Relations Test 37

Result

Relations Test 37

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Question 1
1 / -0

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

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

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

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

D
{(1, 1) (2, 1)}

Solution
Question 2
1 / -0

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

A
R

B
$$R-(\dfrac{1}{3},3)$$

C
$$[\dfrac{1}{3},3]$$

D
$$[-1,\dfrac{5}{3}]$$

Solution

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

So $$\tan x$$ is real in $$x\in(\dfrac{-\pi}{2},\dfrac{\pi}{2})$$

So $$D\ge 0$$

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

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

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

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

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

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

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

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

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

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

A
Reflexive, Symmetric but not Transitive

B
Reflexive, Transitive but not Symmetric

C
Symmetric, Transitive but not Reflexive

D
An Equivalence relation

Solution
Question 4
1 / -0

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

A
$$3$$

B
$$5$$

C
$$7$$

D
$$9$$
Question 5
1 / -0

If $$A$$ and $$B$$ have $$n$$ elements in common, then the number of elements common to $$A\times B$$ and $$B\times A$$ is

A
$$n$$

B
$$2n$$

C
$$n^2$$

D
$$0$$

Solution

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 $$B \times A$$
Question 6
1 / -0

Let $$R$$ and $$S$$ be two non-void relations on a set $$A$$. Which of the following statements is false?

A
$$R$$ and $$S$$ are transitive implies $$R\cap S$$ is transitive

B
$$R$$ and $$S$$ are transitive implies $$R\cup S$$ is transitive

C
$$R$$ and $$S$$ are symmetric imples $$R\cup S$$ is symmetric

D
$$R$$ and $$S$$ reflexive implies $$R\cap S$$

Solution

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
Question 7
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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

A
Neither reflexive nor symmetric

B
Reflexive and symmetric but not transitive

C
Symmetric and transitive but not reflexive

D
An equivalence relation

Solution
Question 8
1 / -0

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

A
$$1000$$

B
$$500$$

C
$$250$$

D
$$125$$
Question 9
1 / -0

If R is a reflexive relation defined on a finite set A and $$n(A)=p$$, and $$n(R)=q$$, then?

A
$$p=q$$

B
$$p\leq q$$

C
$$p\geq q$$

D
None of these

Solution
Question 10
1 / -0

Let $$f:R \to R - \left\{ 3 \right\}$$ be a function such that for some p>0, $$\displaystyle f\left( {x + p} \right) = {{f\left( x \right) - 5} \over {f\left( x \right) - 3}}$$ for all $$x \in R$$. Then, period of $$f$$ is

A
$$2p$$

B
$$3p$$

C
$$4p$$

D
$$5p$$

Solution