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QUESTION: 1

If A = {a, b, c} and R = {(a, a), (a, b) ( b, c), (b, b), (c, c), (c, a)} is a binary relation on A, then which one of the following is correct?

Solution:

Since, (a, a), (b, b), (c, c) ε R

Therefore, R is a reflexive relation

But, (a,b ) ε R and (b, a) R So, R is not a symmetric relation Also, (a, b), (b, cj 6 R.

Implies (a, c)

Hence, R is not a transitive relation

QUESTION: 2

If the cardinality of a set A is 4 and that of a set B is 3, then what is the cardinality of the set A Δ B?

Solution:

Since, the sets A and B are not known thus cardinality of the set A Δ B cannot be determined.

QUESTION: 3

Assertion (A): If events, A, B, C and D are mutually exhaustive, then (A ∪ B ∪ C)^{c} = D.

Reason (R): (A ∪ B ∪ C)^{c} = D implies if any element is excluded from the sets A, B and C, then it is included in D.

Solution:

Both (A) and (R) are true and (R) is the correct explanation of (A).

Since, (A ∪ B ∪ C)^{c} = A^{c} ∩ B^{c} ∩ C^{c} = D

QUESTION: 4

Elements of a population are classified according to the presence or absence of each of 3 attributes A, B and C. What is the number of smallest ultimate classes into which the population is divided?

Solution:

Elements of a population are classified according to the presence or absence of each of 3 attributes A, B and C. Then, the smallest number of smallest ultimate classes into which the population is divided, is 2^{3} = 8

QUESTION: 5

If A and B are subsets of a set X, then what is {A ∩( X - B)} } ∪ B equal to?

Solution:

QUESTION: 6

The total number of subsets of a finite set A has 56 more elements, then the total number of subsets of another finite set B. What is the number of elements in the set A?

Solution:

Let sets A and B have m and n elements, respectively,

Then 2^{m} - 2^{n} = 56 (According to question)

implies2^{n}(2^{m-n} - 1)

= 8 x 7 = (2)^{3 }x 7 = 2^{3}(2^{3}- l )

On comparing implies n = 3

and m-n = 3

implies m = 6 and n = 3

Number of subsets of A = 2^{3} + 5^{6} = 64 = 2^{6}

Hence,Number of elements of in A = 6

QUESTION: 7

If f(x) =

Q. what is the value of (fof) (√3) ?

Solution:

QUESTION: 8

Consider the following statement.

I. Parallelism of lines is an equivalence relation.

II. xRy, if x is a father of y, is an equivalence relation. Which of the statements given above is/are correct?

Solution:

I. Let l, m, n are parallel lines and R is a relation.

Then l|| l, then R is reflexive,

and l || m and m || l, then R is symmetric.

Also, l || m, m || n => l | n, then R is transitive. Hence, R is an equivalence relation.

II. If x is father of y and y is not father of x then relation is not symmetric, thus relation is not equivalence

QUESTION: 9

The function f : R - > R defined by

Solution:

Since ,f(- 1) = f(1) = 2^{35}

i.e., Two real number 1 and -1 have the same image.

So, the function is not one-one and let y = (x^{2} + l )^{35} implies Thus, every real number has no pre-image.

So, the function is not onto.

Hence, the function is neither one-one nor onto.

QUESTION: 10

If A and B are disjoint sets, then A ∩ (A' ∪ B) is equal to which one of the following?

Solution:

Since, A ∩ B = φ (given)

QUESTION: 11

Let U= { 1 , 2 , 3 , ...,20}. Let A, B, C be the subsets of U.

Let A be the set of all numbers, which are prefect squares, B be the set of all numbers which are multiples of 5 and C be the set of all numbers, which are divisible by 2 and 3.

Q. Consider the following statements.

I. A, B, C are mutually exclusive.

II. A, B, C are mutually exhaustive.

III. The number of elements in the complement set of A∪ B is 12.

Q. Which of the statements given above the correct?

Solution:

U = { 1 ,2 ,3 , ...,20 } A = Set of all natural numbers which are perfect square

= {1,4 ,9 ,16 }

B = Set of all natural numbers which are multiple of 5

= {5,10,15,20}

C = Set of all natural numbers which are divisible by 2 and 3

= { 6 ,12 ,18 }

Since, A ∩ B ∩ C = φ

and A ∪ B = {1 ,4 ,9 ,1 6 ,5 ,10,15 ,20}

implies n ( A ∪ B ) = 8

n(A ∪ B)' = 20 - 8 = 12

hence A, B, C are mutually exclusive and the number of elements in the complement set of A ∪ B is 12 .

QUESTION: 12

The function f(x) = e^{x}, x ε R is

Solution:

It is clear form the graph that f(x) = e^{x}, is one- one but not onto. Since, range ≠ codomain, so f(x) is into.

