Sets, Relations and Functions

If the domain of the function \\ $f(x) = \frac{1}{\sqrt{10 + 3x - x^2}} + \frac{1}{\sqrt{x + |x|}}$ is $(a,\ b)$, then $(1 + a)^2 + b^2$ is equal to:

Let $A$ be the set of all functions $f: \mathbb{Z} \to \mathbb{Z}$ and $R$ be a relation on $A$ such that $R = \{(f,g): f(0)=g(1) \text{ and } f(1)=g(0)\}$. Then $R$ is:

Let $A=\{-2,-1,0,1,2,3\}$. Define a relation $R$ on $A$ by $xRy$ if and only if $y=\max(x,1)$. If $l$ is the number of elements in $R$, and $m$ and $n$ are the minimum numbers of ordered pairs that must be added to $R$ to make it reflexive and symmetric respectively, then $l+m+n$ is equal to:

Let $f$ be a function such that $f(x) + 3\,f\bigl(24/x\bigr) = 4x$, $x\neq0$. Then $f(3) + f(8)$ is equal to:

If the domain of the function $f(x)=\log_7\bigl(1-\log_4(x^2-9x+18)\bigr)$ is $(\alpha,\beta)\cup(\gamma,\delta)$, then $\alpha+\beta+\gamma+\delta$ is equal to:

Let $A = \{-3, -2, -1, 0, 1, 2, 3\}$. Let $R$ be a relation on $A$ defined by $xRy$ if and only if $0 \le x^2 + 2y \le 4$. Let $\ell$ be the number of elements in $R$ and $m$ be the minimum number of elements required to be added to $R$ to make it a reflexive relation. Then $\ell + m$ is equal to:

If the domain of the function $f(x)=\log_e\bigl(\tfrac{2x-3}{5+4x}\bigr)+\sin^{-1}\bigl(\tfrac{4+3x}{2-x}\bigr)$ is $[\alpha,\beta]$, then $\alpha^2+4\beta$ is equal to:

Let $A = \{-3,-2,-1,0,1,2,3\}$ and $R$ be a relation on $A$ defined by $xRy$ iff $2x - y \in \{0,1\}$. Let $\ell$ be the number of elements in $R$. Let $m$ and $n$ be the minimum numbers of elements that must be added to $R$ to make it reflexive and symmetric, respectively. Then $\ell + m + n$ is equal to:

Let f, g: (1, \infty) \to \mathbb{R} be defined as f(x) = \frac{2x+3}{5x+2} and g(x) = \frac{2-3x}{1-x}. If the range of the function f\circ g: [2,4] \to \mathbb{R} is [\alpha, \beta], then \frac{1}{\beta - \alpha} is equal to:

Let $A = \{(\alpha, \beta) \in \mathbb{R} \times \mathbb{R} : |\alpha - 1| \le 4 \text{ and } |\beta - 5| \le 6\}$ and $B = \{(\alpha, \beta) \in \mathbb{R} \times \mathbb{R} : 16(\alpha - 2)^2 + 9(\beta - 6)^2 \le 144\}$. Then: