愀氀猀㨀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀쀀⼃㔀∀Ⰰ ∀㈀쀀⼃㔀∀Ⰰ ∀㌀쀀⼃㔀∀Ⰰ ∀쀀∃崀Ⰰ 愀渀猀眀攀爀㨀 ∀쀀⼃㔀∀ 紀 ],਀                    愀猀猀攀爀琀椀漀渀㨀 嬀  { statement1: "The domain of the function defined by f(x) = sin⁻¹√(x-1) is [1,2].", statement2: "The range of the principal value branch of cosec⁻¹ is [-π/2,π/2] - {0}.", options: ["Both true", "Both false", "Only statement 1 is true", "Only statement 2 is true"], answer: "Only statement 1 is true" },਀                        笀 猀琀愀琀攀洀攀渀琀㄀㨀 ∀琀愀渀笀뤠⠀㄀⼀㈀⤀ ⬀ 琀愀渀笀뤠⠀㄀⼀㌀⤀ 㴀 쀀⼃㐀⸀∀Ⰰ 猀琀愀琀攀洀攀渀琀㈀㨀 ∀㈀琀愀渀笀뤠⠀㄀⼀㌀⤀ 㴀 琀愀渀笀뤠⠀㌀⼀㐀⤀⸀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀䈀漀琀栀 琀爀甀攀∀Ⰰ ∀䈀漀琀栀 昀愀氀猀攀∀Ⰰ ∀伀渀氀礀 猀琀愀琀攀洀攀渀琀 ㄀ 椀猀 琀爀甀攀∀Ⰰ ∀伀渀氀礀 猀琀愀琀攀洀攀渀琀 ㈀ 椀猀 琀爀甀攀∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀䈀漀琀栀 琀爀甀攀∀ 紀 ],਀                    琀眀漀䴀愀爀欀㨀 嬀  "Find the principal value of cosec⁻¹(-√2).",਀                        ∀倀爀漀瘀攀 琀栀愀琀㨀 ㌀猀椀渀笀뤠砀 㴀 猀椀渀笀뤠⠀㌀砀 ⴀ 㐀砀대⤀Ⰰ 砀 ࠀ•嬀ⴀ㄀⼀㈀Ⰰ ㄀⼀㈀崀⸀∀Ⰰ "Write the value of tan⁻¹[2sin(2cos⁻¹√3/2)].",਀                        ∀䘀椀渀搀 琀栀攀 瘀愀氀甀攀 漀昀 琀愀渀笀뤠ᨀ㌢ ⴀ 挀漀琀笀뤠⠀ⴀᨀ㌢⤀⸀∀ ],਀                    琀栀爀攀攀䴀愀爀欀㨀 嬀  "Prove that: tan⁻¹(1/2) + tan⁻¹(1/3) = π/4.",਀                        ∀匀漀氀瘀攀 昀漀爀 砀㨀 琀愀渀笀뤠⠀砀ⴀ㄀⤀ ⬀ 琀愀渀笀뤠砀 ⬀ 琀愀渀笀뤠⠀砀⬀㄀⤀ 㴀 琀愀渀笀뤠㌀砀⸀∀Ⰰ "Prove that: 2tan⁻¹(1/2) + tan⁻¹(1/7) = tan⁻¹(31/17)."਀                    崀Ⰰ fiveMark: [ ਀                        ∀倀爀漀瘀攀 琀栀愀琀㨀 琀愀渀笀뤠⠀㄀⼀㔀⤀ ⬀ 琀愀渀笀뤠⠀㄀⼀㜀⤀ ⬀ 琀愀渀笀뤠⠀㄀⼀㌀⤀ ⬀ 琀愀渀笀뤠⠀㄀⼀㠀⤀ 㴀 쀀⼃㐀⸀∀Ⰰ "Solve the equation: cos⁻¹x - sin⁻¹x = cos⁻¹(x√3).",਀                        ∀䘀椀渀搀 琀栀攀 瘀愀氀甀攀 漀昀 砀 椀昀 猀椀渀笀뤠砀 ⬀ 猀椀渀笀뤠⠀㄀ⴀ砀⤀ 㴀 挀漀猀笀뤠砀⸀∀ ],਀                    挀愀猀攀匀琀甀搀礀㨀 嬀  { case: "Two schools A and B want to award their selected students on the values of Honesty, Hard work and Discipline. School A wants to award ₹x each, ₹y each and ₹z each for the three respective values to 3, 2 and 1 students respectively with