QUESTION: 13

If A = P {1, 2} where P denotes the power set, then which one of the following is correct?

Solution:

A = P {1 ,2} = {φ, {1}, {2}, {1 ,2 }}

From above, it is clear that {1 , 2} ε A

QUESTION: 14

Let R and S be two equivalence relations on a set A.

Then,

Solution:

Given, R and S are relations on set A.

Then R ε A x A and S ≤ A x A

=> R ∩ C ≤ A x A

implies R ∩ S is also a relation on A.

**Reflexivity**: Let a be an arbitrary element of A. Then, aA implies (a, a ) R and (a, a) S,

[Since, R and S are reflexive]

implies(a, a) ε R ∩ S

Thus, (a , a ) ε R ∩ S for all a ε A.

So, R ∩ S is a reflexive relation on A.

**Symmetry:** Let a, b ε A such that (a, b) ε R ∩ S.

Then, (a, b) ε R ∩ S

implies (a, b) ε R and (a, b) ε S

implies (b, a) ε R and (b , a ) ε S

[Since R and S are symmetric]

implies (b, a) ε R ∩ S

Thus, (a, b) ε R ∩ S implies

(b, a) ε R ∩ S for all (a, b ) ε R ∩ S .

So, R ∩ S is symmetric on A.

So, R ∩ S is transitive on A.

Hence, R is an equivalence relation on A.

QUESTION: 15

If (1 + 3 + 5 + ...+ p )+ (1 + 3 + 5 + ...+ q) = ( l + 3 + 5 + ... + r) where each set of parentheses contains the sum of consecutive odd integers as shown, what is the smallest possible value o f (p + q + r) where p > 6?

Solution:

QUESTION: 16

Let A = {x | x ≤ 9 , x ε N}. Let B = {a, b, c} be the subset of A where (a + b + c) is a multiple of 3. What is the largest possible number of subsets like B?

Solution:

Here,

A -

= {1,2,3,4,5,6,7,8,9} ;

Total possible multiple of 3 are

= 3,6,9,12,15,18,21,24,27

But 3 and 27 are not possible.

6 → 1 + 2 + 3

9 → 2 + 3 + 4 , 5 + 3 + 1,6 + 2 + 1

12 → 9 + 2 + 1,8 + 3 + 1,7 + 1 + 4,

7 + 2 + 3 ,( 5+ 4 + 2 , 6 + 5 + 1, 5 + 4 + 3

15 → 9 + 4 + 2,9 + 5 + l, 8 + 6 + 1 , 8 + 5 + 2 , 8 + 4 + 3 , 7 + 6 + 2 , 7 + 5 +3 , 6 + 5 + 4

18 → 9 + 8+ 1,9 + 7 + 2,9 + 6 + 3,9 + 5 + 4, 8 + 7 + 3, 8 + 6 + 4,7 + 6 + 5

21→9 + 8 +4, 9 + 7 + 5, 8 + 7 + 6

24 → 9 + 8 + 7

Hence, total number of largest possible subsets are 30.

QUESTION: 17

A mapping f: R → R which is defined a s f( x ) = cos x; x ε R is

Solution:

Given , f(x) = cos x

It is clear from the figure that f(x) is neither one-one nor a onto function.

Since, whenever we drawn a line parallel to x-axis, then it intersects at infinite points to the curve. So, f(x) is not one-one.

And, range o f f ( x ) = [-1, 1]

Codomain of f(x) = R

Range of f(x) ≠ codomain of f(x)

Hence, f(x) is not onto.

QUESTION: 18

If n(A) = 115, n{B) = 326, n(A — B) = 47, then what is n(A ∪ B) equal to?

Solution:

Now,

n(A -B )= n(A) - n(A ∩ B)

implies 47 = 115 - n(A ∩ B)

or n(A ∩ B) = 68

n(A ∪ B) = n(A) + n(B) — n(A ∩ B)

= 115 + 326 - 68 = 373

QUESTION: 19

In a town of 10000 families it was found that 40% family buy newspaper A, 20% buy newspaper B and 10% families buy newspaper C, 5% families buy A and B, 3% buy B and C and 4% buy A and C. If 2% families buy all the three newspaper, then number of families which buy A only is

Solution:

Here.

n(A)= 40% of 10000 = 4000

n(B)= 20% of 10000 = 2000

n(C)= 10% of 10000 =1000

n(A ∩ B) = 5% of 10000 = 500

n(B ∩ C) = 3% of 10000 = 300

n(C ∩ A) = 4% of 10000 = 400

= 4000 - [500 + 400 - 200]

= 4000- 700 - 3300

QUESTION: 20

A survey shows that 63% of the Americans like cheese whereas 76% like apples. If x% of the Americans like both cheese and apples, then

Solution:

Let A denote the set ot Americans who like cheese and let B denote the set of Americans who like apples.

Let population of American be 100.

Then, n(A) = 63, n(B) =76

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