SETS, RELATIONS AND FUNCTIONS
Sets and their representation: Union, intersection and complement of sets and their algebraic properties; Power set; Relation, Type of relations, equivalence relations, functions; one-one, into and onto functions, the composition of functions.
- Let Z be the set of integers. If A = {x ∈ Z : 2(x +2)(x^2 - 5x + 6) = 1} and B {x ∈ Z : -3 < 2x - 1 < 9}, Option 2) 215 is correct 2
- Let S = {1, 2, 3, ... , 100}. The number of non-empty subsets A of S such that the product of elements in A is even is : The correct option (2) 2⁵⁰(2⁵⁰ – 1) Explanation: Product of elements in A is even = (total subjects) – (Number of subject of all whose eleme... 1
- Let X = {n ∈ N : 1 ≤ n ≤ 50}. If A = {n ∈ X : n is multiple of 2} and B = {n ∈ X : n is a multiple of 7}, n(A) = 25 n(B) = 7 n(A ∩ B) = 3 n(A ∪ B) = 25 + 7 - 3 = 29 2
- Let Ui = 1 Xi for i ∈ 50 = Ui = 1 for i ∈ n Yi = T, where each Xi contains 10 elements and each Yi contains 5 elements. Correct answer is 30 EXplaination: n(X_i) = 10. U_(i = 1 )X_i for i ∈ 50 = T, n (T) = 500 each element of T belongs to exactly 20 elements o... 1
- A survey shows that 73% of the persons working in an office like coffee, whereas 65% like tea. Answer is 36 Note**then x cannot be: C > person like coffee T > Person like Tea n(C)=73 n(T) =65 n(C U T) ≤ 100 n(C)+n(T) -n(C ∩ T) ≤ 100 73... 1
- Set A has m elements and Set B has n elements. If the total number of subsets of A is 112 more than the total number of subsets Correct answer is 28 Explaination: No. of subsets of A=2m No. of subsets of B=2n 2^m=2ⁿ+112 2^m−2ⁿ=112 2ⁿ(2^(m−n)−1)=112 2ⁿ(2^(m−n)−1)=2⁴(2³... 1
- In a college of 300 students, every student read 5 newspaper and every newspaper is read by 60 students. Correct option is C) Exactly 25 Let the number of newspapers be x. If every student reads one newspaper, the number of students would be x (... 1
- If A = [(x, y) : x2 + y2 = 25] and B = [(x, y) : x2 + 9y2 = 144], then A ∩ B contains _______ points. A = Set of all values (x, y) : x2 + y2 = 25 = 52 B = [x2 / 144] + [y2 / 16] = 1 i.e., [x2 / (12)2] + [y2 / (4)2] = 1. Clearly, A ∩ B consist... 1
- Let A = {θ : tanθ + secθ = √2secθ} and B = {θ : secθ - tanθ = √2tanθ} be two sets then. (a) A = B (b) A ⊂ B (c) A ≠ B (d) B ⊂ A Correct option is A) A=B Explaination: tan θ + sec θ = √2 sec θ sin θ = √2 – 1 sec θ – tan θ = √2 tan θ sin θ = 1 / (√2 + 1) sin θ = √2 – 1... 1
- In a certain town 30% families own a scooter and 40% on a car 50% own neither a scooter nor a car 2000 families The correct option (d) 1 and 3 Explanation: Suppose x families live in the town A = {families have scooter} B = {families have car} ∴ n(A) =... 1
- Let X = {x, y, z} / x, y, z ∈ N, x + y + z = 10, {x < y < z } and Y = {(x, y, z) / x, y, z ∈ N, y = | x - z |} then X ∩ Y is equal to The correct option (d) {(2, 3, 5), (1, 4, 5)} Explanation: X = {(1, 2, 7), (1, 3, 6), (1, 4, 5), (2, 3, 5)} element of X which belong to Y a... 1
- If two sets A and B are having 43 elements in comman, then the number of elements common to each of the sets A x B and B x A is The correct option (a) 43² Explanation: n((A x B) ∩ (B x A)) = n((A ∩ B) x (B ∩ A)) = n(A ∩ B) . n(B ∩ A) = 43 x 43 = 43² 1
- If X ∪ {3, 4} = {1, 2, 3, 4, 5, 6} the which of the following is true (a) Smallest set X = {1, 2, 5, 6} (b) Smallest set X = {1, 2, 3, 5, 6} (a) Smallest set X = {1, 2, 5, 6} EXplaination: X ∪ {3, 4} = {1, 2, 3, 4, 5, 6} Then the Smallest set of X will be X ∪ {3, 4} – {3, 4} i.e.... 1
- A survey shows that 70% of the Indian like mango where 82% like apple. If x% of Indian like both mango and apples then The correct option (b) 52 ≤ x ≤ 70 Explanation: n(A) = 70 n(B) = 82 n(A ∩ B) = x n(A ∪ B) = n(A) + n(B) – n(A ∩ B) ⇒ n(A ∩ B) = 152 – n(A ∪... 1
- Two finite sets have m and n element respectively. The total number of subsets of first set is 112 more than the total number of sub sets of the second set. (c) 7, 4 No. of subset of first set – No of subset of second set = 112 ∴ 2^m – 2ⁿ = 112 ∴ 2ⁿ(2^(m–n) – 1) = 112 ∴ 2ⁿ(2^(m–n) – 1) = 2⁴ × 7 ∴... 1
- If aN = {ax/x ∈ N} and bN ∩ cN = dN Where b, c ∈ N are relatively prime then (a) d = bc aN = {ax/x ∈ N} bN = {bx/x ∈ N} = the set of positive integers multiple of b cN = {cx/x ∈ N} = the set of positive integers multi... 1