AIEEE - PAPERS

IMPORTANT FACTS, TERMS AND FORMULAE

1) SETS

A set is a collection of well defined and well distinguished objects of our perception or thought.

 The words 'well defined objects' imply that we must given a rule with the help of which we should readily be able to say whether a particular objects 'belongs to' the set or not. The words 'well distinguished objects' imply that if the objects of the collection be named, then in doing so, the number of objects will not increase.

     The sets are usually denoted by capital letters of English alphabet viz. A, B, C, ....

2) ELEMENTS 

 The objects, which constitute the set, are said to be elements of the set.

These are also known as members or points of the set. The elements are usually denoted by the small letters of the English alphabet viz, a, b, c.....

 i) If a is an element of the set A, we write it as a∈ A and is read as " a belongs to A".

ii) If a is not an element of the series A, we write it as a ∉ A and is read as " a does not belongs to A ".

3) REPRESENTATION OF SETS

There are two methods of represent a set.

a) Roaster/Tabulation Method.

In this method, the set is represented by listing all its elements, separating the elements by commas and enclosing them in curvilinear brackets.

b) Defining Property Method.

In this method , the set is represented by specifying the common property of the elements.

Thus the set is represented by A={a: P(A) is true}.

   Here 'a' stands for 'an arbitrary element of the set' and (:) stands for 'such that ' and P(A) stands for 'common property'.

4) FINITE AND INFINITE SETS

a) Finite Set : A set is said to be finite if it has finite number of elements.

b) Infinite Set: A set is said to be infinite if it has an infinite number of elements.

c) Order of a finite set: is the number of elements it contains .

the order of a finite set A is denoted by O(A).

5) EMPTY SET

A set having no element is said to be an empty set.

It is also called Null set or Void set.

The empty set is denoted by ∅ or {  }

6) Singleton Set:

A set having only one element is said to be a singleton set.

7) SUB-SET:

a) Subset: Let A and B be two sets. Then the set A is said to be a subset of the set B if each element of A is also an element of B.

symbolically, we write it as A ⊂ B.

Here, B is superset of A and is written as B ⊃ A.

b) Proper Subset. A set A is a proper subset of B if and only if each element of A is in B and there is at least one element in B, which is not in A.

Symbolically , if A is a proper subset of B, then A ⊂ B and A≠ B or A ⊂ B.

8) COMPARABLE SETS

Two sets are said to be comparable iff either A ⊂ B or B⊂ A.

9)  EQUAL AND EQUIVALENT SETS

a) Equal Sets : Two sets A and B are said to be equal (written as A= B) iff A  ⊂ B and B ⊂ A.

Two sets A and B are said to be equal if they have exactly same elements.

b) Equivalent Sets.  Two sets are said to be equivalent if they have same number of elements.

 family of satis said to be family the set up all possible subset of a state is said to be the power set of a and is the denoted by the main set under discussion or the set containing all possible values in the given frame of reference recept to be Universal set and is denoted by litten b2 set the union and the ab denoted by set up all those elements which are at the Ray in both similarly the union of this setup all the elements which are at least one value let a and b to set the instruction of PNB denoted by setup all those elements which are not both and A and B then the intersection of the denoted by setup all the elements which are fundamental results disjoints said to set to be disjoint any only if they have no common elements that can be two sides and a is the set of those elements of the day which are not in the set symbolically let a and b two sets and their difference is the union of these this is denoted by the compliment of the state of all those elements which are not in the set this is noted by fundamental results in order

SETS

1) Let N be the set of non-negative integers, I the set of integers, Nₚ the set of non-positive integers, E the set of even integers and P the set of prime numbers . Then :

a) I - N =Nₚ b) N ∩ Nₚ = ∅ c) E∩ P= ∅ d) N ∆  Nₚ = I - {0}

2) Let A and B be two sets, then (A U B)ᶜ U (Aᶜ ∩B) equals:

a) Aᶜ B) Bᶜ c) C d) none

3) If A and B are two sets, then A∩ (A U B)ᶜ equal :

a) A B) B c) ∅  d) none

4) The set (A U B)ᶜ U (B U C) equals:

a) Aᶜ U B b) AᶜU Cᶜ c) AᶜUBUC d) none

5) Let U be the universal set and AU B U C = U. Then [(A - B) U (B - C) U (C - A)ᶜ equals:

a) AUBUC b) A∩B ∩C c) AU(B ∩ C) d) A ∩(B U C).

