Relations and Functions

Let Z, be the set of all integers,

 A = $\left\{\left(x,y\right)\in Z\times Z:{\left(x-2\right)}^2+y^2\le 4\right\}$                          

B =  $\left\{\left(x,y\right)\in Z\times Z:x^2+y^2\le 4\right\}$ <!-- [if gte mso 9]><xml> </xml><![endif]--> 

C =  $\left\{\left(x,y\right)\in Z\times Z:{\left(x-2\right)}^2+{\left(y-2\right)}^2\le 4\right\}$

If the total number of relations from A  $\cap$ <!-- [if gte mso 9]><xml> </xml><![endif]--> B to A $\cap$ <!-- [if gte mso 9]><xml> </xml><![endif]--> C is 2^p, then the value of p is:

The domain of the function f(x) = $si{n}^{-1}\left(\frac{x^2-3x+2}{x^2+2x+7}\right)is:$

Let ${\left\{a_n\right\}}_{n=0}^{\infty }$ be a sequence such that a₀ = a₁ = 0 and a_n+2 = 3_an+1 – 2a_n + 1, $\forall n\ge 0.$. Then $a_{25}{a}_{23}-2a_{25}{a}_{22}-2a_{23}{a}_{24}+4a_{22}{a}_{24}$ is equal to

If [t] denotes the greatest integer $\le t,$ then the value of ${\displaystyle {\int }_0^1\left[2x-\left|3x^2-5x+2\right|+1\right]dxis:}$

Let R₁ and R₂ be two relations defined on R by

a R₁ b Û ab $\ge 0$ <!-- [if gte mso 9]><xml> </xml><![endif]--> and a R₂ b $\iff a\ge b$ <!-- [if gte mso 9]><xml> </xml><![endif]--> 

Then,

For $\alpha \in N,$ <!-- [if gte mso 9]><xml> </xml><![endif]--> consider a relation R on N given by R = $\left\{\left(x,y\right):3x+\alpha yisamultipleof7\right\}.$ <!-- [if gte mso 9]><xml> </xml><![endif]--> The relation R is an equivalence relation if and only if:

The number of functions f, from the set A = $\left\{x\in N:x^2-10x+9\le 0\right\}$ <!-- [if gte mso 9]><xml> </xml><![endif]--> to the set B = $\left\{n^2:n\in N\right\}$ <!-- [if gte mso 9]><xml> </xml><![endif]--> such that f(x) $\le$ <!-- [if gte mso 9]><xml> </xml><![endif]-->(x – 3)² + 1, for every x $\in$ <!-- [if gte mso 9]><xml> </xml><![endif]--> A, is…………

Let x * y = x² + y³ and (x * 1) * 1 = x * (1 * 1). Then a value of 2 $si{n}^{-1}\left(\frac{x^4+x^2-2}{x^4+x^2+2}\right)$ <!-- [if gte mso 9]><xml> </xml><![endif]--> is.