Description

This mock test of Test: Real Analysis- 8 for Mathematics helps you for every Mathematics entrance exam.
This contains 20 Multiple Choice Questions for Mathematics Test: Real Analysis- 8 (mcq) to study with solutions a complete question bank.
The solved questions answers in this Test: Real Analysis- 8 quiz give you a good mix of easy questions and tough questions. Mathematics
students definitely take this Test: Real Analysis- 8 exercise for a better result in the exam. You can find other Test: Real Analysis- 8 extra questions,
long questions & short questions for Mathematics on EduRev as well by searching above.

QUESTION: 1

Let R and S be two non-void relations on a set A.

Q. Which of the following statements is false?

Solution:

Let A = {1, 2, 3} and R = {(1, 1), (1, 2)}

S = {(2, 2), (2, 3)} be transitive relations on A.

Then, R ∪ S= {(1,1); (1,2); (2,2); (2, 3)} Obviously, R ∪ S is transitive.

Since, (1, 2) ∈ R ∪ S and (2, 3 ) ∈ R ∪ S but (l , 3) ∉ R ∪ S.

QUESTION: 2

Consider the following statements

For non-empty sets A, B and C

1. A - (B - C) = (A - B) ∪ C

2. A - (B - C) = (A - B) - C

Q. Which of the statements given above is/are correct?

Solution:

In the given statements, statement (I) is correct.

QUESTION: 3

Let N be the set of integers. A relation R on N is defined as R = {(x, y) | xy > 0, x, y ∈ N}. Then, which one of the following is correct?

Solution:

Since, R = {(x, y) | xy > 0, x, y ∈ N}

**Reflexive**

Since, x, y ∈ N

So, x, x ∈ N

implies x^{2} > 0

Hence, R is reflexive.

**Symmetric**

Since, x, y ∈ N and xy > 0 implies yx > 0 Hence, R is also symmetric.

**Transitive**

Since, x,y, z, ∈ N

implies xy > 0, yz > 0

implies xz > 0

So, R is also transitive

Thus, R is an equivalence relation.

QUESTION: 4

Which one of the following function f : R → R is injective?

Solution:

An injective function means one-one.

In option (d),f (x) = -x

For every values of x, we get a different values of f.

Hence, it is injective.

QUESTION: 5

If A is the set of even natural number less than 8 and B is the set of prime number less than 7, then number of relations from A to B is

Solution:

A = {2,4,6}, B = {2,3, 5}

So, A x B contains 3 x 3 = 9 elements.

Hence, number of relations from A to B = 2^{9}.

QUESTION: 6

What does the shaded region in the Venn diagram given following represent?

Solution:

The shaded region in the Venn diagram is represented

QUESTION: 7

Out of 32 persons, 30 invest in National savings Certificates and 17 invest in shares. What is the number of persons who invest in both?

Solution:

n(N ∪ S) = 32, n(N) = 30, n(S) = 17 Since, We know that,

n( N ∪ S) = n(N) + n(S) - n(N ∩ 5)

or 32= 30+ 17 - n(N ∩ S)

or n(N ∩ S ) = 47 - 32= 15

Since, R is an equivalence relation on set A, Therefore, (a, a) ∈ R for all a ∈ A, Hence, R has atleast n ordered pairs.

QUESTION: 8

Let R = {x | x ∈ N ,x is multiple of 3 and x ≤ 10} and S = {x | x ∈ N, x is a multiple of 5 and x ≤ 100} What is the number of elements in (R x S) ∩ (S x R) ?

Solution:

Since, R = {3,6,9,12,15,..., 99}

and S= {5,10,15, ...,95}

Now, (R x S) ∩ ( S x R)

= (R ∩ S ) x ( S ∩ R ) = (15,30,45,60,75,90) x (15,30, 45,60,75,90)

Hence, number of elements in

(R x S) ∩ (S x R) = 6 x 6 = 36

QUESTION: 9

Let X be the set of all graduates in India. Elements x and y in X are said to be related, if they are graduates of the same university. Which one of the following statements is correct?

Solution:

xRy ⇒ x and y are graduates of the same university.

**Reflexive** xRx ⇔ x and y are graduates of the same university.

So, relation is reflexive.

**Symmetric **xRy ⇔ x and y are graduates of the same university.

implies, yRx ⇔ y and x are graduates of the same university.

So, relation is symmetric.

**Transitive** xRy, yRz ⇔ xRz

It means x and y, y and z are graduates of the same university, then x and z are also graduates of the same university.

So, relation is transitive.

Hence, relation is reflexive, symmetric and transitive.

QUESTION: 10

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

Solution:

Let number of newspapers be x.

