MCQ Questions for Class 11 Commerce Applied Mathematics Relations Quiz 14

If n | m means that n is a factor of m, the relation | is

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reflexive and symmetric
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transitive and symmetric
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reflexive, transitive and symmetric
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reflexive, transitive and not symmetric

Let N denote the set of all natural numbers and R a relation on N

$\times$ N. Which of the following is an equivalence relation?

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(a, b, c) R (c, d) if ad (b + c) = bc (a +d)
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(a, b) R (c, d) if a + d = b + c
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(a, b) R (c, d) if ad = bc
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(a, d) R (b, c) if a - d = b - c

The relation R {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)} on set A {1, 2, 3} is

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reflexive but not symmetric
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reflexive but not transitive
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symmetric and transitive
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neither symmetric nor transitive

The relation "is subset of" on the power set P(A) of a set A is

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symmetric
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anti-symmetric
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equivalence relation
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reflexive

The minimum number of elements that must be added to the relation R {(1, 2), (2, )} on the set of natural numbers so that it is an equivalence is

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4
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7
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6
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5

Let X be a non-empty set and P(X) be the set of all subsets of X. For A, B

$\subset$ P(X), ARB if and only if

$A \cap B = \phi$ then the relation

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R is not reflexive
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R is symmetric
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R is not transitive
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R is an equivalence relation

Let

$R_1$ and

$R_2$ be equivalence relations on a set A, then

$R_1 \cup R_2$ may or may not be

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reflexive
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symmetric
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transitive
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anti-symmetric

Let N denote the set of all natural numbers and R a relation on N

$\times$ N. Which of the following is an equivalence relation?

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(a, b) R (c, d) if ad (b + c) = bc (a + d)
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(a, b) R (c, d) if a + d = b + c
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(a, b) R (c, d) if ad = bc
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all the given

Let A and B be finite sets containing m and n elements respectively. The number of relations that can be defined from A to B is

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$mn$
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$2^{mn}$
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$2^{m+n}$
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$n^m$

Let A = {1, 2, 3}. Which of the following is not an equivalence relation on A?

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{(1, 1), (2, 2), (3, 3)}
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{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}
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{(1, 1), (2, 2), (3, 3), (2, 3), (3, 2)}
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{(1, 1) (2, 1)}

Range of the function

$f(x)=\dfrac{sec^2\,x-tanx}{sec^2\,x+tanx}-\dfrac{\pi}{2}<x<\dfrac{\pi}{2}$ , is

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R
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$R-(\dfrac{1}{3},3)$
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$[\dfrac{1}{3},3]$
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$[-1,\dfrac{5}{3}]$

Explanation

Let

$f(x)=y$

$y=\dfrac{sec^2x-\tan x}{sec^2 x+\tan x}$

$\Rightarrow y(sec^2 x+\tan x)=sec^2x-\tan x$

$\Rightarrow y(1+\tan^2x+\tan x)=1+\tan^2x-\tan x$

$\Rightarrow y+y\tan^2x+y\tan x=1\tan^2x-\tan x$

$\Rightarrow \tan^2x(y-1)+\tan(y+1)+y-1=0$

Now it is given

$\dfrac{-\pi}{2}<x< \dfrac{\pi}{2}$

$\tan x$ is real in

$x\in(\dfrac{-\pi}{2},\dfrac{\pi}{2})$

$D\ge 0$

$\Rightarrow (y+1)^2-4(y-1)(y-1)\ge 0$

$\Rightarrow y^2+1+2y-4(y-1)^2\ge 0$

$\Rightarrow y^2+2y+1-4(y^2+1-2y)\ge 0$

$\Rightarrow y^2+2y+1-4y^2-4+8y\ge 0$

$\Rightarrow -3y^2+10y-3\ge 0$

$\Rightarrow 3y^2-910y+3\le 0$

$\Rightarrow 3y^2-9y-y+3\le 0$

$\Rightarrow 3y(y-3)-1(y-3)\le 0$

$\Rightarrow (3y-1)(y-3)\le 0$

1423799_778946_ans_5154f92762a94e3cad89a3bf5317d786.PNG

In the set of all

$3\times 3$ real matrices a relation is defined as follows. A matrix A is related to a matrix B if and only if there is a non-singular

