Relations Test 41

Result

Relations Test 41

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Question 1
1 / -0

If $$k\ \epsilon\ R^+$$ and the middle term of $$(\dfrac{k}{2} + 2)^8$$ is $$1120$$, then value of k is

A
$$3$$

B
$$2$$

C
$$1$$

D
$$4$$

Solution

$$K\epsilon { R }^{ + }$$

The general term, $${ \left( r+1 \right) }^{ th }$$ term of the expansion
$${ \left( \dfrac { K }{ 2 } +2 \right) }^{ 8 }$$ is $${ T }_{ r+1 }={ 8 }_{ Cr }{ \left( \dfrac { K }{ 2 } \right) }^{ 8-r }{ 2 }^{ r }$$

In the above expansion there are $$9$$ terms, hence the middle term will be the $$5th$$ term $$(r=4)$$

$$\therefore$$ $${ T }_{ 5 }={ 8 }_{ { C }_{ 4 } }{ \left( \dfrac { K }{ 2 } \right) }^{ 8-4 }{ 2 }^{ 4 }$$

$$=\dfrac { 8! }{ 4!4! } \dfrac { { K }^{ 4 } }{ { 2 }^{ 4 } } .{ 2 }^{ 4 }$$

$$=\dfrac { 8\times 7\times 6\times 5 }{ 4\times 3\times 2 } { K }^{ 4 }$$

$$=70{ K }^{ 4 }$$

Given,

$${ 70K }^{ 4 }=1120$$

$$\Rightarrow { K }^{ 4 }=112/7$$

$$\Rightarrow { K }^{ 4 }=16$$

$$\Rightarrow { K }^{ 4 }={ 2 }^{ 4 }$$

$$\therefore$$ $$K=2$$
Question 2
1 / -0

If a relation R is defined on the set Z of integers as follows : (a,b) $$\epsilon \quad R\Leftrightarrow { a }^{ 2 }+{ b }^{ 2 }=25,$$ , Then domain (R)=

A
{3,4,5}

B
{0,3,4,5}

C
{0,$$\pm $$3, $$\pm $$4, $$\pm $$5}

D
{3,4}

Solution
Question 3
1 / -0

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

A
Reflexive

B
Symmetric

C
Transitive

D
None of these

Solution
Question 4
1 / -0

Let $$R_1$$ and $$R_2$$ be equivalence relations on a set A, the $$R_1 \cup R_2$$ may or may not be

A
Reflexive

B
Symmetric

C
Transitive

D
Cannot say anything

Solution
Question 5
1 / -0

Let $$A=\left\{ 1,2,3 \right\} ,\quad B=\left\{ 1,3,5 \right\} .$$ If relation R from A to B is given by $$R=\left\{ \left( 1,3 \right) ,\left( 2,5 \right) ,\left( 3,3 \right) \right\} .\quad Then\quad { R }^{ 1 }\quad is$$

A
$$\left\{ \left( 3,3 \right) ,\left( 3,1 \right) ,\left( 5,2 \right) \right\} $$

B
$$\left\{ \left( 1,2 \right) ,\left( 2,5 \right) ,\left( 3,3 \right) \right\} $$

C
$$\left\{ \left( 1,3 \right) ,\left( 5,2 \right) \right\} $$

D
None of these

Solution
Question 6
1 / -0

If n(A)=5 then number of relation on 'A' is:

A
$${ 2 }^{ 5 }$$

B
$${ 2 }^{ 25 }$$

C
$${ 5 }^{ 2 }$$

D
none
Question 7
1 / -0

Consider the set $$A = \{ ( 1,2,3 ) \}$$ and the relation $$R = \{ ( 1,2 ) , ( 1,3 ) \}$$ then $$R$$ is

A
Reflexive

B
Transitive

C
Symmetric

D
Equivalence

Solution
Question 8
1 / -0

If $$A=\{ a,b,c\}$$ then relation $$R=\left\{ \left( b,c \right) \right\}$$ on A is

A
reflexive only

B
symmetric only

C
transitive only

D
reflexive and transitive only

Solution
Question 9
1 / -0

For real numbers x and y, we write $$xRy\Leftrightarrow { x }^{ 2 }-{ y }^{ 2 }+\sqrt { 3 } $$ is an irrational number, then the relation R is

A
Reflexive

B
Symmetric

C
Transitive

D
Equivalence

Solution
Question 10
1 / -0

The number of equivalence relations that can be defined on a set {a,b,c}, is

A
3

B
5

C
7

D
8

Solution