total award money of ₹2200. School B wants to spend ₹3100 to award its 4, 1 and 3 students on the respective values. The total amount of award for one prize on each value is ₹1200.", questions: ["Represent the above situation by a matrix equation.", "Find the award money for each value using matrix method.", "Which value you prefer to be rewarded most and why?"] }਀                    崀 },਀                ⼀⼀ ⸀⸀⸀ ⠀漀琀栀攀爀 挀栀愀瀀琀攀爀猀 昀漀氀氀漀眀 猀椀洀椀氀愀爀 瀀愀琀琀攀爀渀 眀椀琀栀 甀瀀搀愀琀攀搀 挀漀渀琀攀渀琀⤀ "Chapter 13": {਀                    洀挀焀㨀 嬀  { question: "If P(A) = 0.4, P(B) = 0.8 and P(B|A) = 0.6, then P(A ∪ B) equals:", options: ["0.24", "0.72", "0.48", "0.96"], answer: "0.72" },਀                        笀 焀甀攀猀琀椀漀渀㨀 ∀䤀昀 䄀 愀渀搀 䈀 愀爀攀 琀眀漀 攀瘀攀渀琀猀 猀甀挀栀 琀栀愀琀 倀⠀䄀⤀ 怀•　 愀渀搀 倀⠀䈀簀䄀⤀ 㴀 ㄀Ⰰ 琀栀攀渀㨀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀䄀 舀•䈀∀Ⰰ ∀䈀 舀•䄀∀Ⰰ ∀䈀 㴀 였∃Ⰰ ∀䄀 㴀 였∃崀Ⰰ 愀渀猀眀攀爀㨀 ∀䄀 舀•䈀∀ 紀Ⰰ { question: "If P(A|B) > P(A), then which of the following is correct:", options: ["P(B|A) < P(B)", "P(A ∩ B) < P(A).P(B)", "P(B|A) > P(B)", "P(B|A) = P(B)"], answer: "P(B|A) > P(B)" },਀                        笀 焀甀攀猀琀椀漀渀㨀 ∀䤀昀 琀眀漀 攀瘀攀渀琀猀 愀爀攀 椀渀搀攀瀀攀渀搀攀渀琀Ⰰ 琀栀攀渀㨀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀吀栀攀礀 洀甀猀琀 戀攀 洀甀琀甀愀氀氀礀 攀砀挀氀甀猀椀瘀攀∀Ⰰ ∀吀栀攀 猀甀洀 漀昀 琀栀攀椀爀 瀀爀漀戀愀戀椀氀椀琀椀攀猀 洀甀猀琀 戀攀 ㄀∀Ⰰ ∀䈀漀琀栀 ⠀愀⤀ 愀渀搀 ⠀戀⤀∀Ⰰ ∀一漀渀攀 漀昀 琀栀攀猀攀∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀一漀渀攀 漀昀 琀栀攀猀攀∀ 紀Ⰰ { question: "The probability of solving a problem by three students A, B, C are 1/2, 1/3, 1/4 respectively. The probability that the problem will be solved is:", options: ["1/4", "1/2", "3/4", "1/3"], answer: "3/4" }਀                    崀Ⰰ assertion: [ ਀                        笀 猀琀愀琀攀洀攀渀琀㄀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 䄀✀ 愀渀搀 䈀✀ 愀爀攀 愀氀猀漀 椀渀搀攀瀀攀渀搀攀渀琀⸀∀Ⰰ 猀琀愀琀攀洀攀渀琀㈀㨀 ∀䤀昀 䄀 愀渀搀 䈀 愀爀攀 洀甀琀甀愀氀氀礀 攀砀挀氀甀猀椀瘀攀Ⰰ 