If every students reads one newspaper, the number of students would be

x(60) = 60x

Since, every students reads 5 newspapers

Hence, number of students

QUESTION: 11

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 basketball and 40 played cricket and hockey; 24 played all the three games. The number of boys who did not play any game is

Solution:

Here,

QUESTION: 12

Let A = {-1, 2, 5, 8}, B = {0, 1, 3, 6, 7} and R be the relation ‘is one less than’ from A to B, then how many elements will R contain?

Solution:

Given,

A = { - 1 , 2 , 5, 8} and

B = {0 ,1 ,3 ,6 ,7 }

R = { ( - 1 , 0 ) , (2,3), (5,6)}

(Since , R = A is one less than from B) Hence, total number of elements in R is 3.

QUESTION: 13

Consider the function f:R → {0, 1} such that

Q. Which one of the following is correct?

Solution:

Since, on taking a straight line parallel to x-axis, the group of given function intersect it at many points.

So, f(x) is a many one.

And as range of f(x) = codomain

So, f(x) is onto.

Hence, f(x) is many one onto.

QUESTION: 14

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

Solution:

We have (a,b) R (a, b) for all (a, b) ∈ N x N

Since. a + b = b + a.

Hence, R is reflexive.

R is symmetric for we have (a , b) R (c, d)

implies a + d = b + c

implies d + a = c + b

implies c + b = d + a

implies (c, d) R (e, f)

Then, by definition of R,

we have a + d = b + c and c + f = d + e when by addition, we get

a + d + c + f = b + c + d + e

or a + f = b + e

Hence, (a, b) R (e , f)

Thus, (a, b) R( c, d) and (c, d) R (e , f) implies (a , b) R (e , f)

QUESTION: 15

Let N denote the set of natural numbers and A = {n^{2} : n ∈ N} and B = {n^{3/2} : n ∈ N}. Which one of the following is correct?

Solution:

Since, A = { n^{2} : n ∈ N } and

B = {n^{3} : n ∈ N}

A = { 1 ,4 ,9 ,1 6 ,2 5 ,4 9 ,6 4 ,8 1 ,...}

B = {1 ,8 ,2 7 ,6 4 ,1 2 5 ,...}

A ∩ B = { 1 , 6 4 , . .....}

A ∩ B must be a proper subset of {m^{6} : m ∈ N}

QUESTION: 16

If A and. B are two disjoint sets, then which one of the following is correct?

Solution:

Since, A ∩ B = φ (given)

So, A - B = A - ( A ∩ B )

(Since, A - B = A)

Now, B - A' = φ

and B ∩ A = φ

or B ‘- A ' = B ∩ A

and (A- B ) ∩ B = A ∩ B

So, option (d) is correct.

QUESTION: 17

Let M be the set of men and R is a relation ‘is son of’ defined on M. Then, R is

Solution:

M = set of men and R is a relation ‘is son of ’ defined on M.

Reflexive relation _{a}R_{a}.

Since, a cannot be a son o f a.

**Symmetric relation** _{a}R_{b }implies _{b}Ra which is also not possible.

**Transitive relation** _{a}R_{b, }_{b}R_{c} implies _{c}R_{a }which is not possible.

QUESTION: 18

Let f : R --> R be a function defined a s / ( x ) = x | x |; for each x ∈ R, R being the set of real numbers. Which one of the following is correct?

Solution:

Since, f(x) = x |x|

If f(x_{1}) = f(x_{2})

or x_{1} | x_{1}| = x_{2 }| x_{2} |

or x_{1} = x_{2}

So, f(x) is one-one.

Also, range of f(x) = codomain of f(x)

So, f(x) is onto.

Hence, f(x) is both one-one and onto.

QUESTION: 19

What is the range of f(x) = cos 2x - sin 2x?

Solution:

Since, f(x) = cos 2x - sin 2x

[Since, f( x ) = a cos x + b sin x,

QUESTION: 20

Consider the following with regard to a relation R on a set of real number defined by _{x}R_{y} if and only if 3x + 4y = 5.

1. _{0}R_{1 }

2. _{1}R_{1/2}

3. _{2/3}R_{3/4}

Q. Which of the above are correct?

Solution:

Since, on the set of real numbers, R is a relation defined by _{X}R_{y} if and only if 3x + 4y = 5 for which

Hence, both the statements 11 and 111 are correct.

### Real Analysis

Doc | 3 Pages

### Real Analysis (Solved Example)

Doc | 10 Pages

### Determinants Test - 8

Doc | 1 Page

### Solve problems involving real cycle analysis

Doc | 33 Pages

- Test: Real Analysis- 8
Test | 20 questions | 60 min

- Test: Real Analysis - 5
Test | 20 questions | 60 min

- Test: Real Analysis - 6
Test | 20 questions | 60 min

- Test: Real Analysis- 7
Test | 20 questions | 60 min

- Test: Real Analysis- 4
Test | 20 questions | 60 min