$3\times 3$ matrix P such that

$B=P^{-1}AP$ . This relation is

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Reflexive, Symmetric but not Transitive
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Reflexive, Transitive but not Symmetric
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Symmetric, Transitive but not Reflexive
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An Equivalence relation

Let

$R = \left \{(2, 3), (3, 4)\right \}$ be a relation defines on the set of natural numbers. The minimum number of ordered pairs required to be added in

$R$ so that enlarged relation be comes an equivalence relation is

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$3$
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$5$
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$7$
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$9$

$A$ and

$B$ have

$n$ elements in common, then the number of elements common to

$A\times B$ and

$B\times A$ is

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$n$
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$2n$
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$n^2$
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$0$

Explanation

If there are

$m$ elements in

$A$ and

$m$ elements in

$B, A \times B$ and

$B \times A$ both will be having

$m^2$ elements.

Since there are

$n$ elements common to

$A$ and

$B$ , there will be

$n^2$ such pairs in

$A \times B$ and

$B \times A$ , which will have both the elements same.

Since the elements are same, they are commutative and hence there will be

$n^2$ elements common to

$A \times B$ and

$B \times A$

Let

$R$ and

$S$ be two non-void relations on a set

$A$ . Which of the following statements is false?

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$R$ and $S$ are transitive implies $R\cap S$ is transitive
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$R$ and $S$ are transitive implies $R\cup S$ is transitive
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$R$ and $S$ are symmetric imples $R\cup S$ is symmetric
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$R$ and $S$ reflexive implies $R\cap S$

Explanation

Let

$A=\left\{ 1,2,3 \right\}$

$;\quad R=\left\{ (1,1),(2,2) \right\} ;\quad S=\left\{ (2,2),(2,3) \right\}$
be two transitive relations on

$A$
Thus

$R\cup S=\left\{ (1,1),(1,2),(2,2),(2,3) \right\}$
then

$\left( 1,2 \right) \in R\cup S\quad$ and

$\left( 2,3 \right) \in R\cup S\quad$
but

$\left( 1,3 \right) \notin R\cup S$

$R\cup S\quad$ is not transitive

Number of ordered triplets

$(p,q,r)$ where

$1\le p,q,r\le 10$ such that

${ 2 }^{ p }+{ 3 }^{ q }+{ 5 }^{ r }$ is a multiple of

$4\left( p,q,e\in N \right)$

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$1000$
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$500$
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$250$
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$125$

We define a binary relation

$\sim$ on the set of all

$3\times 3$ matrices as

$A\sim B$ if and only if there exist invertible matrices

$P$ and

$Q$ such that

$B=PA{Q}^{-1}$ . The binary relation

$\sim$ is

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Neither reflexive nor symmetric
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Reflexive and symmetric but not transitive
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Symmetric and transitive but not reflexive
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An equivalence relation

Let

$R$ a relation on the set

$N$ be defined by

$\left\{ \left( x,y \right) |x,y\in N,2x+y=41 \right\}$ . Then

$R$ is

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Reflexive
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Symmetric
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Transitive
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None of these

If R is a reflexive relation defined on a finite set A and

$n(A)=p$ , and

$n(R)=q$ , then?

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$p=q$
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$p\leq q$
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$p\geq q$
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None of these

Let R be a relation on the set of integers given by

$aRb \leftrightarrow a = 2^k .b$ for some integer

$k.$ Then

$R$ is

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an equivalence relation
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reflexive and transitive but not symmetric
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reflexive and transitive but not transitive
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symmetric and transitive but nor reflexive

Total number of equivalence relations defined in the set

$S=\left\{a, b, c\right\}$ is?