琀栀攀渀 琀栀攀礀 愀爀攀 椀渀搀攀瀀攀渀搀攀渀琀⸀∀Ⰰ 漀瀀琀椀漀渀猀㨀 嬀∀䈀漀琀栀 琀爀甀攀∀Ⰰ ∀䈀漀琀栀 昀愀氀猀攀∀Ⰰ ∀伀渀氀礀 猀琀愀琀攀洀攀渀琀 ㄀ 椀猀 琀爀甀攀∀Ⰰ ∀伀渀氀礀 猀琀愀琀攀洀攀渀琀 ㈀ 椀猀 琀爀甀攀∀崀Ⰰ 愀渀猀眀攀爀㨀 ∀伀渀氀礀 猀琀愀琀攀洀攀渀琀 ㄀ 椀猀 琀爀甀攀∀ 紀Ⰰ { statement1: "The probability of an event cannot be negative.", statement2: "The probability of a sure event is 1.", options: ["Both true", "Both false", "Only statement 1 is true", "Only statement 2 is true"], answer: "Both true" }਀                    崀Ⰰ twoMark: [ ਀                        ∀䄀 搀椀攀 椀猀 琀栀爀漀眀渀 琀眀椀挀攀 愀渀搀 琀栀攀 猀甀洀 漀昀 琀栀攀 渀甀洀戀攀爀猀 愀瀀瀀攀愀爀椀渀最 椀猀 漀戀猀攀爀瘀攀搀 琀漀 戀攀 㘀⸀ 圀栀愀琀 椀猀 琀栀攀 挀漀渀搀椀琀椀漀渀愀氀 瀀爀漀戀愀戀椀氀椀琀礀 琀栀愀琀 琀栀攀 渀甀洀戀攀爀 㐀 栀愀猀 愀瀀瀀攀愀爀攀搀 愀琀 氀攀愀猀琀 漀渀挀攀㼀∀Ⰰ "Given that the events A and B are such that P(A) = 1/2, P(A ∪ B) = 3/5 and P(B) = p. Find p if they are independent.",਀                        ∀䤀昀 倀⠀䄀⤀ 㴀 　⸀㠀Ⰰ 倀⠀䈀⤀ 㴀 　⸀㔀 愀渀搀 倀⠀䈀簀䄀⤀ 㴀 　⸀㐀Ⰰ 昀椀渀搀 倀⠀䄀 ⤀•䈀⤀⸀∀Ⰰ "A coin is tossed 3 times. Find the probability of getting at least 2 heads."਀                    崀Ⰰ threeMark: [ ਀                        ∀䄀 戀愀最 挀漀渀琀愀椀渀猀 㐀 爀攀搀 愀渀搀 㐀 戀氀愀挀欀 戀愀氀氀猀Ⰰ 愀渀漀琀栀攀爀 戀愀最 挀漀渀琀愀椀渀猀 ㈀ 爀攀搀 愀渀搀 㘀 戀氀愀挀欀 戀愀氀氀猀⸀ 伀渀攀 漀昀 琀栀攀 琀眀漀 戀愀最猀 椀猀 猀攀氀攀挀琀攀搀 愀琀 爀愀渀搀漀洀 愀渀搀 愀 戀愀氀氀 椀猀 搀爀愀眀渀 昀爀漀洀 琀栀攀 戀愀最 眀栀椀挀栀 椀猀 昀漀甀渀搀 琀漀 戀攀 爀攀搀⸀ 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 戀愀氀氀 椀猀 搀爀愀眀渀 昀爀漀洀 琀栀攀 昀椀爀猀琀 戀愀最⸀∀Ⰰ "In answering a question on a multiple choice test, a student either knows the answer or guesses. Let 3/4 be the probability that he knows the answer and 1/4 be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability 1/4. What is the probability that the student knows the answer given that he answered it correctly?",਀                        ∀䄀 挀愀爀搀 昀爀漀洀 愀 瀀愀挀欀 漀昀 㔀㈀ 挀愀爀搀猀 椀猀 氀漀猀琀⸀ 䘀爀漀洀 琀栀攀 爀攀洀愀椀渀椀渀最 挀愀爀搀猀 漀昀 琀栀攀 瀀愀挀欀Ⰰ 琀眀漀 挀愀爀搀猀 愀爀攀 搀爀愀眀渀 愀渀搀 愀爀攀 昀漀甀渀搀 琀漀 戀攀 戀漀琀栀 搀椀愀洀漀渀搀猀⸀ 䘀椀渀搀 琀栀攀 瀀爀漀戀愀戀椀氀椀琀礀 漀昀 琀栀攀 氀漀猀琀 挀愀爀搀 戀攀椀渀最 愀 搀椀愀洀漀渀搀⸀∀ ],਀                    