6) The set (AUBUC)∩(A∩Bᶜ∩Cᶜ) ∩Cᶜ equas:

a) A∩ C b) B U Cᶜ c) B∩ Cᶜ D) none

7) If A and B are two sets, then A∩(AU B) equals:

a) A B) B c) Aᶜ d) Bᶜ

8) If A={1, 3,5,7,9, 11,13, 15,17}, B={2,4.....18} and N is the universal set, then

a) Aᶜ U ((AU B)∩ Bᶜ) is:

a) A B) N c) B d) none

9) if A and B are disjoint non-empty sets, then A - (A - B) equals:

a) A B) B c) ∅  d) A U B

10) Which of the following is empty set?

a) {x: x is a real number and x²-1= 0}

b) {x: x is a real number and x²+1= 0}

c) {x: x is a real number and x²-9= 0}

d) {x: x is a real number and x²= x+2}

11) Which of the following is a singleton set?

a) {x: |x|= 5, x ∈ I}

b) {x: |x|= 6, x ∈ N}

c) {x: x² = 5, x ∈ N}

d) {x: x²+ 3x +2= 0, x ∈ N}

12) Which of the following does not have a proper subset:

a) {x:x∈ N, 4< x <5}

b) {x:x∈ Q}

c) {x:x∈ Q, 4< x <5} d) none

13) If A, B and C are three sets, then A - (B U C) equals:

a) (A- B) U(A - C)

b) (A- B)∩ (A - C)

c) (A- B)U C

d) (A- B)∩ C

14) If A, B and C are three sets, then A ∩ (BU C) equals:

a) (AU B)∩ (A U C)

b) (A ∩B)U (A ∩ C)

c) (AU B) U(A U C) d) none

15) If Q={x: x= 1/y, where x ∈ N}, then:

a) 0∈ Q b) I ∈ Q c) 2∈ Q d) 2/3 ∈ Q

16) If A= {x: x∈ I, -2≤ x ≤2}, B={x: x∈ I, 0≤ x ≤3}, C= {x: x∈ N,  1≤ x ≤2}, {(x,y): (x , y) ∈ N x N, x + y= 8}, then:

a) n(B U C)= 5 b) n(D)= 6 c) n(AU(BUC))= 5 d) none

17) If for α ∈ N, αN = {αx : x ∈ N}, then the set 8N  ∩ 6N is:

a) 8N b) 12 N c) 24N d) 48N

18) Let n(A)= 3 and n(B)= 6 and A ⊂ B. Then the number of elements in A ∩B is:

a) 3 b) 9 c) 6 d) none

19) Sets A and B have 3 and 6 elements respectively. What can be the minimum number of elements in A U B?

a) 3 b) 6 c) 9 d) 18

20) Two finite sets have m and n elements. The total number of subsets of the first set is 56 more than the total number of subsets of second set. The values of m and n are:

a) 7, 6 b) 6, 3 c) 5, 1 c) 8, 7 

21) if A, B and C are any three sets, then A x (B U C) is

a) (A x B) U (A x C)

b) (A U B) x (A U C)

c) (A x B) ∩ (A x C) d) none

22) If A, B and C are any three sets, then A x (B ∩C) is:

a) (A x B) U (A x C)

b) (A x B) ∩ (A x C)

c) (A U B) x (A U C) d) (A ∩B) x (A ∩ C)    ₂₃ ₁₂ ₁₃

23) If S₁={1, 2,3,.....20}, S₂={a b,c,d}, S₃={b,d,e,f}. The number of elements of

   (S₁ x S₂) U(S₁ x S₃) is:

a) 100 b) 120 c) 140 d) 40

24) If A= {1,2,3,6, 11,18, 21}, B={5, 7,9}  and N is the universal set, then Aᶜ U ((A U B)∩Bᶜ) equals:

a) A B) B c) N d) N - A

25) The set (AUBUC) ∩(A∩Bᶜ∩Cᶜ)∩ Cᶜ equal to

a) A∩C b) B ∩Cᶜ c) Bᶜ∩Cᶜ d) none

26) 

27) Consider the set of all determinants of order 3 with entries 0 and 1 only. Let B be the Subset of A consisting of all determinants with value 1. Let C be the Subset of the set of all determinants with value -1, then

a) C is empty

b) B has as many elements as C

c) A= B U c d) B has twice as many elements as C has.

28) If (1,3),(2,5) and (3,3) are three elements of A x B and the total number of elements in A x B is 6, then the remaining elements of A x B are:

a) (1,5),(2,3),(3,5)

b) (5,1),(3,2),(5,3)

c) (1,5),( 2,3),(5,3) d) none 

29) If A, B, C be three sets such that A U B= = C and A ∩ B = A ∩ C, then:

a) A= B b) B= C c) A= C d) A= B = C

30) Let A={(x,y): y= eˣ, x ∈ R}, B={(x,y): y = e⁻ˣx ∈ R}, then

a) A ∩B= ∅ b) A∩B≠  ∅ c) AU B = R d) none

31) Let A={(x,y): y= eˣ, x ∈ R}, B={(x,y): y = x, x ∈ R}, then

a) B ⊂ A b) A ⊂ B c) A∩B = ∅ d) AU B = A

32) If X={4ⁿ - 3n -1; n ∈N} and Y={9(n -1); n ∈ N}, then X U Y is:

a) X b) Y c) N d) none

33) If X={8ⁿ - 7n -1; n ∈N} and Y={49(n -1); n ∈ N}, then 

a) X ⊂ Y b) Y ⊂ X c) X= Y d) none

34) If the set A and B are defined as:

