Relations and F...

The number of ordered pairs $$(x, y)$$ of real numbers that satisfy the simultaneous equations.
$$x + y^{2} = x^{2} + y = 12$$ is.

If $$e^{x} = y + \sqrt {1 + y^{2}}$$, then the value of $$y$$ is

The number of ordered pairs $$\left( m,n \right)$$, where $$ m, n \in \left\{ 1,2,3,\dots ,50 \right\}$$, such that $$ { 6 }^{ m }+{ 9 }^{ n }$$ is a multiple of $$5$$ is

If $$A$$ is a finite set having $$n$$ elements, then the number of relations which can be defined in $$A$$ is

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 $$\varepsilon$$ A and b $$\varepsilon$$ B].
What is RoR$$^{-1}$$ equal to?

The relation R define on the set of natural numbers as {(a, b) : a differs from b by 3} is given.

Let $$f(x) = \dfrac {2025^{x}}{45 + 2025^{x}}$$, then the value of $$f\left (\dfrac {1}{2025}\right ) + f\left (\dfrac {2}{2025}\right ) + f\left (\dfrac {3}{2025}\right ) + ... + f\left (\dfrac {2024}{2025}\right )$$ is equal to

If $$f(x) = \dfrac {a^{x} + a^{-x}}{2}$$ and $$f(x + y) + f(x - y) = k f(x) f(y)$$, then $$k$$ is equal to

$$R$$ is a relation on $$N$$ given by $$R = \left \{(x, y)|4x + 3y = 20\right \}$$. Which of the following doesnot belong to $$R$$?

If $$f\left( 1 \right) =1$$, $$f\left( 2n \right) =f\left( n \right) $$ and $$f\left( 2n+1 \right) ={ \left\{ f\left( n \right)  \right\}  }^{ 2 }-2$$ for $$n=1,2,3,\dots $$, then the value of $$f\left( 1 \right) +f\left( 2 \right) +\cdots +f\left( 25 \right) $$ is