昀椀瘀攀䴀愀爀欀㨀 嬀  "An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accident are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?",਀                        ∀䄀 昀愀挀琀漀爀礀 栀愀猀 琀眀漀 洀愀挀栀椀渀攀猀 䄀 愀渀搀 䈀⸀ 倀愀猀琀 爀攀挀漀爀搀 猀栀漀眀猀 琀栀愀琀 洀愀挀栀椀渀攀 䄀 瀀爀漀搀甀挀攀搀 㘀　─ 漀昀 琀栀攀 椀琀攀洀猀 漀昀 漀甀琀瀀甀琀 愀渀搀 洀愀挀栀椀渀攀 䈀 瀀爀漀搀甀挀攀搀 㐀　─ 漀昀 琀栀攀 椀琀攀洀猀⸀ 䘀甀爀琀栀攀爀Ⰰ ㈀─ 漀昀 琀栀攀 椀琀攀洀猀 瀀爀漀搀甀挀攀搀 戀礀 洀愀挀栀椀渀攀 䄀 愀渀搀 ㄀─ 瀀爀漀搀甀挀攀搀 戀礀 洀愀挀栀椀渀攀 䈀 眀攀爀攀 搀攀昀攀挀琀椀瘀攀⸀ 䄀氀氀 琀栀攀 椀琀攀洀猀 愀爀攀 瀀甀琀 椀渀琀漀 漀渀攀 猀琀漀挀欀瀀椀氀攀 愀渀搀 琀栀攀渀 漀渀攀 椀琀攀洀 椀猀 挀栀漀猀攀渀 愀琀 爀愀渀搀漀洀 昀爀漀洀 琀栀椀猀 愀渀搀 椀猀 昀漀甀渀搀 琀漀 戀攀 搀攀昀攀挀琀椀瘀攀⸀ 圀栀愀琀 椀猀 琀栀攀 瀀爀漀戀愀戀椀氀椀琀礀 琀栀愀琀 椀琀 眀愀猀 瀀爀漀搀甀挀攀搀 戀礀 洀愀挀栀椀渀攀 䈀㼀∀Ⰰ "There are three coins. One is a two headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin?"਀                    崀Ⰰ caseStudy: [ ਀                        笀 挀愀猀攀㨀 ∀䤀渀 愀 最愀洀攀Ⰰ 愀 洀愀渀 眀椀渀猀 愀 爀甀瀀攀攀 昀漀爀 愀 猀椀砀 愀渀搀 氀漀猀攀猀 愀 爀甀瀀攀攀 昀漀爀 愀渀礀 漀琀栀攀爀 渀甀洀戀攀爀 眀栀攀渀 愀 昀愀椀爀 搀椀攀 椀猀 琀栀爀漀眀渀⸀ 吀栀攀 洀愀渀 搀攀挀椀搀攀搀 琀漀 琀栀爀漀眀 愀 搀椀攀 琀栀爀椀挀攀 戀甀琀 琀漀 焀甀椀琀 愀猀 愀渀搀 眀栀攀渀 栀攀 最攀琀猀 愀 猀椀砀⸀ 䰀攀琀 堀 搀攀渀漀琀攀 琀栀攀 愀洀漀甀渀琀 栀攀 眀椀渀猀⼀氀漀猀攀猀⸀∀Ⰰ 焀甀攀猀琀椀漀渀猀㨀 嬀∀䘀椀渀搀 琀栀攀 瀀爀漀戀愀戀椀氀椀琀礀 搀椀猀琀爀椀戀甀琀椀漀渀 漀昀 堀⸀∀Ⰰ ∀䘀椀渀搀 琀栀攀 攀砀瀀攀挀琀攀搀 瘀愀氀甀攀 漀昀 堀⸀∀Ⰰ ∀圀漀甀氀搀 礀漀甀 愀搀瘀椀猀攀 琀栀攀 洀愀渀 琀漀 瀀氀愀礀 琀栀攀 最愀洀攀㼀 䨀甀猀琀椀昀礀 礀漀甀爀 愀渀猀眀攀爀⸀∀崀 紀 ]਀                紀 }਀        紀㬀 ਀        昀甀渀挀琀椀漀渀 最攀琀刀愀渀搀漀洀儀甀攀猀琀椀漀渀猀⠀猀攀氀攀挀琀攀搀䌀栀愀瀀琀攀爀猀Ⰰ 琀礀瀀攀Ⰰ 挀漀甀渀琀⤀ 笀 let allQuestions = [];਀ if (selectedChapters.includes("Full Paper")) {਀                昀漀爀 ⠀氀攀琀 挀栀愀瀀琀攀爀 椀渀 焀甀攀猀琀椀漀渀猀⸀洀愀琀栀攀洀愀琀椀挀猀⤀ 笀 if (questions.mathematics[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⸀洀愀琀栀攀洀愀琀椀挀猀嬀挀栀愀瀀琀攀爀崀嬀琀礀瀀攀崀⤀㬀 }਀                紀 } else {਀                猀攀氀攀挀琀攀搀䌀栀愀瀀琀攀爀猀⸀昀漀爀䔀愀挀栀⠀挀栀愀瀀琀攀爀 㴀㸀 笀 if (questions.mathematics[chapter] && questions.mathematics[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⸀洀愀琀栀攀洀愀琀椀挀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 }਀ function showChapterSelection() {਀            