A={(x,y): y= 1/x, 0≠ x  ∈R}

B={(x,y): y= - x, x∈ R}, then

a)  A∩B = A b)A∩B= B c)  A∩B =∅ d) none 

35) If {∅, {∅}}, then the power set of A is:

a) A b) {∅, {∅}, A} c) {∅, {∅}, {{∅}}, A} d) none

36) If A, B and C are non empty sets, then (A - B)U (B - A) equals to

a) (AU B) - B

b) A - (A ∩B)

c) (A UB) - (A∩B)

d) (A∩B) U(A U B)

37) In a certain town 25% families own a phone and 15% own car, 65% own neither a phone nor a car. 2000 families own both a car and a phone.

Consider the following statements in this regard:

i)  10% families own both a car in the phone

ii) 35% own either a car or a phone

c)  40000 families live in town 

Which of the above statements are correct ?

a) one and two b) one and three c) two and three  d) one, two and three.

38) In a class of 100 students , 55 students have passed in Mathematics and 67 students have passed in Physics . Then the number of students who have passed in Physics only is:

a) 22 b) 33 c) 10 d) 45 

39) Out of 800 boys in a school, 224 played cricket, 240 played hockey and 336 played Basketball. Of the total, 64 played both basketball and hockey. 80 played cricket and hockey, 24 played all the three games. The number of boys which did not play any games is:

a) 128 b) 216 c) 240 d) 160 

40) From 50 students taking examination in Mathematics, Physics and Chemistry. 37 passed Mathematics , 24 Physics , 43 Chemistry . At most 19 passed Mathematics and Physics, atmost 29 Mathematics and Chemistry and atmost 20 physics and chemistry. The largest possible number that could have passed all the three examination is:

a) 9 b) 10 c) 12 d) none

41) Of the members of three athletic team in a school 21 are in and cricket team, 26 are in the hockey team and 29 are in the football team. Among them, 14 play hockey and cricket, 15 play hockey and football. and 12 play football and cricket. Eight play all the three games. The total number of members in the three athletics teams is :

a) 43 b) 49 c) 76 d) none

42) In a college of 300 students, every student reads 5 newspaper and every newspaper is read by 60 students. The number of newspaper is:

a) Atleast 30 b) atmost 20 c) exactly 25  d) none

Answer

1d 2a 3c 4a 5b 6c 7a 8b 9c 10b 11b 12a 13b 14b 15 b 16 d 17 c 18 a 19 b 20 b 21 a 22 b 23 b 24 c 25 b 26 c 27 b 28 a 29 b 30 b 31 c 32 b 33a 34 c 35 c 36 c 37 b,c 38 d 39 d 40 d 41 a 42c

Competitive quotations 

1) If A= {(x,y): x²+ y²=25}, B={(x,y): x²+ 9y²=144}, then A ∩ B contains 

a) 1 point b) 3 point c) 2 points d) 4 points

2) A={A∆a, b}, B={c, d}, C={d, e}, than

a) {(a,c),(a,d),(a,e),(b,c),(b,d),(b,e)} is

a) A ∩(B UC) b) AU (B∩C) c) A x (B UC) d) A x (B ∩ C)

3) In a city 20% of the population travels by car, 50% travels by bus and 10% travels by by both car and bus. Then persons travelling by car or bus is :

a) 80%  b) 40%  c) 60% d) 70%  e) 30%

4) Two finite sets have m and n element. The total number of subsets of the first set is 48 more than the total number of subsets of the second set. The values of m and n are:

a) 7, 6 b) 6,3  c) 6,4 d) 7,4 e) 3,7 

5) A class has 175 students. The following data shows the number of students obtaining one or more subject, 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?

a) 35 b) 48 c) 60 d) 22 e) 30

6) If two sets A and B are having 99 elements in common, then the number of elements common to each of the sets A x B and B x A is:

a) 2⁹⁹ b) 99² c) 100 d) 18 d) 9 

7) Consider n= (U)=20, n(A)= 12,  n(B)=9,  n(A∩B )=4 , where U is the Universal set, A and B are subsets of U, then n+A UB)ᶜ=

a) 17 b) 9 c) 11 d) 3 e) 16

8)  if r, s, t are positive numbers and p, q, r positive integers such that LCM of p, q is r²t⁴s², then the number of ordered pairs (p,q) is:

a) 252 b) 254 c) 225 d) 224 

9) The set S={1,2,3,...... 12} is to be partitioned in to three sets A, B and C of equal size. Thus,

AUBUC = S, A∩ B= B ∩ C= A∩ C= ∅.  

the number of ways to partition S is

a) 12!/3!(3!)⁴ b) 12!/(4!)³ c) 12!/(3!)³ d) 12!/3!(4!)³.

