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 5 CHAPTER HAPTER 1 RELATION AND FUNCTION OBJECTIVE QUESTIONS 1. A function fR R : " defined as fx x x4 5 2 ^ h = - + is (a) injective but not surjective. (b) surjective but not injective. (c) both injective and surjective. (d) neither injective nor surjective. Sol : OD 2024 We have fR R : " where fx x x4 5 2 _ i = - + One-One function : Let x1, x R 2 d , such that f x( )1 = f x( )2 x x 1 4 5 2 - +1 x x 2 4 5 2 = - +2 xx x x 1 4 4 2 2 2 -- +1 2 = 0 ^^ ^ x xx x x x 1 21 2 1 2 - +- - hh h 4 = 0 ^ ^ x xx x 1 21 2 - +- h h4 = 0 Thus x x 1 2 + - 4 = 0 and x x 1 2 = . Both are possible for real numbers. Hence f x^ h is not one-one. Onto function : Now y x x4 5 2 = - + x 2 54 2 = ^ h - +- x 2 1 2 = ^ h - + As x 2 2 ^ h - $ 0 y - 1 $ 0 y $ 1 Range ^ hf ! 6 @ 1,3 and co-domain ! R Since Range ! Co-domain f x^ h is not onto. 2. Select the correct option out of the four given options Let R be a relation in the set N given by R ab a b b = = # - ^ h ,: , - 2 6 > Then, (a) ^ h 8 7, d R (b) ^ h 6 8, d R (c) ^ h 3 8, d R (d) ^ h 8 7, d R Sol : OD 2023 We have R = = $ . ^ h ab a b b ,: , - 2 6 > Here b > 6 . so we substituting use b = 7 and 8. When b = 7, then a = 7 2 - = 5 When b = 8, then a = 8 2 - = 6 So, ^ h 5 7, d R and ^ h 6 8, d R Thus (b) is correct option. 3. Let A = " , 3 5, . Then, number of reflexive relations on A is (a) 2 (b) 4 (c) 0 (d) 8 Sol : OD 2023 Here, n A( ) = 2 The number of reflexive relations are 2n n 2 - , where n is the number of elements in the set. Thus number of reflexive relation are 22 2 2 = - 24 2 = - 22 = = 4 Thus (b) is correct option. 4. The relation R defined in the set A = " , 1234567 ,,,,,, by R ab = { , ^ h : both a and b are either odd or even}. Then, R is (a) symmetric (b) transitive (c) an equivalence relation (d) reflexive Sol : Delhi 2017 Given, any element a in A, both a and a must be either odd or even, so that ^ h aa R , d . Further, ^ h ab R , d & both a and b must be either odd or even & ^ h ba R , . d
Page 6 Relation and Function CHAPTER 1 Similarly, ^ h ab R , d and ^ h bc R , d & all elements abc , , must be either even or odd simultaneously & ^ h ac R , d Hence, R is an equivalence relation. Thus (c) is correct option. 5. Let R be the relation defined in the set A = " , 1234567 ,,,,,, by R ab = { , ^ h : both a and b are either odd or even}. Now, consider the following statements I. All the elements of the subset " , 1357 ,,, are related to each other. II All the elements of the subset " , 246 , , are related to each other. III Some elements of the subset " , 1357 ,,, are related to some elements of the subset " , 246 , , . Choose the correct option (a) I and III are true (b) I and II are true (c) II and III are true (d) All the true Sol : Foreign 2010 All the elements of the subset " , 1357 ,,, are related to each other, as all the elements of this subset are odd. Similarly, all the elements of the subset " , 246 , , are related to each other, as all of them are even. Also, no element of the subset " , 1357 ,,, can be related to any element ot " , 246 , , as elements of " , 1357 ,,, are odd, while elements of " , 246 , , are even. Thus (b) is correct option. 6. Let fI I : " be defined by fx x i ^ h = + where i is a fixed integer, then f is (a) one-one but not onto (b) onto but not one-one (c) non-invertible (d) both one-one and onto Sol : SQP 2020 Let f x^ h1 = f y^ h1 x i 1 + = x i 2 + x1 = x2 and for any integer y , we have y = x i + x = y i - ie, fy i ^ h - = y Hence, f is both one-one and onto. Thus (d) is correct option. 7. Let C be the set of complex numbers. The mapping fCR " given by fz z ^ h = , 6z C d , is (a) one-one and onto (b) one-one but not onto (c) not one-one but onto (d) neither one-one nor onto Sol : Delhi 2015, OD 2011 Here, f z^ h = z 6z C d f^ h 1 = = 1 1 f^ h -1 = -1 1 = f^ h 1 = f^ h -1 But 1 !- 1 Therefore, it is not one-one. Now, let fz z ^ h = . Here, there is not pre-image of negative numbers. Hence, it is not onto. Thus (d) is correct option. 8. Let R be the relation in the set Z of all integers defined by R xy x y = " , ^ h , : int - is an eger .Then R is (a) reflexive (b) symmetric (c) transitive (d) an equivalence relation Sol : Comp 2017 Here, R ={^ h xy x y , : - is an integer} is a relation in the set of integers. Reflexive : Putting y xx x = , - = 0 which is an integer for all x Z d . So, R is reflexive in Z . Symmetric : Let ^ h xy R , , d then ( ) x y - is an integer l (say) and also y x - =-l . ^ h l l d d Z Z &- y x - is an integer & ^ h yx R , , d So, R is symmetric. Transitive : Let ^ h xy R , , d and ^ h yz R , , d , so x y - = integer and y z - = integers, then x z - is also an integer. ^ h xz R , , d So, R is transitive. Thus (d) is correct option. 9. The function fR R : " defined by fx x x ( ) 2 = + is (a) one-one (b) onto (c) many-one (d) None of the above Sol : Foreign 2011 The given function fR R : " defined by f x^ h x x 2 = +
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