Relations Test ...

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

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

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 

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

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

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

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

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

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

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