Answer

1d 2c 3c 4c 5c 6b 7d 8d 9b

RELATIONS

1) Let A={1,2,3}. The total number of distinct relations which can be defined over A is:

a) 6 b) 8 c) 2⁹ d) none

2) The relation R defined A={1,2,3} by aRb if |a²- b²|≤ 5. Which of the following is not true ?

a) domain of R={1,2,3} b) Range of R={5} c) R⁻¹ = R

d) R={(1,1),(2,2),(3,3),( 2,1),(1,2),( 2,3)(3,2)}

3) Let R be a relation in the set of natural numbers defined by R==(1+ x,1+ x²): x≤ 5, x ∈N}. Which of the following is false :

a) domain of R={2,3,4,5,6}

b) Range of R={2,5,10,17,26}

c) R= {(2,2),(3,5),(4,10),(5,17),(6,26)}

d) at least one is false.

4) Assume that R and S are two non empty relations in a set A. Which of the following statement is not true ?

a) If R and S are transitive, then R U S is transitive.

b) If R and S are transitive, then R  ∩S is transitive.

c) If R and S are symmetric, then R US is symmetric.

d) If R and S are reflexive, then R  ∩S is reflexive.

5) let A={1, 2,3,4} and R={(2,2),(3, 3),(4,4),(1,2)} be a relation on A. Then A is

a)  reflexive  b) symmetric  c) transitive d) none

6) The void relation on a set is:

a) reflexive b) symmetric and transitive c) reflexive and symmetric d) reflexible and transitive 

7) The relation 'is subset of' on the power set P(A) of a set A is:

a) symmetric b) anti-symmetric c) equivalence relation d) none

8) The relation 'congruence modulo m' is:

a) reflexive only b) symmetric only c) transitive only d) an equivalence relation

9)  Let R be the relation over the set N x N and is defined by (a,b) R (c,d)=>a + d= b + c. Then R is:

a) reflexive only b) symmetric only c) transitive only d) an equivalence relation

10) R is a relation over the set of real numbers and it is given by mn> 0.  Then R is:

a) reflexive and symmetric b) symmetric and transitive c) an equivalence relation d)  partial order relation

11) R is a relation over the set of integers and it is given by (x,y) ∈ R | x-y|≤ 1.  then R is:

a) reflexive and symmetric  b) reflexive and transitive  c) symmetric and transitive d) an equivalence relation

12) let P={(x,y): x²+ y²=1, x,y  ∈R}. Then P is:

a) reflexive b) symmetric c) transitive d) anti symmetric

13) Let R be a relation on a set A such that R= R⁻¹. Then R is:

a) reflexive b) symmetric c) transitive d) none

14) Let L be the set of all straight lines in the Euclidean plane. Two lines l₁ , l₂ are said to be related by the relation R iff l₁ || l₂ . Then the relation R is 

a) reflexive b) symmetric c) transitive d) equivalence

15) let R be the relation over the set of straight lines in a plane such that l R m <=> perpendicular to m. Then R is 

a) reflexives b) symmetric c) transitive d) an equivalence relation.

16) let R be the relation over the set of integers such that l R m <=> is a multiple of m. Then R is

a) reflexive  b) symmetric  c) an equivalence relation d) none

17) let {(x,y): x² + y²=1, x,y ∈R} be a relation in R.  Then the relation R is

a) reflexive b) symmetric c) transitive d) anti systematic

18) Let a relation R in the set of natural numbers be defined by (x,y)   ∈R <=> x²- 4xy + 3y²= 0 for all x,y   ∈N. Then the relation R is 

a) reflexive b) symmetric c) transitive d) an equivalent relation.

19) Which one of the following relation on R is an equivalence relation?

a) a R₁ b <=> |a|= |b|

b) a R₂ b <=> a≥ b

c) a R₃ b <=> a divides b

d) aR₄ b <=> a< b

20) Let R={(x,y): x, y ∈ A, x+ y=5}, where A={1,2,3,4,5}. Then:

a) R is reflexive , symmetric but not transitive.

b) R is not reflexive, not symmetric but transitive

c) R is not reflexive , symmetric and not transitive

d) R is an equivalence relation.

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Revision test (SETS AND RELATION)

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FUNCTIONS

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