Sets-1-Quiz-10

SETS-1 Quiz-10

Dear Readers,

In mathematics a set is a collection of distinct elements. The elements that make up a set can be any kind of things: people, letters of the alphabet, numbers, points in space, lines, other geometrical shapes, variables, or even other sets. Two sets are equal if and only if they have precisely the same elements. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century..

Q1. Let Z denote the set of all integers and A={(a,b):a^2+3b^2=28, a,b∈Z} and B={(a,b):a>b,a,b∈Z}. Then, the number of elements in A∩B is

•  2
•  3
•  4
•  6

Solution
(d) ∵A={(a,b):a^2+3b^2=28,a,b∈Z} ={(5, 1), (-5, -1), (5, -1), (-5, 1), (1, 3), (-1, -3), (-1, 3), (1, -3), (4, 2), (-4, -2), (4, -2), (-4, 2)} And B={(a,b):a>b,a,b ∈Z} ∴ A∩B={(-1,-5),(1,-5),(-1,-3),(1,-3),(4,2),(4,-2)} ∴ Number of elements in A∩B is 6.

Q2.If A={4,6,10,12} and R is a relation defined on A as “two elements are related iff they have exactly one common factor other than 1”. Then the relation R is

•  Antisymmetric
•  Only transitive >
•   Only symmetric
•  Equivalence

Solution
(c) Clearly, R={(4,6),(4,10),(6,4),(10,4)(6,10),(10,6),(10,12),(12,10)} Clearly, R is symmetric (6,10)∈R and (10,12)∈R but (6,12)∉R So, R is not transitive Also, R is not reflexive

Q3.  Let A={1,2,3},B={3,4},C={4,5,6}. Then, A∪(B∩C) is

•   {3}
•  {1,2,3,4}
•  {1,2,5,6}
•  {1,2,3,4,5,6}

Solution
B

Q4. If R={(a,b):a+b=4} is a relation on N, then R is

•  Reflexive
•  Symmetric
•  Antisymmetric
•  Transitive

Solution
(b) Clearly, R={(1,3),(3,1),(2,2)} We observe that R is symmetric only

Q5.If A={1,2,3},B{3,4},C{4,5,6}. Then, A∪(B∩C) is

•  {1, 2}
•  {ϕ}
•  {4, 5}
•   {1, 2, 3, 4}

Solution
 (d) B∩C={4}. ∴A∪(B∩C)={1,2,3,4}

Q6. If relation R is defined as : aRb if "a is the father of b". Then, R is

•  Reflexive
•  Symmetric
• Transitive
•  None of these

Solution
(d) Clearly, R is neither reflexive, nor symmetric and not transitive

Q7. If A={1,2,3,4,5,6}, then how many subsets of A contain the elements 2,3 and 5?

•  4
•  8
•  16
•  32

Solution
(b) The number of subsets of A containing 2, 3 and 5 is same as the number of subsets of set {1,4,6} which is equal to 2^3=8

Q8. If a N={a x∶x∈N} and b N∩c N=d N, where b,c∈N then

•  d=bc
•  Tc=bd
•  b=cd
•  None of these

Solution
(d) We have, b N={b x│x∈N}= Set of positive integral multiples of b c N={c x│x∈N}= Set of positive integral multiples of c ∴c N={c x | x∈N}= Set of positive integral multiples of b and c both ⇒d=1.c.m.of b and c

Q9. In a class of 175 students the following data shows the number of students opting one or more subjects. Mathematics 100; Physics 70; Chemistry 40; Mathematics and Physics 30; Mathematics and Chemistry 28; Physics and Chemistry 23; Mathematics, Physics and Chemistry 18. How many students have offered Mathematics alone?

•  35
•  48
•  60
•  22

Solution
(c) We have, c+e+f+g=100 a+d+e+g=70 b+d+f+g=40 e+g=30 g+f=28 d+g=23 g=18 ∴g=18,f=10,e=12,d=15,a=35,b=7,c=60

Q10. If A={n:(n^3+5n^2+2)/n is an integer and itself is an integer}, then the number of elements in the set A, is

•  1
•  2
•  3
• 4

Solution
(d) We have, (n^3+5n^2+2)/n=n^2+5n+2/n ∴(n^3+5n^2+2)/n is an integer, if 2/n is an integer ⇒n=±1,±2 ⇒ A consists of four elements viz.-1,1,-2,2