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$5$
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$3!$
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$2^3$
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$3^3$

Let

$A={1,2,3}$ . Then number of equivalence relations containing

$(1,2)$ is

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$1$
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$2$
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$3$
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$4$

Explanation

$A=\left\{1,2,3\right\}$

For equivalence relation containing

$(1,2)$

For symmetric, it must consists

$(1,2)$ and

$(2,1)$ .

For transitivity, it must consists

$(1,3)$ and

$(3,2)$ and

$(1,2),(2,1),(2,3),(3,1)$

For reflexivity, it must consists

$(1,1)$ and

$(2,2),(3,3)$

$\Rightarrow R=\left\{(1,1),(2,2),(2,3),(3,3),(1,2)(1,3),(2,1),(3,1),(3,1)\right\}$

$\Rightarrow$ Only

$1$ such relation is possible.

If R is a reflexive relation on a finite set A having n-elements, and there are m ordered pairs in R. then?

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$m \geq n$
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$m\leq n$
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$m = n$
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$m\neq n$

Let R be a relation from

$A=\{1,2,3,4\}$ to

$B=\{1,3,5\}$ such that R=[(a,b):a<b where a\in{A} and b\in{B}].What is

$RoR^{-1}$ equal to

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$(1,3),(1,5),(2,3),(2,5),(3,5),(4,5)$
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$(3,1),(5,1),(3,2)(5,2),(5,3),(5,4)$
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$(3,3),(3,5),(5,3),(5,5)$
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$(3,3),(3,4),(4,5)$

$f:\left( { - \infty ,\infty } \right) \to \left( {0,1} \right]$ defined by

$f\left( x \right) = \frac{1}{{{x^2} + 1}}$ is

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one -one but not onto
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onto but not one-one
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bijective
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neither one-one nor onto

Let

$X=\left\{ x \epsilon {R};\cos(\sin\ x)=\sin(\cos\ x)\right\}$ . The number of elements in

$X$ is

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$0$
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$2$
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$4$
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$not\ finite$

If the relation

$R:A \to B$ where

$A = \left\{ {1,2,3,4} \right\},B = \left\{ {1,3,5} \right\}$ is defined by

$R = \left\{ {\left( {x,y} \right):x < y,x \in A,y \in B} \right\}$ then

$Ro{R^{ - 1}} =$

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$\left\{ {\left( {1,3} \right),\left( {1,5} \right),\left( {2,3} \right),\left( {2,5} \right),\left( {3,5} \right),\left( {4,5} \right)} \right\}$
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$\left\{ {\left( {3,1} \right),\left( {5,1} \right),\left( {3,2} \right),\left( {5,2} \right),\left( {5,3} \right),\left( {5,4} \right)} \right\}$
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$\left\{ {\left( {3,3} \right),\left( {3,5} \right),\left( {5,3} \right),\left( {5,5} \right)} \right\}$
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$\left\{ {\left( {3,3} \right).\left( {3,4} \right),\left( {4,5} \right)} \right\}$

For real number

$x$ and

$y$ , define a relationship

$R$ {;

$x\ Ry$ if and only if

$x-y+\sqrt{2}$ is an irrational number }. Then the relation

$R$ is

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Reflexive
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symmetric
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transitive
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an equivalence relation

Let

$X$ be a family of sets and

$R$ be a relation on

$X$ defined by

$'A$ is disjoint from

$B ^ { \prime }$ . Then

$R$ is _________.

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Reflexive
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Symmetric
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Anti-symmetric
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Transitive

$p \Rightarrow \,q$ can also be written as

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$p \Rightarrow \, \sim q$
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$\sim p \vee q$
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$\sim q \Rightarrow \, \sim p$
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None of these

CBSE Questions for Class 11 Commerce Applied Mathematics Relations Quiz 14 - MCQExams.com

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Practice Class 11 Commerce Applied Mathematics Quiz Questions and Answers

CBSE Questions for Class 11 Commerce Applied Mathematics Relations Quiz 14 - MCQExams.com

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Practice Class 11 Commerce Applied Mathematics Quiz Questions and Answers

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