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✀⤀⸀挀氀愀猀猀䰀椀猀琀⸀愀搀搀⠀✀栀椀搀搀攀渀✀⤀㬀 document.getElementById('chapterSelection').classList.remove('hidden');਀        紀 ਀        昀甀渀挀琀椀漀渀 琀漀最最氀攀䌀栀愀瀀琀攀爀猀⠀⤀ 笀 const fullPaperCheckbox = document.getElementById('fullPaper');਀            挀漀渀猀琀 挀栀攀挀欀戀漀砀攀猀 㴀 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∀崀㨀渀漀琀⠀⌀昀甀氀氀倀愀瀀攀爀⤀✀⤀㬀 checkboxes.forEach(cb => cb.disabled = fullPaperCheckbox.checked);਀        紀 ਀        昀甀渀挀琀椀漀渀 戀愀挀欀吀漀匀琀愀爀琀⠀⤀ 笀 document.getElementById('generatedPaperContainer').classList.add('hidden');਀            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 document.getElementById('chapterSelection').classList.add('hidden');਀            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✀⤀⸀挀氀愀猀猀䰀椀猀琀⸀爀攀洀漀瘀攀⠀✀栀椀搀搀攀渀✀⤀㬀 questionCounter = 1;਀        紀 ਀        昀甀渀挀琀椀漀渀 瀀爀漀洀瀀琀匀挀栀漀漀氀一愀洀攀⠀⤀ 笀 schoolName = prompt("Write your school/institute name").toUpperCase();਀            椀昀 ⠀℀猀挀栀漀漀氀一愀洀攀⤀ 笀 alert("School/Institute name is required!");਀            紀 攀氀猀攀 笀 promptSession();਀            紀 }਀ function promptSession() {਀            猀攀猀猀椀漀渀夀攀愀爀 㴀 瀀爀漀洀瀀琀⠀∀䔀渀琀攀爀 琀栀攀 愀挀愀搀攀洀椀挀 猀攀猀猀椀漀渀 ⠀攀⸀最⸀Ⰰ ㈀　㈀㌀ⴀ㈀㐀⤀∀⤀⸀琀漀唀瀀瀀攀爀䌀愀猀攀⠀⤀㬀 if (!sessionYear) {਀                愀氀攀爀琀⠀∀䄀挀愀搀攀洀椀挀 猀攀猀猀椀漀渀 椀猀 爀攀焀甀椀爀攀搀℀∀⤀㬀 } else {਀                最攀渀攀爀愀琀攀倀愀瀀攀爀⠀⤀㬀 }਀        紀 ਀        昀甀渀挀琀椀漀渀 最攀渀攀爀愀琀攀倀愀瀀攀爀⠀⤀ 笀 document.getElementById('startContainer').classList.add('hidden');਀            挀漀渀猀琀 最攀渀攀爀愀琀攀搀倀愀瀀攀爀䌀漀渀琀愀椀渀攀爀 㴀 搀漀挀甀洀攀渀琀⸀最攀琀䔀氀攀洀攀渀琀䈀礀䤀搀⠀✀最攀渀攀爀愀琀攀搀倀愀瀀攀爀䌀漀渀琀愀椀渀攀爀✀⤀㬀 generatedPaperContainer.classList.remove('hidden');਀            最攀渀攀爀愀琀攀搀倀愀瀀攀爀䌀漀渀琀愀椀渀攀爀⸀椀渀渀攀爀䠀吀䴀䰀 㴀 ✀✀㬀 ਀            氀攀琀 瀀愀瀀攀爀䌀漀渀琀攀渀琀 㴀 怀 ਀                㰀搀椀瘀 挀氀愀猀猀㴀∀挀漀渀琀攀渀琀ⴀ愀爀攀愀∀㸀 ਀                    㰀搀椀瘀 挀氀愀猀猀㴀∀瀀愀瀀攀爀ⴀ洀攀琀愀∀㸀 ਀                        㰀搀椀瘀㸀吀椀洀攀 䄀氀氀漀眀攀搀㨀 ㌀ 栀漀甀爀猀㰀⼀搀椀瘀㸀