Tiêu đề Maths 12th.pdf ✅

CONTENTS CHAP 1. Relation and Function 5-35 CHAP 2. Inverse Trigonometric Functions 36-57 CHAP 3. Matrices 58-83 CHAP 4. Determinants 84-129 CHAP 5. Continuity and Diff erentiability 130-180 CHAP 6. Application of Derivatives 181-235 CHAP 7. Integration 236-298 CHAP 8. Application of Integrals 299-309 CHAP 9. Diff erential Equations 310-356 CHAP 10. Vector Algebra 357-397 CHAP 11. Three Dimensional Geometry 398-428 CHAP 12. Linear Programming 429-455 CHAP 13. Probability 456-505 ********


CHAPTER 1 Relation and Function Page 7 Now, for x = 0 and -1 we have f^ h 0 = 0 and f^ h -1 = 0 Hence, f^ h 0 = f^ h -1 but 0 !- 1 Thus f is not one-one. Function f is not many-one. Thus (c) is correct option. 10. If R is the relation defined in the set " , 123456 ,,,,, as R ab b a = = $ . ^ h ,: , + 1 then R is (a) reflexive (b) symmetric (c) transitive (d) None of these Sol : OD 2010, Comp 2007 Let A = " , 123456 ,,,,, Reflexive : A relation R is defined on set A is R = = $ . ^ h ab b a ,: . + 1 Therefore, R = " , ^^^^^ 12 23 34 45 56 ,,,,,,,,, hhhhh Now, 6 d A but ^ h 6 6, . z R Therefore, R is not reflexive. Symmetric : It can be observed that ^ h 1 2, d R but ^ h 2 1, z R. Therefore, R is not symmetric. Transitive : Now, ^ ^ 12 23 ,,, h h d R but ^ h 1 3, z R. Therefore, R is not transitive. Hence, R is neither reflexive nor symmetric nor transitive. Thus (d) is correct option. 11. Let A = " , 123 , , and R = " , ^ ^ 12 23 ,,, h h be a relation in A. Then, the minimum number of ordered pairs may be added, so that R becomes an equivalence relation, is (a) 7 (b) 5 (c) 1 (d) 4 Sol : OD 2018 The given relation is R = " , ^ ^ 12 23 ,,, h h in the set A = " , 123 ,, . Now, R is reflexive, if ^^^ 11 22 33 ,,,,, . hhh d R R is symmetric, if ^ ^ 21 32 ,,, . h h d R R is transitive, if ^ h 1 3, and ^ h 3 1, . d R Thus, the minimum number of ordered pairs which are to be added, so that R becomes an equivalence relation, is 7. Thus (a) is correct option. 12. A function fx y : " is said to be one-one , if for every xx X 1 2 , d , (a) fx fx x x ^ ^ 1 2 12 h h = = & (b) fx fx x x ^ ^ 1 2 12 h h = & ! (c) fx fx x x ^ ^ 1 2 12 h h ! & = (d) None of these Sol : Foreign 2014, Delhi 2012 A function fx y : " is defined to be one-one (or injective), if the images of distinct elements of x under f are distinct, i.e., for every x x xfx fx 12 1 2 , , d ^ ^ h h = implies x x 1 2 = . Otherwise, f is called many-one. Thus (a) is correct option. 13. The function f 4 defined by (a) one-one only (b) onto only (c) bijective (d) many-one Sol : OD 2016 Since, distinct elements of x1 have distinct images in x4 and every element in x4 has a unique pre image in x1, the function f 4 is both one-one and onto. Thus f 4 is bijective. Thus (c) is correct option. 14. The function fN N : " given by fx x ^ h = 2 is (a) surjective (b) bijective (c) injective (d) many-one Sol : Delhi 2010 The function f is one-one, as f x^ h1 = f x^ h2 2x1 = 2x2 x1 = x2. Further, f is not onto, as for 1 d N , there does not exist any x in N such that fx x ^ h = = 2 1. Thus (c) is correct option. 15. The function fX Y : " defined by fx x ( ) sin = is one- one but not onto, if X and Y respectively equal to